Physics Reports vol.313

INSTABILITY IN DYNAMIC FRACTURE

J. FINEBERG , M. MARDER
The Racah Institute of Physics, The Hebrew University of Jerusalem Jerusalem, 91904, Israel
Center for Nonlinear Dynamics and Department of Physics, The University of Texas at Austin, Austin TX 78712, USA

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 313 (1999) 1}108

Instability in dynamic fracture J. Fineberg *, M. Marder
 The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Center for Nonlinear Dynamics and Department of Physics, The University of Texas at Austin, Austin TX 78712, USA Received July 1998; editor: I. Procaccia

Contents 1. Introduction 1.1. Brief overview of the paper 1.2. Scaling arguments 2. Continuum fracture mechanics 2.1. Structure of fracture mechanics 2.2. Linear elasticity 2.3. The Inglis solution for a static crack in Mode III 2.4. Linear elastic equations for moving solutions in Mode I fracture 2.5. Mode I, structure near the tip, stress intensity factors 2.6. Mode I, example of speci"c loading 2.7. The J integral and the equivalence of the Irwin and Gri$th points of view 2.8. The general equation for the motion of a crack in an in"nite plate 2.9. Crack paths 3. Experimental methods in dynamic fracture 3.1. Application of stress 3.2. Direct measurement of the stress intensity factor

4 4 5 9 9 13 16 19 21 22 24 27 36 39 39 41

3.3. Crack velocity measurements 3.4. Measurements of heat generation and the temperature rise near a crack 3.5. Acoustic emissions of cracks 4. Phenomenology of dynamic fracture 4.1. Comparisons of theory and experiment 4.2. Phenomena outside of the theory 5. Instabilities in isotropic amorphous materials 5.1. Introduction 5.2. Experimental observations of instability in dynamic fracture 6. Theories of the process zone 6.1. Cohesive zone models 6.2. Continuum studies 6.3. Dynamic fracture of lattice in Mode I 6.4. Dynamic fracture of a lattice in antiplane shear, Mode III 6.5. The generality of the results in a ideal brittle crystal 6.6. Molecular dynamics simulations 7. Conclusions References

42 44 44 45 45 49 58 58 59 74 74 76 79 84 96 97 101 102

* Corresponding author.  Supported by the US, Israel Binational Science Foundation, Grant 95-00029/1.  Theoretical work supported by the National Science Foundation, DMR-9531187, the US, Israel Binational Science Foundation, Grant 95-00029//1, and the Exxon Education Foundation; experimental work supported by the Texas Advanced Research Program; supercomputing supported by the Texas Advanced Computation Center, and the National Partnership for Academic Computing Infrastructure.

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 8 ) 0 0 0 8 5 - 4

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Abstract The fracture of brittle amorphous materials is an especially challenging problem, because the way a large object shatters is intimately tied to details of cohesion at microscopic scales. This subject has been plagued by conceptual puzzles, and to make matters worse, experiments seemed to contradict the most "rmly established theories. In this review, we will show that the theory and experiments "t within a coherent picture where dynamic instabilities of a crack tip play a crucial role. To accomplish this task, we "rst summarize the central results of linear elastic dynamic fracture mechanics, an elegant and powerful description of crack motion from the continuum perspective. We point out that this theory is unable to make predictions without additional input, information that must come either from experiment, or from other types of theories. We then proceed to discuss some of the most important experimental observations, and the methods that were used to obtain the them. Once the #ux of energy to a crack tip passes a critical value, the crack becomes unstable, and it propagates in increasingly complicated ways. As a result, the crack cannot travel as quickly as theory had supposed, fracture surfaces become rough, it begins to branch and radiate sound, and the energy cost for crack motion increases considerably. All these phenomena are perfectly consistent with the continuum theory, but are not described by it. Therefore, we close the review with an account of theoretical and numerical work that attempts to explain the instabilities. Currently, the experimental understanding of crack tip instabilities in brittle amorphous materials is fairly detailed. We also have a detailed theoretical understanding of crack tip instabilities in crystals, reproducing qualitatively many features of the experiments, while numerical work is beginning to make the missing connections between experiment and theory.  1999 Elsevier Science B.V. All rights reserved. PACS: 68.35.Ct; 83.50.Tq; 62.20.Mk; 46.50.#a; 81.40.Np

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1. Introduction 1.1. Brief overview of the paper Fracture mechanics is one of the most heavily developed branches of engineering science and applied mathematics [1}9]. It emerged from mathematical exercises in the early part of the 20th century into a closely knit collection of theoretical concepts and experimental procedures that are now widely used to ensure the safety of critical structures, ranging from airplanes to the housings of microelectronic devices. Some practitioners of this "eld feel that its development is essentially "nished. However, a group of physicists has recently begun to work in the area, and our number appears to be growing. The "rst task is therefore to explain the new developments that have created excitement in this mature area, and induced physicists to work on it after an absence, with few exceptions [10], of many decades. There are two separate lines of inquiry pulling people into fracture of brittle materials. The "rst is a set of puzzles about the dynamics of cracks. It is often stated that cracks do not reach the terminal velocity predicted by theory, and that they have an unexplained instability at a critical velocity. We will show that these puzzles are real, but have been di$cult to solve because they are not correctly stated. The real puzzle concerns energy dissipation at the crack tip, and the answer we will present is that when energy #ux to a crack tip passes a certain critical value, e$cient steady motion of the tip becomes unstable to the formation of microcracks that propagate away from the main crack. As it undergoes a hierarchy of instabilities, the ability of the crack tip to absorb energy is enormously increased. The second line of inquiry is the attempt to understand how things break from an atomic level. There is a broad consensus that this problem is best met by a direct attack from molecular dynamics simulations, watching cracks move one atom at a time. However, before moving to large computer simulations, we believe it is important to study analytical results so as to understand the qualitative e!ect of atomic discreteness upon crack motion. With these results in hand, many laboratory experimental results become comprehensible, the relation between simulations and laboratory experiments becomes clearer, and molecular dynamics simulations can be made much more e$cient. In fact, the puzzles in fracture dynamics, at both macroscopic and atomic scales, are manifestations of the same underlying phenomena. Our purpose in this review is to explain why the conventional puzzles have arisen, how to recast them, and how to explain them. We do not pretend to provide a general overview of fracture mechanics as a whole. For example, we focus upon brittle materials, and do not deal with ductile fracture [11], although dynamic elastic}plastic fracture is very well developed [7]. We will emphasize the dynamics of cracks once they start moving, rather than the most important engineering topic, which is the matter of reliably determining the point of initiation [9]. We are not challenging conventional fracture mechanics; all our "ndings must be compatible with it. Instead, we are answering some questions that conventional continuum fracture mechanics does not ask. The basic structure of this paper is E A summary of some of the principal features of dynamic fracture mechanics. E An explanation of why dynamic fracture mechanics appears to fail, why it does not actually do so, and why dynamic fracture mechanics is incomplete without certain crucial information about the nature of the crack tip that must be provided by computations at scales too small for continuum mechanics to follow.

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1.2. Scaling arguments 1.2.1. All objects are far from mechanical equilibrium The world is farther from equilibrium than most of us realize. Consider a piece of rock, of area A and height h. According to equilibrium principles the rock should not be able to sustain its own weight under the force of gravity if it becomes tall enough. We begin with a simple estimate of what the critical height should be. The gravitational potential energy of the rock is oAhg/2 where o is the density. By cutting the rock into two equal blocks of height h/2 and setting them side by side, this energy can be reduced to oAhg/4, for an energy gain of oAhg/4. The cost of the cut is the cost of creating new rock surface, which characteristically costs per unit area G"1 J/m. Taking the density to be o"2000 kg/m, the critical height at which it pays to divide the rock in two is h"(4G/og&1.4 cm .

(1)

So every block of stone more than 2 cm tall is unstable under its own weight. A similar estimate applies to steel or concrete. Yeats' observation that `Things fall aparta is a statement about equilibrium which fortunately takes a long time to arrive. 1.2.2. The energy barrier seems immense, but this is experimentally wrong If most objects are out of mechanical equilibrium, the next task is to estimate the size of the barriers holding them in place. An easy way to obtain a rough value is by imagining what happens to the atoms of a solid as one pulls it uniformly at two ends. At "rst, the forces between the atoms increase, but eventually they reach a maximum value, and the solid falls into pieces, as shown in Fig. 1. Interatomic forces vary greatly between di!erent elements and compounds, but the forces typically reach their maximum value when the distance between atoms increases by around 20% of their original separation. The force needed to stretch a solid slightly is (Fig. 1) F"EAd/¸ ,

(2)

Fig. 1. Mechanically stable con"gurations are often far from their lowest energy state. For example, a solid completely free of #aws would only pull apart when all bonds as long a plane snapped simultaneously, despite the fact that it would generally be energetically advantageous for the solid to be split.

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Table 1 The experimental strength of a number of materials in polycrystalline or amorphous form, compared to their theoretical strength (from [12,13]) Material

Young's modulus (GPa)

Theoretical strength (GPa)

Practical strength (GPa)

Practical/theoretical strength

Iron Copper Titanium Silicon Glass Plexiglas

195}205 110}130 110 110}160 70 3.6

43}56 24}55 31 45 37 3

0.3 0.2 0.3 0.7 0.4 0.05

0.006 0.005 0.009 0.01 0.01 0.01

where E is the material's Young's modulus, so the force per area needed to reach the breaking point is around p "F/A"E/5 . (3)  As shown in Table 1, this estimate is in error by orders of magnitude. The problem does not lie in the crude estimates used to obtain the forces at which bonds separate, but in the conception of the calculation. The "rst scaling argument greatly underestimated the practical resistance of solid bodies to separation, while this one greatly overestimates it. The only way to uncover correct orders of magnitude is to account for the actual dynamical mode by which brittle solids fail, which is by the propagation of a crack. 1.2.3. Cracks provide the most ezcient path to equilibrium As we shall see in Section 2.3, the presence of a crack in an otherwise perfect material leads to a stress singularity at its tip. For an atomically sharp crack tip, a single crack a few microns long is su$cient to explain the large gaps between the theoretical and experimental material strengths as shown in Table 1. The theoretical strength of glass "bers, for example, can be approached by means of acid etching of the "ber surface. The etching process serves to remove any initial microscopic #aws along the glass surface. By removing its initial #aws, a 1 mm glass "ber can lift a piano (but during the lifting process, we would not advise standing under the piano). The stress singularity that develops at the tip of a crack serves to focus the energy that is stored in the surrounding material and e$ciently use it to break one bond after another. Thus, the continuous advance of the crack tip, or crack propagation, provides an e$cient means to overcome the energy barrier between two equilibrium states of the system having di!erent amounts of mechanical energy. 1.2.4. The scaling of dynamic fracture The "rst analysis of rapid fracture was carried out by Mott [14], and then slightly improved by Dulaney and Brace [15]. It is a dimensional analysis which clari"es the basic physical processes, despite being wrong in many details, and consists of writing down an energy balance equation for crack motion. Consider a crack of length l(t) growing at rate v(t) in a very large plate where a stress p is applied  at the far boundaries of the system, as shown in Fig. 2. When the crack extends, its faces separate, causing the plate to relax within a circular region centered on the middle of the crack and with

J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

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Fig. 2. As a crack of length l expands at velocity v in an in"nite plate, it disturbs the surrounding medium up to a distance on the order of l. Fig. 3. The energy of a plate with a crack as a function of its length. In the "rst part of its history, the crack grows quasi-statically, and its energy increases. At l the crack begins to move rapidly, and energy is conserved. 

diameter of order l. The kinetic energy involved in moving a region of this size is Mv/2, where M is the total mass, and v is a characteristic velocity. Since the mass of material that moves is proportional to l, the kinetic energy is guessed to be of the form KE"C lv . (4) ) The region of material that moves is also the region from which elastic potential energy is being released as the crack opens. Therefore, the potential energy gained in releasing stress is guessed to be of the form PE"!C l . (5) . These guesses are correct for slowly moving cracks, but fail qualitatively as the crack velocity approaches the speed of sound, in which case both kinetic and potential energies diverge. This divergence will be demonstrated later, but for the moment, let us proceed fearlessly. The "nal process contributing to the energy balance equation is the creation of new crack surfaces, which takes energy Cl. The fracture energy C accounts not only for the minimal energy needed to snap bonds, but also for any other dissipative processes that may be needed in order for the fracture to progress; it is often orders of magnitude greater than the thermodynamic surface energy. However,

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for the moment, the only important fact is that creating new surface scales as the length l of the crack. So the total energy of the system containing a crack is given by E"C lv#E (l) , ) 

(6)

with E (l)"!C l#Cl . (7)  . Consider "rst the problem of quasi-static crack propagation. If a crack moves forward only slowly, its kinetic energy will be negligible, so only the quasi-static part of the energy, E , will be  important. For cracks where l is su$ciently small, the linear cost of fracture energy is always greater than the quadratic gain of potential energy, and in fact such cracks would heal and travel backwards if it were not for irreversible processes, such as oxidation of the crack surface, which typically prevent it from happening. That the crack grows at all is due to additional irreversible processes, sometimes chemical attack on the crack tip, sometimes vibration or other irregular mechanical stress. It should be emphasized that the system energy E increases as a result of these processes. Eventually, at length l , the energy gained by relieving elastic stresses in the body exceeds  the cost of creating new surface, and the crack becomes able to extend spontaneously. One sees that at l , the energy functional E (l) has a quadratic maximum. The Gri$th criterion for the onset of   fracture is that fracture occurs when the potential energy released per unit crack extension equals the fracture energy, C. Thus fracture in this system will occur at the crack length l where  dE /dl"0 . (8)  Using Eq. (7) we "nd that l "C/(2C ) .  . Eq. (7) now becomes

(9)

E (l)"E (l )!C (l!l ) . (10)    .  This function is depicted in Fig. 3. Much of engineering fracture mechanics boils down to calculating l , given things such as  external stresses, which in the present case have all been condensed into the constant C . Dynamic . fracture starts in the next instant, and because it is so rapid, the energy of the system is conserved, remaining at E (l ). Using Eqs. (6) and (10), with E"E (l ) gives     l l C 1!  . (11) v(t)" . 1!  "v

 l l C ) This equation predicts that the crack will accelerate until it approaches the speed v . The

 maximum speed cannot be deduced from these arguments, but Stroh [16] correctly argued that v should be the Rayleigh wave speed, the speed at which sound travels over a free surface.

 A crack is a particularly severe distortion of a free surface, but assuming that it is legitimate to represent a crack in this way, the Rayleigh wave speed is the limiting speed to expect. This result was implicit in the calculations of Yo!e [17]. In this system, one needs only to know the length at which a crack begins to propagate in order to predict all the following dynamics. As we will see presently, Eq. (11) comes astonishingly close to







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anticipating the results of a sophisticated mathematical analysis developed over 15 years. This result is especially puzzling since the forms (4) and (5) for the kinetic and potential energy are incorrect; these energies should actually diverge as the crack begins to approach the Rayleigh wave speed. The reason that the dimensional analysis succeeds anyway is that to "nd the crack velocity one takes the ratio of kinetic and potential energy; their divergences are of exactly the same form, and cancel out.

2. Continuum fracture mechanics In this section we will attempt to review brie#y the background, basic formalism and underlying assumptions that form the body of continuum fracture mechanics. We will "rst schematically describe the general `game plana pursued within fracture mechanics. The following subsections will introduce the main `playersa or concepts on which the current continuum description of dynamic fracture is based. As the work described in this review mainly deals with the dynamics of a crack in thin plates or quasi-2D media, we will "rst discuss the reduction of linear elasticity to two dimensions. Some basic concepts common to both static and dynamic cracks will then be discussed. These will include the creation of a singular stress "eld at the tip of a static crack together with criteria for the onset and growth of a moving crack. We then turn to a description of moving cracks. We will "rst describe the formalism used to describe a crack moving at a given velocity and, using this formalism, look at the dependence of the near tip stress "elds as a function of the crack's velocity. The stage will then be set for a description of the general formalism used to determine an equation of motion for a moving crack in an in"nite medium. The central result of dynamics fracture mechanics, which is obtained by equating the (velocity dependent) elastic energy #owing into the tip of a moving crack with the dissipative mechanisms absorbing it, will be described. This result provides an equation of motion for a crack moving along a straight-line trajectory. We then discuss criteria for determination of a crack's path and conclude the section by discussing a number of points that the theory cannot address. 2.1. Structure of fracture mechanics The structure of fracture mechanics follows the basic ideas used in the scaling arguments described in the previous section. The general strategy is to solve for the displacement "elds in the medium subject to both the boundary conditions and the externally applied stresses. The elastic energy transported by these "elds is then matched to the amount of energy dissipated throughout the system, and an equation of motion is obtained. For a single moving crack, as in the scaling argument above, the only energy sink existing in the system is at the tip of the crack itself. Thus, an equation of motion can be obtained for a moving crack if one possesses detailed knowledge of the dissipative mechanisms in the vicinity of the tip. 2.1.1. Dissipation and the process zone Unfortunately, the processes that lead to dissipation in the tip vicinity are far from simple. Depending on the type of material, there is a large number of complex dissipative processes ranging from dislocation formation and emission in crystalline materials to the complex unraveling and

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fracture of intertangled polymer strands in amorphous polymers. At "rst glance, the many di!erent (and in many cases, poorly understood) dissipative processes that are observed would appear to preclude a universal description of fracture. A way around this problem was proposed by Irwin and Orowan [18,19]. Fracture, together with the complex dissipative processes occurring in the vicinity of the tip, occurs due to intense values of the stress "eld that occur as one approaches the tip. As we will show shortly, if the material surrounding the tip were to remain linearly elastic until fracture, a singularity of the stress "eld would result at the mathematical point associated with the crack tip. Since a real material cannot support singular stresses, in this vicinity the assumption of linearly elastic behavior must break down and material dependent, dissipative processes must come into play. Irwin and Orowan independently proposed that the medium around the crack tip be divided into three separate regions as follows (Fig. 4). E ¹he process zone: In the region immediately surrounding the crack tip, called the process zone (or cohesive zone), all of the nonlinear dissipative processes that ultimately allow a crack to move forward, are assumed to occur. Fracture mechanics avoids any sort of detailed description of this zone, and simply posits that it will consume some energy C per unit area of crack extension. The size of the process zone is material dependent, ranging from nanometers in glass to microns in brittle polymers. The typical size of the process zone can be estimated by using the radius at which an assumed linear elastic stress "eld surrounding the crack tip would equal the yield stress of the material. E ¹he universal elastic region: Everywhere outside of the process zone the response of the material can be described by continuum linear elasticity. In the vicinity of the tip, but outside of the process zone, the stress and strain "elds adopt universal singular forms which solely depend on the symmetry of the externally applied loads. In two dimensions the singular "elds surrounding the process zone are entirely described by three constants, called stress intensity factors. The stress intensity factors incorporate all of the information regarding the loading of the material and are related, as we shall see, to the energy #ux into the process zone. The larger the overall size of the body in which the crack lives, the larger this region becomes. In rough terms, for given values of the stress intensity factors, the size of this universal elastic region scales as (¸, where ¸ is the macroscopic scale on which forces are applied to the body. Thus the assumptions of fracture mechanics become progressively better as samples become larger and larger. E Outer elastic region: Far from the crack tip, stresses and strains are described by linear elasticity. There is nothing more general to relate; details of the solution in this region depend upon the locations and strengths of the loads, and the shape of the body. In special cases, analytical solutions are available, while in general one can resort to numerical solution. At "rst glance, the precise linear problem that must be solved might seem inordinately complex. How can one avoid needing explicit knowledge of complicated boundary conditions on some complicated loop running outside the outer rim of the process zone? The answer is that so far as linear elasticity is concerned, viewed on macroscopic scales the process zone shrinks to a point at the crack tip, and the crack becomes a branch cut. Replacing the complicated domain in which linear elasticity actually holds with an approximate one that needs no detailed knowledge of the process zone is another approximation that becomes increasingly accurate as the dimensions of the sample, hence the size of the universal elastic region, increase. The assumption that the process zone

J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

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Fig. 4. Structure of fracture mechanics. The crack tip is surrounded by a region in which the physics is unknown. Outside this process zone is a region in which elastic solutions adopt a universal form.

in a material is encompassed within the universal elastic region is sometimes called the assumption of small-scale yielding. The dissipative processes within the process zone determine the fracture energy, C, de"ned as the amount of energy required to form a unit area of fracture surface. In the simplest case, where no dissipative processes other than the direct breaking of bonds take place, C is a constant, depending on the bond energy. In the general case, C may well be a complicated function of both the crack velocity and history and di!er by orders of magnitude from the surface energy de"ned as the amount of energy required to sever a unit area of atomic bonds. No general "rst principles description of the process zone exists, although numerous models have been proposed (see e.g. [20]). 2.1.2. Conventional fracture modes It is conventional to focus upon three symmetrical ways of loading a solid body with a crack. These are known as modes, and are illustrated in Fig. 5. A generic loading situation produced by some combination of forces without any particular symmetry is referred to as mixed mode fracture. Understanding mixed-mode fracture is obviously of practical importance, but since our focus will be upon the physics of crack propagation rather than upon engineering applications, we will restrict attention to the special cases in which the loading has a high degree of symmetry. The fracture mode that we will mainly deal with in this review is Mode I, where the crack faces, under tension, are displaced in a direction normal to the fracture plane. In Mode II, the motion

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Fig. 5. Illustration of the three conventional fracture modes, which are characterized by the symmetry of the applied forces about the crack plane.

of the crack faces is that of shear along the fracture plane. Mode III fracture corresponds to an out of plane tearing motion where the direction of the stresses at the crack faces is normal to the plane of the sample. One experimental di$culty of Modes II and III is that the crack faces are not pulled away from one another. It is unavoidable that contact along the crack faces will occur. The resultant friction between the crack faces contributes to the forces acting on the crack, but is di$cult to measure precisely. For these reasons, of the three fracture modes, Mode I corresponds most closely to the conditions used in most experimental and much theoretical work. In two-dimensional isotropic materials, Mode II fracture cannot easily be observed, since slowly propagating cracks spontaneously orient themselves so as to make the Mode II component of the loading vanish near the crack tip [21], as we will discuss in Section 2.9. Mode II fracture is however observed in cases where material is strongly anisotropic. Both friction and earthquakes along a prede"ned fault are examples of Mode II fracture where the binding across the fracture interface is considerably weaker than the strength of the material that comprises the bulk material. Pure Mode III fracture, although experimentally di$cult to achieve, is sometimes used as a model system for theoretical study since, in this case, the equations of elasticity simplify considerably. Analytical solutions, obtained in this mode, have provided considerable insight to the fracture process. 2.1.3. Universal singularities near the crack tip As one approaches the tip of a crack in a linearly elastic material, the stress "eld surrounding the tip develops a square root singularity. As "rst noted by Irwin [22], the stress "eld at a point (r,h) near the crack tip, measured in polar coordinates with the crack line corresponding to h"0, takes the form p "K (1/(2pr) f ? (v,h) , GH GH ?

(12)

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where v is the instantaneous crack velocity, and a is an index running through Modes I}III. For each of these three symmetrical loading con"gurations, f ? (v,h) in Eq. (12) is a known universal GH function. The coe$cient, K , called the stress intensity factor, contains all of the detailed informa? tion regarding sample loading and history. K will, of course, be determined by the elastic "elds ? that are set up throughout the medium, but the stress that locally drives the crack is that which is present at its tip. Thus, this single quantity will entirely determine the behavior of a crack, and much of the study of fracture comes down to either the calculation or measurement of this quantity. One of the main precepts of fracture mechanics in brittle materials is that the stress intensity factor provides a universal description of the fracture process. In other words, no matter what the history or the external conditions in a given system, if the stress intensity factor in any two systems has the same value, the crack tip that they describe will behave in the same way. The universal form of the stress intensity factor allows a complete description of the behavior of the tip of a crack where one need only carry out the analysis of a given problem within the universal elastic region (see Section 2.1.1). For arbitrary loading con"gurations, the stress "eld around the crack tip is given by three stress intensity factors, K , which lead to a stress "eld that is a linear combination of the pure Modes: ?  1 f ? (v,h) . (13) p " K GH ? (2pr GH ? 2.1.4. The relation of the stress intensity factor to energy yux How are the stress intensity factors related to the #ow of energy into the crack tip? Since one may view a crack as a means of dissipating built-up energy in a material, the amount of energy #owing into its tip must in#uence its behavior. Irwin [23] showed that the stress intensity factor is related to the energy release rate, G, which is de"ned as the quantity of energy #owing into the crack tip per unit fracture surface formed. The relation between the two quantities has the form  1!l A (v)K , (14) G" ? ? E ? where l is the Poisson ratio of the material and the three functions A (v) depend only upon the ? crack velocity v. This relation between the stress intensity factor and the fracture energy is accurate whenever the stress "eld near the tip of a crack can be accurately described by Eq. (12). The near-"eld approximation of the stress "elds surrounding the crack tip embodied in Eq. (13) becomes increasingly more accurate as the dimensions of the sample increase. 2.2. Linear elasticity 2.2.1. Reductions to two dimensions Most of the theoretical work that we will describe in this review is performed in 2-D (or quasi-2-D) systems. In this subsection we will perform a reduction of the full 3-D elastic description of a crack to two dimensions in three important cases; for Mode III fracture and Mode I fracture in very thin and very thick plates. The "rst case, Mode III fracture, is an important model system where much analytic work can be performed with the corresponding gain in intuition of qualitative

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features in fracture. The second case, Mode I fracture of a thick plate, describes stress and strain conditions of importance in describing fracture in the immediate vicinity of the crack tip. The third case, Mode I fracture in thin plates, corresponds to much of the experimental work that we will describe. Our starting point is the equation of motion for an isotropic elastic body in the continuum limit, o Ru/Rt"(j#k) ( ) u)#k u ,

(15)

originally found by Navier. Here u is a "eld describing the displacement of each mass point from its original location in an unstrained body and o is the density. The constants k and j are called the LameH constants, have dimensions of energy per volume, and are typically of order 10 erg/cm. We de"ne the linear elastic strain tensor [24]






1 Ru Ru G# H . e " (16) GH 2 Rx Rx H G When a linear stress}strain relation exists in a homogeneous isotropic medium, the stress tensor, p , is de"ned by GH p "jd e #2ke . GH GH II GH I

(17)

2.2.2. Mode III The simplest analytical results are for pure Mode III, illustrated in Fig. 5. The only non-zero displacement is u , and it is a function of x and y alone. The only non-vanishing stresses in this case X are p "k Ru /Rx VX X

(18)

and p "k Ru /Ry . WX X The equation of motion for u is X 1 Ru X" u , X c Rt

(19)

(20)

where c"(k/o .

(21)

Therefore, the vertical displacement u obeys the ordinary wave equation. X 2.2.3. Mode I: plane strain Consider a sample that is extremely thick along z (see Fig. 5), and where all applied forces are uniform in the z direction. Since all derivatives with respect to z vanish, all "elds can be viewed as functions of x and y alone. This situation is called plane strain. The reduction to two dimensions is quite simple, but this geometry is rarely convenient for experiments.

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2.2.4. Mode I: plane stress A third case in which the equations of elasticity reduce to two dimensions corresponds to pulling on a thin plate in Mode I, and is called plane stress. If the scale over which stresses are varying in x and y is large compared with the thickness of the plate along z, then one might expect that the displacements in the z direction will come quickly into equilibrium with the local x and y stresses. When the material is being stretched, (think of pulling on a balloon), the plate will contract in the z direction, and when it is being compressed, the plate will thicken. Therefore, one guesses that u "zf (u ,u ) , (22) X V W and that u , and u are independent of z. One can deduce the function f by noticing that p must V W XX vanish on the face of the plate. This means that j





Ru Ru Ru V# W #(j#2k) X"0 Ry Rz Rx

(23)

at the surface of the plate, which implies that





(24)





(25)

Ru j Ru Ru V# W . f (u ,u )" X"! V W j#2k Rx Ry Rz So Ru Ru Ru Ru 2k Ru V# W# X" V# W , Rx j#2k Rx Ry Rz Ry and one can write






Ru Ru Ru ?# @ , A #k p "jI d ?@ ?@ Rx Rx Rx @ ? A where jI "2kj/(j#2k) ,

(26)

(27)

and a and b now range only over x and y. Therefore, a thin plate obeys the equations of two-dimensional elasticity, with an e!ective constant jI , so long as u is dependent upon u and X V u according to Eq. (24). In the following discussion, the tilde over jI will usually be dropped, with W the understanding that the relation to three-dimensional materials properties is given by Eq. (26). The equation of motion is still Navier's equation, Eq. (15), but restricted to two dimensions. A few random useful facts: materials are frequently described by the Young's modulus E and Poisson ratio l. In terms of these constants, El , j" (1#l)(1!2l)

E E jI " , k" . 2(1!l) 2(1#l)

(28)

The following relation will be useful in discussing two-dimensional static problems. First note that e ) u"R (j#2k)p . ? ??

(29)

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J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

Second, taking the divergence of Eq. (15), one "nds that o Rp ??" p . ?? j#2k Rt

(30)

Therefore, ) u obeys the wave equation, with the longitudinal wave speed c "(j#2k)/o . J Similarly, ;u also obeys the wave equation, but with the shear wave speed c "(k/o . 

(31)

(32)

2.2.5. The transition from 2D to 3D Near the tip of a crack in a plate, stresses become severe enough that the approximations leading to two-dimensional plane stress elasticity fail. Nakamura and Parks [25] have discussed the way this happens; if the thickness of the plate along z is denoted by d, then at distances from the crack tip much larger than d all "elds are described by equations of plane stress. At distances from the crack tip much less than d, and away from the x}y surfaces of the plate, the "elds solve the equations of plane strain. 2.3. The Inglis solution for a static crack in Mode III Why do cracks have a profound e!ect on the strength of materials? As we observed in Table 1, a huge discrepancy exists between the practical and theoretical strengths of materials. The reason for the discrepancy has been understood since the "rst decades of this century. If one takes a plate, puts an elliptical hole in it, and pulls, then as "rst found by Inglis in 1913, the stresses at the narrow ends of the hole are much larger than those exerted o! at in"nity, as shown in Fig. 6. We see then that a crack acts as an `ampli"era of stresses, thereby causing elastic energy to be preferentially focused into its tip. Thus, the existence of a crack will lead to a large decrease in the e!ective strength of a material. The ratio of maximum to applied stress is l Maximum Stress "2 , o Applied Stress

(33)

where l is the length of the crack and o the radius of curvature at its tip. This means that if one assumes that typical solids have cracks with tips of size 1 A> , and of length 10A> , then one can account for the discrepancies in Table 1. We will now derive Eq. (33). Solving even the static equations of linear elasticity is frequently a complicated and di$cult a!air. To simplify matters as much as possible, assume that the stresses applied to the plate coincide to the conditions of anti-plane shear stress, as shown in Fig. 5, so the only nonzero displacement is u . Returning to Eq. (20), one sees that the static equation of linear elasticity is now X simply

u "0 . X

(34)

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17

Fig. 6. The stresses at the tips of an elliptical hole in a solid are much greater than those applied o! at in"nity.

Eq. (34) is Laplace's equation, so the whole theory of complex variables can be brought to bear in order to "nd solutions. For the boundary problem at hand, conformal mapping is the appropriate technique. Since u is a solution of Laplace's equation, it can be represented by X u " ( (f)# (f)) , (35) X  where is analytic, and f"x#iy. One can easily write down the asymptotic behavior of . Far from the hole, the displacement u (x,y) just increases linearly with y, so X

"!iRf . (36) In Eq. (36), the constant R is dimensionless, but one can think of it measuring the stress in units of the LameH constant k. How does the presence of the hole a!ect the stress "eld? Because the edges of the hole are free, the stress normal to the edge must vanish. If s is a variable which parameterizes the edge of the hole, so that [x(s),y(s)] (37) travels around the boundary of the hole as s moves along the real axis, then requiring the normal stress to vanish means that Ru Rx Ru Ry X ! X "0 Rx Rs Ry Rs

(38)

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Using the representation of u in Eq. (35), one "nds that X R /Rs"R M /Rs

(39)

on the boundary, or since is arbitrary up to a constant,

(f)" (f)

(40)

when f lies on the boundary. To illustrate the use of Eq. (40), let us de"ne u such that f"l/2(u#m/u) ,

(41)

When u lies on the unit circle, in other words u"e F ,

(42)

with h real, f traces out an elliptical boundary. When m"0, the boundary is a circle of radius l/2, and when m"1, the boundary is a cut along the real axis extending from !l to #l. Considering

as a function of u, one has

(u)" (u)" M (1/u) ,

(43)

since uN "1/u on the unit circle. Eq. (43) can now be analytically continued o! the unit circle. Outside of the hole must be completely regular, except for the fact that it diverges as !iRf for large f. But when f is large, u+f, so from Eq. (41),

&!iRu as uPR .

(44)

Consulting Eq. (43) one is forced to conclude that

M (1/u)&!iRu ,

(45)

which means that

M (u)&!iR/u as uP0

(46)

but has no other singularities within the unit circle. Having determined all the possible singularities of , it is determined up to a constant. It must be given by

(u)"!iRu#iR/u

(47)

and, substituting Eq. (41), we have f f

(f)"!iR (1#1!ml/f)#iR (1!1!ml/f) . l lm

(48)

The case in which mP1 is particularly interesting. The hole becomes a straight crack along [!l,l]. Notice that has a branch cut over exactly the same region. The displacement u is "nite X approaching the tip of the crack, but the stress p "k Ru /Ry&1/(z!l as zPl , WX X diverges as one approaches the tip of the crack.

(49)

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2.3.1. Remarks about the singularity Although this calculation concerns a particular case, the existence of a square root stress singularity at the tip of a crack is of general validity, in accord with Eq. (12). Giving a crack a "nite radius of curvature is one way to cut o! this singularity. A useful application of this idea is, for example, a common mechanic's trick to arrest the advance of a crack in a damaged engine block. To arrest the crack, the mechanic will drill a small hole at its tip. It may seem counter-intuitive at "rst to `"xa a hole by making a larger one, but by increasing the radius of curvature at the tip, the mechanic, in e!ect, is canceling the singularity in the stress "eld and thus considerably strengthening the engine block. 2.3.2. Static cracks in Mode I The method of conformal mapping we described for Mode III cracks has been extended to Mode I by Muskhelishvili [26]. Matters are somewhat more complicated, since one must solve the biharmonic equation rather than Laplace's equation, and solve for two complex functions not one. Muskhelishvili alone solved enough problems with these techniques to "ll several hundred pages, and hundreds of publications have since been devoted to solutions of fracture problems using these methods. 2.4. Linear elastic equations for moving solutions in Mode I fracture Eq. (49) shows that in an elastic medium where uniform stress is applied at the boundaries, the stress "eld at the tip of a static crack becomes singular. We now turn to the case of a moving crack and examine the structure of the stress "eld at the tip of a moving Mode I crack. So as to be able to compare eventually with experiment, and because this review focuses upon crack dynamics, we carry out the analysis in the complicated case of Mode I loading. The "rst step is to develop the general form of the stress and displacement "elds for a moving crack. Begin with the dynamical equation for the strain "eld u of a steady state in a frame moving with a constant velocity v in the x direction, (j#k) ( ) u)#k u"ov Ru/Rx .

(50)

Divide u into transverse and longitudinal parts so that u"u #ul , 

(51)

with










Rv Rv ul"evl and u " ,!  .  Rx Ry

(52)

It follows immediately that





R R (j#2k) !ov ul"! k !ov u ,f (x,y) . Rx Rx 

(53)

In the end, it will be possible to set f to zero, but some intermediate steps are needed to see why this is legitimate. Acting on the left-hand side of Eq. (53) with the operator [R/Ry,!R/Rx] gives zero,

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while acting on u with [R/Rx,R/Ry] also gives zero. Therefore  Rf Rf W! V"0 , Rx Ry Rf Rf V# W"0 , Ry Rx

(54a) (54b)

meaning that f obeys Cauchy's equations, and f !if is an analytic function of x#iy. We then V W have

 

 

a

R R #

vl"0 , Rx Ry

(55)

b

R R # Rx Ry

(56)

v "0 , 

where ov v a"1! "1! , j#2k cl v ov b"1! "1! . c k  Therefore, the general form of the potentials is vl"vl (z)#vl (z)#vl (x#iay)#vl (x#iay)

(57a) (57b)

(58)

(59) v "v(z)#v(z)#v(x#iby)#v(x#iby) ,      subject to the constraint of Eq. (53), which gives a relation between vl and v.  In fact, the purely harmonic pieces vl and v disappear entirely from expression (51) for u. They  result from the freedom one has to add a harmonic function to vl and v simultaneously, and can be  neglected; f could have been set to zero from the beginning. De"ning (z)"Rvl (z)/Rz and t(z)"Rv(z)/Rz we have for u,  (60a) u " (z )# (z )#ib[t(z )!t(z )] , @ @ V ? ? u "ia[ (z )! (z )]![t(z )#t(z )] , (60b) W ? ? @ @ where z "x#iay, z "x#iby . (61) ? @ Eq. (60) gives a general form for elastic problems which are in steady state moving at velocity v. De"ne also U"R (z)/Rz and W"Rt(z)/Rz. Then the stresses are given by p #p "2(j#k)[U(z )#U(z )](1!a) , VV WW ? ? p !p "2k+(1#a)[U(z )#U(z )]#2ib[W(z )!W(z )], , @ @ VV WW ? ? 2p "2k+2ia[U(z )!U(z )]!(b#1)[W(z )#W(z )], . VW ? ? @ @

(62a) (62b) (62c)

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21

It is worth writing down the stresses directly as well: (62d) p "!k(1#b)[U(z )#U(z )]!2ibk[W(z )!W(z )] , @ @ WW ? ? p "k(1#2a!b)[U(z )#U(z )]#2ibk[W(z )!W(z )] . (62e) VV ? ? @ @ The de"nitions of a and b in Eq. (57) have been used to simplify the expressions. To solve a general problem, one has to "nd the functions and t which match boundary conditions. It is interesting to notice that when vP0, the right-hand side of Eq. (62a) goes to zero as well. Since one will be "nding the potentials from given stresses at the boundaries, U must diverge as 1/v, and the right-hand side of Eq. (62) will turn into a derivative of U with respect to a. The static theory has, therefore, a di!erent structure than the dynamic theory, although the dynamic theory is, in fact, more straightforward. 2.5. Mode I, structure near the tip, stress intensity factors As a "rst application of Eqs. (60)}(62), we will "nd the form of the stresses around the tip of a crack moving under Mode I loading. Assume the crack to lie along the negative x-axis, terminating at x"0, and moving forward. The problem is assumed symmetric under re#ection about the x-axis, but no other assumption is needed. We know that in the static case, the stress "elds have a square root singularity at the crack tip. We will assume the same to be true in the dynamic case also (The assumption is veri"ed in all cases that can be worked explicitly.). Near the crack tip, we assume that

(z)&(B #iB )z , P G t(z)&(D #iD )z . P G We "rst appeal to symmetry. Since the crack is loaded in Mode I, the displacements obey

(63) (64)

u (!y)"u (y), u (!y)"!u(y) . (65) V V W Placing Eq. (64) into Eq. (60) and using Eq. (65), we "nd immediately that B "D "0. Thus G P U(z)&B /z, W(z)&iD /z . (66) P G We also observe that the square roots in Eq. (64) must be interpreted as having their cuts along the negative x-axis, corresponding to the crack. On the crack surface the stresses are relaxed. We thereby have two boundary conditions which require that p and p vanish on the surface of VW WW a crack. Upon substituting Eq. (66) into Eq. (62) we "nd that the condition upon p is satis"ed WW identically for x(0, y"0. However, substituting into Eq. (62d) with y"0 we "nd that p "ki+2aB !(b#1)D ,+1/(x!1/(x, . VW P G

(67)

Thus D /B "2a/(b#1) . G P This relation is enough to "nd the angular structure of stress "elds around a crack.

(68)

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Using Eq. (68) to substitute Eq. (66) into Eq. (62) we "nd that

       



K 1 1 ' p " (b#1)(1#2a!b) # VV (2p D (z? (zN ? 1 1 !4ab # , (z (z @ @ 1 1 K 1 1 ' p " 4ab # !(1#b) # WW 2(2pD (z (z (zN (zN @ ? @ ? 1 K 1 1 1 ' 2ia(b#1) p " ! ! # , VW 2(2p D (z (zN ? (z (zN ? @ @





(69a) (69b)



,

(69c) (69d)

with D"4ab!(1#b) .

(69e)

Di!erent features of Eq. (69) have varying degrees of signi"cance. The feature of greatest physical importance is the overall scale of the stress singularity, which is characterized by the Mode I stress intensity factor (70) K " lim (2pxp . WW ' V> W As we mentioned in Section 2.1.3, and will calculate in the following Subsection, the stress intensity factor is directly related to energy #ux into a crack tip. In addition, Eq. (69) carries information about the angular structure of the stress "elds. This information can be used in two ways. Experimentally, it can be used to check the predictions of fracture mechanics, and to obtain measurements of the stress "elds surrounding rapidly moving cracks, as we will discuss in Section 3.2. Theoretically, it can be used to make predictions about the direction of crack motion, and the conditions under which a crack will bifurcate. We discuss these uses further in Section 2.9. It is important to note that although Eq. (69) was derived for cracks moving at a constant speed, the same expressions are valid for cracks that accelerate and decelerate, just so long as the change in crack velocity is small during the time needed for sound to travel across the region of the universal elastic singularity. A demonstration of this claim is discussed by Freund [7]. 2.6. Mode I, example of speci,c loading We now brie#y sketch the solution of a speci"c case in which the "elds about the crack tip can be worked out explicitly. A source in which to "nd problems of this type worked out in more explicit detail is Ref. [26]. Suppose that a crack is loaded by two delta-function stresses, located a distance l behind the crack tip, moving with it in steady state at velocity v, and of strength !p , as shown   in Fig. 7. That is, lim p (x,y)"!p d(x#l ) for x(0 . WW   W>

(71)

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23

Fig. 7. A crack is loaded by the application of two delta-function stresses of strength p behind the tip. 

Taking the tip of the crack to sit at the origin, the stress and displacement "elds are continuous and di!erentiable apart from a branch cut starting at the origin and running backwards along the negative x-axis. Denote by U (x) the functions ! (72) U (x), lim U(x$iy) , ! > W with W (x) de"ned similarly. Because of the branch cut, for x(0, U (x)"!U (x), and ! > \ W (x)"!W (x). For Mode I loading, p "0 for yP0> and all x. From Eq. (62c) > \ VW 2ia[U !UM ]"(b#1)[W #WM ] , (73) > \ > \ using the fact that W(x#ie)"WM (x!ie). A bar over a function means that if one expresses the original function as a power series, one replaces all the coe$cients with their complex conjugates. The function F (x)"2iaW (x)!(1#b#1)W (x) (74) > > > is de"ned for all x, and can be analytically continued above the x-axis, where it is related to stresses, and must be free of singularities. Similarly, F must be free of singularities below the real axis. \ Whenever two complex functions are equal, one without singularities above the x-axis, and the other without singularities below the x-axis, the two must individually equal a constant. The constant must be zero, since all stresses die o! to zero far from the crack. It follows that F "F "0, and one has "nally > \ 2iU (x)"(1#b)W (x) and 2iUM (x)"(1#b)WM (x) . (75) > > \ \ We now turn to the boundary condition on p , which from Eq. (62d) is, for x(0, WW p "!k(1#b)(U #UM )!2ibk(W !WM )"!p d(x#l ) . (76) WW > \ > \  

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J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

Employing Eqs. (75) and (69), Eq. (76) becomes p "!p d(x#l )"(kD/2ia)(W !WM ) . WW   > \ Representing the delta function as

(77)

1 i d(x#l )" , (78)  p x#l #ie  one can argue that the only complex function that decays properly at in"nity, has a singularity no worse than a square root at the origin, and obeys Eq. (77) is



ia p l  . W (x)" (79) > Dkp x#l #ie x  The function W(z) can now be obtained by analytical continuation of W (x). In particular, the stress > p can easily be found for x'0 from Eq. (77) and it is WW 1 l p   . p " (80) WW p x x#l  The stress intensity factor associated with this stress "eld is



K "p (2/pl . '   This result will be useful in our discussions of cohesive zone models in Section 6.1.

(81)

2.7. The J integral and the equivalence of the Irwin and Gri.th points of view What are the conditions under which a crack propagates? In the calculations above we have calculated the value of the stress "elds at the tip of a moving crack, but have not addressed the conditions under which a crack will actually move. In 1920, A.A. Gri$th [27] proposed that fracture occurs when the energy per unit area released by a minute extension of a crack is equal to that necessary to create new fracture surface, C. This idea is the "nal simplifying assumption of fracture mechanics. In general form, it states that the dynamics of a crack tip depend only upon the total energy #ux G per unit area into the process zone. All details about the spatial structure of the stress "elds are irrelevant. The energy G creates new fracture surfaces, and also is dissipated in numerous ways near the crack tip. The form the crack velocity v takes can be a general function, v(G). Conventionally, response of the crack is expressed in a di!erent way. It is usual to write !(v) to represent the energy consumed in the process zone by a crack as a function of its velocity, in which case the equation of motion for a crack is G"C(v) .

(82)

Engineering fracture mechanics is mainly concerned with the conditions under which a static crack begins to move. The critical fracture energy C is the minimal energy per unit area needed for  a crack to move ahead, irrespective of velocity. It is usually assumed that the velocity consuming the minimal energy is a very small one, but this assumption is not necessarily correct. One can

J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

25

equivalently de"ne a critical stress intensity factor K at which the crack "rst begins to move. The ' equivalence is provided by Eq. (14), which we will now derive. In the following calculation, adopt the summation convention for repeated indices. Energy #ux may be found from the time derivative of the total energy. We have

 



d 1 Ru o d ?p [¹#;]" dx dy uR uR # , (83) ? ? dt 2 Rx ?@ 2 dt @ where ¹ and ; are, respectively, the total kinetic and potential energies within the entire medium. The spatial integral in Eq. (83) is taken over a region which is static in the laboratory frame. So

 



d RuR [¹#;]" dx dy ou( uR # ? p , (84) ? ? Rx ?@ dt @ where the symmetry of the stress tensor under interchange of indices is used for the last term. Using the equation of motion ou( "(R/Rx )p , ? @ ?@ we have

   

(85)



R RuR p uR # ? p Rx ?@ ? Rx ?@ @ @ R " dxdy [p uR ] ?@ ? Rx @

dxdy

(86) (87)

uR p n , (88) ? ?@ @ .1 where the integral is now over the boundary, RS, of the system, and nL is an outward unit normal. As we see from Eq. (88), energy is transported by a #ux vector j whose components are "

j "p uR . (89) ? ?@ @ The total energy #ux J per unit time into the crack tip is called the J integral. A convenient contour for the integration is indicated in Fig. 8. For a crack loaded in pure Mode I, using the asymptotic forms, Eq. (69), for p and the corresponding expressions for u from Eq. (69a), one "nds that J is WW W 1 a K , (90) J"v(1!b) 2k 4ab!(1#b) ' where K is the stress intensity factor de"ned by Eq. (70), with a subscript I to emphasize that the ' result is speci"c to Mode I loading. Thus, the energy release rate G in the case of pure Mode I loading is G"J/v"(1!b)

1 a 1!l K, A (v)K . ' ' 2k 4ab!(1#b) ' E

(91a)

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J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

Fig. 8. The energy #owing to a crack tip can be computed by integrating the energy #ux across a contour that surrounds it. The most convenient contour is depicted by the dotted line in this diagram; it runs along the x-axis just below the crack, closes at in"nity, and runs back along the axis just above the crack. This contour is easier to handle analytically than one that has a vertical segment at some distance to the right of the crack tip.

The corresponding result for pure Mode II loading is G"(1!b)

1 b 1!l K, A (v)K , '' '' '' 2k 4ab!(1#b) E

(91b)

while for Mode III loading it is G"vK /2ak . (91c) ''' It is valuable to look at the quasistatic limit of Eq. (91) where vP0. In this limit each of the functions A (v)P1 and Eq. (91a), for example, becomes ? G "((1!l)/E)K . (92) T ' In the general case of mixed mode fracture, the energy release is given by linear combinations of Eq. (91), as in Eq. (14). 2.7.1. Signixcance and limitations The functions A (v), in Eq. (91), are universal function in the sense that they are independent of ? most details of the system's loading or geometric con"guration. They do depend upon the instantaneous velocity of the crack. Eq. (91) is of great signi"cance in the study of fracture. Assuming that there is no energy sink in the system other than at the tip of the crack, it relates the total #ow of energy from the entire elastic medium to the tip of the crack. Setting the #ow equal to the energy dissipated in the process zone determines an equation of motion for the crack. It is important to emphasize assumptions that were tacitly made in the derivation of this relation. The "rst is that near "eld descriptions of stress and displacement "elds given by Eq. (69), Eq. (60) is valid. If, for example, the process zone is on the order of 1 mm in a sample whose dimensions are a few centimeters, the value of the stress "eld on the contour RS used in Eq. (88) will not be approximated well by the asymptotic forms of the stress and displacement "elds, invalidating Eq. (91).

J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108

27

In obtaining an equation of motion for a crack, an additional ingredient is missing. Energy balance carries no information about a crack's path. We have assumed that the crack travels in a straight line. We will return to this issue in Section 2.9. The rules determining paths of slowly moving cracks are known, but for rapidly moving cracks, the problem is largely unsolved. 2.8. The general equation for the motion of a crack in an inxnite plate 2.8.1. General formalism An equation of motion for a crack now comes down to the calculation of either the energy release rate, G or, equivalently, the dynamic stress intensity factor, K. We will now show how to "nd a general expression for K as a function of the loading history of a crack, its length, and its velocity. This general expression will then be used to derive an equation of motion for a moving crack in a number of important situations. The calculation, due to Eshelby [28], Freund [7], Kostrov [29,30], and Willis [31] is somewhat limited. It applies only to a perfectly straight semi-in"nite crack in an in"nite plate, with loads applied to the crack faces. Given these restrictions the calculation is exact, and holds with remarkable generality. The calculation is somewhat heavy going, and in the end reproduces the scaling result of Eq. (11) with almost no modi"cation, as a special case. Our presentation most closely follows Willis [31]. Because this calculation is performed in the context of linear elasticity, it must be posed as a boundary-value problem. The most general problem to which a solution has been found is 1. The crack is a semi-in"nite branch cut running along a straight line an in an in"nite isotropic two-dimensional elastic plate. 2. The velocity of the crack v(t) is not required to be constant. The position of the tip is l(t)"Rdt v(t). In the boundary-value problem, l(t) and v(t) are assumed to be known. The only restriction is that v(t) must be less than relevant sound speeds at all times. 3. External stresses p are permitted only along the crack line; They are allowed arbitrary time  and space dependence otherwise. The most faithful experimental realizations of this restriction load cracks by placing wedges between the crack faces. 4. The calculation is carried out in the three symmetric loading modes, I, II and III. The symbols u, p, and c will denote a displacement, a stress, and a sound speed in each case, as shown in Table 2. In all cases, u(x,t)"0 for x'l(t), by symmetry. 5. The end result of the calculation is the energy #ux G as a function of the position l(t) of the crack, of the instantaneous velocity v(t), and as a functional of the load p (x,t).  Table 2 See Fig. 5 for a de"nition of the coordinatesystem. c is the 0 Rayleigh wave speed, determined by the zero of D in Eq. (69e), usually around 90% of the transverse wave speed c 

u denotes p denotes c denotes

Mode I

Mode II

Mode III

u (x,y"0>,t) W p (x,y"0>,t) WW c 0

u (x,y"0>,t) V p (x,y"0>,t) WV c 0

u (x,y"0>,t) X p (x,y"0>,t) WX c 

28

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The strategy is to look for a Green function, G operating on the displacement "eld such that:



G*u, dx dt G(x!x,t!t)u(x,t)"p(x,t) .

(93)

The fracture problem is soluble if the Green function G has a special form. Writing



G(k,u)" dx dt e IVY\ SRYG(x,t) ,

(94)

the special form requires that G be decomposed as G(k,u),G\(k,u)/G>(k,u) .

(95)

Upon transforming back to real space, the two functions G! need to have the properties that G>(x,t) vanishes for x(c t, and G\(x,t) vanishes for x'!c t, where c is the Rayleigh wave 0 0 0 speed. That is, G> is nonzero only for x so large that a pulse beginning at the origin at t"0 could never reach it in the forward direction, and G\ is de"ned similarly. In fact, for the cases to be discussed below G>Jd(x!c t) and G\Jd(x#c t) . (96) 0 0 It is far from immediately obvious that G can be decomposed in this way, or even that a function G must exist to satisfy Eq. (93), but for the moment we simply assume these facts and examine their consequences. Decompose p into two functions p"p>#p\ ,

(97)

and let us de"ne u"u\ ,

(98)

where p> vanishes for x(l(t), p\ vanishes for x'l(t), and u\ vanishes for x'l(t). Thus, p\ corresponds to the known function which describes the stresses along the crack faces and p> is an, as yet unknown, function. The function u\, on the other hand, is an unknown function along the crack faces and vanishes ahead of the crack tip. Using Eqs. (93) and (95), we can write G*u"p

(99)

NG(k,u)u(k,u)"p(k,u)

(100)

NG\(k,u)u(k,u)"G>(k,u)p(k,u)

(101)

PG>*p"G\*u .

(102)

From Eq. (102) one can solve formally for the stress and strain "elds as follows. We "rst show that for x(l(t) G>*p>"0. Consider x'l(t). Because p> is zero behind the crack



G>*u>" dxdtG>(x!x,t!t)p>(x,t)

(103)

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29

is certainly zero whenever x(l(t). The only chance for the integrand to be nonzero is for x'l(t). In this case x!x'l(t)!l(t)"lQ (t夹)(t!t) ,

(104)

where t夹 is some time between t and t. However, this means that x!x((t!t)[c ] , (105) 0 since c is the largest value the crack velocity can have. Eq. (105) is precisely the condition for 0 G>(x!x,t!t) to vanish. The conclusion is that



dxdtG>(x!x,t!t)p>(x,t)"0 for x(l(t) .

(106)

An identical argument shows that



dx dtG\(x!x,t!t)u\(x,t)"0 for x'l(t) .

(107)

De"ning H(x,t)"h(x!l(t)) ,

(108)

where h is a Heaviside step function, one can deduce from Eq. (102) that G>*p>"![G>*p\]H(x,t) ,

(109)

which has now been shown to be true both for x'l(t), and for x(l(t). But Eq. (109) can be inverted to give p>"!G>\*+[G>*p\]H, .

(110)

Since p\ is a known stress to the rear of the crack tip, Eq. (110) provides a formal solution in terms of the decomposed Green function. The most interesting thing to pull from this formal expression is the stress intensity factor (111) K" lim (2pe p(e#l,t) , C> which means identifying the terms that lead to a divergence of the form 1/(e as x"l(t)#e approaches l(t) from above. The detailed analysis to follow will demonstrate that G>\ has a singularity for eP0 going as 1/(e, while G>*p> is "nite. Therefore, in searching for the singularity in Eq. (110), G>*p> can be evaluated at x"l(t) and pulled outside the convolution as a multiplicative factor. The stress intensity factor can therefore be written K"KI (l(t)) ) K(v) ,

(112a)

where KI (l),![(2G>*p\] JR

(112b)

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and K(v), lim +(pe G>\*H, . (112c) J>CR C> The stress intensity factor is therefore the product of two terms. The "rst, KI (l,p) is the stress intensity factor that would emerge at the tip of a static crack sitting at l for all times, subjected to the load p\(t). KI does not depend upon crack velocity at all. The second term, K turns out to depend only upon instantaneous velocity v of the crack, but is otherwise completely unaware of the crack's history. 2.8.2. Application to Mode III We shall now apply the general result (112) to the particular case of anti-plane shear, and recover the result "rst found by Kostrov [29] and Eshelby [28]. Along the way it will be possible to verify the various claims about the structure of the Green function G. By calculating the stress intensity factor, we will, using Eq. (91), equate the energy release rate to the fracture energy thereby obtaining the equation of motion for a Mode III crack. The qualitative features of the resulting equation of motion are, as we shall see, applicable to Mode I. The starting point is Eq. (20), the wave equation for u . Fourier transforming in both space and X time by



dx dt e IV> SR

(113)

one has that Ru /Ry"[k!u/c!2ibu]u , (114) X X where a tiny amount of damping, b, has been added to take care of some convergence problems that will arise later. Only the solutions that decay as a function of y are allowed in an in"nite plate, so Eq. (114) is solved by u (k,y,u)"e\W(I\SA\ @Su(k,u) . X Right on the x-axis, taking u"u (y"0) and p"p (y"0), one has that X WX G(k,u)"p/u"!k(k!u/c!2ibu .

(115)

(116)

Decompose G as G"G\/G>

(117)

G\"!k(ik!iu/c#b

(118)

G>"1/(!ik!iu/c#b .

(119)

with

and

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To verify that this decomposition satis"es the conditions of the preceding section, "nd

 

G>(x,t)"

"

dk du e\ IV\ SR 2p 2p (!ik!iu/c#b

(120)

dp du e\ NV\ SR\VA , 2p 2p (!ip#b

(121)

with p"k#u/c "d(t!x/c)



dp e\ NV . 2p (!ip#b

(122)

When x(0, one must close the contour in the upper half plane, and as the branch cut is in the lower half plane one gets zero. When x'0, deform the contour to surround the branch cut, and get



(123)

G>(x,t)"d(t!x/c)h(x)/(px .

(124)

 dp 2e\NV 1 " . 2p (p#b (px  Therefore

To "nd G>\ one must do



G>\(x,t)"

dk du e\ IV\ SR(!ik!iu/c#b 2p 2p

"d(t!x/c)



dp e\ NV(!ip#b . 2p

(125) (126)

One cannot legitimately deform the contour to perform this integral, but can instead write that





dp e\ NV R dp e\ NV(!ip#b" , 2p Rx 2p (!ip#b

(127)

obtaining in this way G>\(x,t)"d(t!x/c)

 

R h(x) . Rx (px

(128)

Having now calculated G>\(x,t), we are now in a position to "nd the stress intensity factor, K(l,t), by using the general relation Eq. (112a). Using the expression for G>\ derived in Eq. (128), the velocity dependent, singular integral, Eq. (112), becomes







R h(x )  h(l(t)#e!x !l(t!t )) K(v)"(pe dx dt d(t !x/c)      Rx (px  

(129)

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dx R h(x )   h(e/[1!v/c]!x ) ,  (p Rx (x  since only very small x are important  dx h(x )   d(e/[1!v/c]!x ) "(pe  (p (x  "(1!v/c . "(pe



(130)

(131) (132)

We will now use the expression obtained for G> (Eq. (124)) to calculate the expression for KI (l(t)) in Eq. (112a):

 

h(x )  p\(l(t)!x ,t!t ) KI "!(2 dx dt d(t !x /c) (133)     (px    h(x )  p\(l(t)!x ,t!x /c) . (134) "!(2 dx (px    This is as far as one can take matters without an explicit expression for p\. However, for the particular case where p\ is time independent and p(x)"p h(x) ,  one gets

(135)

(136) KI "!p (4/(2p)(l .  The reason for the minus sign is that stresses ahead of the crack tip always counteract those on applied on the crack faces. Notice that Eq. (132) reduces to unity when vP0. This means that in the case of timeindependent loading, KI is indeed the stress intensity factor one would have had if the crack had been located unmoving at l for all time. For the moving crack, we have K"(1!v/c KI (l(t)) .

(137)

One computes the stress singularity that would have developed if one had a static crack of the present length, l(t), and multiplies by a function of the instantaneous velocity. It should be stressed that all details of the history of the crack motion are irrelevant, and only the velocity and loading con"guration are needed to "nd the stress "elds su$ciently close to the tip. As a consequence, one can use Eq. (91c) to determine the energy #ow to the tip of the crack. It is J"v(1!v/c)KI /2ak .

(138)

Finally, one can present the equation of motion for the crack. The rate at which energy enters the tip of the crack must be equal to vC(v). There is nothing to prevent the fracture energy C from being a function of velocity, but the notions of local equilibrium which have prevailed until now strongly suggest that it should not depend upon anything else. So one must have C(v)"(1!v/c)KI (l)/2ak ,

(139)

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which by writing out a and KI may be rewritten as kCp/4lp"((1!v/c)/(1#v/c)  or de"ning

(140)

l "kCp/4p  

(141)

l /l"((1!v/c)/(1#v/c) . 

(142)

as

2.8.3. Application to Mode I The same analysis may be carried out for thin plates under tension. Everything proceeds as before, except that it is not possible to display simple analytical expressions, although there are excellent approximations that can be put in simple form. We will just record the "nal result, discussed in more detail by Freund [7], that gives the energy #ux to the tip of the crack. The result we quote is for plane stress, which is characterized by a LameH constant jI de"ned by Eq. (27). After computing K(v), and multiplying by the function A (v) from Eq. (91), Freund "nds that the energy ' #ux per unit length extension of the crack is to a good approximation (1!v/c )KI (l) 0 G(v)"C(v)" 2jI

(143)

EC(v) v N "1! , (144) (1!l)KI (l) c 0 where c is the Rayleigh wave speed (the speed at which the function D given in Eq. (69c) vanishes), 0 KI is still given by Eq. (134), using p on the x-axis for p. In the case of time-independent loading WW described by Eq. (135) one gets l /l"1!v/c ,  0

(145)

l "pCjI /4p  

(146)

with

or v"c (1!l /l) . (147) 0  Surprisingly enough, the elaborate analysis above reproduces the result of the simplest scaling arguments, Eq. (11). Placing the result in the form of Eq. (147) is a bit misleading, since it hides the possibility that C and hence l can depend strongly upon the crack velocity v. Large apparent  discrepancies between theory and experiment have been due to nothing more than assuming that l could be treated as a constant. We will discuss this subject more in Section 4.1.  2.8.4. Practical considerations We now brie#y discuss some of the considerations implied by Eq. (143) in the design of experiments. We discuss three experimental geometries; one where presumptions of the theory are

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met well, a second where they are satis"ed in an approximate fashion, and a third where they clearly fail. 1. A thin plate has a crack running half-way through, and driven by wedging action in the middle. For times less than that needed for sound to travel from the point of loading to system boundaries and back to the tip of the crack, all the assumptions of the theory are obeyed. 2. A thin plate has a long crack as before, but now uniform static stresses p are applied at the outer  boundaries, while the faces of the crack are stress-free. This problem is equivalent to one in which the upper and lower outer boundaries are stress-free, but uniform stresses !p are applied along  the crack faces. The reason for the equivalence is that an uncracked plate under uniform tension p is a solution of the equations of elasticity, so this trivial static solution can be subtracted from  the "rst problem, to obtain the second equivalent one. Unfortunately, in this new problem, stresses are being applied to the crack faces all the way back to the left-hand boundary of the sample. The problem needs to be mapped onto one where stresses are applied to the faces of a semiin"nite crack in an in"nite plate, and the correspondence is only approximate. 3. Finally, consider a semi-in"nite crack in an in"nitely long strip, shown in Figs. 9 and 10. The strip is loaded by displacing each of its boundaries at y"$=/2 by a constant amount d. Far behind the crack tip as xP!R all the stresses within the strip have been relieved by the crack. Far ahead of the crack tip, as xP#R, the medium is una!ected by the crack with the

Fig. 9. Stress p in a polar plot for various values of v/c . For v/c , the maximum tensile stress clearly lies ahead of the FF   crack at h"0. For v/c "0.7, the maximum has moved away from h"0, but the change is so slight it is scarcely visible,  while by v/c "0.85 the maximum tensile stress is clearly far o! the axis. The small inner loops near the crack tip result  from compressive stresses near h"p.

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Fig. 10. Some typical experimental loading con"gurations.

stress "eld linear in y. Thus, the energy per unit extension far ahead of the crack has a constant value of 2Ed/=(1!l) ,

(148)

where E and l are, respectively, Young's modulus and Poisson ratio of the material. The translational invariance of the system along x suggests that the crack should ultimately propagate at a constant velocity v, for a given extension d. Balancing energy as usual, G"C"2Ed/=(1!l) .

(149)

Now assume that it is still valid to use Eq. (143). The stress intensity factor KI of a static crack in a strip loaded with constant displacements d cannot depend upon where the crack is located, so KI is a constant, and Eq. (143) would predict (1!l)KI  (1!v/c ) . G"C" 0 E

(150)

The velocity dependence on the right-hand side of Eq. (150) contradicts Eq. (149), and is simply wrong. The reason for the failure of Eq. (143) in this case is that the assumption that the crack tip does not feel the presence of the system's boundaries is obviously not valid. The translational invariance of the system is in fact, crucially dependent on the presence of its vertical boundaries. Energy is continuously re#ected back into the system as the amount of kinetic energy reaches a steady state. In contrast, the kinetic energy within a system of in"nite extent increases inde"nitely as ever farther reaches of material learn about the moving crack, while elastic waves carrying the information propagate outward.

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2.9. Crack paths We now brie#y discuss the path chosen by a moving crack. Energy balance provides an equation of motion for the tip of a crack only when the crack path or propagation direction is assumed. Although criteria for a crack's path have been established for a slowly moving cracks, no such criterion has been proven to exist for a crack moving at high speeds. 2.9.1. Quasistatic crack paths A slow crack is one whose velocity v is much less than the Rayleigh wave speed c . The 0 path followed by such cracks obeys the `the principle of local symmetrya, "rst proposed by Goldstein and Salganik [32]. This criterion states that a crack extends so as to set the component of Mode II loading to zero. One consequence is that if a stationary crack is loaded in such a way as to experience Mode II loading, upon extension it forms a sharp kink and moves at a new angle. A simple explanation for this rule is that it means the crack is moving perpendicular to the direction in which tensile stresses are the greatest. Cotterell and Rice [21] have shown that a crack obeying this principle of local symmetry is also choosing a direction so as to maximize the energy release rate. The distance over which a crack needs to move so as to set K to zero is on the order of the size of the process zone; Hodgdon '' and Sethna [33] show how to arrive at this conclusion using little more than symmetry principles. Cotterell and Rice also demonstrated that the condition K "0 has the following consequences '' for crack motion. Consider an initially straight crack, propagating along the x-axis. The stress "eld components p and p have the following form: VV WW p "(K /(2pr))#¹#O(r) , (151a) VV ' p "(K /(2pr))#O(r) . (151b) WW ' The constant stress ¹ is parallel to the crack at its tip. Cotterell and Rice showed that if ¹'0, any small #uctuations from straightness cause the the crack to diverge from the x direction, while if ¹(0 the crack is stable and continues to propagate along the x-axis. They also discuss experimental veri"cation of this prediction. Yuse and Sano [34] and Ronsin et al. [35] have conducted experiments by slowly pulling a glass plate from a hot region to a cold one across a constant thermal gradient. The velocity of the crack, driven by the stresses induced by the nonuniform thermal expansion of the material, follows that of the glass plate. At a critical pulling velocity, the crack's path deviates from straight-line propagation and transverse oscillations develop. This instability is completely consistent with the principle of local symmetry [36,37]; the crack deviates from straightness when ¹ in Eq. (151) rises above 0. Adda-Bedia and Pomeau and Ben-Amar [38] have extended the analysis to calculate the wavelength of the ensuing oscillations. Hodgdon and Sethna [33] have generalized the principle of local symmetry to three dimensions. They show that an equation of motion for a crack line involves, in principle, nine di!erent constants. It would be interesting for an experimental study to follow upon their work and try to measure the many constants they have described, but we are not aware of such experiments. Larralde and Ball [39,40] analyzed what such equations would imply for a corrugated crack, and

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found that the corrugations should decay exponentially. They also performed some simple experiments and veri"ed the predictions. Thus, the principle of local symmetry is consistent with all experimental tests that have been performed so far on slowly moving cracks. Nevertheless, it does not rest upon a particularly solid foundation. There is no basic principle from which it follows that a crack must extend perpendicular to the maximum tensile stress, or that it must maximize energy release. 2.9.2. Dynamic crack paths The absence of a rigorous basis on which to decide which direction a crack will move becomes particularly perplexing in the case of rapid fracture. A variety of path selection criteria for rapid fracture has been proposed in the literature. They can be divided into two types, those proposing that a crack propagate in the direction of a maximal stress and those that are based on a maximum dissipation of energy. In contrast to the case of quasistatic fracture, these criteria are not equivalent, and none of them is broadly supported by experiment. 2.9.3. Yowe instability Yo!e [17] proposed that one check the stability of a rapidly moving crack by returning to the dynamic stress "elds recorded in Eq. (69), approaching the tip of the crack along a line at angle h to the x-axis, and computing the stress perpendicular to that line. One wants to choose z "r cos h#ria sin h, z "r cos h#rib sin h , ? @ and to evaluate the stress

(152)

p "[cosh]p #[sinh]p ![sin(2h)]p . (153) FF WW VV VW Below a velocity of about 0.61c , depending upon the Poisson ratio, the maximal tensile stress  occurs for h"0. Yo!e noticed that above this critical velocity, the tensile stress p develops FF a maximum in a direction h'0. Above this velocity, the angle of maximum tensile stress smoothly increases until "nally it develops a maximum at about $603 relative to the x-axis. Thus, above the critical velocity a crack might be expected to propagate o!-axis. The physical reason for this spontaneous breaking of the axial symmetry of the problem stems from purely kinetic e!ects. In an elastic medium information is conveyed at the sound speeds. The stress "eld at the tip of a rapidly moving crack is analogous to the electric "eld surrounding a point charge moving at relativistic velocities. The stress "eld then experiences a Lorentz contraction in the direction of propagation as the crack's velocity approaches the sound speed. As a result, symmetric lobes around the x-axis of maximal tensile stress are formed above a critical velocity. Yo!e's critical velocity was "rst considered to provide a criterion for crack branching until it was noted, experimentally, that large-scale branching occurs in a variety of materials at velocities much less than this one. In addition, branching angles of order 10}153 instead of the 603 angle predicted by Yo!e are generally observed. 2.9.4. Other crack branching criteria To address the failure of the Yo!e criterion as a prediction for crack branching, a number of other criteria have been proposed [7,41]. The form of the stress "eld at the boundary of the process zone around the crack tip has been used to derive criteria for the branching angle of a crack

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[42,43]. Criteria of this sort are based on the determination of the direction in which the local energy density, evaluated at the edge of the process zone, is maximal. The underlying assumption for this criterion, which was originally suggested by Sih [44], is that crack propagation occurs in the radial direction along which the local energy density possesses a stationary value. Variants of this criterion have yielded branching angles consistent with those experimentally measured. On the other hand, these same criteria predict critical velocities for crack branching that are nearly identical to the one that is predicted by the Yo!e criterion. The most recent addition to this class of criteria was proposed by Adda-Bedia et al. [45]. They propose that one draw contours of constant principal stress, and look for points where these contours are perpendicular to lines drawn from the crack tip. The crack may choose to travel along such lines. According to this criterion, there are two critical speeds, a "rst at which the crack must choose between three possible directions, and a second at which it must choose between "ve possible directions. This proposal is perfectly reasonable, but there is no argument or experimental test that has shown it is to be preferred to other proposals. One rationale for all these models is the idea that microscopic voids ahead of the crack would tend to initiate growth near the edge of the process zone. Their direction would then be governed by the form of the stress "eld at the process zone edge. The actual branching angles predicted by the various branching criteria are not substantially di!erent than those trajectories determined by the following `statica condition. Let us look at the stress "eld formed ahead of a single moving crack. From this "eld one can compute [46,47] the trajectories that satisfy the quasi-static K "0 condition. If one now looks at the angle '' determined by this trajectory at a distance r from the crack tip, where r is the typical size of the   process zone, one also obtains relatively good quantitative agreement with observed branching angles. 2.9.5. Questions fracture mechanics cannot answer Throughout the last section we have endeavored to provide an overview of continuum fracture mechanics. In general, we have seen that by balancing the energy #owing into the vicinity of a crack's tip with that required to create new surface we can predict the motion of a straight, smooth crack. Fracture mechanics predicts both the strength and functional form of the near-"eld stresses. These can be measured and, as we will show, agree well with the predicted values in a variety of situations. What then don't we know? 1. What are the ingredients of the fracture energy in brittle materials? How should it be expected to vary with crack velocity? 2. What sorts of processes are happening in the process zone? 3. When a rapidly moving crack follows a macroscopically curvy path, what determines its direction of motion? 4. What are the conditions under which a crack bifurcates into two macroscopic cracks? 5. Fracture surfaces can go through transitions from smooth to rough appearance. Why? In the "nal sections of this paper, we will describe a dynamic instability that occurs at a critical energy #ux to a smooth, initially straight crack. We will demonstrate that this instability and its resulting development may address many of the questions above, and thus present seemingly disparate phenomena as a single coherent picture of fracture.

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3. Experimental methods in dynamic fracture In this section, we will brie#y review some of the main experimental methods used in the study of dynamic fracture. Depending on both the speci"c problem being investigated and on the experimental resources at hand, experimental methods vary greatly. In a given experiment stress is applied via externally controlled boundary conditions and the resulting behavior of the crack is observed. Some of the quantities that can be measured as the crack progresses are the crack's position and velocity, the instantaneous stress "eld at its tip, and the acoustic emissions resulting from its motion. On completion of an experiment, the resulting fracture surface can be measured and correlated with any of these dynamical measurements. Below, we will touch on typical ways by which the various quantities can be measured. 3.1. Application of stress 3.1.1. Static stress application The externally imposed stress distribution determines the stresses in the immediate vicinity of the tip (or, equivalently, the stress intensity factor) and hence drive a crack. There are two general types of loading that are typical in fracture experiments, static and dynamic. In experiments using static loading either the boundary conditions or applied stresses are constant throughout the duration of an experiment. Static loading conditions essentially imprint an initial static stress distribution into the sample. Depending on the loading and boundary conditions that are chosen, the stress intensity factor (or stored energy density) along the prospective path of a crack can increase, resulting in a continuously accelerating crack, or decrease, leading to a decelerating or possibly arrested crack. Some examples of some common loading con"gurations used are shown in Fig. 10 where `single edge notcheda (SEN) `double cantilever beama (DCB), and `in"nite stripa loading con"gurations are shown. The `single edge notcheda con"guration is sometimes used to approximate crack propagation in a semi-in"nite system. When the loading is performed via the application of constant stress at the vertical boundaries of the sample, for a large enough sample K Jp(l and therefore the energy release rate, GJpl, becomes ' a linearly increasing function of the crack length. This con"guration could be used, for example to study the behavior of an accelerating crack. In the `in"nite stripa con"guration, loading of the sample is performed by displacing the vertical boundaries by a constant amount. In this con"guration, G is constant for a crack that is su$ciently far from the horizontal boundaries of the sample. This loading would be amenable to the study of a crack moving in `steady-statea, where the energy release rate, controlled by the initial displacement of the boundaries, is constant. In the DCB con"guration, when a constant separation of the crack faces is imposed at l"0, GJ1/l is a decreasing function of l and could be used to cause crack arrest. How can one use con"gurations like the DCB to study dynamic fracture? An initially imposed seed crack of length l"l would propagate the moment that G exceeds the limit imposed by the Gri$th   condition (8). Under ideal conditions, the crack should propagate for an in"nitesimal distance and immediately come to a stop, since in this con"guration G is a decreasing function of l. The Gri$th criterion, however, assumes that the initial crack is as sharp as possible. However, initial seed cracks prepared in the laboratory by cutting rarely prepare a tip that is as conducive as possible to

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fracture. One can think of the initially created seed crack having a "nite radius at its tip thereby blunting the stress singularity. A blunt crack tip allows a substantially higher energy density to be imposed in the system prior to fracture than that allowed by a `sharpa crack. This excess elastic energy can then drive the crack beyond the constraints imposed by an initially sharp crack. In the case of the DCB con"guration with constant separation imposed, the excess energy stored in the sample prior to fracture initiation can cause a crack to propagate well into the sample before crack arrest occurs. Blunting of the singularity around a crack tip can also arise dynamically. This can be caused in the tip region by mechanisms such as nonlinear material deformation around the tip [9], by plastic #ow induced by the large stress build-up, or from crack-tip shielding that results from the formation of either micro-cracks or minute bridges across the crack faces in the near vicinity of the tip [20]. The DCB con"guration can also be used to generate an accelerating crack by imposing the condition of constant stress instead of constant separation at the crack faces. Under these conditions, the (quasi-static) energy release rate will increase quadratically with the crack length [20] as G"12Pl/Ewd

(154)

where P is the stress applied at opposite points on the crack faces at the edge of the sample, and w and d are, respectively, the thickness and half-width of the sample. 3.1.2. Fracture initiation Due to the stress singularity at the tip of a crack as the radius of the crack at its tip approaches zero, fracture initiation for static loading con"gurations is strongly dependent on the initial crack tip radius and therefore on the preparation of the initial crack. The stress build-up preceding fracture initiation can be used to advantage to load a system with an initial energy density prior to the onset of fracture. Unfortunately, unless one is extremely careful, experimental reproducibility of the stress at fracture initiation is di$cult. In Plexiglas we can achieve a reproducible stress at fracture initiation by "rst bringing the system to the desired stress and then either waiting several minutes for the material to fracture as a result of noise-induced perturbations or by `sharpeninga the initial crack by gentle application of a razor blade at its tip, once the desired initial conditions have been reached. These tricks do not work so well in more brittle materials, such as ceramics. 3.1.3. Dynamic stress application In some applications (e.g. the study of crack initiation before the material surrounding the crack tip has had time to react to the applied stress) very high loading rates are desirable. A common way to achieve this is by loading an initially seeded sample by collision with a guided projectile. In this way loading rates as high as KQ &10 MPa (ms\ [48] have been achieved. An alternative way to ' produce high rate loading has been achieved by means of sending a very large current through a folded conducting strip, inserted between the two faces of an initial crack. In this con"guration magnetic repulsion between adjacent parts of the strip is induced by the current. This method enables direct loading of the crack faces. A high loading rate can be produced by the discharge of a capacitor-inductor bank through the strips. This method has been used to produce a pressure pulse with a step function pro"le on the crack faces having loading rates on the order of

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KQ &10 MPa (ms\ [49] in experiments designed to investigate the response of a moving crack ' to rapidly changing stresses. 3.2. Direct measurement of the stress intensity factor Optical methods can be used for a direct measurement of the stress intensity factor and energy release rate. Two common methods are the `shadow-spota or `method of causticsa and photoelasticity. The method of caustics was originally derived by Manogg [50] with signi"cant contributions by Theocaris [51] and Kaltho! [52] in transparent materials and by Rosakis [53] in opaque materials. The method, applicable in thin quasi-2D plates, utilizes the de#ection of an incident collimated beam of light as it either passes through (transparent materials) or is re#ected by (opaque materials) the material surrounding the tip of a crack. Due to Poisson contraction generated by the high tensile stresses near the tip, the initially #at faces of a plate will deform inwardly. This deformation of the medium creates a lensing e!ect and diverts light away from the crack tip. The diverted rays will form a three dimensional surface in space in which no light propagates. When this light is imaged on a screen, a shadow (`shadow-spota) will be observed. The shadowed region will be bounded by a caustic surface or a region of high luminescence formed by the locus of the diverted rays. From the shape of the caustic surface, which can be recorded by a high speed camera, the `instantaneousa value of the stress intensity factor may be derived. The method works well with the caveat that the derivation of the stress intensity factor is based on the assumption that the angular structure of the stresses is given by Eq. (69). In the immediate vicinity of the tip, this assumption must break down as the yield stress of the material is approached. Thus, care must be taken that the curve on the material that maps onto the caustic is well away from the process zone surrounding the crack tip [54]. Photoelasticity coupled with high-speed photography can also be used to measure the stress distribution, hence the stress intensity factor, induced by a moving crack [55]. This technique is based on the birefringence induced in most materials under an imposed stress. Birefringence causes the rotation of the plane of polarization light moving through the material. The induced polarization must depend upon features of the stress tensor which are rotationally invariant, and therefore can depend only upon the two principal stresses, which are the elements in a reference frame where the stress tensor is diagonalized. In addition, there should be no rotation of polarization when the material is stretched uniformly in all directions, in which case the two principal stresses are equal. So the angular rotation of the plane of polarization must be of the form C(p !p ) , (155)   where p and p are the principal stresses at every point (eigenvalues of the stress tensor), and C is   a constant that must be determined experimentally. Whenever stresses of a two-dimensional problem are calculated analytically, the results can be placed into Eq. (155), and compared with experimental fringe patterns obtained by viewing a re#ected or transmitted beam of incident polarized light through a polarizer. The observed intensity is dependent on the phase di!erence picked up while traversing the material and hence provides a quantitative measure of the local value of the stress "eld. As in the method of caustics, quantitative interpretation of these measurements is limited to the region outside of the plastic zone. In transparent materials the

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application of this method is straightforward. Photoelastic methods have also been extended to opaque materials by the use of birefringent coatings which, when su$ciently thin, mirror the stress "eld at the surface of the underlying material [56]. Surveys of experimental results obtained by the methods of caustics and photoelasticity can be found in [52] and [57], respectively. 3.2.1. Direct measurement of energy A direct measure of the energy release rate as a function of the velocity of a moving crack can be obtained by constraining a crack to propagate along a long and narrow strip as shown in Fig. 10. This method has the advantage that it relies solely on symmetry considerations and therefore does not require additional assumptions regarding, for example, the size or properties of the process zone, as we discussed in Section 2.8.4. A series of experiments using a long strip geometry and varying the value of d, will result in a direct measure of G(v). In experiments performed in polymethylmethacrylate (PMMA) [58] steady-state mean velocities were indeed attained once the crack length surpassed roughly half the strip height, and the measurements of G(v) obtained agreed well with results previously obtained in PMMA by means of the methods of caustics [59]. 3.3. Crack velocity measurements In dynamic fracture, the tip of a crack will generally accelerate to velocities on the order of the sound speed in the sample. As the duration of a typical experiment is of order &100 ls, relatively high-speed measurement techniques are necessary. Common methods, based on either high-speed photography, resistance measurements, or the interaction of a moving crack with ultrasonic waves are brie#y reviewed below. 3.3.1. Optical methods The most straightforward method of velocity measurement is based on high-speed photography of a moving crack. This method may be used in conjunction with instantaneous measurements of the stress intensity factor by means of the method of caustics or photoelasticity. High-speed photography has some major drawbacks as a velocity measurement method. Although the frame rates of high speed cameras are typically between 200 kHz and 10 MHz, these cameras are capable of photographing only a limited number of frames (&30). Thus, by this method one can either provide measurements of the mean velocity (averaged over the interval between frames) at a few points, or, at the highest photographic rates, provide a detailed measurement of the crack velocity over a short (&3 ls) interval. An additional shortcoming of this method is that its precision is limited by the accuracy at which the location of the crack tip can be determined from a photograph. In the method of caustics, for example, the location of the crack tip falls within the rather large area of the (asymmetric) shadow-spot. These problems can partially be o!set by using a streak camera [60]. In this mode, "lm is pulled past the camera's aperture at high speed. The sample is illuminated from behind so that, at a given instant, only the light passing through the crack will be photographed. Since a crack can be made to essentially propagate along a straight line, the exposed "lm provides a continuous record of its length as a function of time. The intrinsic resolution of the measurement depends on the velocity of the "lm and the resolution of the high-speed "lm used. The "nal resolution obtained is also

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dependent both on the post-processing performed on the "lm in order to extract the velocity measurement, and the stability of the "lm's travel velocity. The same type of measurement has also been performed by high-speed measurement of the total beam intensity that penetrates the sample [61]. Assuming that the crack does not change its shape, this intensity is linearly dependent on the crack's length. 3.3.2. Resistive measurements The velocity of a rapidly moving crack can also be measured without the use of a high-speed camera. A simple way to do this is to adhere a grid of thin electrically conductive strips to a sample prior to fracture. As a crack propagates, the crack faces and therefore the conducting strips, will separate. If, for example, the strips are connected in parallel to a current source, measurement of the total electric resistance of the grid with time will provide a jump at each instant that the crack tip traverses the end of a strip. In this way the precise location of the crack tip at a number of discrete times is obtained. The thickness of the strips used must be at least an order of magnitude less than the crack face separation in order to be certain that the crack tip is not signi"cantly ahead of the fracture of the strip. For small samples of materials such as glass, where, at fracture, the total extension of the sample can be less than &1 lm, the thickness of the resistive coating should be on the order of 200}300 A> to ensure precise measurement. The disadvantage of the conductive strip method is that the discrete measurements inherent in the method can only provide a measure of the mean velocity between strips. This method can be extended to the use of a continuous coating in place of discrete strips. The advantage of using a continuous coating is, of course, that the location of the crack tip is obtained as quickly as the voltage drop across the coating can be digitized. The change in the resistance of the coating as a function of the crack length can be either measured or calculated for any given sample geometry and electrode placement. This method has been used in experiments where the samples themselves were highly conducting with either DC excitation [62] or with skin e!ect conduction using RF "elds [63]. In experiments on nonconducting materials a conductive coating must be used. In polymers and glass a 30 nm thick evaporated aluminum coating was successfully used [64,65] to measure the crack velocity to a precision of better than 20 m/s with a spatial resolution of order 0.5 mm. Using this method, the precision of the measurement is only limited by the background noise and the uniformity of the coating. With an evaporated coating, precise velocity measurements are actually obtained only near the sample faces. When the sample is e!ectively two dimensional, this does not present a limitation. This property can actually be used to advantage when one wants to correlate the instantaneous velocity with localized features formed on the fracture surface. 3.3.3. Ultrasonic measurements A velocity measurement method called `stress wave fractographya was developed by Kerkho! [66] and used in early studies of brittle fracture. In this method, a running crack is perturbed by an ultrasonic wave generated from a sample boundary in a direction orthogonal to the direction of crack propagation. The interaction of the sound with the crack tip causes it to be de#ected periodically as it traverses the sample. The trace of this de#ection is imprinted on to the resulting fracture surface. Since the temporal frequency of the modulation is that of the ultrasonic driving, measurement of the distance between neighboring surface modulations yields a nearly continuous measurement of the instantaneous velocity of the crack tip. This method has been used both in

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glass and brittle polymers. Its precision is only limited by the ultrasonic frequency used (typically in the MHz range) and the precision of the surface measurement. A disadvantage of this method relative to other techniques is that the method is a perturbative one, since the de#ection of the crack is accomplished by altering the stress "eld at the crack tip. Thus, externally induced oscillations can potentially mask intrinsic, time dependent e!ects. 3.4. Measurements of heat generation and the temperature rise near a crack As fracture occurs, a moving crack transforms the elastic energy stored in the elastic "eld of a material to either kinetic energy, the breaking of bonds or to heat. Two basic types of measurements have been performed to measure the heat that is generated by a crack. The most straightforward method has been to place small temperature sensors at a given distance from the path of a crack and measure the temperature rise in the material as a function of time after fracture has occurred. Since the time scale of fracture is orders of magnitude shorter (&100}300 ls) than the typical thermal di!usion times within the material, the problem can be approximated by assuming that an instantaneous planar heat source is created along the fracture plane. Assuming that the radiative losses are negligible over the period of measurement, the measured temperature variation with time at a single point can be "t to the solution of the heat conduction equation. Measurements of this sort were performed in Polymethylmethacrylate (PMMA) by Doll [67] in glass by Weichert [68] and in steel by Zimmerman et al. [69]. The precision of these measurements varied between 6 and 20%. In addition to the heat radiated from a moving crack, it is possible, with the assumption of a black-body radiation spectrum, to estimate of the temperature rise in the vicinity of the crack tip by the use of IR detectors. Such experiments have been conducted by Fuller et al. [70] in careful experiments on cracks propagating in PMMA and polystyrene (PS) and in AISI 4340 carbon steel [71] and Beta-C titanium [72] by Rosakis and Zehnder. In these experiments, data were obtained by focusing the infrared radiation emitted by cracks moving at di!erent velocities into an indium}antimonide infrared detector. The temperature of the crack was then obtained using the assumption that the emission spectrum of a crack corresponds to a black body spectrum. The assumption of a black body spectrum may be suspect. This assumption can be checked in molecular dynamic simulations of fracture in a crystalline material, where, at least in the immediate vicinity of the tip, it does not appear to be even approximately true. As will be shown in Section 6.6 the energy excited by a moving crack is taken up by discrete phonon frequencies and there is no sign that these modes are being rapidly thermalized. Both the measurement of emitted heat and the local temperature of the crack tip can be correlated with the crack's velocity. The results of these measurements will be discussed in the next section. 3.5. Acoustic emissions of cracks Measurement of acoustic emissions has long been used as a tool to detect either the onset or precursors to fracture, where the existence, frequency of events and event locations can be measured (see e.g. [73]). These techniques, however, have not been used extensively in dynamic fracture experiments since, generally, both the existence and location of a crack are determined

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more accurately by other methods. On the other hand, monitoring the acoustic emissions of a moving crack is a sensitive method to determine whether changes in the stress "eld are occurring during fracture, because any rapid changes will invariably broadcast stress waves. Most acoustic emissions used for #aw detection and fracture onset measurements have utilized arrays of resonant acoustic transducers since the advantage of their high relative sensitivity more than o!sets the loss of information of the spectral content of the signal. In dynamic fracture, we shall see that the spectral content of the acoustic signal broadcast by a moving crack carries important information. Therefore, broadband transducers should be used together with relatively high ampli"cation to o!set the transducers' lack of sensitivity. Gross et al. [74] used calibrated NIST-type transducers [75] with a #at response from 0.1 to 1 MHz, and Boudet et al. [76,77] used relatively broadband `pinducera probes to monitor acoustic emissions during rapid fracture, and correlate them with velocity and fracture surface measurements. In both experiments, although de#ections of the two dimensional sample normal to its surface were measured, the probe was sensitive to both longitudinal and shear waves due to mode conversion [78]. In the experiments by Gross et al., the acoustic probes were calibrated for shear waves so that a quantitative estimate of the amount of acoustic energy released by the crack could be obtained.

4. Phenomenology of dynamic fracture 4.1. Comparisons of theory and experiment How do the predictions of linear elastic fracture mechanics compare with experimental measurements? As long as the basic assumptions of fracture mechanics hold, the theory is quite successful in predicting both the motion of a crack and the behavior of the stress "eld throughout the medium. Once these assumptions break down, we will see that the linear theory loses its predictive power. Fracture mechanics has been highly successful in predicting the value of the stress intensity factor at the tip of both stationary and moving cracks for both static and dynamically applied loads. An example is work by Kim [79] where the measured transient behavior of the stress intensity factor was compared quantitatively to the predictions of Eq. (112). In this experiment, step function loading was applied to the crack faces in a sheet of Homalite-100 that was large enough to approximate an in"nite medium. The stress intensity factor was measured optically, using a method developed by Kim where the relation of the transmitted light through the crack tip to the stress intensity factor is used. Results of the experiment compared well with the calculated time dependence of the stress intensity factor [7]. Experiments on crack arrest by Vu et al. in polymethylmethacrylate (PMMA) [80] indicated similar agreement with theoretical predictions of the transient relaxation of the stress "eld within the medium. In these experiments, strain gauges, with a temporal resolution of &1 ls, were placed throughout the sample and were used to measure the temporal behavior of the stress "eld surrounding a crack at times immediately following crack arrest. As predicted by Freund [7], the stress "eld at a point directly ahead (behind) the crack was seen to reach its equilibrium value (to within a few percent) as soon as the shear (Rayleigh) wavefront passed.

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Fig. 11. Data of Kobayashi et al. [81] in PMMA. The data seem to indicate that cracks accelerate much more slowly than linear elastic theory predicts.

How do theory and experiment compare at high crack velocities? Experiments have often seemed to disagree with Eq. (147). One of the "rst experiments to test these predictions quantitatively was by Kobayashi et al. [81] in polymethylmethacrylate (PMMA). These data are reproduced in Fig. 11. Similar data have now been duplicated many times in a variety of brittle amorphous materials, with similar results. Theory clearly predicts that if the fracture energy is not a strong function of velocity, a crack should smoothly accelerate from rest to the Rayleigh wave speed, c . As observed in the data of 0 Kobayashi et al., although the crack initially accelerates rapidly, it becomes increasingly sluggish and "nally reaches a "nal velocity well below the Rayleigh wave speed. A brief look at Eq. (147) shows that there is a way out of this di$culty. One has only to suppose that the fracture energy C is a function of velocity. Specifying in Eq. (146) that l is de"ned in terms  of the minimal C(0) at which crack propagation "rst occurs, one obtains instead of Eq. (147): v"c (1!C(v)l /C(0)l) . (156) 0  By allowing the possibility that the fracture energy C(v) increases rapidly as velocity increases, one can obtain almost any pro"le of velocity versus crack length desired. Indeed, one can view Eq. (156) as a way to extract the velocity dependence of fracture energy from experimental velocity measurements. On the other hand, if one is interested in the question of whether the theory is valid for all crack velocities, then this way out is rather unsatisfactory since it only demonstrates the theory's plausibility. Validation of the theory cannot be accomplished without an independent measurement of the fracture energy. Even with such validation, fracture mechanics provides no fundamental explanation of the origin of any measured velocity dependence of the fracture energy.

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The "rst comparison of theory and experiment where the velocity dependence of fracture energy was explicitly taken into account was performed by Bergqvist [60]. He performed a series of beautiful experiments on crack arrest in PMMA which a!orded direct comparison of calculated energy release rates with experiment for crack velocities below 0.2c ((200 m/s). In these experi0 ments 50;20 cm samples were loaded by the application of constant stress on opposing boundaries. After fracture initiation, a high speed camera in streak mode was used to obtain a continuous record of the crack tip location with a temporal resolution of about 1 ls. To compare the velocity data to theory, Bergqvist used independent measurements of the fracture energy of PMMA as a function of the crack's velocity that were obtained in a strip geometry. By equating the measured value of the fracture energy to the calculated value of the energy release rate, he predicted values of the crack velocity. Comparison of the predicted and measured velocities showed agreement of the two to within 10%. As the main goal of the experiments was to investigate crack arrest, comparisons between theory and experiment for velocities higher than 0.2c were not attempted. 0 Sharon and Fineberg [82] have also performed such a comparison between theory and experiment for PMMA. To accomplish it, they "rst performed an independent measurement of the fracture energy of a crack by the use of the strip geometry. An additional series of experiments was then carried out in 40;40 cm samples, set up as in case 2 of Section 2.8.4. The velocity measurements in this set of experiments were then input into Eq. (156) and the derived values of C(v) thus obtained were then compared to the direct measurements. The results of these experiments are shown in Fig. 12. As in the experiments of Bergqvist, the data agree with Eq. (156) the theory for low velocities, less than about 400 m/s&0.4c . Above this velocity there is a sharp divergence between observed and 0 predicted values of C(v). The reasons for this divergence, as we will see, are due to the growth of the process zone around the crack tip to a scale where it invalidates the assumptions of fracture mechanics in the samples used.

Fig. 12. A comparison of directly measured values of the velocity dependence of the fracture energy, C(v), in PMMA with values predicted by fracture mechanics from Eq. (156), [83]. Note the good agreement until velocities of approximately 400 m/s&0.4c . 0

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J. Fineberg, M. Marder / Physics Reports 313 (1999) 1}108 Table 3 The maximal crack velocities observed in a number of brittle materials Material

v /c

 0

LiFl Rolled tungsten Single-crystal tungsten MgO Weak interface PMMA Grooved PMMA Glass PMMA Homalite

0.63 [84] 0.85 [85] 0.88 [86] 0.88 [85] 0.9 0.8 0.47}0.66 0.58}0.62 0.33}0.41

4.1.1. The limiting velocity of a crack An additional, rather robust, theoretical prediction is that of the limiting velocity of a crack. Eq. (144) predicts that, barring divergent behavior of C as a function of v, a crack should accelerate until asymptotically arriving at the Rayleigh wave speed, c . As Table 3 indicates, in amorphous 0 materials such as PMMA and glass, the maximal observed velocity of a crack barely exceeds about 1/2 of the predicted value. On the other hand, in strongly anisotropic materials such as LiF [84] tungsten [86,85] and MgO [85], a running crack indeed attains up to 90% of the predicted asymptotic velocity as cleavage through a weak plane occurs. Is strong anisotropy necessary to attain the limiting velocity of a crack? An interesting experiment by Washabaugh and Knauss [87] indicates that this may indeed be the case. In this experiment, plates of PMMA were "rst fractured and then rehealed to form a preferred plane in the material that was substantially weaker than the material on either sides of it. Although the interface did weaken the PMMA, the re-healed material used still had between 40 and 70% of the strength of the virgin material. Fracture was performed by impulsively loading the faces of an initial `seeda crack by means of the electromagnetic loading technique "rst introduced by Ravi-Chandar and Knauss. Using an interferometer together with a high-speed rotating mirror camera, interferograms of the crack tip were recorded at 20 ls intervals. In this way, the crack's maximum velocity was determined and velocities of up to 0.9c were observed. Similar velocities were also observed in 0 PMMA which was weakened by drilling a line of holes spaced 0.5 mm apart across the sample. The authors noted evidence of non-steady crack propagation suggestive of a `jerking acceleratingdeceleratinga process in both virgin PMMA and the re-healed material but with the severity of the uneven motion greatly reduced in the latter case. In the experiments performed with the re-healed material, Washabaugh and Knauss noted that none of the cracks propagating along the weakened interfaces produced branches beyond the point of fracture initiation. The same type of behavior takes place in strongly anisotropic crystalline materials. Field et al. [85] noted that in experiments on MgO and rolled tungsten (where the rolling in the preparation of tungsten induces a preferred orientation in the material) branching of a crack is suppressed until very high velocities. Thus, in strongly anisotropic materials (either crystalline or arti"cially weakened materials), where microscopic crack branching is inhibited, cracks approach

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Fig. 13. A comparison of the energy release rate G, with the measured heat #ux Q (from Doll [88]) in PMMA. 50}60% of the energy #owing into the tip of a crack in PMMA ends up as heat in the immediate region of the crack's tip.

the predicted limiting velocity. This fact is an important veri"cation of claims we will later make about the consequences of dynamic instabilities for isotropic materials. 4.2. Phenomena outside of the theory We will now brie#y mention some of the phenomena that occur in fast fracture that lie outside the predictive power of linear elastic fracture mechanics. 4.2.1. The heat dissipated by fracture We described in Section 3.4 the methods used to measure heat generated by a running crack. These heat #ow measurements were then correlated with simultaneous velocity measurements in PMMA by Doll and Zimmerman et al. [67,69] in glass by Weichert et al. [68] and in steel by Zimmerman et al. [69]. Results of these experiments showed that heating accounted for most of the elastic energy driving the crack. The experiments of Doll and Weichert indicated that the measured heat #ux accounts for 50}60% of the energy release, for crack velocities ranging from 0.1 to 0.6c 0 (see Fig. 13). Later experiments by Zimmerman et al. for lower velocities 0.1 to 0.3c in both 0 PMMA and steel estimated that the measured heat #ux accounted for virtually the entire energy release. Although these results indicate that nearly all of the elastic energy "nds its way into the formation of heat, a central question is where this dissipation occurs within the sample. Is all of the energy converted to heat within the process zone or does the heating occur as elastic waves propagating away from the crack are attenuated within the sample? The answer to this question was obtained by real-time infrared visualization of the crack tip during propagation. These experiments, "rst performed by Fuller et al. [70] on PMMA and polystyrene, indicated that temperatures at the crack tip in both materials were approximately constant as a function of the crack's velocity with a temperature rise of order 500 K. Similar temperature rises were recently measured in AISI 4340 carbon steel [71] and Beta-C titanium [72] by Zehnder et al. Besides the

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large temperature rise (in PMMA and polystyrene the temperatures at the crack tip were well above the equilibrium melting temperature!) these experiments also established that, in PMMA, the source of the heating was within a few lm of the crack's path or well within the process zone, as de"ned by the material's yield stress. Thus, these experiments indicate that nearly all of the dissipation in the material occurs in the near vicinity of a crack. The mechanism for the heat release appears to be the extreme plastic deformation induced by the fracture process in the near vicinity of the tip. The statement that nearly all of the dissipation occurs within the plastic zone is supported by the experiments of Kusy and Turner [89] in their investigation of the fracture energy of PMMA. These authors found that the fracture energy of high ('10) molecular weight PMMA (the typical molecular weight of commercially available PMMA is order 10) can be over two orders of magnitude larger than the surface energy; i.e. the energy needed to break a unit area of bonds. This large increase in the fracture energy was explained in terms of plastic deformation of the polymer chains. Their model [89,90] predicts that below a molecular weight of about 10, no signi"cant plastic deformation occurs in fracture and the fracture energy becomes comparable with the surface energy. These predictions were borne out by a series of careful experiments where long polymers were exposed to high energy radiation whose e!ect was to reduce systematically the mean molecular weight of the polymer as the exposure duration was increased. As a result of the radiation, the fracture energy of PMMA was observed to decrease by over two orders of magnitude as the molecular weight of the molecule was reduced from 10 to about 3;10. How do the heat and temperature measurements of the process zone in#uence basic understanding of the fracture process? From the point of view of continuum fracture mechanics, the fracture energy is in any case an external input into the theory. Thus, neither the extreme temperature rise observed within the process zone nor the causes for this have any e!ect on the predictions of the theory. What is important for fracture theories is that the observed dissipation is localized within the process zone and not spread out throughout the medium. If the latter were to occur, the entire rationale behind the balance leading to Eq. (112) would be invalidated. In Section 5 we will demonstrate that in brittle amorphous materials the dissipation described above has a well-de"ned structure related to the dynamic behavior of a crack. Thus, although the fracture energy is a material dependent quantity which carries with it `baggagea such as plastic deformation within the process zone, the total fracture energy observed is related to the amount of microscopic surface actually created by the fracture process. The total surface created by the process will, in turn, be intimately related to instabilities that occur to a single crack as a function of the energy that it dissipates. 4.2.2. Analysis of fracture surfaces There has been a great deal of work invested in the analysis of fracture surfaces, and the amount of literature on this subject is correspondingly large (see, for example, Ref. [91]). Much of the interest in this branch of engineering, called fractography, is concerned with the determination of the location of the onset of fracture of a given structure together with the probable cause for its failure. The visual inspection of a fracture surface is accomplished by using either optical or electron microscopes, depending on the scale needed for analysis. Although every fracture surface is di!erent, the proven utility of fracture surface analysis in the determination of di!erent fracture processes stems from the fact that, empirically, a close relation exists between the deterministic

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dynamics of a crack and the fracture surface that it leaves behind. In many cases the mechanisms leading to characteristic surface features are not known, but the fact that these features are at all general is strong evidence that they are created by a deterministic process, independent of details of the loading or initial conditions (e.g. initial defect structure or distribution) of the object under study. It is one of the goals of the study of fracture to uncover these fundamental mechanisms and to understand their generality. 4.2.3. Proxlometry Besides the imaging of a fracture surface, it is often useful to measure its topography quantitatively. Depending on the required scale, there are various methods to perform measurements of the local fracture surface amplitude. For scales ranging from 1 to 100 lm commercial contact type scanning pro"lometers can be used to measure properties such as the rms roughness of a surface. Due to the tip size of the contact probe, however, surface features that are under 10 lm in size may not be properly resolved. To this end, specialized x}y contact [64,64] and optical [76] pro"lometers have been used. The study of fracture surfaces at submicron scales has recently been performed using both scanning tunneling and atomic force microscopy [92,93] where 1 nm resolution is attainable. 4.2.4. Mirror, mist, hackle From many studies of fracture surfaces formed in brittle materials, it was found that the surface created by the process of dynamic fracture has a characteristic structure in brittle amorphous materials. This structure, called `mirror, mist, hacklea, has been observed to occur in materials as diverse as glass and ceramics, noncrosslinked glassy polymers such as PMMA and crosslinked glassy polymers such as Homalite 100, polystyrene and epoxies. Inspection of the fracture surface in a given experiment shows that near the location of fracture onset, the fracture surface appears smooth and shiny, and is thereby called the `mirrora region. As a crack progresses further, the fracture surface becomes cloudy in appearance, and is referred to as `mista. As the `generica crack progresses still further, its surface progressively roughens. When it becomes extremely rough, the fracture surface is said to be in the `hacklea region. 4.2.5. Microscopic crack branches Is there any speci"c structure observed within these regions? In early studies of the fracture of glass rods, Johnson and Holloway [94], by progressive etching of the fracture surface in the `mista region, demonstrated the existence of microscopic cracks that branch away from the main crack. Similar microscopic branched cracks (`micro-branchesa) were later observed by Hull [95] in polystyrene, and Ravi Chandar and Knauss [96] in Homalite 100 by visualizing the fracture surface in a direction normal to the faces of the fracture sample. Microscopic ((100 lm) branched cracks have also been observed to result from the rapid fracture of brittle tool steel [97]. As we will discuss in Section 5, the formation and evolution of micro-branches are a major in#uence on the dynamics of a crack (Fig. 14). 4.2.6. Parabolic surface markings In addition to the appearance of small branches in the mist regime, small parabolic markings as shown in Fig. 15 are commonly observed upon the fracture surface of amorphous materials. These

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Fig. 14. A typical view of microscopic branches as observed in PMMA. The main crack has propagated from left to right along the center of the photograph. The arrow, indicating the propagation direction, is of length 125 lm.

Fig. 15. Parabolic markings typically observed on the fracture surface of PMMA.

markings appear in all three fracture regimes and are interpreted as being the result of microscopic defects opening up ahead of the main crack front. To see how these markings come about, let us imagine a microscopic void situated directly ahead of a crack. The intense stress "eld, generated at the crack's tip, may cause the void to propagate some distance before the main crack catches up with it. The intersection of the main crack front with the front initiated by the void will then lead to the parabolic markings appearing on the fracture surface. As observed by Carlsson et al. [98], the number of the parabolic markings increases with the crack velocity. As the stress intensity factor increases with velocity, this observation is consistent with the above picture since an increasing

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Fig. 16. A photograph of typical rib-like patterns observed on the fracture surface of PMMA. The distance between ribs is approximately 1 mm. On smaller scales parabolic markings can be discerned.

number of voids should be activated ahead of the crack tip as the stress at the tip increases. Shioya et al. found the depth of the parabolic markings in PMMA [99] to be approximately 1 lm and Gross [100] was able to measure small velocity #uctuations resulting from the interaction of the main crack with these voids (of order 10}25 m/s) in PMMA for velocities between 150 and 330 m/s which correlated with the increase in microvoid density observed by Carlsson et al. Ravi-Chandar et al. have recently completed a comprehensive study of the development of the parabolic markings as a function of the velocity of a crack for four polymers; PMMA, Homalite-100, Solithane-113 and polycarbonate [101]. As in the study by Carlsson, they "nd that the parabolic markings in all of these materials increase in density with increasing values of the stress intensity factor. 4.2.7. Patterns on the fracture surface In the `mista and `hacklea regions of many brittle polymers, rib-like patterns, as demonstrated in Fig. 16, on the fracture surface are commonly observed. Similar patterns have been observed in polystyrene [95], PMMA [65], Solithane-113 and polycarbonate [101]. In these materials the typical distance between markings is on the order of 1 mm, so that they can easily be seen by eye. In PMMA, where extensive work has been performed to characterize these patterns, the patterns initiate within the `mista regime. Initially, the width of these structures is much less than the thickness of the sample, but the typical width of these structures increases with the crack velocity and eventually, within the `hacklea zone extend across the entire thickness of the sample [102]. These patterns are not smooth undulations along the fracture surface but, instead, are discrete bands of jagged cli!-like structures. As the crack velocity increases, these structures increase in height and exist up to the point where a crack undergoes macroscopic crack branching.

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In PMMA, the spacing between ribs was found to be strongly related to the molecular weight of the monomers used to form the material. Kusy and Turner [89], by varying the exposure time to gamma-ray radiation, managed to control precisely the average molecular weight of their PMMA samples. The results of these experiments clearly showed that the rib spacing was a strongly increasing function of the mean molecular weight. The typical spacing was shown to increase by over two orders of magnitude as molecular weight was varied between 1;10 and 1;10. These authors also observed that the fracture energy is a strongly increasing function of the rib spacing. 4.2.8. Roughness of the fracture surface Much recent e!ort has been directed to characterizing the fracture surface in terms of a `roughnessa exponent. This idea was pioneered by Mandelbrot et al. [103] who measured the fracture surface of steel samples for a number of di!erent heat treatments. They then demonstrated that the fracture surfaces obtained were scale-invariant objects with a self-a$ne character. Later studies [104}106] of aluminum alloys, steel, ceramics and concrete indicated that the local height, z, of the fracture surface scales as z&lD where l is the scale of observation within the fracture `planea and z the local height of the fracture surface. These studies indicated that for both quasistatic and dynamic fracture a `universala roughness exponent of f&0.8 is obtained for values of l greater than a material-dependent scale, m [107,93]. For values of l(m , a di!erent roughness exponent of   f&0.5 is observed [92]. This latter has been interpreted as the result of crack front pinning by microscopic material inhomogeneities in very slow fracture [108]. In many of the experiments where the roughness of the fracture surface was measured, the typical scales where scaling behavior was observed were many orders of magnitude smaller that the typical sample size. For example, the largest scale observed in recent measurements performed on soda-lime glass [93] was order 0.1 lm (or well within the `mirrora regime). Thus, in the context of continuum dynamic fracture, this roughness does not constitute a departure from `straight-linea propagation. It is conceivable, though, that the scaling structure observed may e!ect the observed value of the fracture energy. To this date, no systematic measurements of how the roughness scales with the velocity of a crack (at velocities of interest to dynamic fracture) have been performed. Although it is known that the rms surface roughness increases with the velocity of a crack within the `mista and `hacklea regions in PMMA [64,76], Homalite-100 [109], and crystals when cleaved at high velocities [85,110] it is not known whether the above scaling persists in this regime. The scales at which self-a$ne behavior of the fracture surface has been measured in these experiments are, in general, well within the process zone. As a crack accelerates, however, the surface structure within the `mista and `hacklea regimes may, depending on the overall system size, become larger than scales at which the singular contribution to the stress "eld in the medium is dominant. At this point the structure within the fracture surface may no longer be `swallowed upa within the process zone, and the description of the dynamics of a crack goes beyond the realm of linear fracture mechanics. 4.2.9. The velocity dependence of the fracture energy The fracture energy, or the energy needed to create a unit fracture surface, is of tremendous practical and fundamental importance. For this reason, measurements of the fracture energy, C, as a function of the velocity of a crack have been performed for many di!erent materials. The most common way to perform these measurements is by use of the method of caustics. In single crystals

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Fig. 17. The velocity dependence of the fracture energy, C(v) for (a) AISI 4340 steel from [53], (b) Homalite-100 from [112], and (c) PMMA from [113]. The data are also shown in dimensionless form, with the velocity scaled by the shear wave speed displayed versus the dimensionless measure of loading D"K /K . ' 

measured values of the fracture energy necessary to initiate crystal cleavage agree well with theoretical predictions [20]. In amorphous or polycrystalline materials, however, experiments indicate that C(v) is a sharply increasing function of a crack's velocity, whose form is only empirically known. Most of the fracture energy, as we have already seen, eventually ends up as heat within the process zone. Additional sinks for this energy are the acoustic energy radiated from the crack (i.e. the kinetic energy within the material) and `fracto-emissiona or the emission of photons from excited molecules along the fracture surface [111]. In Fig. 17 we present some typical measurements of fracture energy C versus crack velocity v for PMMA, Homalite-100 and AISI 4340 steel. We also present the data in dimensionless form. The variable D is a dimensionless measure of loading, equal to K /K "(G/G , the stress intensity ' '  factor divided by the critical value at which fracture "rst occurs. Velocity is made dimensionless by dividing through by the Rayleigh wave speed c . We will present relationships between fracture 0 energy and velocity in this same dimensionless form throughout the rest of this paper; see Figs. 40, 42, 47 and 49. A common feature in all of these, quite di!erent, materials is the sharp rise that occurs in C as the crack velocity increases. In the case of steels, the rise in fracture energy can be explained by modeling the process zone as a plastically deforming region, and calculating the change in plastic dissipation as a function of crack velocity. These calculations are described by Freund [7]. However, in brittle amorphous materials, such as PMMA and Homalite, dislocations are immobile, and there is no reason to believe that the classical theory of plasticity can be used to describe deformations near the crack tip. The origin of enhanced dissipation in these materials must therefore be sought in other mechanisms.

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Fig. 18. High-speed photographs of the tip of a moving crack in Homalite 100 (from [109]). The photographs were taken in the (a) mirror, (b) mist and (c) hackle regimes. Note that in the mirror region a single crack tip develops. In the mist regime small caustics, indicative of multiple crack tips, develop at either side of the crack tip. In the hackle regime the strength of the multiple caustics increases, and small crack branches behind the crack tip are in evidence.

4.2.10. The observation of a multiple-crack front A view of fracture, suggested by experiments performed by Ravi-Chandar and Knauss [109], is that dynamic fracture is not due to the propagation of a single crack but is due to the formation and coalescence of microscopic voids ahead of a crack front. This view is based on a series of experiments on Homalite-100. Fracture was induced via the electromagnetic loading technique described in 3.1.3 where a trapezoidal pressure pro"le with a 25 ls rise time and 150 ls duration was applied to the faces of a seed crack. The Homalite sheet used was large enough so that the "rst re#ected waves from the sample boundaries would not interact with the crack throughout the 150 ls duration of the experiment. Within the mist and hackle regions, a front of multiple microscopic parallel cracks instead of a single crack was observed. In the mirror region, as shown in Fig. 18, the authors noted that cracks tended to propagate within a single crack plane. As the crack propagated within the mist region (Fig. 18b) caustics due to the formation of multiple crack tips were observed. These increased in intensity within the hackle regime (Fig. 18c) as the secondary cracks increased in size. The authors then went on to show that the stress intensity factor, measured by means of the method of caustics, was correlated with the depth of the parabolic markings observed at the same spatial location (see Fig. 15). The authors interpreted the multiple micro-cracks, whose caustics were observed in the high speed photographs, as being due to the nucleation of microscopic material #aws, whose traces were indicated by the parabolic markings left on the fracture surface. These voids, as had been proposed earlier by Broberg [114], are nucleated by the high stresses ahead of the crack front. In this picture the motion of the crack is then dictated by the interactions between these growing #aws and the front. 4.2.11. Microscopic and macroscopic crack branching The question of when a crack has branched is rather a subtle one. If a crack begins to emit branches that remain small enough relative to the size of the sample, they can simply be viewed as part of the process zone. As shown in Fig. 18, above a certain energy #ux, cracks in the brittle plastic Homalite-100 are actually composed of multiple microscopic cracks propagating in unison. Section 4.2.2 shows that microscopic crack branches are observed in a variety of di!erent materials within the `mista and `hacklea zones. However, in samples of any given size, an increase in size of microbranches with energy release rate, G, must eventually bring the process zone to a size that invalidates the assumptions of fracture mechanics.

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When the #ow of energy to the tip of a crack increases su$ciently, the crack branches into two or more macroscopic cracks. Once a crack bifurcates, single crack models are, of course, no longer valid. Therefore, a theory describing a single crack can, at best, provide a criterion for when crack branching occurs. A number of such criteria for the onset of crack branching have been proposed. The criterion due to Yo!e and extremal energy density criteria have already been mentioned in Section 2.8.4. These criteria all su!er from the common problem that the velocities predicted for the onset of branching are much too high. Additional criteria such as postulating a critical value of the stress intensity factor, have not been consistent with experiments [41,115] since measurements at the point of branching show considerable variation of the stress intensity factor K . Eshelby ' [116] suggested that a crack should branch when the energy going into the creation of a single propagating crack is enough to support two single cracks. One problem with this criterion is that if C were not a strongly increasing function of v, once branching occurred, one would necessarily observe a large decrease in the branch velocities relative to the single crack velocity preceding the branching event. In glass, however, post branching velocities are either observed not to decrease at all [117] or at most undergo a decrease of &10% [66]. If C is a strongly varying function of v, as presented in Fig. 17, the slight drops in velocity after branching would not necessarily present a problem. At any rate, the Eshelby criterion is certainly a necessary one for crack branching. The fact that a critical value for the stress intensity factor does not seem to work would preclude the use of the Eshelby criterion as a su$cient condition. Experimentally, there seems to be no basis for a critical velocity for macroscopic crack branching, as predicted by the Yo!e criterion. For example, experiments in glass [97] have yielded branching velocities between 0.18 and 0.35c . In PMMA [118] branching velocities are consis0 tently about 0.78c , and in Homalite branching velocities between 0.34 to 0.53c have been 0 0 observed [115]. When using data on branching velocities to examine intrinsic properties, one must ensure that branching occurred in a given experiment at locations that are far enough away from the lateral boundaries to ensure that the system is e!ectively `in"nitea in extent. Otherwise, studies by Ravi-Chandar and Knauss [119] have shown that branching can be induced by the arrival of waves generated at the onset of fracture and re#ected at the lateral boundaries of the system back into the crack tip. Although the existence of a well-de"ned criterion for the onset of branching is not apparent in experiments, the consistent values of branching angles that have been observed in many di!erent materials suggest that there may be a degree of universality in the macroscopic branching process. The branching angles, as quoted in the literature, are generally determined by measurement of the tangent of a branched crack at distances of order a fraction of a millimeter from the crack tip. Angles of 103 in PMMA [118] and glass [94], 143 in Homalite, 153 in polycarbonate [41] and &183 in steel [97] have been reported for samples under pure uniaxial tension. 4.2.12. `Non-uniquenessa of the stress intensity factor In an additional series of experiments on Homalite-100, Ravi-Chandar and Knauss [96] discovered another apparent discrepancy between theory and experiment. In these experiments, high-speed (5 ls between exposures) photographs of the caustic formed at the tip of a crack initiated by electromagnetic loading at high loading rates were performed. The velocity of the crack, as deduced from the position of the crack tip in the photographs, was then compared with the instantaneous value of the stress intensity factor, which was derived from the size of the caustic.

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At low velocities (below &300 m/s"0.3c ) a change in the value of the stress intensity factor (e.g. 0 as induced by a wave re#ected from the boundaries) resulted in an `instantaneousa change in the crack's velocity, exactly as the theory predicts. On the other hand, at higher velocities (above &300 m/s) signi"cant changes in the stress intensity factor produced no discernible corresponding change in the crack's velocity. These experiments have been interpreted as indicating that the stress intensity factor is not a unique function of crack velocity. The puzzling questions that these experiments posed on the relation between fracture velocity and fracture energy interested us and many others in the "eld of dynamic fracture. In the next section we will provide an explanation for these questions that builds on this work. 5. Instabilities in isotropic amorphous materials 5.1. Introduction Dally [112] spent many years studying dynamic fracture in amorphous polymers, and in steels. In summarizing these experiments, he concluded that 1. The proper way to characterize a dynamic fracture experiment is through two dimensionless numbers; the crack velocity divided by a wave speed, v/c, and the dynamic stress intensity factor divided by the stress intensity factor at onset, K /K , which is identical to the dimensionless ' ' parameter D used in characterizing our experiments and theories. The functional relation between these two numbers carries most of the dynamical information about fracture, and can be measured reproducibly for a wide variety of geometries and loading histories. 2. The energy needed for fracture of brittle amorphous materials increases sharply past a certain velocity, where the straight crack becomes unstable to frustrated branching events. We will provide detailed experimental evidence for this point of view in what follows, and show that it allows one to bring together many apparently con#icting experimental and theoretical results. When we began experiments on dynamic fracture of amorphous brittle materials there was a general perception that theory and experiment did not agree. Freund [7, pp. 37}38], speci"cally mentions in a short list of phenomena `not yet completely understooda the `apparent terminal crack speed well below the Rayleigh wave speed in glass and some other very brittle materialsa. In retrospect, the divergence between theory and experiment was not as great as it seemed. Unease with the results of theory arose from the expectation that motion of cracks should be predictable on the basis of linear elastic fracture mechanics alone. In a brittle material it seemed plausible that fracture energy should not vary much as a function of crack velocity, despite unambiguous experimental evidence that large variations in fact occurred. Thus, the apparent di$culty was that cracks did not accelerate to the expected limiting speed or obey expected equations of motion, while in fact the question needing an answer was why the fracture energy varied so strongly with crack velocity. Besides the question of the terminal speed of a crack, we were bothered by the apparent lack of explanation of the many characteristics of brittle fracture, mentioned in the last section, which elastic fracture mechanics could not answer. No satisfactory answers were available for basic questions such as how and why structure arises on a fracture surface or why macroscopic

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Fig. 19. A schematic view of a typical experimental system used for monitoring the velocity of a crack (from [102]). The center of the sample is coated with a thin (&30 nm wide) conductive coating. As a crack progresses across the sample, it cuts the conductive layer thereby increasing its resistance. The resistance change is converted to a voltage by means of a bridge circuit and the voltage is then digitized to 8}12 bit resolution at high rates (typically 10}20 MHz). This provides a `continuousa record of the position of the crack's tip with time.

branching occurs. We now have an answer to many of these questions which has arisen from experimental work on isotropic amorphous materials, and is complemented by theoretical and numerical work on crystals. The answer is incomplete since a compelling theory of amorphous materials is absent, as are decisive experiments in brittle crystals. However, many of the experimental and theoretical observations now "t neatly into a coherent scheme, which we now will endeavor to explain. 5.2. Experimental observations of instability in dynamic fracture We began performing experiments in dynamic fracture [64] in the hopes of obtaining detailed dynamical records of crack motion, and correlating them with features left above and below the crack surface. We measured crack velocities using the conductive strip technique described in Section 3.3.2. A sketch of the experimental setup used in this type of velocity measurement is shown in Fig. 19. With this measurement system it was possible to obtain high resolution measurements of the crack's velocity at 1/20 ls intervals for about 10 000 points throughout the duration of an experiment. The velocity resolutions obtained in the "rst experiments were order $25 m/s. This resolution was later improved to about $5 m/s by analog di!erentiation of the signal [100,120] prior to digitization. Thus, it became possible to follow the long-time dynamics of a crack in

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Fig. 20. A typical measurement of the velocity of a crack tip as a function of its length in PMMA. After an initial jump to about 150 m/s the crack accelerates smoothly up to a critical velocity v (dotted line). Beyond this velocity, strong  oscillations in the instantaneous velocity of the crack develop and the mean acceleration of the crack slows.

considerable detail. In experiments on the fracture of PMMA a spatial resolution between measurements of order 0.2 mm was obtained [65,58]. An additional important advantage of this technique is that information can constantly be stored into bu!ers and discarded. The crack can be loaded very slowly, and when it "nally moves, simply refrain from throwing away the contents of the bu!er. Thus, the initial motion of a crack can be recorded, although it begins to move rapidly at an uncertain time. These high-resolution measurements have yielded a rather di!erent picture of a crack's dynamics than had previously been obtained. A typical measurement of a crack propagating in PMMA is shown in Fig. 20, where the velocity of a crack as a function of its length is presented. The "gure highlights the following features of the crack's dynamics. The crack "rst accelerates abruptly, over a time of less than 1 ls, to a velocity on the order of 100}200 m/s. Beyond v the dynamics of the  crack are no longer smooth, as rapid oscillations of the crack's velocity are apparent. As the crack's velocity increases, these oscillations increase in amplitude. 5.2.1. Initial velocity jump It is natural to wonder whether the initial velocity jump seen in Fig. 20 is an intrinsic feature of the crack dynamics, or whether it is related to the initial conditions. The crack begins at rest, and the tip has ample time to become slightly blunted and make it di$cult for the crack to begin moving. Hauch [121] performed experiments where the energy available per unit length decreased

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Fig. 21. Crack velocity versus time in geometries designed to produce gradual crack arrest. In PMMA and Homalite, cracks decelerate slowly towards zero velocity, on the scale of ls, while in glass, cracks are able to accelerate slowly from zero velocity, and travel stably at very low velocities. These data show that jumps in velocity from zero are not intrinsic properties of fracture dynamics in amorphous materials. Data of J. Hauch [121].

slowly through the length of the sample. We show some results from these experiments in Fig. 21. In both PMMA and Homalite-100, cracks decelerated gradually to zero velocity, strongly indicating that initial trapping rather than any intrinsic dynamical e!ect was responsible for the velocity jumps always seen when cracks begin to move. In the case of glass, it was possible to prepare very sharp seed cracks so that crack motion could initiate gradually, and the crack would then propagate steadily at velocities only a few percent of the Rayleigh wave speed. These data are also shown in Fig. 21. 5.2.2. Velocity oscillations beyond a critical velocity Are the oscillations evident in Fig. 20 randomly #uctuating or periodic in time? A blow-up of a typical time series of velocity measurements of a crack moving within PMMA, after the onset of the oscillations, is shown in Fig. 22. In the "gure, it is apparent that, although the oscillations of the crack velocity are not perfectly periodic, a well-de"ned time scale exists. In PMMA the value of this scale is typically between 2 and 3 ls. Power spectra of experiments in which the crack accelerated throughout the experiment indicate that the location of the peak in the frequency domain is constant despite changes of up to 60% of the mean velocity [65]. Evidence for the existence of a critical velocity for the onset of the oscillations is presented in Figs. 23 and 24. The data presented in the "gure were obtained by plotting the velocity at which the

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Fig. 22. Measurements of the velocity of a crack propagating in PMMA as a function of time (upper). The crack is propagating in steady-state above v . The corresponding power spectrum is shown in the lower plot. Note the existence of  a well-de"ned time scale (from [122]).

Fig. 23. Critical velocity as a function of the crack length at the moment of appearance of surface structure. Triangles, 1.6 mm wide extruded PMMA surrounded by air; circles, 3.2 mm wide cell-cast PMMA surrounded by air; squares, 3.2 mm wide cell-cast PMMA surrounded by helium gas (data from [65]).

Fig. 24. Geometry of the fracture instability. The crack travels from left to right, creating a ribbed structure on the fracture surface, the x}z plane, and microbranches beneath it, in PMMA.

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Fig. 25. Photographs of typical fracture surfaces in PMMA for v(v (upper left), v&v (upper right) and v'v (lower    "gure). The photographs are all to scale where the width of the surfaces shown is 3 mm. Note that the pattern starts to develop in the vicinity of v (from [122]). 

"rst deviation from a #at fracture surface was obtained [65]. As the value of v obtained for PMMA  was suspiciously close to the speed of sound in air, experiments were also performed in helium gas (where the sound speed is 965 m/s) where no discernible change in the value of v was obtained.  Experiments have since indicated that v is independent of sample geometry, sample thickness,  applied stress, and the acceleration rate of as the crack. Whenever, in PMMA, the critical steady-state velocity of 0.36c is surpassed, both oscillations in the crack velocity and an increase in 0 the fracture surface area result.

5.2.3. The creation of surface structure via the instability As mentioned previously, at high fracture velocities characteristic surface features are observed along the fracture surface of brittle polymers. At low velocities, on the other hand, the characteristic featureless `mirrora fracture surface is obtained. How does the fracture surface structure relate to the dynamical behavior of the crack that was described above? In Fig. 25 typical photographs of the fracture surface in PMMA are presented for values of v(v , v&v , and v'v . From    the "gure it is evident that the surface structure appears in the near vicinity of v . Pro"lometer  [65] measurements were performed to map out the topology of the fracture surface. These were then correlated with measurements of the crack velocity taken when the fracture surfaces were formed. As Fig. 25 indicates, in the near vicinity of v structure starts to appear. Initially,  the surface structure is apparent on only a relatively small amount of the fracture surface. In order to characterize the amplitude of this structure, the average height of the points not found in the mirror-like regions within the fracture surface was plotted as a function of the mean velocity of the crack. This plot is reproduced in Fig. 26. The following features are evident in the "gure.

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Fig. 26. The rms value of the surface height (obtained as explained in the text) as a function of the mean crack velocity in PMMA. The di!erent symbols indicate experiments performed with di!erent stresses and sample geometries (from Ref. [65]).

E At v "0.36c , in PMMA, a sharp, well-de"ned transition occurs where surface structure is  0 created. E The surface structure formed is a well-de"ned, monotonically increasing function of the mean velocity of the crack. E Both the transition point and functional form of the graph are independent of the details (e.g. initial and boundary conditions) used in the experiment, and are therefore intrinsic to the fracture process. The dependence of the rms surface deviations with the mean velocity has also been measured over a larger range of velocities in PMMA by Boudet et al. [76,77]. These authors have measured the pure rms deviations of the surface (without accounting specially for the mirror-like regions as described above). Qualitatively, these latter measurements agree with those in Fig. 26 although, in the region of the transition the data look quantitatively di!erent, although it is reasonable to assume that the di!erence is due to the fact that a pure rms calculation of the data gives little weight to the areas where the fracture surface is non-trivial, since, for v&v the mirror-like regions  dominate the fracture surface. Fineberg et al. [64,65] also measured the cross-correlation between the local surface structure and the instantaneous measurements of the velocity. The two types of measurements were compared along the faces of the sample, where the local velocity of the crack was, in essence, measured. The cross-correlation function obtained revealed the temporal oscillations at the 2}3 ls scale observed in the velocity measurements, although the degree of correlation (about 0.3) was quite low. A higher degree of correlation (about 0.5}0.6) was later observed by Boudet et al. [77] between the #uctuating part of the velocity and acoustic emissions of a crack. In summary, both the existence of a sharp critical velocity for the onset of oscillatory behavior of the crack together with the well-de"ned monotonic dependence of the surface structure created by the crack beyond v point to the existence of an instability of a moving crack beyond this critical  velocity. The behavior of the system is not in#uenced either by the boundary or initial conditions

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and is seen to be solely a function of the mean velocity of the crack, or, equivalently, the energy release rate. This indicates that the instability is intrinsic to the system and dynamical in nature. Although the quantitative measurements described above have been performed in PMMA, the similarities of the fracture surface formed in such diverse amorphous brittle materials as PMMA, glass and brittle ceramics suggest that the dynamical instability is not con"ned to this single material, but is a general instability of brittle fracture. In this section, we will describe additional experiments that will verify this conjecture. The following sections will describe both numerical calculations and analytical models where behavior bearing close resemblance to the experiments is observed. 5.2.4. Micro-branching as the instability mechanism What is the mechanism that gives rise to the dynamic instability described above? An answer to this question was provided by experiments performed by Sharon et al. [58]. As detailed in Section 4.2.2, microscopic branches have been observed within the `mista region in a variety of brittle materials ranging from PMMA to hardened steels. Sharon et al. analyzed these structures as a function of crack velocity. In Fig. 27 we show photographs of the transverse structure observed in PMMA for velocities prior to, at and beyond v .  As Fig. 27 indicates, below v no micro-branches are observed. At v , branches begin to appear   and, as the mean velocity of the crack increases, the branches become both longer and more numerous. In Fig. 28 a the mean length of a microbranch is plotted as a function of the mean velocity of the crack. As in Fig. 26 we see that, although at a given crack length the size of micro-branches varies widely, the mean branch length of a micro-branch is a smooth, well-de"ned function of the mean velocity. As in the case of the surface amplitude, at v"v there is a sharp  transition from a state having no branches to a state where both the main crack and daughter branches are observed. As Fig. 29 indicates, the value of v can be measured quite reproducibly for  a wide variety of initial conditions. Whether the experiment begins with a blunted seed crack that accelerates rapidly to high velocities, or with a long sharp crack that stabilizes at lower velocities, once v is attained, we "nd micro-branches appearing. Presumably, a critical energy #ux to the tip  is achieved at the same time as well, but we did not measure energy #ux directly. The same value of v describes both the transition to micro-branches and the appearance of surface structure.  How does the appearance of micro-branches relate to the structure formed on the fracture surface? In Fig. 28 the mean microbranch length (a) as a function of the mean velocity is compared to the surface amplitude measurements (b). Although the two "gures are markedly similar, the typical size of micro-branches, for a given velocity, is about 2 orders of magnitude larger than the scale of the surface structure. This indicates that the surface structure is a result of the microbranching process. These measurements imply that the structure observed on the fracture surface is, essentially, the initial stage of a micro-branch which subsequently continues into the sample in a direction transverse to the fracture plane. This is illustrated in Fig. 24 where a section of the fracture surface is shown together with a cut that reveals the branched structure beneath it. As Fig. 28c demonstrates, the increase in the size of the velocity #uctuations is also a direct result of the micro-branching instability. The #uctuations in the velocity can now be understood as follows. As a crack accelerates, the energy released from the potential energy that is stored in the surrounding material is channeled into the creation of new fracture surface (i.e. the two new faces created by the crack). When the velocity of the crack reaches v , the energy #owing into the crack 

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Fig. 27. Images of local crack branches in the x}y plane in PMMA. The arrow, of length 250 lm, indicates the direction of propagation. All "gures are to scale with the path of the main crack in white. (top) v(v (center) v"1.18v (bottom)   v"1.45v (from [58]). 

tip is now sub-divided between the main crack and its `daughtersa which are formed by the branching events. As a result, less energy is directed into each crack and the velocity of the crack `ensemblea diminishes. The daughter cracks, which compete with the main crack, have a "nite lifetime. This presumably occurs due to screening of the micro-branches by the main crack as it `outrunsa them, because of its geometrical advantage of straight line propagation. The daughter cracks then die and the energy that had been diverted from the main crack now returns to it. This causes it to accelerate until, once again, the scenario repeats itself. 5.2.5. Micro-branch pro,les As Fig. 27 indicates, at a given mean velocity both the lengths and distances between consecutive micro-branches are broadly distributed. Sharon et al. [102] have shown that in PMMA, these quantities are characterized by log normal distributions whose mean and standard deviation values are linearly increasing functions of the mean crack velocity. As an example, the branch

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Fig. 28. (a) Mean branch length, (b) the rms value of the fracture surface amplitude, and (c) the rms velocity #uctuations as functions of the mean crack velocity, v, in PMMA. The arrows indicate the critical velocity of 340 m/s. The data in (a) and (b) were obtained by measurements of accelerating cracks in plates having di!erent geometries and loading stresses. Note the nearly two order of magnitude di!erence in scales between the mean branch length and fracture surface amplitudes for the same values of v. The velocity #uctuation data were measured at steady-state velocities. Although velocity #uctuations are observed prior to v , a sharp rise in their amplitudes occurs beyond v (from [113,102]).  

Fig. 29. The critical velocity for the appearance of micro-branches (squares) as a function of the energy density stored in the sample far to the right of the crack. For comparison, the terminal steady-state velocity (triangles) for the same experiments are shown. The data, taken from [102], are from experiments performed on PMMA in the strip geometry.

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Fig. 30. A typical probability distribution of micro-branch lengths at a constant velocity of 585 m/s in PMMA. Note the "t to a log normal distribution (solid line) together with a minimum branch length of about 30 lm. (inset) the mean length, 1¸2, (squares) and standard deviation (triangles) of the micro-branch length distributions in PMMA as a function of the mean crack velocity. Both increase linearly with the same 1 ls slope. Thus, the entire distribution scales linearly with the mean crack velocity (from [102]).

length distribution at a single velocity together with (inset) the dependence of the mean length and standard deviation of the length distributions with the mean crack velocity are shown in Fig. 30. Although it is obvious from the broad distributions above that the instantaneous lengths of micro-branches have a random character, one might want to look at the path that a given branch takes as it moves away from the main crack. Although a given branch may choose its length from a broad distribution, all micro-branches will propagate along an extremely well-de"ned trajectory. This is demonstrated in Fig. 31a, where the pro"les of a number of micro-branches, formed at the same mean velocity, are superimposed and shown to trace out a well-de"ned function. Log}log plots of these trajectories in both PMMA and glass (Fig. 31b) show that the micro-branch pro"les in both materials follow a power-law of the form y"0.2x  ,

(157)

where x and y are, respectively, the directions parallel and perpendicular to the direction of propagation of the main crack, and the origin is taken to be the point at which the micro-branch begins. Surprisingly, both glass and PMMA, two materials with greatly di!erent microscopic structure, have nearly identical values for both the exponent and prefactor [83]. The nearly identical branch trajectories observed in Fig. 31 in such highly di!erent materials suggests a universality of micro-branch pro"les in brittle materials whose origin is dictated by the universal behavior of the stress "eld surrounding the crack tip. This conjecture is supported by the observations by Hull [95] of the same type of trajectory in polystyrene, where the micro-branch pro"les were attributed to craze formation. Hull's analysis of these structures indicated that the branch pro"les follow the trajectory of maximum tangential stress of the singular "eld created at

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Fig. 31. Demonstration of the power-law scaling of micro-branch trajectories (from [102,83]). Shown are the mean micro-branch pro"le in PMMA for (squares) v"374 m/s (triangles) v"407 m/s, and (circles) v"470 m/s. All pro"les are described by the same function: y"0.2x  where x and y are, respectively, the distance from the branching point, in mm, in the directions parallel and perpendicular to the direction of propagation of the main crack. (inset) log}log plot of micro-branch trajectories measured in PMMA (triangles) and glass (squares). Note that in both materials the microbranch pro"les are described by the same law.

the tip of the main crack. Using this criterion, this trajectory is also described by Parleton's [46] numerical calculation of the stress "eld of a single static crack. How are micro-branches related to macroscopic branches? Let us examine the `branching anglea predicted by Eq. (157). The power-law form of Eq. (157) would predict a branching angle of p/2 as x approaches the point of branching. The maximum branching angle that has actually been observed was limited, experimentally, by di!raction. The largest angle observed by Sharon et al. [102] in PMMA was reported to be 303 at a 3 lm distance from the bifurcation point. On the other hand, the values of the `branching anglea for macroscopic branching that are quoted in the literature are measured at distances typically of order 100}300 lm from the branch onset. The angles that are tangent to the branch trajectory at these distances range from 15 to 113. This is in excellent agreement with the values of the branching angle (see Section 4.2.11) quoted in the literature for brittle materials ranging from polymers to hardened steel. These observations suggest that a smooth transition between microscopic and macroscopic crack branches occurs in brittle materials and that the characteristic features of crack branches exhibit, at these scales, a high degree of universality. If this picture is correct, the criterion for the formation of macroscopic crack branches simply coincides with the onset of the micro-branching instability. 5.2.6. The transition from 3D to 2D behavior Although the origin of crack branching may now be understood to result from the microbranching instability, a new question now arises: under what circumstances will a branched crack

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Fig. 32. The mean width of coherent zones on the fracture surface of PMMA as a function of the mean crack velocity. The data shown were obtained from both 0.8 mm wide (circles) and 3.0 mm wide (squares and triangles) samples. At v"550 m/s the pattern width is on the order of the sample thickness, indicating a transition from a 3D to 2D state (from Ref. [102]). Fig. 33. The surface ratio of a running crack as a function of its mean velocity, v. The surface ratio is de"ned as (S !S )/(S !S ) where S and S are, respectively, the surface areas created by micro-branches along the faces       and center planes of the sample. The width of the PMMA sample used was 3 mm. At velocities above 1.65v , S "S    indicating a homogeneity of microcrack production across the entire width of the sample (from [102]).

survive and continue to propagate away from the main crack? A necessary condition for a microbranch to develop into a macroscopic crack is the coherence of the micro-branch over the entire thickness of the sample. As shown in experiments on PMMA [102], near the instability onset, the width of a micro-branch is quite small (on the order of 100 lm). As the crack velocity increases beyond v , the branch width increases along with the branch length.  Sharon et al. [102] quanti"ed the increase of the `coherence widtha of the branches in two ways. The "rst method used was to study the width of the patterns formed by the branches along the fracture surface as a function of v. As can be seen in Fig. 25, the width of coherent `islandsa formed along the fracture surface increases with v. The results of this analysis are reproduced in Fig. 32. From microscopic widths near v the pattern width increases sharply with the mean velocity of the  crack, until, at a velocity of about 1.7v the pattern becomes coherent across the entire thickness of  the sample. Shortly above this velocity, macroscopic branching occurs. An additional measure of this 3D to 2D transition can be seen in the micro-branch distribution across the sample width. Near the onset of the instability, the distribution of the micro-branches as a function of the distance away from the sample faces is very nonuniform. In PMMA the production of micro-branches near onset falls o! sharply with the distance from the faces of the sample. To quantify this fall-o!, Sharon et al. [102] measured the ratio of the amount of total fracture surface produced by the crack and branches located at the sample faces with that produced at the center of the sample (see Fig. 33). As Fig. 33 shows, the di!erence in surface production between the outer and center planes decreases continuously until, at v"1.65v , this di!erence  vanishes, indicating a homogeneity of micro-branch production across the sample. Curves similar in behavior to Fig. 32 have also been observed in studies of the rms surface amplitude and total sound emission of the crack by Boudet et al. [77]. In these experiments, the acoustic emission and rms surface amplitude of cracks propagating through PMMA were measured as a function of the crack's velocity. Both the sound emissions and surface roughness were

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observed to diverge as the mean velocity approached 1.8v (600 m/s). These measurements, together  with the divergent behavior of the pattern widths at nearly the same velocity, suggest that a second transition may be occurring at v&1.7v . As the divergence of surface roughness observed by  Boudet et al. [77] is indicative of macroscopic branching, the transition from 3D to 2D behavior described above may also be a su.cient condition for macroscopic branching to occur. 5.2.7. Energy dissipation by a crack As we mentioned in Section 4.2.9, the fracture energy, C, increases sharply with the crack velocity in a variety of di!erent brittle materials. In light of the micro-branching instability, can this sharp increase in the energy needed to form a crack be explained? In PMMA, as demonstrated in Fig. 17, the energy release rate increases by nearly an order of magnitude as the mean velocity of a crack surpasses v . Beyond v we know that the micro-branching instability occurs and, as a result the   total amount of fracture surface created by the crack `fronta, must increase. This surface increase must thereby lead to an increase in C. Sharon et al. [113,102] quanti"ed the amount of surface formed by the crack ensemble, by de"ning the relative surface area as the total area per unit crack width created by both the main crack and micro-branches, normalized by that which would be formed by a single crack. Their measurement, in PMMA, of the relative surface area as a function of the mean velocity of the crack is shown in Fig. 34. As only a single crack propagates prior to the instability the relative surface area is unity for v(v . Beyond this velocity, the total surface created  indeed rises steeply with v, with the surface contribution due to micro-branches a factor of six greater than the surface contributed by a single crack at 600 m/s. The data shown in Fig. 34 were obtained in experiments performed both in strip geometries, which yielded steady-state propagation, and large square aspect ratio samples where continuously accelerating cracks were observed. The velocity dependence of the relative surface area formed in both systems was the same, indicating the intrinsic character of the micro-branching instability.

Fig. 34. The relative surface area (de"ned as the total area per unit width created by both the main crack and micro-branches normalized by that which would be created by a single crack) as a function of the mean crack velocity, v. The data in the "gure were obtained from cracks moving at steady-state velocities in the strip con"guration (crosses) as well as from accelerating cracks (other symbols) driven by stored energy densities varying between 3.2;10 to 5.1;10 erg/cm. Note that the total surface area at high velocities is many times greater than that created by the main crack (from Ref. [113]).

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Fig. 35. The relative surface area as a function of the energy release rate, G, in PMMA. The data in the "gure were obtained from experiments in the strip geometry, for steady-state mean crack velocities. After an initial jump (see text) the relative surface area is linearly dependent on G. The inverse slope of the graph leads to a constant energy cost per unit surface of 5;10 erg/cm. This is equal to the fracture energy immediately preceding the branching instability (from Ref. [113]).

In the experiments using the strip geometry, simultaneous measurements of the energy release rate, G, and the relative surface area were possible. The relationship between these quantities is shown in Fig. 35. The "gure shows that after an initial jump which occurs at around v'v , the amount of  surface area formed is a linear function of the energy release rate. This means that both before and after the instability onset, the fracture energy, given by the inverse slope of the curve, is nearly constant. The fracture energy `increasea observed in Fig. 17 and Fig. 13 in PMMA is thus explained entirely as a direct result of the micro-branching instability. The fracture energy e!ectively rises because increasingly more surface is created due to micro-branch formation. The cost, however, to create a unit fracture surface, remains constant and is close to the value of the fracture energy immediately preceding the instability onset. Although, at this writing, experiments verifying the constancy of C have only been performed in PMMA, it is conceivable that the universally observed increase in the fracture energy of brittle materials, demonstrated in Fig. 17, may be generally explained by this mechanism. Some comments are in order. First, the micro-branching mechanism does not entirely explain the rise with crack velocity of the fracture energy. A close look at Fig. 17 reveals that even prior to the instability onset an increase in C with v is apparent. This increase, in itself, is not entirely surprising. PMMA is a polymer consisting of long, tangled molecular strands. In this material, the value of C may be determined by complex, rate-dependent processes which include plastic deformation of the material together with the heat dissipation involved in craze formation [90]. However, the micro-cracking instability contributes much more to changes in dissipation than these other processes. An additional question arising from Fig. 35 is the origin of the jump in relative surface area that occurs at approximately the instability onset in PMMA. The `jumpa in this "gure would indicate that, initially, additional surface area is formed for `freea, i.e. at a very small energy cost. We surmise that the reason for this jump is due to the size of the process zone surrounding the crack tip

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in PMMA. As the vast majority of the fracture energy goes into the deformation of the area around the tip, the additional fracture energy contributed by a micro-branch within the process zone formed by the main crack would be negligible. Thus, the initial part of a micro-branch, whose additional area is counted in Fig. 35, dissipates nearly no additional energy. This conjecture is supported by the observations, in PMMA, of the non-zero minimum length of a micro-branch seen in Fig. 30 and that the minimum length is indeed consistent with estimates of the process zone size [102]. In light of these remarks, the long-standing puzzle of why a crack never seems to approach the Rayleigh wave speed in isotropic material, can now be answered. A crack does not need to dissipate increasing amounts of energy by accelerating, thereby increasing the amount of kinetic energy in the system. Beyond v a crack now has the option of dissipating energy by creating an increased  amount of fracture surface at the expense of a reduction of the total kinetic energy throughout the system. As increasing amounts of energy #ow to its tip, a crack forms a corresponding amount of surface via microscopic branching. As the energy #ux to the crack tip increases further, the mean length of the micro-branches increases, as Fig. 28 shows. If G is increased still further, a second generation of micro-branches has been observed [102] to bifurcate o! of the "rst generation of micro-branches. If this process continues, we see that the micro-branching instability may lead to a well-de"ned mechanism for the creation of a fractal structure. As in the Kolmogorov theory of turbulence, the smallest scale in this structure (analogous to the Kolmogorov or dissipation scale in turbulence) may be determined by the size of G. 5.2.8. Evidence for the universality of the instability Most of our experimental work on the micro-branching instability has been performed in PMMA. An important question is whether the instability is a universal feature of dynamic fracture or is limited to some types of brittle polymers, such as PMMA. Both experimental data together with theoretical work, which will be described in later sections, indicate that the instability is indeed of a general nature in brittle fracture. As noted previously, patterns on the fracture surface, similar to those observed in PMMA, have been observed within the `mista region in a number of brittle polymers such as polycarbonate and polystyrene. This observation would seem to, at least, suggest the existence of a similar, pattern-forming instability in brittle polymers. To what extent is the micro-branch aspect of the instability is universal? As described in Section 4.2.2, microscopic branches have also been observed in all of the above materials. In addition, micro-branches were also seen within the mist region in both hardened steels, and glass as well as in brittle polymers. The claim for universality is also strengthened by the fact that micro-branches in both glass and PMMA develop nearly identical trajectories, as noted in Fig. 31. Additional experimental support for the universality of the functional form of micro-branches is given by the description of micro-branches, identi"ed as `crazesa in early experiments in polystyrene [95]. As noted earlier, a universal trajectory for micro-branches would explain the consistent value of the macroscopic `branching anglea observed in a large variety of brittle materials. Is the existence of a critical velocity for the instability also a universal aspect of brittle fracture? Irwin et al. [123], Ravi-Chandar and Knauss, 4.2.10) and Hauch and Marder [121] observed that microbranches are initiated in Homalite beyond 0.37v , which is within 2% of the critical velocity 0 observed in PMMA.

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A critical velocity also exists in glass, but its value is di!erent. Gross et al. [74], measured the acoustic emission of both glass and PMMA as a function of v for accelerating cracks. In these experiments, the acoustic spectra of these two very di!erent materials were measured within windows centered at increasing velocities for both materials. The results of these experiments showed that the behavior of the acoustic spectra changed dramatically in both materials at values of v/c "0.36"v in PMMA and v/c "0.42 in soda lime glass. Below this value, the acoustic 0  0 spectra were featureless whereas above the critical velocity, high intensity peaks in the acoustic emissions were observed. In PMMA, the frequency of these peaks was on the order of the characteristic time scale observed in the velocity oscillations (about 2}3 ls) and were Dopplershifted as a function of v. The spectra in glass were higher in frequency (2}3 MHz) and had no observable Doppler shifts. The lack of Doppler shifts in glass might be understood in light of work by Lund [124], who shows that the acoustic spectrum in glass is dominated by plate resonances. This prediction is plausible in glass, where acoustic attenuation is relatively small. In PMMA, where attenuation is much larger, the plate resonances are damped, and only the frequencies generated by moving crack tip are picked up by the transducers. Experiments by Sharon et al. [83] have recently measured the "rst velocity at which microbranches were observable in soda-lime glass, for both accelerating and steady-state cracks. As in PMMA, this velocity, in glass, is equal to the same value, v"0.42c that corresponds to the onset 0 of acoustic emissions. 5.2.9. An unanswered question Many features of the instability in PMMA show up as a rough oscillation with a frequency scale of 600 kHz. We believe that this frequency is closely connected with the distance a micro-branch travels before it is screened by the main crack and stops traveling; the characteristic crack speed of 500 m/s divided by the micro-branch length scale of 1 mm gives a frequency scale of 500 kHz. The oscillations observed in crack velocity as well as on the fracture surface are all re#ections of the birth and death of micro-cracks However, we do not know why 1 mm is the characteristic distance the micro-cracks travel, so the origin of the 600 kHz oscillations is still unexplained. 6. Theories of the process zone The experiments described in the previous section have established the existence and some of the characteristics of the micro-branching instability in the dynamic fracture of amorphous materials. In order to describe theoretical work related to this instability, it is necessary to introduce some of the concepts that have been used to describe the inner workings of the process zone. A huge fraction of this work has been devoted to metals, where the process zone involves plastic deformation. As our focus is upon brittle materials, we will not discuss this area. 6.1. Cohesive zone models The stress "elds of linear elastic fracture mechanics diverge as 1/(r approaching the crack tip. In reality stress cannot diverge. The actual phenomena in the process zone that limit the stress are

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unquestionably complicated, so a variety of simple models has been proposed to show how the apparent elastic singularity actually joins smoothly onto a region around the crack tip where all "elds are "nite. The cohesive zone of Barenblatt [125,126] and Dugdale [127] is the simplest possible view of the process zone. Assume that from the tip of the crack back to a distance K, a uniform stress p acts  between the crack surfaces, and then drops abruptly to zero at the point where the separation between the surfaces reaches a critical separation of s, as shown in Fig. 36. If the crack moves in a steady state so that the cohesive zone and all the elastic "elds translate in the x direction without changing form, it is simple to "nd the energy absorbed by cohesive forces. Under these circumstances, the result of translating the crack by a distance dx is to increase by dx the length of material that has passed through the cohesive zone. To bring a length dx of material through the cohesive zone costs energy (per unit length along z)



dx

Q

dy p "dxsp .  

(158)

 If all the energy going into the crack tip is dissipated by the cohesive forces, then the energy release rate G equals sp . The idea of the cohesive zone models is to choose s and p precisely so as to   eliminate any singularities from the linear elastic problem. If any stress singularities remain after introduction of the cohesive zone, they will deliver a #ux of energy into a mathematical point at the crack tip. Therefore, the condition G"sp (159)  must coincide exactly with the condition for eliminating stress singularities. Using Eq. (81), it is possible to determine the length of the cohesive zone K. The cohesive zone can be viewed as a superposition of delta function stresses of the sort considered in that equation, but with tensile stresses p , rather than the compressive stresses used before. The stress intensity factor due to this  superposition is



K "! '

 K

dl p (2/pl "!p (8K/n .    

(160)

\ It is negative because the cohesive zone is pulling the crack faces together, and canceling out the positive stress intensity factor being produced by other forces outside the crack. According to Eq. (91), at distances from the crack tip very large compared to the cohesive zone size K, the energy release rate due to the apparently singular stress "eld is 1!l 8K G" A (v)p , '  E n

(161)

which in combination with Eq. (159) determines K. As the velocity v of a crack approaches the Rayleigh wave speed, A (v) diverges, so the width of the cohesive zone, K, must drop to zero. The ' reason is that the crack opens more and more steeply as the crack speeds up, and reaches the critical separation s sooner and sooner. Cohesive zones of this sort are frequently observed in the fracture of polymers, since behind the crack tip, there still are polymers arrayed in a `craze zonea stretching between the two crack faces

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Fig. 36. Sketch of cohesive zone model of a crack tip. The faces of the crack are pulled together by a cohesive force p hich acts until the faces are separated by a critical distance s. The crack is traveling from left to right, and the shaded  region is the cohesive zone. Because the cohesive zone cancels out any externally generated stress intensity factor, the tip does not open up quadratically, and all stresses are "nite. Fig. 37. Crack velocities measured in "nite element simulations of Xu and Needleman [129]. When the crack is allowed to branch, crack motion becomes unstable, and the mean velocity saturates at around half the Rayleigh wave speed. When branching is suppressed, the crack reaches speeds very close to the Rayleigh wave speed (dotted line).

and pulling them together. In the case of metals, the cohesive zone is viewed as a very simple representation of plastic #ow around the crack tip. In cases where the ingredients of the cohesive zone model have a clear physical interpretation, it provides a helpful way to describe the physics of the fracture energy. However, for the experimental phenomena we have described in the previous section, it is not obviously helpful. The phenomenological parameter C has been replaced by two phenomenological parameters p and s, and one  is no nearer than before to having a "rm sense of how dissipation should vary with crack velocity. Cohesive zone models have played an important role in many attempts to understand brittle fracture instabilities from a continuum viewpoint, as we now describe. 6.2. Continuum studies 6.2.1. Finite element simulations The closest correspondence with experiments in brittle amorphous materials has been obtained by Johnson [128], and Xu and Needleman [129], in "nite element calculations. Both sets of authors observe frustrated branching, oscillations in crack velocities, and limiting velocities well below the Rayleigh wave speed. Johnson [128] performed "nite element calculations of fracture in an isotropic elastic material. He implements a model for the process zone intended to simulate material weakening around the crack that results from the nucleation of defects. In this model, the process zone is not a given size

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but is adaptive, changing its size and character in accordance with the behavior of the crack. Cracks were driven by loading the crack faces with a number of di!erent loads. These resulted in maximal crack velocities of 0.29, 0.44, and 0.55c . At the lowest velocities (lowest driving), smooth 0 acceleration of a crack was observed. As the loading of the crack faces was increased, multiple attempts at crack branching were observed. As in the experiments, the length of the attempted branches increased with the crack loading. These observations were not dependent on the details of the model. One of the goals of these simulations was to explore Ravi-Chandar and Knauss's observation that the stress intensity factor, beyond a certain crack velocity, is no longer a unique function of v. Similar behavior was indeed observed, and the cases where non-uniqueness of the stress intensity factor occurred were, as in the experiments, accompanied by attempted crack branching beyond a model-dependent threshold velocity. Xu and Needleman also carried out "nite element simulations, but with a di!erent model of the process zone, and with more extensive results. To model the crack tip, Xu and Needleman implement a cohesive zone model similar to the one described in Section 6.1 which takes into account both tensile and shear stresses, and allows for the creation of new fracture surface with no additional dissipation added to the system. In order to permit cracks to branch o! the main crack line, there is an underlying grid of lines on which material separation is permitted if a critical condition is obtained. Therefore, this code combines features of "nite element models with lattice models to be described below, but is more realistic in many respects than the lattice models. The parameters used in the simulation were made to correspond to an isotropic elastic material with the properties of PMMA. Results of these simulations looked much like the experiments in PMMA. Beyond a critical velocity of 0.45c , crack velocity oscillations together with attempted crack 0 branching were observed. Branching angles of 293 were obtained, which are close to the maximal branching angles of 323 seen in experiment. Additional simulations were also conducted in a fashion that constrained the crack to move in a straight line. In this case, the crack accelerates to velocities close to c , just as in the experiments of Washabaugh and Knauss [87] that were 0 discussed in Section 4.1. The crack velocities measured in the simulations are shown in Fig. 37. These computations come closer than any other approach to describing instabilities in the fracture of PMMA in a realistic fashion. However, they do not fully resolve the conceptual questions raised, for example, in Section 6.2.3. 6.2.2. Three-dimensional crack propagation There is a number of calculations that explore the possibility that crack tip instabilities naturally result from a wiggly crack front moving through a heterogeneous medium. Rice et al. [130] analyzed the stability of a straight-line crack front in a scalar model of a crack propagating through such a heterogeneous solid. They found that although a single perturbation of the crack front was damped, the decay of a perturbation in this model was algebraic, decaying as t\ with time. Building on this result, Perrin and Rice [131,132] showed that the scalar model predicts that a crack propagating through a heterogeneous medium, where the crack front continually interacts with randomly distributed asperities, will never reach a statistically steady state. Instead, heterogeneities in the fracture energy lead to a logarithmic divergence of the rms deviations of an initially straight crack front. This result led to the suggestion that perhaps the roughness observed along a fracture surface may be the direct result of a continuous roughening of the surface that is driven by small inhomogeneities within the material. More recently Willis and Mochvan [133] calculated

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the coupling of the energy release rate to random perturbations to the crack front in the case of planar perturbations to the crack in Mode I fracture. This analysis was recently extended by the same authors to out of plane perturbations [134]. Ramanathan and Fisher [135] used the Willis and Mochvan result to calculate the dynamics of planar perturbations to a tensile crack front. They found that, in contrast to the case of the scalar model where Perrin and Rice observed logarithmic instability of the crack front, in Mode I fracture weak heterogeneity of the medium can lead to a non-decaying unstable mode that propagates along the crack front. This propagating mode is predicted to occur in materials having RC/Rv40, where a constant value of C is a marginal case. For RC/Rv'0, the propagating mode is predicted to decay. The propagation velocity of this new mode is predicted to be between 0.94}1.0c . Numerical 0 simulations of Mode I fracture in a three-dimensional medium with a constant C by Morrissey and Rice [136,137] support these results, indicating that the propagating mode is highly localized in space and indeed propagates at the predicted velocities. Both Ramanathan et al. and Morrissey et al. show that these localized modes lead to linear growth of the rms deviations of an initially straight crack with its distance of propagation. This may, as the authors suggest, provide a new mechanism for the roughness produced by a propagating crack in materials where the fracture energy does not increase rapidly with the velocity of a crack. Both the calculations and simulations have been performed for in-plane disturbances to a crack front. Disturbances of this type cannot, of course, create the out-of-plane roughness typically observed along a fracture surface. It should be interesting to see whether fully threedimensional perturbations to the crack front produce analogous e!ects. 6.2.3. Dynamic cohesive zone models A large body of calculations has been carried out by Langer and collaborators on dynamic cohesive zone models. The cohesive zone is de"ned in a manner similar to that of Section 6.1, but cracks are not presumed to move at a constant rate, or always in a straight line, and the cohesive zone therefore becomes a dynamical entity, interacting with the crack in a complex fashion. The question posed by these authors is whether crack tip instabilities will emerge from such an analysis, and the results have been extremely puzzling. In a "rst set of calculations, Barber et al. [138], Langer [139,140], and Ching [141] investigated the dynamics of cracks con"ned to straight lines. Such cracks always traveled in a stable fashion, consistent also with the results in Ref. [142], although there were various tantalizing hints of instabilities. Therefore, attention turned to the dynamics of cracks allowed to follow curvy, out of plane, paths [143}145]. Pathological short-wavelength instabilities of cracks now began to emerge from the analysis, for a reason that has a simple underlying explanation. The logic of the principle of local symmetry says that bonds under the greatest tension break "rst, and therefore cracks loaded in Mode I move straight ahead, at least until the velocity identi"ed by Yo!e when a crack is predicted to spontaneously break the symmetry inherent in straight-line propagation. This logic is called into question by a very simple calculation, "rst described by Rice [146]. Let us look at the ratio p /p right on the crack line. From Eq. (69) it is VV WW (b#1)(1#2(a!b)!4ab p VV" (162) 4ab!(1#b) p WW

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2(b#1)(a!b) " !1 . 4ab!(1#b)

79

(163)

The Taylor expansion of Eq. (163) for low velocities v is v(c#cl )  #2 , (164) 1# 2cl c(cl!c )(cl#c )    and in fact Eq. (163) greater than unity for all v. This result is surprising because it states that, in fact, the greatest tensile forces are perpendicular to a crack tip and not parallel to it, as soon as the crack begins to move. Therefore, it is hard to understand why a crack is ever supposed to move in a straight line. These remarks do not do justice to the calculations performed with the cohesive zone models. In their most elaborate versions, the crack is allowed to pursue an oscillating path, and the cohesive zone contains both tensile and shear components. In most, although not all of these models, crack propagation is violently unstable to very short-length oscillations of the tip. A summary of this work has been provided by Langer and Lobchevsky [147]. They consider a large class of models, and correct subtle errors in previous analyses. They do "nd some models in which a crack undergoes a Hopf bifurcation to an oscillation at a critical velocity. However, their `general conclusion is that these cohesive-zone models are inherently unsatisfactory for use in dynamical studies. They are extremely di$cult mathematically and they seem to be highly sensitive to details that ought to be physically unimportant.a One possibility is that cohesive zone models must be replaced by models in which plastic yielding is distributed across an area, and not restricted to a line. Dynamic elastic}plastic fracture has not, to our knowledge, considered cracks moving away from straight paths. Another possibility is that these calculations signal a failure of continuum theory, and that the resolution must be sought at the atomic or molecular scale. It is not possible right now to decide conclusively between these two possibilities. However, the experimental observation that the dynamic instabilities consist of repeated frustrated branching seems di$cult to capture in a continuum description of the process zone. 6.3. Dynamic fracture of lattice in Mode I In contrast to cohesive zone models, where the correct starting equations still are not yet known with certainty, and instabilities in qualitative accord with experiment are di$cult to "nd, calculations in crystals provide a context where the starting point is unambiguous, and instabilities resembling those seen in experiment arise quite naturally. In this section we will record some theoretical results relating to the instability. We will "rst focus on a description of brittle fracture introduced by Slepyan [148], and Marder and Liu [149,120]. The aim of this approach to fracture is to "nd a case where it is possible to study the motion of a crack in a macroscopic sample, but describing the motion of every atom in detail. In this way, questions about the behavior of the process zone and the precise nature of crack motion can be resolved without any additional assumptions. This task can be accomplished by arranging atoms in a crystal, and adopting a simple force law between them, one in which forces rise linearly up to a critical separation, and then abruptly drop to zero. We call a solid built of atoms of this type an ideal brittle crystal. A force law of this type is

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not, of course, completely realistic, but has long been thought a sensible approximation in brittle ceramics [20]. It is more realistic in brittle materials than, for example, Lennard}Jones or Morse potentials. A surprising fact is that this force law makes it possible to obtain a large variety of analytical results for fracture in arbitrarily large systems. Furthermore, the qualitative lessons following from these calculations seem also to be quite general. A summary of results from the ideal brittle crystal is E ¸attice trapping: For a range of loads above the Gri$th point, a crack can be trapped by the crystal, and does not move, although it is energetically possible [150,10]. E Steady-states: Steady-state crack motion exists, and is a stable attractor for a range of energy #ux. E Phonons: Steadily moving cracks emit phonons whose frequencies can be computed from a simple conservation law. E Fracture energy The relation between the fracture energy and velocity can be computed. E <elocity gap: The slowest steady state runs at around 20% of the Rayleigh wave speed; no slower-moving steady-state crack exists. E Instability: At an upper critical energy #ux, steady-state cracks become unstable, and generate frustrated branching events in a fashion reminiscent of experiments in amorphous materials. One may wonder about the motivation for formulating a theory for the dynamic behavior of a crack in a crystal. Is the lattice essential or can one make the underlying lattice go away by taking a continuum limit? So far as we know, all attempts to describe the process zone of brittle materials in a continuum framework have run into severe di$culties [147]. These problems do not arise in an atomic-scale description. The simplicity of the ideal brittle crystal is somewhat misleading in a number of respects. Therefore, before introducing the mathematics, we will comment on a number of natural questions regarding the generality of the predictions it makes. E Does the simple force law employed between atoms neglect essential aspects of the dynamics? We will demonstrate that the same qualitative results observed in the ideal brittle crystal occur in extensive molecular dynamic simulations using realistic potentials. E ¹he calculations are in a strip. Is this geometry too restrictive? To this question the general formalism of fracture mechanics provides an answer. Fracture mechanics tells us that as long as the conditions of small-scale yielding are satis"ed, the behavior of a crack is entirely governed by the structure of the stress "elds in the near vicinity of the crack tip. These "elds are solely determined by the #ux of energy to the crack tip. A given energy #ux can be provided by an in"nite number of di!erent loading con"gurations, but the resultant dynamics of the crack will be the same. As a result, the speci"c geometry used to load the system is irrelevant and no generality is lost by the use of a speci"c loading con"guration. This fact is borne out by the experimental work described in the previous section. E Have lattice trapping and the velocity gap ever been seen experimentally? No, they have not, but appropriate experiments have not yet been performed. Molecular dynamics simulations indicate that lattice trapping disappears at room temperature. New experiments are needed to obtain a detailed description of dynamic fracture in crystals at low temperatures. E Are results in a crystal relevant for amorphous materials? This is still an open question. However, the results of the lattice calculations seem to be remarkably robust. Adding quenched noise to

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the crystal has little qualitative e!ect. The e!ects of topological disorder are not known. However, a certain amount is known about the e!ects of increasing temperature. When the temperature of a brittle crystal increases above zero, the velocity gap closes [151], and its behavior is reminiscent of that seen in experiments performed on amorphous materials. Kulakhmetova et al. [152] "rst showed that it is possible to "nd exact analytical expressions for Mode I cracks moving in a square lattice. They found exact relationships between the energy #ux to a crack tip and crack tip velocity. They also observed that phonons must be emitted by moving cracks, and calculated their frequencies and amplitudes. Later calculations extended these results to other crystal geometries, allowed for a general Poisson ratio, showed that there is a minimum allowed crack velocity, found when steady crack motion is linearly stable, calculated the point at which steady motion becomes unstable to a branching instability, and estimated the spacing between branches [120]. These calculations are extremely elaborate. The analytical expressions are too lengthy to "t easily on printed pages, and most of the results were obtained with the aid of symbolic algebra. For this reason, we will simply summarize some of the results, and then proceed to describe in detail how the calculations work in the case of anti-plane shear, Mode III, where the algebra is much less demanding but most of the ideas are the same. 6.3.1. Mode I equations Let u describe the displacement of a mass point from its equilibrium location. Assume that the G energy of the system is a sum of terms depending upon two particles at a time, and linearize the energy to lowest order in particle displacements. Translational invariance demands that the forces between particles 0 and 1 depend only upon u !u , which will be de"ned to be D . However, the    force can be a general linear functional of D . A way to write Figs. 38 and 39 such a general linear  functional is to decompose the force between particles 0 and 1 into a component along dK , and , a component that is along dK . The "rst component corresponds to central forces between atoms, , while the second component is a non-central force. Non-central forces between atoms are the rule

Fig. 38. A sketch showing steady-state motion of crack moving in an ideal brittle crystalline strip loaded in Mode I.

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Fig. 39. The geometry of the lattice used for fracture calculations.

rather than the exception in real materials, as "rst appreciated by Born. Their quantum-mechanical origin is of no concern here, only the fact that they are not zero. Suppose that the restoring force parallel to the direction of equilibrium bonds is proportional to K , while that perpendicular to , this direction is proportional to K . The force due to the displacement of the particle along , D "u !u is  G\H> GH K dK (D ) dK )#K dK (D ) dK ) (165) , ,  , , ,  ,




"K ,




!1 (3 , 2 2







!1 (3 1 (3 DV , DW #K ,   , 2 2 2 2



(3 1 DV , DW .  2 2 

(166)

Adding up contributions from other particles in this way we get for the force due to neighbors  F(m,n)" K dK (D (m,n) ) dK ) . (167) O OH H OH H O,, By varying the constants K and K , one can obtain any desired values of shear and longitudinal , , wave speeds, which are given by 3a (K #3K ) , cl " , 8m ,

(168a)

3a c" (3K #K ) , , ,  8m

(168b)

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Fig. 40. Crack velocity versus loading parameter D"K /K for strips 80 and 160 atoms high, in limit of vanishing ' ' dissipation, computed for Mode I crack by methods of following section. D will be de"ned precisely in Eq. (178). The spring constants have values K "2.6 and K "0.26, so that the non-central forces are relatively small. The fact that , , the results have become independent of the height of the strip for such small numbers of atoms in the vertical direction suggests that relatively small molecular dynamics simulations can be used to obtain results appropriate for the macroscopic limit.

where m is the mass of each particle. In addition to forces between neighbors, it is possible to add complicated dissipative functions depending upon particle velocities. In Ref. [120], a term was added to the equations so as to reproduce the experimentally measured frequency dependence of sound attenuation in Plexiglas. There is a slight technical restriction in the calculations of Ref. [120]; right on the crack line, forces are required to be central. E. Gerde has found a way to overcome this technical limitation, and the results do not change noticeably. There is no universal curve describing Mode I fracture. Many details in the relation between loading and crack velocity depend upon ratios of the sound speeds, and upon the frequency dependence of dissipation. In the limit of central forces, K "0, it turns out to be di$cult but , not impossible to have cracks in steady state. This means that the range of loads for which cracks can run in a stable fashion is small, and depends in detail upon the amount of dissipation. Crack motion is greatly stabilized by having nonzero K . Fig. 40 displays a representative , result with some implications for the design of molecular dynamics simulations. In the limit of vanishing dissipation, the relation between crack speed and dimensionless loading D"K /K ' ' becomes nearly independent of the number of vertical rows of atoms for strips around 40 rows high. The de"nition of D is that it describes the vertical displacement imposed upon the top and bottom of the strip, but scaled so that D"1 when these boundaries have been stretched just far enough apart so that the energy stored per length to the right of the crack tip just equals the energy cost per length of extending the crack. It is de"ned precisely by Eq. (178).

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Rather than plunging into a series of graphs similar to Eq. (40), which would serve little purpose but to demonstrate that Mode I lattice fracture can indeed be solved analytically in a wide range of cases, we will move on to a simpler geometry in which the analytical methods can be displayed in full detail, and in which almost all the ideas needed to understand Mode I can be explored with much less elaborate algebra. 6.4. Dynamic fracture of a lattice in anti-plane shear, Mode III 6.4.1. Dexnition and energetics of the model We now work in detail through some of the analytical results available for a crack moving through an ideal brittle crystal loaded in Mode III. The main results are the relation between loading and crack velocity, the prediction of a velocity gap, and the calculation of the point at which steady crack motion becomes unstable. Linear stability of the steady states will not be considered here; the result obtained in [120] is that the steady states, when they exist, are always linearly stable. Consider a crack moving in a strip composed of 2(N#1) rows of mass points, shown in Fig. 41. All of the bonds between lattice points are brittle}elastic, behaving as perfect linear springs until the instant they snap at a separation of 2u , from which point on they exert no force. The location of D each mass point is described by a single spatial coordinate u(m,n), which can be interpreted as the height of mass point (m,n) into or out of the page. The force between adjacent masses is determined

Fig. 41. Dynamic fracture of a triangular crystal in anti-plane shear. The crystal has 2(N#1) rows of atoms, with N"4, and 2N#1 rows of bonds. Heights of spheres indicate displacements u(m,n) of mass points out of the page once displacements are imposed at the boundaries. The top line of spheres is displaced out of the page by amount u(m,N#1/2)" ; "D(2N#1, and the bottom line into the page by amount u(m,!N!1/2)"!; "!D(2N#1. Lines , , connecting mass points indicate whether the displacement between them has exceeded the critical value of 2u ; see D Eq. (169b).

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by the di!erence in height between them. The index m takes integer values, while n takes values of the form 1/2,3/2,2,N#1/2. The model is described by the equation 1 u( (m,n)"!buR # 2

F[u(m,n)!u(m,n)] ,   KYLY  


(169a)

with F(u)"uh(2u !"u") (169b) D describing ideal brittle springs, h the step function, and b the coe$cient of a small dissipative term. There is no di$culty involved in choosing alternative forms of the dissipation, if desired. The boundary condition which drives the motion of the crack is u(m,$[N#1/2])"$; . (169c) , It is important to "nd the value of ; for which there is just enough energy stored per length to the , right of the crack to snap the pair of bonds connected to each lattice site on the crack line. For m<0 one has u(m,n)"n; /(N#1/2) , , and the height di!erence between mass points with adjacent values of n is

(170)

; "; /(N#1/2) .   , Therefore, the energy stored per unit length in the 2N#1 rows of bonds is

(171)

;[2 Upper Bonds/Site];[Rows];[Spring Constant];;    ;  , " 2(2N#1)    N#1/2

(172)

"2Q (; ) ,  ,

(174)






(173)

with Q "1/(2N#1) .  The energy required to snap two bonds each time the crack advances by a unit length is

(175)

;[2 Bonds/Site];[Spring Constant];[Separation at fracture] (176)  ";2;;(2u )"2u . (177)   D D Therefore, equating Eqs. (174) and (177) the proper dimensionless measure of the external driving is (178) D"; (Q /u , ,  D a quantity which reaches 1 as soon as there is enough energy to the right of the crack to snap the bonds along the crack line, and which is linearly related to the displacements imposed at the edges of the strip.

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6.4.2. The detailed solution The techniques used to solve problems of this type were found by Slepyan [148,152]. There are di!erences between details of his solution and ours because Eq. (169) describes motion in a strip rather than an in"nite plate, and in a triangular rather than a square lattice. The strip is preferable to the in"nite plate when it comes time to compare with numerical simulations, while reducing to the simpler in"nite plate results in certain natural limits. The velocity gap and nonlinear instabilities were "rst found in [149,120]. 6.4.3. Symmetries Assume that a crack moves in steady state, so that one by one, the bonds connecting u(m,1/2) with u(m#1,!1/2) or u(m,!1/2) break. They break because the distance between these points exceeds the limit set in Eq. (169b) and as a consequence of the driving force described by Eq. (169c). Assuming that the times at which bonds break are known, the original nonlinear problem is immediately transformed into a linear problem. However, one has to come back at the end of the calculation to verify that 1. Bonds break at the time they are supposed to. Imposing this condition determines a relation between crack velocity v and loading D. 2. No bonds break when they are not supposed to. Imposing this condition leads to the velocity gap on the low-velocity end, and to crack tip instabilities above a critical energy #ux. Steady states in a crystal are more complicated than steady states in a continuum. In a continuum, steady state acts as u(x!vt). The closest one can come in a triangular crystal is by having the symmetry u(m,n,t)"u(m#1,n,t#1/v)

(179a)

and also u(m,n,t)"!u(m,!n,t![1/2!g ]/v) L which implies in particular that u(m,1/2,t)"!u(m,!1/2,t!1/2v) .

(179b)

(179c)

We have de"ned



0

g" 1 L mod(n!1/2,2)

if n"1/2,5/22 if n"3/2,7/22 ,

(180)

in general .

6.4.4. Eliminating the spatial index m Assuming that a crack is in steady state, we can therefore eliminate the variable m entirely from the equation of motion, by de"ning u (t)"u(0,n,t) , L

(181)

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and write the equations of motion in steady state as





#u (t!(g !1)/v) #u (t!g /v) L> L> L> L> 1 u( (t)" #u (t#1/v) !6u (t)#u (t!1/v) !buR L L L L L 2 #u (t!(g !1)/v) #u (t!g /v) L\ L\ L\ L\ if n'1/2, and

(182a)





#u (t) #u (t!1/v)   1 u( (t)" #u (t#1/v)!4u (t) !buR #u (t!1/v)      2 #[u (t)!u (t)]h(!t) #[u (t!1/v)!u (t)]h(1/(2v)!t) \  \  (182b) if n"1/2. The time at which the bond between u(0,1/2,t) and u(0,!1/2,t) breaks has been chosen to be t"0, so that by symmetry the time the bond between u(0,1/2,t) and u(1,!1/2,t) breaks is 1/2v. 6.4.5. Equations solved in terms of a single mass point on crack line Above the crack line, the equations of motion (182a) are completely linear, so it is simple to "nd the motion of every atom with n'1/2 in terms of the behavior of an atom with n"1/2. Fourier transforming Eq. (182a) in time gives





 u (u)[e SEL>\T#e SEL>T]  L> . !uu (u)"ibu# # u (u)[e ST!6#e\ ST]  L L # u (u)[e SEL\\T#e SEL\T]  L\ Let

u (u)"u (u)eIL\\ SELT . L  Substituting Eq. (184) into Eq. (183), and noticing that g #g "1 gives L L>  u (u)eI[e SEL>>EL\T#e SEL>>ELT]   !uu (u)"ibuu (u)# # u (u)[e ST!6#e\ ST]     # u (u)e\I[e SEL\>EL\T#e SEL\>ELT]   Nu#ibu#2cosh(k)cos(u/(2v))#cos(u/v)!3"0 .



(183)

(184)



(185) (186)

De"ning 3!cos(u/v)!u!ibu z" , 2cos(u/2v)

(187)

one has equivalently that y"z#(z!1 ,

(188)

y"eI .

(189)

with

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One can construct a solution which meets all the boundary conditions by writing



u (u)"u (u)e\ SELT L 



y ,>\L !y\ ,>\L

; (n!1/2) 2a # , . y,!y\, a#u N

(190)

This solution equals u for n"1/2, and equals ; 2a/(a#u) for n"N#1/2. The reason to  , introduce a is that for n"N#1/2, u(m,n,t)"; . The Fourier transform of this boundary , condition is a delta function, and hard to work with formally. To resolve uncertainties, it is better to use instead the boundary condition u

(t)"; e\?R , (191) ,> , and send a to zero the end of the calculation. In what follows, frequent use will be made of the fact that a is small. The most interesting variable is not u , but the distance between the bonds which will actually  snap. For this reason de"ne u (t)!u (t) u (t)#u (t#1/2v) \ "   ;(t)"  . 2 2

(192)

Rewrite Eq. (182b) as





#u (t) #u (t!1/v)   1 u( 1/2(t)" #u (t#1/v)!4 u (t)#u (t!1/v) !buR .     2 !2;(t)h(!t) !2;(t!1/2v)h(1/(2v)!t) Fourier transforming this expression using Eq. (190) and de"ning



;!(u)"



due SR;(t)h($t) , \

(193)

(194)

now gives u

; 2a (u)F(u)!(1#e ST);\(u)"! , ,  N u#a

(195)

with



F(u)"



y ,\ !y\ ,\

!2z cos(u/2v)#1 y,!y\,

(196)

Next, use Eq. (192) in the form ;(u)" (1#e\ ST)u (u)   to obtain ; 2a . ;(u)F(u)!2(cosu/4v);\(u)"! , N u#a

(197)

(198)

Writing ;(u)";>(u)#;\(u)

(199)

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"nally gives





89

1 1 ;>(u)Q(u)#;\(u)"; Q # , ,  a#iu a!iu

(200)

F(u) Q(u)" . F(u)!2cosu/4v

(201)

with

To obtain the right hand side of Eq. (200), one uses the facts that F(0)"!1/N, and that a is very small, so that the right-hand side of Eq. (200) is a delta function. The Wiener}Hopf technique [153] directs one to write Q(u)"Q\(u)/Q>(u) ,

(202)

where Q\ is free of poles and zeroes in the lower complex u plane and Q> is free of poles and zeroes in the upper complex plane. One can carry out this decomposition with the explicit formula

 



du ln Q(u) . (203) Q!(u)"exp lim 2p iuGe!iu C Now separate Eq. (200) into two pieces, one of which has poles only in the lower half plane, and one of which has poles only in the upper half plane: 1 1 Q ; ;\(u) ;>(u) Q ; !  , "  , ! . Q>(u) Q\(0) (!iu#a) Q\(0) (iu#a) Q\(u)

(204)

Because the right- and left-hand sides of this equation have poles in opposite sections of the complex plane, they must separately equal a constant, C. The constant must vanish, or ;\ and ;> will behave as a delta function near t"0. So Q Q\(u)  ;\(u)"; , Q\(0)(a#iu)

(205a)

Q Q>(u)  . ;>(u)"; , Q\(0)(a!iu)

(205b)

and

One now has an explicit solution for ;(u). Numerical evaluation of Eq. (203), and ;(t) from Eq. (205) is fairly straightforward, using fast Fourier transforms. However, in carrying out the numerical transforms, it is important to analyze the behavior of the functions for large values of u. In cases where functions to be transformed decay as 1/iu, this behavior is best subtracted o! before the numerical transform is performed, with the appropriate step function added back analytically afterwards. Conversely, in cases where functions to be transformed have a step function discontinuity, it is best to subtract o! the appropriate multiple of e\Rh(t) before the transform, adding on the appropriate multiple of 1/(1!iu) afterwards. A solution of Eq. (205) constructed in this manner appears in Fig. 43. 6.4.6. Relation between D and v Recall that making the transition from the nonlinear problem originally posed in Eq. (169a) to the linear problem in Eq. (182) relies on supposing that bonds along the crack line snap at time

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intervals of 1/2v. Because of the symmetries in Eq. (179), it is su$cient to guarantee that u(t)"u at t"0 . (206) D All displacements are simply proportional to the boundary displacement ; , so Eq. (206) "xes , a unique value of ; , and its dimensionless counterpart, D. Once one assumes that the crack moves , in steady state at a velocity v, there is a unique D to make it possible. To obtain Eq. (206), one needs to require that



du lim e\ SR;\(u)"u . D 2p \ R This integral can be evaluated by inspection. One knows that for positive t'0,



du exp[!iut];\(u)"0 ,

(207)

(208)

and that any function whose behavior for large u is 1/iu has a step function discontinuity at the origin. Therefore, Eqs. (207) and (205a) become u "; Q Q\(R)/Q\(0) . D ,  Since from Eq. (201) it follows that Q(R)"1, one sees from Eq. (203) that Q\(R)"Q>(R)"1 .

(209)

(210)

As a result, one has from Eq. (209) and the de"nition of D given in Eq. (178) that D"Q\(0)/(Q . (211)  To make this result more explicit, use Eq. (203) and the fact that Q(!u)"QM (u) to write

   



du1 ln Q(u) ln Q(!u) # e#iu 2p 2 e!iu

Q\(0)"exp

(212)

      

du 1 Q(u) e ln # ln Q(0) 2p !2iu QM (u) e#u

"exp

du 1 Q(u) NQ\(0)"(Q exp ! ln  2p 2iu QM (u)

.

(213)

Placing Eq. (213) into Eq. (211) gives

 

D"exp !

 

du 1 Q(u) ln 2p 2iu QM (u)

.

(214)

In order to record a "nal expression that is correct not only for the Mode III model considered here, but for more general cases, rewrite Eq. (214) as

 

D"C exp !



du 1 +ln Q(u)!ln Q(u), , 2p 2iu

(215)

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where C is a constant of order unity that is determined by the geometry of the lattice, equaling 1 for the triangular lattice loaded in Mode III, but 2/(3 for a triangular lattice loaded in Mode I [120]. When written in this form, Eq. (215) is suitable for numerical evaluation, since there is no uncertainty relating to the phase of the logarithm. When b becomes su$ciently small, Q is real for real u except in the small neighborhood of isolated roots and poles that sit near the real u-axis. Let r> be the roots of Q with negative G imaginary part (since they belong with Q>), r\ the roots of Q with positive imaginary part, and G similarly p! the poles of Q. Then one can rewrite Eq. (215) as G (216) D"C(“r\p>/r>p\, for bP0 . G G G G One may derive Eq. (216) as follows: away from a root or pole of Q, the integrand of Eq. (215) vanishes. Consider the neighborhood of a root r> of Q which falls to the real axis from the negative side as bP0. For the sake of argument, take the imaginary part of this root to be !ib. In the neighborhood of this root, say within a distance (b, the integral to compute for Eq. (215) is







1 P>>(@ du ln(u!(r>!ib)) ! . 2p > ( 2iu !ln(u!(r>#ib)) P\ @ De"ning u"u!r>, and integrating by parts gives



(217)

(

@ du 2ib ln[u#r>] (218) 2i u#b ( \ @ (219) "! lnr>#O(b .  Similar integrals over other roots and poles of Q "nally produce Eq. (216). Together with Eq. (205), Eqs. (215) and (216) constitute the formal solution of the model. Since Q is a function of the steady-state velocity v, Eq. (214) relates the external driving force on the system, D, to the velocity of the crack v. The results of a calculation appear in Fig. 42. 1 ! 2p

6.4.7. Phonon emission Right at D"1 just enough energy is stored to the right of the crack tip to break all bonds along the crack line. However, all steady states occur for D'1, so not all energy stored to the right of the crack tip ends up devoted to snapping bonds. The fate of the remaining energy depends upon the amount of dissipation b, and the distance from the crack tip one inspects. In the limit of vanishing dissipation b, traveling waves leave the crack tip and carry energy o! in its wake; the amount of energy they contain becomes independent of b. Such a state is depicted in Figs. 43 and 44, which shows a solution of Eq. (205) for v"0.5, N"9, and b"0.01. For all nonzero b, these traveling waves will eventually decay, and the extra energy will have been absorbed by dissipation, but the value of b determines whether one views the process as microscopic or macroscopic. The frequencies of the radiation emitted by the crack have a simple physical interpretation as Cherenkov radiation. Consider the motion of a particle through a lattice, in which the phonons are described by the dispersion u (k) . ?

(220)

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Fig. 42. The velocity of a crack v/c, scaled by the sound speed c"(3/2, is plotted as a function of the driving force D. The calculation is carried out using Eq. (216) for N"9.

Fig. 43. A plot of ;(t) for v"0.5, N"9, and b"0.01, produced by direct evaluation of Eq. (205). Note that mass points are nearly motionless until just before the crack arrives, and that they oscillate afterwards for a time on the order of 1/b.

Fig. 44. Graphical solution of Eq. (223), showing that for low velocities, a large number of resonances may be excited by a moving crack.

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If the particle moves with constant velocity *, and interacts with the various ions according to some function I, then to linear order the motions of the ions can be described by a matrix D which describes their interactions with each other as u( J "! D (RJ!RJY)uJY# I (RJY!*t) . (221) I IJ J I JJY JY Multiplying everywhere by e k  RJ, summing over l, letting K be reciprocal lattice vectors, and letting X be the volume of a unit cell gives 1 mu( (k)" D (k)u (k)# e * k>KRI (k#K) . (222) I IJ J I XK J Inspection shows that the lattice frequencies excited in this way are those which in the extended zone scheme [154] obey u(k)"* ) k .

(223)

Pretending that the crack is a particle, one can use Eq. (223) to predict the phonons that the crack emits. There is another version of this argument that is both simpler and more general. The only way to transport radiation far from a crack tip is in traveling waves. However, in steady state, the traveling waves must obey symmetry (179a). In a general crystal with lattice vectors R and primitive vectors a, applying this requirement to a traveling wave exp[ik ) R!iut] gives e k R>La\ SkR>L?T "e k  R\ SkR .

(224)

Assuming a and * are parallel, Eq. (223) results again. There are two phonon dispersion relations to consider. One gives the conditions for propagating radiation far behind the crack tip, and the other gives the conditions for propagating radiation far ahead of the crack tip. Far behind the crack tip, all the bonds are broken. Finding waves that can travel in this case is the same as repeating the calculation that led to Eq. (198), but with ;\ set to zero, since all bonds are broken, and with ; "0, since phonons can propagate without any , driving term. Examining Eq. (198), one sees that the condition for surface phonons to propagate far behind the crack tip is F(u)"0. Similarly, far ahead of the crack tip no bonds are broken ;\ should be set equal to ;, and the condition for phonons is F(u)!2cosu/4v"0. According to Eq. (201), the roots and poles of Q(u) are therefore the phonon frequencies behind and ahead of the crack, and these are the quantities appearing in Eq. (216). We do not know if Eq. (216) is more than approximately correct for particle interactions more general than ideally brittle bonds. 6.4.8. Forbidden velocities After making sure that bonds along the crack line break when they are supposed to, it is necessary to verify that they have not been stretched enough to break earlier. That is, not only must the bond between u and u reach length 2u at t"0, but this must be the "rst time at which > \ D that bond stretches to a length greater than 2u . For 0(v(0.32 (the precise value of the upper D limit varies with b and N) that condition is violated. The states have the unphysical character shown in Fig. 45. Masses rise above height u for t less than 0, the bond connecting them to the D lower line of masses remaining however intact, and then they descend, whereupon the bond snaps.

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Since the solution of Eq. (182) is unique, but does not in this case solve Eq. (169a), no solutions of Eq. (169a) exist at all at these velocities. Once the crack velocity has dropped below a lower critical value, all steady states one tries to compute have this character. This argument shows that no steady state in the sense of Eq. (179a) can exist. It is also possible to look analytically for solutions that are periodic, but travel two lattice spacings before repeating. No solutions of this type have yet been found. Numerically, one can verify that if a crack is allowed to propagate with D right above the critical threshold, and D is then very slowly lowered through the threshold, the crack stops propagating. It does not slow down noticeably; suddenly the moving crack emits a burst of radiation that carries o! its kinetic energy, and stops in the space of an atom. That is why Fig. 45 shows a velocity gap. 6.4.9. Nonlinear instabilities It was assumed in the calculations predicting steady states that the only bonds which break are those which lie on the crack path. From the numerical solutions of Eq. (205), one can test this assumption; it fails above a critical value of D. The sound speed c equals (3/2, and velocities will be scaled by this value. For N"9, at a velocity of v /c"0.6662, D "1.1582, the bond between   u(0,1/2) and u(1,1/2) reaches a distance of 2u some short time after the bond between u(0,1/2) and D u(0,!1/2) snaps. The steady-state solutions strained with larger values of D are inconsistent; only dynamical solutions more complicated than steady states, involving the breaking of bonds o! the crack path, are possible. To investigate these states, one must return to Eq. (169) and numerically

Fig. 45. Behavior of ;(t) for v"0.2, b"0.05, and N"9. Notice that at t"0, indicated by the dashed line, u is decreasing, and that it had already reached height 1 earlier. This state is not physical. Fig. 46. Pictures of broken bonds left behind the crack tip at four di!erent values of D. The top "gure shows the simple pattern of bonds broken by a steady-state crack. At a value of D slightly above the critical one where horizontal bonds occasionally snap, the pattern is periodic. All velocities are measured relative to the sound speed c"(3/2. Notice that the average velocity can decrease relative to the steady state, although the external strain has increased. As the strain D increases further, other periodic states can be found, and "nally states with complicated spatial structure. The simulations are carried out in a strip with half-width N"9, of length 200 and b"0.01. The front and back ends of the strip have short energy-absorbing regions to damp traveling waves. The simulation was performed by adding unbroken material to the front and lopping it o! the back as the crack advanced.

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solve the model directly. These simulations have been carried out [149,155] and some results are contained in Fig. 46. The diagram shows patterns of broken bonds left behind the crack tip. Just above the threshold at which horizontal bonds begin to break, one expects the distance between these extra broken bonds to diverge. The reason is that breaking a horizontal bond takes energy from the crack and slows it below the critical value. The crack then tries once more to reach the steady state, and only in the last stages of the approach does another horizontal bond snap, beginning the process again. This scenario for instability is similar to that known as intermittency in the general framework of nonlinear dynamics [156]; the system spends most of its time trying to reach a "xed point which the motion of a control parameter has caused to disappear. Here is a rough estimate of the distance between broken horizontal bonds. Let u (t) be the length  of an endangered horizontal bond as a function of time. Actually, one needs to view matters in a reference frame moving with the crack tip, so every time interval 1/v, one shifts attention to a bond one lattice spacing to the right. When D is only slightly greater than D , the length of such  a bond viewed in a moving frame should behave before it snaps, as u &2u #(Ru /RD)(D!D )!due\@R . (225)  D   Here Ru /RD means that one should calculate the rate at which the steady-state length of u would   change with D if this bond were not allowed to snap, and du describes how much smaller than its steady-state value the bond ends up after the snapping event occurs. Ref. [120] shows that deviations from steady states die away at long times as exp[!bt]. From this expression, one can estimate the time between snapping events by setting u to 2u and solving Eq. (225) for t. The result is that the  D frequency l with which horizontal bonds snap should scale above the critical strain D as  1 Ru  [D!D ] , (226) l&!b/ln  du RD






a result that is consistent with the numerics, but hard to check conclusively. One can calculate numerically that Ru /RD"5.5 for the conditions of Fig. 46, but du is hard to "nd independently.  Assuming that Eq. (226) is correct, one "nds from the second picture of Fig. 46 that du"0.04. Further increasing the external strain D makes a wide variety of complicated behavior possible, including dendritic patterns, in the lowest panel of Fig. 46 that are reminiscent of experiment. 6.4.10. The connection to the Yowe instability The basic reason for the branching instability seen above is the crystal analog of the Yo!e instability, working itself out on small scales. The Mode III calculation "nds that the critical velocity for the instability to frustrated branching events is indeed close to the value of 0.6c 0 predicted by Yo!e in the continuum. The critical velocity seen experimentally in amorphous materials is 1/3 of the wave speed, not 2/3. This discrepancy could be due to some combination of three factors. 1. The force law between atoms is actually much more complicated than ideal snapping bonds. Gao [157], has pointed out that the Rayleigh wave speed in the vicinity of a crack tip may be signi"cantly lower than the value of c far away from the tip because material is being stretched 0 beyond the range of validity of linear elasticity.

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Fig. 47. The two lines, as indicated by the legend, compare analytical results for steady crack velocities in a strip 20 atoms high with simulation results in a strip 20 high and 60 wide. The symbols display the e!ect of random bond strengths on crack velocities. The springs all snap at the same extension, but have spring constants that vary randomly by the amounts indicated in the legend.

2. The experiments are at room temperature, while the calculations are at zero. 3. The experiments are in amorphous materials, while the calculations are in crystals. 6.5. The generality of the results in a ideal brittle crystal It is easy to carry out numerical simulations that add various e!ects to the solvable model that are di$cult to include analytically. For example, one can vary the interparticle potentials from the simple form given in Eq. (169b), one can make the bond strengths in the lattice vary randomly, or watch the crack propagate in a lattice maintained at nonzero temperature. A brief summary of these numerical searches would be that small changes in the basic model, Eq. (169) do not alter the qualitative conclusions obtained so far. For example, in Fig. 47, one can see the e!ect of making bond strengths random by varying amounts. The randomness is implemented by setting the spring constants between neighbors in Eq. (169) to (1#Krr)1/2, where r and r are nearest neighbors,   and Krr is a random variable ranging with uniform probability from !K to K. The springs still snap at the extension of 2u . The qualitative changes introduced by the randomness are that it D becomes possible for the crack to jump up or down by a lattice spacing, as shown in Fig. 48, and it also becomes possible for the crack tip to encounter a particularly tough bond, and arrest prematurely. Fig. 47 was produced by obtaining a moving crack at D"1.2, and then very slowly ramping D down to 1, with velocities calculated by measuring the time needed for the crack to progress 20 lattice spacings. The #uctuations visible in Fig. 47 are to be understood as resulting from this procedure; if enough averaging were carried out, no #uctuations in velocity would be visible despite the presence of randomness. We have also carried out studies of the e!ect of temperature. We later found in molecular dynamics simulations that three-dimensional calculations are quite di!erent from two-dimensional ones, so we will not present the results here. There is a qualitative di!erence between static

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Fig. 48. In the presence of randomness, cracks no longer travel along straight-line paths. This simulation is carried out with K"0.2, for a strip 20 atoms high and 80 long

randomness and thermal #uctuations that is quite important for fracture. If a crack encounters a tough spot in a material, it can halt and sit there forever. Thermal #uctuations might halt a crack temporarily, but are just as likely to take a static crack and give it energy to start moving. Lattice-trapped cracks are not completely static in the presence of thermal #uctuations; they creep ahead with some probability [158]. When the rate of creep rises to speeds on the order of the sound speed, the distinction between creeping and running cracks vanishes, and the velocity gap vanishes. The robustness of the branching phenomena described in this model was further illustrated by recent numerical studies of Eq. (167) in the nonlinear regime, where micro-branches are observed. Hieno and Kaski [159] have investigated some e!ects of changing the model parameters governing Mode I fracture. A number of features similar to those observed in the experiments on amorphous materials were observed. Disorder of the perfect crystal lattice, for example, was introduced by imposing a random distribution of the local values of Young's moduli. The e!ect of the disorder was to disrupt the periodicity of the frustrated branching events, and thus broaden the otherwise sharp branching frequency spectrum. Additional numerical work in this model by Astrom and Timonen [160] observed power-law dependence of the micro-branch trajectories with the same 0.7 power observed in experiments in both PMMA and glass. 6.6. Molecular dynamics simulations The fracture of brittle solids is a physical process which naturally connects large and small scales [7,8]. Stresses and strains which cause the fracture are applied on macroscopic scales, while the end result is the severing of bonds on an atomic scale. Therefore, it seems reasonable to assume that computer simulations of the fracture process that account for phenomena at the atomic level must be very large. Several such simulations are now being carried out, in systems involving as many as hundreds of millions of atoms [161}163]. 6.6.1. Scalable molecular dynamics simulations However, the goal in computer simulation should not be to build the largest simulation possible, but to build the smallest one capable of answering speci"c physical questions. Many features of

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brittle fracture may pro"tably be studied in simulations that are comparatively small, involving only thousands of atoms. The essential problem facing the simulations is one of length and time scales. We are now in the midst of carrying out fracture experiments in silicon. The samples are several centimeters long, several high, a millimeter thick and contain about 10 atoms. The duration of an experiment is around 50 ls. The largest simulations now being performed follow 10 atoms for around 10\ s. Direct atomic simulation of our samples therefore requires a 10-fold increase in computer power over what we now have. How then is comparison to be performed? A "rst thought is to merge atomistic and continuum simulations. Use atoms in the vicinity of the crack tip, and continuum elasticity everywhere else. This approach has the potential to solve the problem of length scales, but not the problem of time scales. We describe below what we believe to be a more elegant approach to the problem that takes care of length and time scales at once. The basic idea is to make greatest possible use of the conceptual framework of fracture mechanics. Fracture mechanics allows the entire crack problem to be followed in the context of continuum mechanics if only one is supplied with a relationship between the fracture energy, C, and the velocity, v. The goal is therefore to obtain this relationship from an atomic simulation that is absolutely as small as possible. The results can then be employed as input to continuum simulations making all the macroscopic predictions desired. The analytical results of Sections 2 and 6.3 provide tremendous help in designing these simulations. The "rst suggestion of the theoretical framework is that one look for steady states: con"gurations obeying Eq. (179a). Since states of this sort repeat inde"nitely, observation over "nite time allows one to make predictions for an in"nite time, solving the time scaling problem. The only question to be settled is how long it takes a numerical simulation to pass through transients and reach the steady state. In a strip of height ¸ this time is proportional to the time needed for sound to travel from the crack tip to the top of the system and return. In practice, steady states establish themselves fully after about 100 such transit times. The taller a strip becomes, the longer this time, so one wants a strip that is as narrow as possible. The analytical work is particularly useful in establishing the minimum height of a strip. Fig. 40 shows the relation between v and D for strips of height 80 and 160. Above a height of 50, the curves become almost indistinguishable. The di!erences are certainly not measurable in experiment. We conclude that for ideal brittle fracture in steady state, the relationship between v and D may be established in a strip 50 atoms high. We have performed our molecular dynamic simulation runs in strips 200 atoms high and 50 atoms high to make sure the results do not change. The number of atoms in the simulations ranges between 100 000 and 200 000. In order to keep the simulation running for the times needed to achieve steady states, we put it on a conveyor belt [151]. Whenever the crack approaches within a certain distance (typically 100 A> ) of the right end, we glue a slab of new unbroken material to that end, and chop an equal amount o! the back. Finding the length needed in the x direction to obtain satisfactory results has been a matter of numerical trial and error: a total system length of 500 A> makes the results independent of length. The "nal test consists in obtaining a steady state. The three dimensional system has spontaneously adopted a dynamical state with the periodicity of a unit cell (#at front) along the z direction, obeying the symmetry Eq. (179a) where the behavior of atoms near the top is independent of the length of the system along x, and the relationship between v and D is

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independent of the height of the strip. Therefore, this steady state provides one point on the curves relating v and D, which we believe to be correct in the limit of in"nite size systems and long times. The particular solid on which we have focused so far for our numerical studies is silicon. The reason for this choice is that silicon single crystals with a variety of orientations are inexpensive, thus making numerical predictions amenable to experimental veri"cation. Silicon is very brittle, the crystal structure is well known, and considerable e!ort has been expended in developing classical three-body potentials suitable for use in molecular dynamics. We employ Stillinger}Weber potentials, with a modi"ed version of the three-body term, but keep an open mind as to whether some other potential might provide better results. Indeed, given the severely nonequilibrium nature of fracture, it is possible that no classical potentials provide a realistic account, and in view of the emission of electrons and light that is observed in the vicinity of the crack tip, it is entirely possible that density functional theory could fail as well. Only a detailed and patient comparison of theory and experiment, which has not yet been performed, will be able to settle such doubts. Two additional points concerning the simulations: 1. In steady state, the energy consumed by the crack per unit length must equal the energy stored per unit length to the left. This claim remains true even for strains large enough that applicability of linear elasticity could be called into question. This claim relies on symmetry rather than the matched asymptotics of fracture mechanics. 2. The simulation contains a complete description of the process zone. As more energy is fed to the crack tip, as temperature increases, or as one moves to inhomogeneous or ductile materials, the size of the process zone increases, and the simulation size will have to increase accordingly. We do not expect molecular dynamics to provide easy predictions for materials where the process zone is on the order of microns, let alone millimeters. The amorphous polymers on which our experiments have focused provide a particularly great challenge. 6.6.2. Sample results in silicon Simple predictions for the zero-temperature fracture of silicon on the (111) plane in the (110) direction appear in Fig. 49. Not only is there a velocity gap of 1500 m/s but there are also plateaus and hysteretic loops that are essentially unexplained. It is natural to wonder how much of this intricate structure should be preserved at room temperature. We have therefore performed simulations focusing on the "rst hysteresis loop at temperatures between 0 and 300 K. An interesting result of these simulations is that lattice trapping is predicted to disappear at a temperature of 150 K, and experiments will have to proceed to temperatures lower than this to see it. Although this result was obtained in silicon, it implies an explanation for why no lattice trapping has yet to be observed experimentally in either crystalline or amorphous materials. Cracks at zero temperature display a clear dynamic instability at a critical energy #ux corresponding to D"2. Up until this point, phonons are able to carry away all excess energy. This instability does not take the form of a simple micro-branching instability, partly because bonds can easily rejoin above the main crack line in a single component solid with no environmental impurities available. Instabilities at room temperature have not yet been explored either numerically or analytically.

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Fig. 49. Steady state crack velocities in silicon as a function of the dimensionless loading parameter D for fracture along (111) and (110) planes [151].

6.6.3. Large size molecular dynamic simulations A number of very large-scale molecular dynamic simulations of dynamic fracture have been performed [161}164]. The "rst of these is the work by Abraham et al. where the dynamics of a crack were investigated within a 10 atom crystal, where the atoms are attached via a Lennard Jones 6}12 potential. Stress was applied to the system by displacing opposing boundaries of the system at a strain rate that is approximately equivalent to that obtained in explosive loading applied to crack faces. These conditions were necessary to achieve su$cient acceleration of the crack tip so as to achieve high enough crack velocities over the duration of the simulation to be able to detect the existence of instabilities of the crack's motion. The results of these simulations were surprisingly close to experimental observations in amorphous materials. The crack was observed to accelerate smoothly until reaching a velocity of 0.32c . At velocities beyond this the 0 instantaneous velocity of the crack tip was observed to become erratic, as large velocity #uctuations occurred. These #uctuations were coupled with a `zig-zaga motion of the crack tip, which formed in its wake a rough fracture surface. These interesting simulations highlight the robust and general nature of the instability. The robust nature of the instability was further highlighted by the work of Zhou et al. [165] where crack propagation was investigated in a 400 000 atom crystal where the atoms were coupled via a Morse potential. By varying the applied strain rates the maximal velocity that a crack could achieve over the duration of a simulation was varied between 0.18 and 0.36c , where the strain rates 0 used corresponded, as in the Abraham et al. work, to explosive loading of the system. At a velocity of 0.36c , instability of the crack was observed to occur by branching of the crack. The branching 0 process was seen to be immediately preceded by the nucleation of a dislocation in the crystal together with a build up of the phonon "eld in the vicinity of the crack tip. In both of the above large-scale simulations, the entire fracture process (from initiation to the instability onset) occurred over a time corresponding to approximately 1 ns. For this reason both

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the strain rates used and the amount of strain in the material at the onset of fracture (approximately an order of magnitude larger than observed in real materials) had to be extremely large. Thus, the close correspondence of the results of the simulations with the laboratory results obtained in amorphous materials is rather surprising. The short time scales used in these simulations, of course, preclude examination of steady-state properties of the system. It should therefore be interesting to compare the results obtained in these experiments with steady-state results obtainable in the smaller simulations described in Section 6.6.1. 7. Conclusions We began working in the "eld of fracture with the misconception that there was a problem with the terminal velocity of cracks needing to be explained. Gradually, we came to understand that the di$culty in explaining the terminal velocity of cracks had been so persistent because the problem had not been properly posed. The real question concerned the nature of dissipation near the crack tip. In a su$ciently brittle material, it is surprising to need to consider dissipation. It seems natural to suppose that energy will mainly be consumed by the act of snapping bonds to create new surface, and that this process should depend only weakly upon crack velocity. It should have been obvious all along that this view cannot be correct. By loading cracks in di!ering fashions, greatly varying quantities of energy can be forced into the crack tip. The tip must "nd some mechanism for dealing with the energy not needed to break a minimal set of bonds, and once transfer into phonons and other tame mechanisms has been exhausted, the crack tip begins a sequence of dynamical instabilities, designed to digest energy by creating rami"ed networks of broken surface on small scales. These ideas o!er a detailed account of aspects of fracture that had been considered too complicated to describe quantitatively, or were simply ignored. There is no con#ict with conventional fracture mechanics. One of the basic tenets of fracture mechanics is that if the system of interest is su$ciently large that the assumption of small-scale yielding is justi"ed, all of the complex dissipative processes that go into creating a new fracture surface can be thrown together into a phenomenological function of velocity called the fracture energy. In large enough systems, the crack tip instabilities occur within the process zone, where the descriptive power of linear elastic fracture mechanics does not operate. Thus, from the viewpoint of conventional fracture mechanics, the instabilities simply rede"ne the value of C. On the other hand, without fundamental understanding of the structure of the process zone that yields C, the theory loses both its descriptive as well as its predictive power. We have now achieved a good understanding of the structure and dynamics of mechanisms for dissipation within the near vicinity of the tip of a moving crack in a brittle material. We "nd that fracture in brittle materials is governed by a dynamic instability which leads to repeated attempted branching of the crack. The major features of this instability are summarized below. 1. There seems to be no material so brittle that the process zone always remains featureless. Once energy #ux to the tip of a crack exceeds the maximum value that can safely be transported away by phonons or other dissipative mechanisms, the tip undergoes a progression of instabilities.

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2. Beyond the instability threshold, an initial propagating crack changes its topology by creating short-lived microscopic crack branches. The micro-branching process give rise to oscillations of the velocity of the leading crack. In addition, the branching process forms the non-trivial surface structure that is observed on the fracture surface of amorphous materials. 3. Once the micro-branching instability occurs, the amount of energy dissipated by the system increases by an amount that is simply the total length formed by the micro-branches and main crack times the fracture energy of a single crack. Experimentally, this can increase the total crack length (and with it the dissipation) by up to an order of magnitude. 4. The mean length of the frustrated branches increases as a function of the energy #ux into the crack. Eventually, the micro-branches evolve into macroscopic crack branches. The onset of the micro-branching instability therefore provides a well-de"ned criterion for the process that eventually culminates in macroscopic crack branching. There is evidence that a second transition may occur at velocities larger than v where the width of the micro-branches appears  to diverge. This transition may be a su$cient condition for macroscopic crack branching to occur. 5. In crystalline materials, theory can account for the instability in some detail, and makes the added prediction that a forbidden band of velocities exists for cracks. A crack may only propagate stably above a "nite minimum velocity. Molecular dynamics simulations of crystalline silicon indicate that this forbidden band of states may disappear by room temperature, but should be observable in low temperature experiments. Thus the picture is still incomplete. The most detailed experiments are in amorphous materials at room temperature, while the most detailed theory applies only to crystals at very low temperatures. A theoretical description of fracture general enough to encompass the full range of brittle materials, or even to provide a precise description of what `brittlea means has not yet been obtained. Attempts to describe the process zone within some sort of continuum framework have not yet been successful, but the limitations in the atomic point of view provide ample motivation for continued e!ort along the continuum line. Although the instability appears in dynamic "nite element simulations, it has no analytical explanation in a continuum framework. In fact, many classical models of the process (cohesive) zone have been shown to be ill-posed in that they admit a continuum of possible states under identical conditions. Theories formulated on a lattice, on the other hand, do not exhibit these di$culties. It remains to be seen whether a simple continuum limit exists, or whether a crucial ingredient in understanding fracture is the discreteness of the underlying atoms. References [1] T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, CRC Press, Boca Raton, FL, 1991. [2] H.L. Ewalds, R.J.H. Wanhill, Fracture Mechanics, Edward Arnold, London, 1984. [3] D. Broek, Elementary Engineering Fracture Mechanics, 2nd ed., Sijtho! and Noorho!, Alphen aan den Rijn, 1978. [4] J.F. Knott, Fundamentals of Fracture Mechanics, Butterworth, London, 1973. [5] G.P. Cherepanov, Mechanics of Brittle Fracture, McGraw-Hill, New York, 1979. [6] J.G. Williams, Fracture Mechanics of Polymers, Wiley, New York, 1984. [7] L.B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, Cambridge, 1990.

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INTERACTION OF VACUUM ULTRAVIOLET PHOTONS WITH MOLECULES. FORMATION AND DISSOCIATION DYNAMICS OF MOLECULAR SUPEREXCITED STATES

Y. HATANO Department of Chemistry, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 313 (1999) 109}169

Interaction of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states Yoshihiko Hatano* Department of Chemistry, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan Received April 1998; editor: J. Eichler

Contents 1. Introduction 2. Electron impact studies of superexcited molecules 2.1. General 2.2. Production of excited fragments by electron impact 2.3. Absolute cross sections for molecular photoabsorption and photoionization processes by fast electron impact 2.4. Translational spectroscopy of dissociation fragments produced by electron-impact excitation of molecules 3. Photon impact studies of superexcited molecules 3.1. General

112 115 115 117

119

123 129 129

3.2. Laser-multiphoton studies of superexcited molecules 3.3. Synchrotron radiation studies of superexcited molecules 3.4. Other examples of synchrotron radiation studies on superexcited molecules 4. Related topics 4.1. Superexcited states as reaction intermediates in various collision processes 4.2. Superexcited states in reaction kinetics 4.3. Superexcited states in the condensed phase 5. Conclusions and future perspectives Acknowledgements References

131 132 158 159 159 160 161 162 163 163

Abstract A survey is given of recent progress in experimental studies of photoabsorption, photoionization, and photodissociation of molecules in the wavelength range of photons in the vacuum ultraviolet. Corresponding photon energies are 10}50 eV, at which most of the optical oscillator strength of molecules are distributed. Examples of molecules chosen here

* Tel.: #81-3-5734-2235; fax: #81-3-5734-2655; e-mail: [email protected]. 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 8 ) 0 0 0 8 6 - 6

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range from simple diatomic and triatomic molecules to polyatomic molecules such as hydrocarbons, alcohols, ethers, and some Si-containing compounds in the gas phase. Some remarks are presented on molecules in the condensed phase. A particular emphasis in the survey is placed on current understanding of the spectroscopy and dissociation dynamics of molecules in the superexcited states which are produced in the interaction of photons in this wavelength (or energy) range with molecules. A comparative survey is also given of the spectroscopy and dynamics of superexcited states which are produced in electron}molecule collisions. Most of the observed superexcited states are assigned to high Rydberg states which are vibrationally (or/and rotationally), doubly, or inner-core excited, and converge to each of ion states. Non-Rydberg superexcited states are also observed. Some remarks are also presented of superexcited states as reaction intermediates or collision complexes in electron}ion recombination, electron attachment, and Penning ionization processes.  1999 Elsevier Science B.V. All rights reserved. PACS: 33.80.!b Keywords: Oscillator strength; Photoabsorption cross section; Photoionization cross section; Photoionization quantum yield; Synchrotron radiation; High-Rydberg states

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1. Introduction The interactions of photons with molecules are classi"ed into absorption, scattering, and pair production. In this article, photons of moderate energies particularly in the vacuum ultraviolet (VUV) region are discussed, and therefore only the absorption process is considered. The absorption of a single photon by a molecule in the electronically ground state changes its electronic state from the ground state 0 to a "nal excited or ionized state j. Its transition probability is expressed in terms of the optical oscillator strength f [1,2]. H It is in turn expressed in terms of E /R, the transition energy measured in units of the Rydberg H energy R"me/(2h)"13.6 eV and of M, the dipole matrix element squared as measured in a, H  where a "h/me"0.0529 nm, as  f (E )"(E /R)M . (1) H H H H A set of E and f characterizes a discrete spectrum. To discuss a continuous spectrum, one H H expresses the oscillator strength in a small region of the excitation energy between E and E#dE as (df/dE)dE, and calls df/dE the oscillator-strength distribution, or, more precisely, the spectral density of the oscillator strength. The total sum of the oscillator strength including discrete and continuous spectra is equal to the total number Z of electrons in the molecule, viz.,



 (2) f (E )# (df/dE)dE"Z , H H ' H where I represents the ("rst) ionization potential. Eq. (2) is called the Thomas}Kuhn}Reiche (TKR) sum rule. The oscillator-strength distribution is proportional to the cross section p for the absorption of a photon of energy E. Explicitly one may write df df df p"4naaR "8.067 ) 10\ (nm)"0.01098 (eV\ nm) ,  dE d(E/R) dE

(3)

where a"e/hc"1/137.04. A decisive step in the physical and physicochemical stages of the action of any ionizing radiation on matter is collisions of secondary electrons in a wide energy range with molecules. Ionization and excitation of molecules in collisions with electrons in the energy range higher than about 10 eV are well elucidated by the Born}Bethe theory [3,4] and the cross section Q (¹) at the electron energy H ¹ to form the state j at least for optically allowed transitions is given by 4C ¹ 4paR  M ln H , Q (¹)" H H R ¹

(4)

where C is a constant and M is given by H H  R df dE , (5) M" (E) H H E dE ( where J is the threshold energy of the state J formation and (E) the probability of the state j H formation upon excitation of the energy E. Using Eq. (4) and further assumptions, Platzman [5,6]



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showed that the number of product species j formed per 100 eV absorbed, G , is approximately H proportional to M. With the use of the = value, viz., the mean energy, measured in eV, for the H production of an ion pair, one may write 100 M H Gj" = M G where M is the dipole matrix element squared for ionization, i.e., G  R df M" g(E) dE , G E dE ' where g(E) is the quantum yield for ionization [7]. Since



(6)

(7)

g(E)"p (E)/p (E) , (8) G R where p (E) and p (E) are, respectively, the photoionization cross section and the photoabsorption G R cross section, Eq. (6) means that the radiation chemical yield G is obtained from optical crossH section data. Eq. (6) is therefore called the optical approximation. Optical cross-section data are, thus, of fundamental importance in understanding not only the interactions of photons with molecules, but also the actions of any ionizing radiation with matter. Oscillator-strength values have long been measured for various molecules in the wavelength regions at least longer than the near-ultraviolet (UV) region, whereas until recently measurements in the wavelength region shorter than the LiF cuto! at 105 nm (at which the photon energy is 11.8 eV) were relatively few because of experimental di$culties in obtaining appropriate photon sources and because no suitable window materials were available [8}11]. The cuto! energy of 11.8 eV corresponds, roughly speaking, to the ionization potentials of commonly occurring molecules. The sum of the oscillator strengths below 11.8 eV amounts to only a few percent [8}11] of the total sum, which is equal to the total number of electrons in a molecule, according to the Thomas}Kuhn}Reiche sum rule. The absorption in the vacuum ultraviolet and soft X-ray (VUVSX) region is much stronger than in all other wavelength regions. Since there have been remarkable advances in synchrotron radiation (SR) research and related experimental techniques in the VUV-SX region, many measurements in this region are now available [8}11]. Such a situation of SR as a photon source is summarized in Fig. 1. Fig. 1 shows the wavelengths of electromagnetic radiation from the infrared to the c-ray regions and corresponding photon energies [8,9]. Characteristic X-rays, Co-c rays, and VUV light from discharge lamps are indicated by the arrows. The shaded areas indicate regions for which photon sources, apart from SR, are available and correspond to photochemistry and radiation chemistry. Fig. 1 clearly demonstrates that SR bridges the wide gap in the photon energy between

 The quantity g was named by Platzman either `ionization e$ciencya or `ionization probabilitya. They have proposed recently that g should be called `ionization quantum yielda instead of `ionization e$ciencya in the case of photon-impact experiments in order to prevent terminological confusion with mass spectrometrical `ionization e$ciencya curves. In cases other than photon-impact experiments, therefore, it is recommended that `ionization probabilitya be also used instead of `ionization e$ciencya.

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Fig. 1. Synchrotron radiation (SR) chemistry as a bridge between radiation chemistry and photochemistry. The dipole oscillator strength df/dE is shown as a function of wavelength j and photon energy E. Note the relation E ) j"1.24;10 (eV nm). The intensity of SR is also shown, as are the energies of photons from several line sources. VUV, EUV, SX, and HX stand for vacuum ultraviolet, extreme ultraviolet, soft X-ray, and hard X-ray, respectively [8,9].

photochemistry and radiation chemistry, i.e., `photon-collision chemistrya and `electron-collision chemistrya, respectively. Platzman [5,12] considered theoretically the interactions of ionizing radiation with atoms and molecules, and pointed out that atoms and molecules receive a large fraction of energy from ionizing radiation with a spectrum determined by their optical oscillator strength distributions according to the optical approximation. He also pointed out the following important features, although very few experimental data were available at that time. (a) The value of an oscillator strength distribution shows generally its maximum at the energy of 10}30 eV, which is larger than the "rst ionization potential (I). (b) Ionization e$ciency (g) values of molecules are much smaller than unity in the energy range just above I. (c) There exists a hydrogen isotope e!ect in g values. By combining all available information, (a)}(c), Platzman indicated an important role of highly excited electronic states in the primary action of ionizing radiation as follows: ABPAB>#e\ direct ionization ,

(9)

PAB superexcitation ,

(10)

PABH excitation ,

(11)

ABPAB>#e\ autoionization ,

(12)

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PA#B dissociation ,

(13)

PAB processes other than Eqs. (12) and (13) .

(14)

When a molecule AB receives energy which is larger than its I, AB may be directly ionized (9) and may be excited (10) to form AB which was named by Platzman a `superexciteda molecule. The superexcited AB can ionize (12) or dissociate into neutral fragments (13). Since g is de"ned by Eq. (8), the value of 1!g shows the importance of the dissociation process in the total decay channels of AB because process (14) does not seem to be important in the decay of such extremely highly excited states. As one of process (14), ion-pair formation, ABPA>#B\, has been extensively studied to clarify the dynamics of superexcited molecules although its cross section is much smaller than those of processes (12) and (13) [13}17]. The value of g may therefore be smaller than unity in the energy range even above I, which means that the molecules excited into this energy range are not always ionized but use their energy for processes other than ionization. To express dissociation process (13), therefore, the dipole matrix element squared for the dissociation, M, is given by   R df dE (15) M" [1!g(E)]  E dE ' which corresponds to



(E)"1!g(E) (16) H and J"I in Eq. (5). The Platzman's concept of the superexcited state has made a profound in#uence on the science of excited states and motivated researchers in a wide "eld to "nd new objectives of research. In order to study experimentally the superexcited states or to substantiate Platzman's ideas, scientists had to "nd the way to produce such superexcited states. Electron beams were mainly used in 1960s and 1970s, and even now, for this purpose, whereas after 1980s laser multiphotons and SR have also been used. Details of the electronic states of superexcited molecules and the mechanism of autoionization and dissociation have been quickly clari"ed. The theoretical studies have also been advanced greatly and recently surveyed elsewhere [18}24], in which a uni"ed view of the superexcited states in a variety of molecular collision dynamics is summarized as shown in Fig. 2 [18,19]. This article gives a survey of recent experimental studies of photoabsorption, photoionization, and photodissociation cross sections of molecules in the VUV-SX region, and also a summary of current understanding in spectroscopy and dynamics of molecules in superexcited states.

2. Electron impact studies of superexcited molecules 2.1. General There have been three major experimental studies of superexcited molecules by means of an electron beam as an excitation source to form superexcited states. In the following sections a survey

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Fig. 2. Uni"ed view of various dynamic processes involving superexcited states [18,19].

is given of these studies, as brie#y introduced below, with particular emphasis on a comparison with photon impact studies. Ionization and excitation of molecules in collisions with electrons in the energy range higher than about 10 eV are well elucidated by the Born}Bethe theory [3,4] and the cross section Q (¹) H to form the state j at least for optically allowed transitions is given by Eq. (4). In general, it is di$cult to measure M experimentally for every excited state j of interest. However, when H a superexcited molecule produces electronically excited, in some cases optically emissive, dissociation fragments, M may be obtained experimentally from the measurement of optical emission H cross section as a function of ¹. de Heer and coworkers [25}31] have extensively investigated the dissociative excitation of molecules by electron impact and obtained the values of M. They have H made a plot of Q ¹/4paR vs. ln ¹, which is called the Fano plot or in some articles also called the H  Bethe plot, and obtained M from the slope of its plot. A straight line with a positive slope at H su$ciently high electron energies indicates an optically allowed excitation process, while a horizontal line indicates an optically forbidden excitation process. In a way similar to that for the M measurement, the values of M have been obtained for a variety of molecules by electron} H G molecule collision experiments. In electron energy loss experiments of molecules the extrapolation of energy loss intensities for incident electron energies much higher than 10 eV to zero momentum transfer, i.e., to the intensity at the forward scattering, should give, according to the Born}Bethe theory, the corresponding optical oscillator strength. In this way, absolute cross sections for molecular photoabsorption and photoionization also of a variety of molecules have been obtained by fast electron impact [32}34].

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It is concluded therefore that the important information on the interaction of photons with molecules and also on the dynamics of superexcited molecules is obtained from electron}molecule collision experiments. To understand further the electronic structure of superexcited molecules and their dissociation processes, it has been indispensable to develop the translational spectroscopy of molecular dissociation, i.e., to measure the kinetic energy of dissociation fragments and their angular distribution. The translational spectroscopy of molecular dissociation by measuring the Doppler pro"le of optical emission spectra of dissociation fragments has been combined with the measurements of excitation spectra of molecular dissociation, i.e., with those of yield spectra of dissociation fragments, leading to comprehensive understanding of the electronic structures and dissociation dynamics of molecular superexcited states [9,10,35,36]. 2.2. Production of excited fragments by electron impact Optical emission from excited atoms or free radicals has been observed in electron-impact dissociative excitation of a variety of molecules. Illustrative examples are chosen and brie#y described in this section. Absolute emission cross sections for production of atoms in the excited states with the principal quantum number n"2}6 have been determined for dissociative excitation of molecular hydrogen and deuterium by electron impact in the energy range from 0.05 to 6 keV [28]. The excited atoms have been observed by detection of their #uorescence, Lyman-a radiation in the case of n"2 and Balmer-a, b, c, and d radiation in the case of the higher states. The Fano plot of H(2p) formation is shown in Fig. 3. The obtained results of the dipole matrix elements are summarized in Table 1. Fig. 3 clearly shows that H(2p) is produced from optically allowed excitation of H . A strong  isotope e!ect is observed for the production of excited atoms from molecular hydrogen and

Fig. 3. Emission cross sections for Lyman-a radiation presented in the form of the Fano plot for electron impact on H and D [28].   Fig. 4. Emission cross sections for Balmer b radiation presented in the form of the Fano plot for electron impact on acetylene, methane and ethylene [30].

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Table 1 M values for the Balmer and Lyman emissions from H and D [28] Q   Hydrogen

Deuterium

M ? M @ M A M B

0.00344 0.000184 0.0000432 0.0000223

0.00222 0.0000267 2 2

M N M Q

0.120 0.0572

0.0839 0.0411

deuterium. Approximately 20% fewer excited atoms are formed in the case of the heavier isotope. This di!erence arises from a competition in the decay of superexcited states between dissociation and autoionization. The superexcited states were assigned, after a brief discussion, to the vibrationally excited D% state.  de Heer and coworkers extended measurements to other molecules, HF, HCl, H O and  hydrocarbons [25}27,29}31]. The results for hydrocarbons showed a marked contrast in the Fano plot for the production of excited hydrogen atoms as compared with the results of H , HF, HCl and  H O. Fig. 4 shows the Fano plot for the Balmer-b radiation produced from the electron impact on  simple hydrocarbons [30]. The slopes of the straight lines at higher electron energies are almost horizontal, which shows that the production of excited hydrogen atoms is due to optically forbidden excitation. In Fig. 5, however, the Fano plot for CH(A*PX%) radiation shows clearly the positive slope of the straight line for each hydrocarbon molecule, indicating that CH(A*) is produced by optically allowed transitions in the excitation which may involve superexcited states of the parent hydrocarbon molecule [30]. de Heer and coworkers attempted to compare their obtained conclusion with the results of photoexcitation experiments. However, it was very di$cult for them at that time to make such comparison because of relatively very few results of photoexcitation experiments in the energy range above the "rst ionization potential. In a way similar to that for the measurement of excited fragments, the values of M have been G extensively measured also in fast-electron impact experiments [37}39]. Schram et al. [37] measured absolute total ionization cross sections for 0.6}12 keV electrons in a variety of hydrocarbons, analyzed the obtained M values by making a correlation diagram G as shown in Fig. 6 between M and the number of C atoms in a hydrocarbon molecule, and G also attempted to compare the obtained M values with those in photoionization experiments. G The paper by Schram et al. points to two important items for future investigations; one is to "nd some theoretical explanation of the interesting correlation as presented in Fig. 6; the other is photoionization experiments of hydrocarbons and other polyatomic molecules using synchrotron radiation. Rieke and Prepejchal [38] also obtained M values for a large number of gas G molecules from rare gases and simple diatomic and triatomic molecules to hydrocarbons and some polar organic molecules by measuring ionization cross sections for 0.1}2.7 MeV electrons.

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Fig. 5. Emission cross sections for CH(A*PX%) radiation in the form of the Fano plot for electron impact on acetylene, methane and ethylene [30]. Fig. 6. Values of M for hydrocarbons against the number of C-atoms [37]. G

2.3. Absolute cross sections for molecular photoabsorption and photoionization processes by fast electron impact In electron energy loss experiments of molecules extrapolation of energy loss intensities for incident electron energies much higher than 10 eV to zero momentum transfer, i.e., to an intensity at the forward scattering, should give, according to the Born}Bethe theory, the corresponding optical oscillator strength [3]. Brion and coworkers [32}34] have extensively investigated such an experimental approach using fast electrons as the virtual photon source, which van der Wiel named `the poor man's synchrotrona. They have pointed out some expected characteristics of SR in comparison with virtual photons in understanding ionization and excitation of molecules and made clear some necessary assumptions to virtual photons instead of real photons. It should be noted here, as described in detail later in this article, that these two methods, real- and virtualphoton experiments, have complementary roles with each other to substantiate the dynamics of superexcited molecules as well as to understand essential features of the interaction of VUV photons with molecules [10].

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Table 2 Photon and electron-impact experiment [34] Photon experiment

Equivalent electron-impact experiment

Total photoabsorption Total photoionization Photoelectron spectroscopy Photoionization mass spectrometry

Electron-energy-loss spectroscopy, dipole (e, e) Dipole (e, 2e) or (e, e#ion) (from sums of partials) Electron energy loss-ejected electron coincidence, dipole (e, 2e) Electron}ion coincidence,dipole (e, e#ion)

Fig. 7. Absolute oscillator strength for the photoabsorption of H S; solid circles (䢇), dipole (e, e) [40]; open circles (*),  synchrotron radiation [41]. The insert shows the photoionization quantum yield [40].

Brion's approach to real-photon experiments using the virtual photon source is called the electron impact dipole-simulation method, in which some coincidence techniques have been developed for measurements of the total photoabsorption, electronic partial photoionization, and ionic photofragmentation cross sections. These measurements are summarized in Table 2, in comparison with those in real-photon experiments. These simulation techniques have provided a large body of data for comparison with photon experiments of absolute cross sections [34]. The photon experiments using synchrotron radiation, which are greatly in progress, can be compared with the simulation measurements. Examples are chosen in the following for this comparison. The absolute p values for H S have   been measured [40], as shown in Fig. 7, using low-resolution dipole (e, e) spectroscopy in the equivalent photon energy range up to 90 eV and compared with that obtained from SR experiments [41]. Fig. 7 shows good agreement in the p values between the two di!erent experiments  and con"rms the necessary assumptions, such as the sum-rule normalization used in the simulation

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Fig. 8. Photoabsorption cross section of H S in the wavelength region of 67}102 nm [41]. The circles indicate the dipole  (e, e) data [40].

Fig. 9. Absolute oscillator strengths for molecular and dissociative photoionization of H S [40]. 

experiments, are reasonable at least for the present molecule. Fig. 8 shows another comparison between the two experiments, which clearly illustrates a large di!erence in the wavelength or energy resolution between the two experiments. Photoionization mass spectra have also been measured [40], as shown in Fig. 9, over the energy range up to 40 eV using dipole (e, e#ion) spectroscopy. Absolute oscillator strengths for molecular photoionization and all dissociative photoionization channels have been obtained from the total photoabsorption and the

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Fig. 10. Schematic diagram of the dipole-induced breakdown of H S for energetic radiation below 40 eV [40]. 

photoionization branching ratios. The inset of Fig. 7 shows the photoionization quantum yield as estimated indirectly from the ratio of the summed photoionization signals in Fig. 9 and the photoabsorption signal in Fig. 7. In this estimation it is assumed that the ionization quantum yield is unity in the energy range above about 20 eV. Earlier measurements of photoelectron branching ratios have been used together with the photoionization oscillator strengths to generate absolute partial oscillator strengths for production of the four valence shell electronic states of H S>.  A consideration of all the data sets provides insight into the dipole-induced breakdown of H S  using radiation up to 40 eV energy. The obtained results are summarized in Fig. 10. An important role of superexcited states in the excitation and ionization of H S is shown in  Fig. 7; the ionization quantum yield curve deviates from unity in the energy range just above the "rst ionization potential, though this is obtained indirectly from experimental data with assumptions. It is di$cult, however, to derive from this result more detailed information about superexcited H S molecule, i.e., its electronic state and dissociation or autoionization processes. Instead,  the present typical results of the simulation experiment show the gross features of photoabsorption, -ionization, and -ionic fragmentation of H S in a broad energy range, and provide  complementary information about superexcited states in comparison with the information obtained from SR experiments. This is not an essential comparison between the simulation and the real photon experiments to understand superexcited states. In an SR experiment in the vacuum ultraviolet (VUV) region as shown in Fig. 8 an energy resolution of 20 meV which corresponds to a wavelength resolution of 0.1 nm at 80.0 nm ("15.5 eV) is easily obtained, while in most of the simulation experiments the energy resolution was about 1 eV. However, Brion's group has developed, as was expected by the present author [9], a dipole (e, e) simulation experiment with a higher energy resolution of about 50 meV [42}47]. Another example is shown in Fig. 11 of a comparison between a dipole (e, e) simulation experiment and a real-photon experiment. There existed a marked discrepancy between the two in the absolute p values [9,48}51]. In a recent high-resolution dipole (e, e) experiment of SiH [47],  

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Fig. 11. Comparison of the photoabsorption cross sections of SiH . * Low-resolution dipole (e, e) experiment [47],  * high-resolution dipole (e, e) experiment [47], and 22 SR experiment [49].

however, as shown in Fig. 11, good agreement has been obtained with a real-photon SR experiment [49]. The former discrepancy has been ascribed to the interaction of a target gas with the oxide cathode of the electron gun in the simulation experiment [47]. Although this simulation experiment has still somewhat lower energy resolution in the p curve, it has a distinctive advantage  in that it shows the gross features of photo-absorption, -ionization, and -ionic fragmentation of a molecule over a wide energy range, and provides information about superexcited states complementary to that obtained from SR experiments. 2.4. Translational spectroscopy of dissociation fragments produced by electron-impact excitation of molecules For a deeper understanding of the electronic structure of superexcited molecules and their dissociation processes, it is indispensable to measure the kinetic energy of dissociation fragments and their angular distribution as well as to measure the threshold energy of dissociation [35,36]. The former measurement, which is called translational spectroscopy of molecular dissociation [35], is described in this section, while the latter one is described in Section 2.2. The former one has been made not only for charged particles, for long lived metastables or high Rydberg atoms by means of time-of-#ight spectroscopy, but also for emissive excited atoms with short lifetimes by means of Doppler-pro"le spectroscopy [35]. In the mid 1970s, several groups [52}56] independently observed anomalous Doppler pro"les of Balmer emission lines produced by electron impact on simple molecules. The observed pro"les could not be assigned to any of several well-known factors to describe the broadening of an atomic line spectrum both in shape and in the magnitude of the width, but to a Doppler pro"le due to the translational energy of fragment hydrogen atoms with n"3 formed from the dissociative excitation of molecules. Similar anomalous Doppler

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pro"les due to molecular dissociation had been observed earlier in other types of experiments [57,58]. The velocities of H atom fragments with n"3 were also measured using anticrossings [59]. The observation of such anomalous pro"les in e\-H collision experiments [53,55] has made  a physically clear contribution to directly understanding the dissociation potential of molecules into fragments because other essential information, such as excitation spectra of Balmer emission and theories of dissociation potentials, was already presented in detail for molecular hydrogen. An apparatus developed in early 1970s exclusively for measuring the anomalous Doppler pro"les of optical emission in electron}molecule collision experiments has shown the promising versatility and usefulness of this method for further substantiating the dissociation potentials or processes of superexcited molecular states [35,36,55]. A schematic diagram of this apparatus is shown in Fig. 12 [35,55]. The apparatus should give a high wavelength resolution even for very weak photon signals originating from electron}molecule single collisions, and, therefore, this is composed of a specially devised photon counting system combined with an etalon-grating monochromator giving a wavelength resolution j/*j510 at 650 nm. This apparatus has been used to

Fig. 12. Block diagram for the measurement of Balmer-a emission [35,36,55]. EG: electron gun, FC: Faraday cup, G: grating, M: mirrors, PM: photomultiplier. Fig. 13. Schematic of the dissociative excitation, which shows the relation of the kinetic energy distribution of the fragment H atoms to the potential energy curves of the excited states [63]. Potential curve (a) corresponds to the excited state which produces fast H atoms, and potential curve (b) to that which produces slower H atoms.

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study H and D [55], the molecules such as H O, D O, CH , NH and HF isoelectronic with Ne,       the molecule H S isosymmetric with H O, and other simple hydrocarbons [60}63].   In the experimental condition as shown in Fig. 12, it is easy to eliminate possibilities of atomic line broadening other than Doppler broadening due to molecular dissociation. The Doppler pro"le of an optical emission line due to a dissociation fragment atom is in#uenced by the following three factors [63]: (1) the kinetic energy distribution of fragment atoms. (2) the angular distribution of fragment atoms. (3) the polarization of fragment atoms with respect to the dissociation axis. The e!ect of factors (1) and (2) on the pro"le is theoretically simulated for a modeled potential curve of dissociation as shown in Fig. 13 [63]. The pro"le is largely dependent on factor (2) except the pro"le observed at 54.73, the so-called magic angle. Since in this simulation an equal population of the magnetic sublevels of a fragment atom is assumed, the e!ect of factor (3) on the pro"le is also investigated. The polarization of atomic radiation excited by molecular dissociation has been pointed out theoretically [64], and in fact some polarization measurements have been carried out on the Balmer radiation under electron impact on simple molecules [65}67]. The result of the simulation shows that the pro"le is again largely a!ected by factor (3). It is concluded from these simulations that the shape of the pro"le depends on all the factors. In fact, however, the translational energy of a fragment atom can be obtained from a simple analysis of a pro"le, i.e., from its half-width. It should be noted here that there exists another analysis method of the pro"le, namely di!erentiating the intensity with respect to the wavelength, giving the translational energy distribution, instead of the average energy obtained from the half-width of the pro"le, of fragment atoms [54,68}70]. This method gives precise information about the velocity distribution of a fragment atom only when the e!ect of factors (2) and (3) on the Doppler pro"le is experimentally con"rmed. Practically, however, it is di$cult to con"rm them. The experiments on H (and D ) determined the kinetic energy of H*(n), where n is the principal   quantum number of a hydrogen atom formed from doubly excited states (2pp )(nlj) as well as from  vibrationally excited states (1sp )(nlj) [53,55]. This is typical evidence for Platzman's prediction of  the dissociation of superexcited states of molecules. For molecular hydrogen excess energies beyond the ionization threshold of a molecule result from vibrational or double excitation. For other molecules one can easily estimate another type of excess energy resulting from inner valence excitation. The same experimental technique has been applied to H O, D O, CH , NH , HF and     other more complex molecules, giving direct evidence for the dissociation potential of inner valence excited states as well as doubly excited states [60}63]. In a series of these investigations the internal energies of these highly excited states have been found to be in good agreement with the energies of corresponding molecular ions obtained from photoelectron or ESCA spectra and (e, 2e) experiments [35,62]. The agreement shows that superexcited states are molecular high-Rydberg states converging to corresponding ionic states, and supports the core-ion model proposed for high-Rydberg atomic dissociation fragments. Almost the same conclusion has been obtained recently by other authors using an apparatus similar to that in Fig. 12. They have measured not only Balmer-a but also b, c, d and e produced from H and other molecules [68}75]. 

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Fig. 14. Electron energy dependence of the Doppler pro"les of the Balmer-a emission spectra by electron impact on (a) H and (b) D [55].  

2.4.1. H and D   The Doppler pro"les of Balmer-a emission are shown in Fig. 14 for H and D at di!erent   electron energies [55]. The relative intensities between the two components, a narrow one and a broad one, vary remarkably with incident electron energies. The pro"les for H and D show   similar dependence on the electron energy. In the case of D , however, the pro"les are narrower  than those for H . At 25 eV the observed pro"les consist of only one component (the narrow  component: NC) for both H and D . With an increase in the electron energy, the second   component (the broad component: BC) shows up gradually. The threshold for BC exists at the energies between 25 and 30 eV. Above 40 eV the observed pro"les clearly consist of two components. The average kinetic energies of H*(3) are estimated from the pro"les to be 0.2 and 7 eV, respectively, for NC and BC [55]. A similar result has been obtained also for H*(4) and H*(5) by observing Balmer b and c, respectively [74].

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The excitation energies required to produce H*(3) with kinetic energies of 0.2 and 7 eV are estimated to be 17.0 and 30 eV, respectively, which are in good agreement with the threshold energies of Balmer-a emission by electron impact on H [27,28,53]. It is concluded, therefore, that  NC corresponds to the predissociation of vibrationally excited molecular hydrogen that belongs to the Rydberg series converging to H>(1sp ) and BC to the direct dissociation (or partially involving   predissociation) of the doubly excited molecular hydrogen that belongs to the Rydberg series converging to H>(2pp ) [36,55].   2.4.2. H O and D O   An experiment similar to that for H and D was carried out also for H O and D O [60]. The     Doppler pro"les were decomposed into two components, which were not expected because the dissociation mode of a highly excited water molecule is much more complicated than that of a hydrogen molecule. The average kinetic energies of H*(3) were obtained from the two components of the pro"le to be 0.4 and 4 eV, from which the internal energies of the precursor excited water molecules were determined and compared with the threshold energies of the two components or the emission cross sections for both Balmer-a and OH(A&>PX%). The obtained results are summarized as follows: H OH (18 eV) P HH(3) (KE"0.4 eV ) # OH (X%)  H OHH(25}30 eV)PHH(3) (KE"4 eV )#O(2p)#H(1)  PHH(3) (KE"4 eV )#OHH(B&>) .

(17) (18) (19)

Since there has been little information about the electronic structure of a neutral water molecule at the energy above its ionization threshold, the following procedure is useful for estimating the electronic states of H O* and H O**. By a comparison of the above results with the correl  ation diagram of the dissociation of H O> ion states, it is concluded that H O* is the inner   valence-excited state of the water molecule which belongs to the Rydberg series converging to the BB (lb )\ ion state and that H O** is one of the doubly excited molecular states    which belong to the Rydberg series converging to eight doubly excited ion states in this energy region. 2.4.3. HF The spectral pro"les of the Balmer-a emission consist of three components, which indicate that there are three kinds of precursors leading to H*(3) contributing to the Balmer-a emission [61]. The average kinetic energy of H*(3) is obtained for each component. The dissociation modes corresponding to the observed three components of the pro"le are HFH(18 eV) P HH(3) (KE"0.3 eV)#F(2p) ,

(20)

HFHH(23 eV) P HH(3) (KE"5.2 eV)#F(2p) ,

(21)

HFHHH(40 eV) P HH(3) (KE"6.5 eV)#FH or F>(2p)#e\

(22)

or HF>(40 eV)PHH(3) (KE"6.5 eV)#F>(2p) .

(23)

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Fig. 15. Binding energy spectra of Ne, HF, H O, NH and CH in isoelectronic sequence [35,62]. The vertical bars do    not give the intensities of the spectra but denote the position of the vertical ionization potentials for each ionic state. The elliptical marks show the approximate positions of precursors of H*(3). The meaning of the capital letters A, B, C and D is explained in the text.

It is concluded that HF* is the inner valence-excited state which belongs to the Rydberg series converging to A&>(2pp)\ ionic state, that HF** is the doubly excited state which belongs to the Rydberg series converging to the B%(2pp)\(3pp) ionic state and that HF> and HF*** at 40 eV are respectively the &\(2s)\ ionic state and the Rydberg series converging to the same state. 2.4.4. General features of the dissociation modes The dissociation processes and the precursor states of H*(3) have been obtained also for CH  and NH . All the results obtained for HF, H O, NH , and CH are thus compared systematically     with the data on the corresponding ionic states to determine whether a physically important correlation exists between them [62], as discussed already in the theory of superexcited states of these molecules [76]. The binding energy spectra of Ne, HF, H O, NH , and CH are shown in Fig. 15 with the    approximate energy locations of the precursors of H*(3) denoted by the elliptical marks [35,62]. The ionic states of this isoelectronic sequence can be classi"ed into three groups, namely the ionic states correlating with Ne(2p)\, Ne(2s)\ and Ne(1s)\. The last group of ionic states is not shown, since it has much higher energy. Fig. 15 does not include doubly excited ionic states which

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have been observed by (e, 2e) experiments [32,33], although these ionic states are also very closely related to precursors of H*(3). The precursors labeled A are the Rydberg states converging to the ionic states correlating with Ne(2p)\. The precursors labeled C are the Rydberg states converging to the ionic states correlating with Ne(2s)\. In short, the A and C precursors are one-valence-electron transition states. The precursors of H*(3) labeled B and D are two-valence-electron transition states. The B precursors do not exceed energetically the ionic states correlating with Ne(2s)\ and do not result from the transition of the innermost valence electron. Those labeled D do exceed energetically the ionic states correlating with Ne(2s)\, and most of them would result from the transition of the innermost valence electron. The precursors of H*(3) from NH and CH are di!erent from those of H*(3) from HF and H O.    For NH and CH the precursors labeled A could not be observed for the following reasons. The   minimum energy to form H*(3) from both molecules is 16.5 eV, which lies within the (1e)\ Franck}Condon region for NH and near the upper limit of the (1t )\ Franck}Condon region   for CH .  Interestingly, the D precursors of H*(3) could not be observed in HF and H O. This result agrees  well with the fact that in the binding energy spectra of HF and H O obtained by binary (e, 2e), few  satellite structures could be observed above (2sp)\ and (2a )\, respectively, while in NH and   CH clear satellite structures were observed above (2a )\. The agreement between the results in   the correlation of the precursor electronic states and those in the binary (e, 2e) experiments implies that they have the same origin and that the molecules isoelectronic with Ne are essentially classi"ed into two groups: one is HF and H O, the other NH and CH .    It is concluded [9,10,35,36] that the superexcited states of molecules are molecular high-Rydberg states converging individually to each ionic state and classi"ed into the following three types: (1) vibrationally (or rotationally) excited states, (2) doubly excited states, and (3) inner-core excited states.

3. Photon impact studies of superexcited molecules 3.1. General Excitation photon sources which have been used for photon impact studies of superexcited molecules are classi"ed into discharge lamps, lasers, and SR sources. Photochemistry of simple molecules in the gas phase was comprehensively summarized in a book published in 1978 [77], in which the excitation wavelength was relatively shorter than that for complex molecules but restricted to that longer than their "rst ionization potentials. As was pointed out already by Platzman [12], this book was not concerned with the description of autoionization or dissociation of molecules excited to the energy range above the "rst ionization potential, i.e., molecules in the superexcited states. The experimental data surveyed in this book were obtained mainly from discharge lamp experiments. Most of the review articles [78,79] published more recently on the photochemistry of gas-phase molecules have also been restricted almost to consequences of photoabsorption below the "rst ionization potential. These articles show, however, that the used photon sources have been

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drastically changed from discharge lamps to laser-multiphoton and -single photon sources. Interestingly from the viewpoint of the subject of the present section, there is a brief description in a few chapters of these articles concerning the competitive destruction of an excited molecule above its "rst ionization potential between autoionization and dissociation. It is concluded that very few review articles on the photochemistry of gas-phase molecules have been concerned with the dynamics of superexcited molecules. This conclusion is almost the same as that pointed out by Platzman [12]. In research "elds other than photochemistry, however, there are several useful sources of information on photon impact studies of superexcited molecules. Berkowitz [80] compiled the experimental data on photoabsorption, photoionization and photoelectron spectroscopy in the photon energy range much wider than that in photochemistry, i.e., from absorption thresholds to the VUV}SX region in some cases even to hard X-ray region, summarized theoretical backgrounds, and brie#y introduced a general scheme of the dynamics of superexcited states. The most important point in this book from the viewpoint of superexcited states is a comprehensive survey of the data on photoionization quantum yields obtained up to late 1970s. A similar survey of the data published more recently [34] has clari"ed that photoionization quantum yields and other photon impact cross-section data should be measured for a variety of molecules using a real photon source, i.e., synchrotron radiation, instead of a virtual photon source. There have been other reports of the compilation of photon cross-section data (references cited in Ref. [34] and Refs. [81}83]). Two recently published conference records [84,85] include studies of superexcited states. The fact that such a higher energy range "rst attracted general attention is valuable for future progress in studies of superexcited states. Most of the papers therein, however, were concerned not with the dynamics of superexcited molecules, but with either the spectroscopy of formed ions and electrons or the dissociation of excited molecules below the "rst ionization potential. As an exception to all the papers in the two conference records, a comprehensive review was presented by Leach in Ref. [84] on the decay dynamics of superexcited molecules with particular emphasis on molecular autoionization and ion-dissociation. However, he did not discuss in detail neutral fragmentation in competing with ionic processes, but presented general comments on the scheme. The proceedings [84,85] also contain several original papers on similar topics. A book with similar topics has been published recently [86]. Photophysics and photochemistry in the vacuum ultraviolet were also summarized by Guyon and Nenner [87], and by Nenner and Beswick [88] as related with the formation and decay of superexcited states. They demonstrated an important role of synchrotron radiation as a promising excitation source in the experiments of molecular photoionization and dissociative ionization, presented a comprehensive survey of recent progress in both experimental and theoretical studies of these processes, and chose typical examples mainly from the experimental results obtained at Orsay. Concerning the neutral photodissociation of superexcited molecules, two examples, i.e., H and H O, have been chosen to describe in some detail their dissociation mechanisms. In the   case of H O, #uorescence excitation spectra of OH* (A, B and C) and H* (n"2, 3 and 4) have been  measured in the photon excitation region of 9}35 eV. A more detailed discussion in comparison with that in electron-impact dissociation has been presented of the electronic structure and corresponding dissociation mechanism of the superexcited H O molecule [89]. In the case of H ,   special emphasis has been placed on experimental results and their theoretical backgrounds of predissociation line shapes as observed in Lyman-a excitation spectra in the excitation wavelength

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of 86}68 nm [90]. Further recent progress in studies on H superexcited states will be discussed in  Section 3.3. The most comprehensive survey of the spectroscopy and dynamics of neutral dissociation of superexcited molecules has been presented recently by the present author as studied using both electron and SR photon impact methods [8}11,35,36,91]. Further surveys of these studies have been presented recently also by the present author and his coworkers in more detail [36,92,93].

3.2. ¸aser-multiphoton studies of superexcited molecules A compilation of experimental data published since 1970 has been presented recently on the photodissociation of molecules mainly excited with laser photons [94]. Most of them are, however, with the excitation photon energy below the ionization potential of molecules. There are still very few experimental studies of the spectroscopy and dynamics of superexcited states excited with laser photons. Since, however, NO is a simple diatomic molecule with a relatively low ionization potential, this molecule has been intensively studied in the entitled experiments. The autoionization dynamics of NO has been studied in detail using photoelectron spectroscopy combined with laser multiphoton ionization, providing conclusive evidence for the electronic autoionization due to continuum}continuum interaction via dissociative valence-excited states [95}97]. If a molecule is excited to one of its superexcited states from a speci"c lower excited state by double-resonance excitation, then it is possible to study autoionization with the measurement of

Fig. 16. Schematic potential curves relevant to (2#1) resonant ionization of NO molecule [97].

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the photoelectron energy distribution. Schematic potential curves relevant to (2#1) resonant ionization of NO molecule are shown in Fig. 16 [97]. The autoionization branching ratio for producing di!erent electronic and vibrational states of ions is determined from the branching ratio of photoelectrons emitted from NO autoionization states produced via the Rydberg C state. From photoelectron spectra, conclusive evidence is obtained for the electronic autoionization, as shown in Fig. 16, due to continuum}continuum interaction via dissociative valence-excited states. The photoelectron spectra have been analyzed theoretically by employing the multichannel quantum defect theory [96]. It is concluded that the autoionization is dominated by the two-step electronic mechanism in which a dissociative superexcited state plays a role of an intermediate state. Potential curves and electronic coupling strengths of the relevant dissociative states are estimated from this analysis. Neutral fragments, N(D) and N(S), formed from a superexcited NO molecule in Rydberg states have been detected recently with a resonance-enhanced multiphoton ionization (REMPI) technique [98,99]. The use of lasers for detecting products as well as for producing superexcited states has provided precise information on the spectroscopic features of these states, which is certainly helpful also to understand their dynamic features. However, more quantitative discussion on the branching ratios among processes (9)}(14), particularly on g, is needed in the laser studies for deeper understanding of the dynamics of superexcited molecules. 3.3. Synchrotron radiation studies of superexcited molecules 3.3.1. General Recently, synchrotron radiation has been used as a new photon source to clarify the electronic structures of superexcited states and dissociation processes [8}11,36,92,93]. Spectroscopy and dynamics studies of superexcited molecules have greatly progressed by applying SR as a new excitation source to form superexcited states. As pointed out in preceding sections, there were relatively few studies until about 1990 on superexcited molecules using SR in comparison with those using electron impact or dischargelamp photon impact even in the case of H , and much less in the case of chemically important  complex molecules. Recently, however, such studies have been quickly motivated and accelerated by the development of new dedicated SR facilities [8}11,36,92,93]. As selected topics in this section, we will discuss, "rst of all, recent progress in SR research on the dissociation dynamics of superexcited molecular hydrogen, particularly doubly excited states, secondly in the development of a new experimental technique, the two-dimensional spectroscopy of dissociation dynamics of superexcited diatomic and triatomic molecules, and thirdly in SR research on superexcited states of chemically important polyatomic molecules, i.e., measurements of photoabsorption, photoionization and photodissociation cross sections. An approach to polyatomic molecules is the choice of molecules to be studied, which are in a stereo-isomer series. Since measurements of the absolute values of photoionization quantum yields are the most basic to the study of the neutral dissociation of superexcited molecules, recent progress in those measurements will also be described in detail. To clarify the neutral dissociation of superexcited molecules, we need to compare in detail a photoionization quantum yield with cross sections for photoabsorption, photoionization and optical emission from neutral dissociation fragments as a function of

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photon energy. Selected examples in such comparison will be presented for C H , SiH and    CH OCH .   3.3.2. Dissociation of superexcited states of H  Dissociation of superexcited states of H has been extensively studied both experimentally and  theoretically [26,35,53,55,75,88,100}106], and summarized recently in a topical review article [36]. The observation of #uorescence from excited fragments is highly informative as a method for studying these states and their dissociation processes. It has been known, as summarized in a preceding section, from the translational spectroscopy [35] of excited hydrogen atoms, H*, formed by electron impact on H that the slow and fast H* atoms are produced. The slow H* is  produced by the predissociation of vibrationally excited Rydberg states (1sp ) (nlj) converging to  the ground state of H> X&> (1sp ) and the fast H* is produced by the dissociation of doubly    excited states (2pp ) (nlj) converging to the excited state of H>. The former states have already been   investigated extensively in photon-impact experiments by the observation of #uorescence from the fragment H* atoms [90,107}114], while the latter states have not been investigated in detail. The "rst observation of the neutral dissociation of doubly excited molecular hydrogen has been successful by the measurement of Lyman-a emission in the photodissociation of H using SR and  compared with theoretically predicted potential curves. The results are shown in Fig. 17a and b [104]. The excitation wavelength in Fig. 17a was scanned from 89 to 70 nm with about 0.2 nm bandwidth, which is much broader than that for the measurement of the Lyman-a excitation spectrum in the energy region only corresponding to the single excitation. It was needed to normalize the intensity of the Lyman-a emission from doubly excited states to that from single excited states for the determination of the absolute cross section for the former emission which was considered to be much weaker than the latter emission. Lyman-a #uorescence in this region is attributed to the slow H(2l) atoms produced by the predissociation of vibrationally excited molecular npp and npp states that belong to the Rydberg series converging to the ground state of H>. The main peaks in Fig. 17a are due to the progression of bands from the D(3pp) state.  Lyman-a #uorescence shown in Fig. 17a has already been observed with higher wavelength resolution and the predissociation mechanisms of H have been discussed in detail [90,107}114].  In the shorter-wavelength region corresponding to the double excitation of H , as shown in  Fig. 17b, Lyman-a #uorescence has been observed obviously in this experiment, which is the "rst observation of the neutral fragmentation of the optically formed doubly excited states. The excitation wavelength is scanned from 72 to 35 nm with the further broader bandwidth of about 0.8 nm. Lyman-a #uorescence observed in the longest-wavelength region in Fig. 17b, which corresponds to the #uorescence in the shortest-wavelength region in Fig. 17a, is due to the predissociation of the single electron excited states of H . No #uorescence was observed between 63  and 47 nm. This result agrees with the theoretical result that there are no optically allowed molecular excited states in the Franck}Condon region of the ground state of H . In the wavelength  region below 47 nm, Lyman-a #uorescence is again observed. This is attributed to the production of the doubly excited states. Comparing the Lyman-a excitation spectrum measured by Mentall and Gentieu [107] in the same wavelength region and normalizing the intensity in Fig. 17b with that in Fig. 17a in the range from 70 to 72 nm, the cross section of Lyman-a #uorescence in the dissociation of the double electron excited states is estimated to be about 10\ cm at 30 eV, which is much smaller than

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Fig. 17. Lyman-a excitation spectra (a) and (b) in the wavelength ranges of 70}89 and 35}72 nm, respectively [104].

that, about 10\ cm at the main peaks in the excitation spectrum, in the dissociation of single-electron excited states. This result agrees qualitatively with that of a preliminary estimation [115], and quantitatively with that recently measured in detail by Glass-Maujean [116]. It is concluded that the cross section for the photon-impact neutral fragmentation of doubly excited states of H is much smaller than that of the single-electron excited states, which clari"es  experimentally remarkable features of the doubly excited states of H . Since, in the case of  electron-impact excitation at low energies, the cross section for neutral fragmentation from the doubly excited states does not di!er so much from the single-electron excited states [26,29], the doubly excited states, which are optically forbidden from the ground state, seem to have a signi"cant role in the neutral fragmentation of H . It is greatly needed to theoretically explain such 

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Fig. 18. An attempt to "nd threshold energies in the Lyman-a excitation spectrum [104]. Fig. 19. Calculated potential energy curves of optically allowed doubly excited states of H and potential energy curves  of H> [119,120,122]. 

a large di!erence in the dissociation cross-section values for doubly excited states of H between  photon impact and electron impact. Fig. 18 shows an attempt to "nd threshold energies in the obtained Lyman-a excitation spectrum [104]. There are three thresholds, as indicated by the arrows, i.e., E(1)"26.6$0.5 eV, E(2)"29.2$0.7 eV and E(3)"30.9$0.6 eV, and also exists a dip at E(4)"34.1$0.5 eV. The threshold values have been compared with theoretical ones [117}121]. Fig. 19 shows the potential energy curves of optically allowed doubly excited states of H calculated by Takagi and Nakamura [120] and by Guberman [119] together with curves of  H> states published by Sharp [122]. There have been several calculations of the potential energy  curves of doubly excited states [117}121]. The calculations for each state, however, are not always consistent with one another. For the Q &>(1)(2pp , 2sp ), Q % (1)(2pp , 3dp ), and         Q &>(2)(2pp , 3sp ) states, the calculated curves by Guberman [119] are in good agreement with     those by Takagi and Nakamura [120] except for the internuclear distance larger than 1.2 A> . For the Q % (2)(2pp , 4dp ) state, the calculation by Hazi [118] agrees with that by Guberman [119]     quite well. However, the curve given by Bottcher [117] lies higher than that by others. On the

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contrary, the energy curves of the Q % (2pp , 2sp ) state calculated by Bottcher [117] is in good     agreement with that by Guberman [119] within 0.1 eV in the Franck}Condon region. The discrepancy between the calculated potential curves of Q % by Takagi and Nakamura [120] and   by Guberman [119] is larger than that of the Q states.  By comparing the experimental results shown in Fig. 18 with the theoretical potential curves shown in Fig. 19 and assuming that the observed thresholds correspond to the excitation of the state in some small distances outside the classical Franck}Condon region, these three threshold values can be attributed to the successive excitation of doubly excited states. The "rst E(1) threshold is attributed to the excitation to the Q &>(1) state. The second E(2) threshold is   assigned to the Q % (1) and/or &>(2) states. These states lie so closely and cross so steeply over    the Franck}Condon region that they cannot be isolated in the excitation spectrum. These Q states  should dissociate to H(1s)#H(2l) directly or by crossing with single electron excited states, which have a dissociation limit of H(1s)#H(2l). As mentioned above, there exists no remarkable discrepancy among the calculation for each Q state in the Franck}Condon region except for Bottcher's calculation [117] for the Q % state.    The E(1) and E(2) thresholds are in good agreement with theories at an internuclear distance of 0.9 A> . So it seems reasonable to discuss in the following also the other thresholds of doubly excited states at the same internuclear distance, i.e., 0.9 A> . Then the third threshold E(3) can be attributed to the threshold of the Q % (2pp , 2sp ) state calculated by Takagi and Nakamura [120]. Since     E(3) is above the potential energy of H>&>(2pp ), it may be reasonable to consider that this    threshold is ascribed to the Q % state. This state has the dissociation limit of H(2l)#H(2l), and   makes a considerable contribution to the formation of H(2l) atoms. The interpretation of the dip at 34.1$0.5 eV is not so simple as those of other thresholds [104]. The dip was ascribed at least partly to the opening of a new channel, i.e., the threshold of a state strongly coupled with autoionization or neutral fragmentation leading to H(nl) (n53) atoms. The opening of the excitation to such a state was expected to decrease the intensity of Lyman-a #uorescence considerably. A threshold for such a state has been found in the observation of H> and high Rydberg H atoms as mentioned later. Glass-Maujean [116] has measured the Lyman-a excitation spectrum in the energy range higher than that near the dip, clarifying overall features of the excitation spectrum from the threshold to the energy up to 60 eV. The dip in Fig. 18 is well explained by the presence of the peak in her measurement, which is assigned to the maximum of the contribution from the Q % (1) state at about 35 eV. In the overall features of   the Lyman-a excitation spectrum, the assignment of the spectrum in the energy range just above the threshold has been made to the Q % (1) state, which is di!erent from the Q &>(1) state     assigned in Fig. 18. The assignment of the spectrum in the energy range higher than that at the threshold is almost the same as those in Fig. 18. To clarify the discrepancy in the assignment it is necessary to study more the doubly excited states of H both experimentally and theoretically [36].  It is of considerable interest in understanding the doubly excited states of H to compare the  observed thresholds of Lyman-a #uorescence with those of the dissociative autoionization of the doubly excited states of H producing energetic protons. Strathdee and Browning [123,124] have  measured the kinetic energy spectra of protons resulting from photoionization of H using Ne II  radiation at the energies of 26.9 and 30.5 eV. In the 26.9 eV photoionization the spectrum is ascribed to the dissociative autoionization of the Q &>(1) state. This conclusion agrees well with   the result of the "rst threshold E(1) in Fig. 18. In the 30.5 eV photoionization they have considered

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an additional contribution from Q % (1) and some other higher states and contribution to the   proton formation at this energy. To clarify further the doubly excited states of H , Lyman-a Lyman-a coincidence has been  measured in the following dissociation process [103]: H #hlPH**PH(2p)#H(2p) ,  

(24)

where H** is the doubly excited molecular hydrogen formed by an optically allowed transition.  A schematic view of the gas cell with Lyman-a detectors and a block diagram of the electric circuit for the coincidence detection are shown in Figs. 20 and 21, respectively. The excitation spectrum of process (24) obtained from coincidence signals by changing the photon energy is shown in Fig. 22. In the excitation spectrum the cross section has the threshold at the energy just below 29 eV and increases with increasing energy. The observed threshold is clearly larger than the "rst threshold E(1) in the Lyman-a excitation spectrum in Fig. 18 and is close to the E(2) and E(3) thresholds. The

Fig. 20. Schematic view of a gas cell and Lyman-a detectors for Lyman-a Lyman-a coincidence detection in the photoexcitation of H [103].  Fig. 21. Block diagram of the electric circuit for Lyman-a Lyman-a coincidence detection in the photoexcitation of H [103]. 

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Fig. 22. Excitation spectrum of the process: H #hlP H**P H(2p)#H(2p) [103].  

precursor H** of the observed two H(2p) atoms in process (24) is the doubly excited Q state, which   dissociates into two H(2p) atoms. The Q states dissociate into H(1s)#H(nl) and cannot be the  precursor of the observed two H(2p) atoms. The Q % (1) state must be a precursor of process (24)   because it is the lowest Q state and is expected to dissociate into the lowest limit of the Q state,   i.e., H(2p)#H(2p) at 24.9 eV. The experimentally obtained threshold (about 29 eV) is smaller than the threshold energy (about 30 eV) as expected from the calculated potential curves in Fig. 19 of the Q % (1) state in the Franck}Condon region. The reason for this discrepancy is not clear at   present but might be ascribed either to the fact that the excitation to the doubly excited states occurs slightly outside the Franck}Condon region, or to some uncertainty in the theoretical potential curves. 3.3.3. Dissociation of superexcited states of diatomic and triatomic molecules Primary and secondary processes following photoabsorption of molecules in the VUV region are strongly in#uenced by superexcited states. Among various decay processes of molecular superexcited states, the dissociation, especially neutral dissociation, in competition with the autoionization has been shown to play an important role in the VUV photophysics and photochemistry above ionization thresholds [9,10,36,92,93]. The di$culties of detecting neutral products, however, hampered the experimental investigation of neutral dissociation. Thus, in the region far beyond the ionization threshold, most of the experimental studies have been restricted to those of photoionization and the measurements of photoabsorption spectra. Since, however, they are not so sensitive to superexcitation owing to a huge contribution from direct ionization, most of superexcited states,

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especially those having repulsive potential curves or surfaces, have not yet been revealed even in diatomic molecules. The observation of neutral dissociation of simple molecules by detecting undispersed #uorescence from excited fragments using optical "lters has been shown to be a useful tool for studying the decay of superexcited states [9,10,36,92,93]. This #uorescence technique has made it possible to get the information on repulsive potential structures of molecular superexcited states as well as on well-de"ned vibrational levels of bound superexcited states. However, it has not provided detailed information on particular dissociation products and precise dissociation limits, which is indispensable for understanding the dissociation and autoionization dynamics of superexcited states. A powerful new experimental method, the two-dimensional #uorescence spectroscopy, has been recently developed to reveal a bird-eye view of the neutral dissociation dynamics of a superexcited molecule in a wide excitation energy range [125]. In the two-dimensional #uorescence spectroscopy, the yield spectra of dispersed #uorescence radiation or the dispersed #uorescence excitation spectra emitted from neutral fragments produced by the dissociation of molecular superexcited states have been simultaneously measured as a function of the excitation wavelength of SR in a wide range. This method has been applied to O , N , CO, and N O. Some examples are chosen    from these to describe the method brie#y in the following. Fig. 23 shows a comprehensive summary of the potential energy curves of O , in which one may  realize that there exists almost no information on superexcited states, i.e., on neutral excited states in the energy range above the lowest ionization potential. Recently, a careful and comprehensive survey of photon and electron impact studies of O [126,127] has shown that there has been still  very few information on the potential energy curves of the superexcited molecular oxygen in particular on their dissociation dynamics. The "rst experimental measurement has been reported of the dispersed #uorescence excitation spectra of dissociation fragments produced from superexcited molecular oxygen [128]. Using a secondary VUV monochromator #uorescence spectra of fragment oxygen atoms have been measured by changing the excitation wavelength of SR. Some of the obtained results are shown in Fig. 24, from which the new information has been obtained on the neutral dissociation dynamics of state-selected superexcited molecular oxygen. A detailed investigation of the measurements of absolute p , p and g values of O has been   recently reported by Holland et al. [129]. Fig. 25 shows a composite set of p of O in a wide   wavelength range comprehensively compiled using reported data and their sum rule analysis. Fig. 26 also shows a composite set of g of O compiled by Berkowitz [80]. In Figs. 23, 25 and 26,  the wavelength or energy range where the present new experimental method has been applied to is shown by the bold solid lines. In the present wavelength range there is, as shown in Fig. 25, an extremely important part of the total oscillator strength distribution of O in terms of its absolute  magnitude and there is also, as shown in Fig. 26, clear evidence for the neutral dissociation of superexcited molecular oxygen, i.e., clear discrepancy from unity in the g-value. As the extension of the "rst experimental measurement [128] of the dispersed #uorescence excitation spectra of dissociation fragments produced from superexcited molecular oxygen, the experiment of the two-dimensional spectroscopy was performed at the Photon Factory in Tsukuba [125]. Synchrotron radiation from a 2.5 GeV positron storage ring was monochromatized with a 3 m normal incidence monochromator. Fig. 27 shows the outline of the secondary monochromator [130], which consists of a collision chamber and a grating chamber. The

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Fig. 23. A comprehensive summary of the potential energy curves of O [126,127]. 

monochromatized SR was focused into a gas cell that contained O gas. Vacuum ultraviolet  #uorescence was dispersed by the secondary monochromator equipped with a holographic grating. A #uorescence spectrum was recorded using a resistive anode-type position sensitive detector (PSD) behind a microchannel plate (MCP) coated with CsI. Fig. 28 shows an example of the two-dimensional #uorescence excitation spectra emitted from dissociatively excited neutral O atoms, O*, as a function of the excitation wavelength k for 

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Fig. 24. Vacuum ultraviolet #uorescence spectra from photodissociation fragments of O at the incident photon  wavelengths of 62, 65, 69, 71, and 74 nm. The abscissa in channel number corresponds to the #uorescence wavelength [128].

Fig. 25. Absolute photoabsorption cross sections of O [129]. 

504k 460 nm (the photon energy 24.8}20 eV) [125]. Each of the horizontal line structures in  Fig. 28 presents the partial #uorescence yield from individual O* atoms produced in the neutral dissociation of superexcited molecular oxygen O**. Fig. 28 shows clearly the following two  striking behavior of the dissociation dynamics of vibrationally resolved superexcited molecular

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Fig. 26. The photoionization quantum yields of O [80]. 

Fig. 27. Experimental apparatus for the two-dimensional #uorescence excitation spectroscopy [130]. The right side is a collision chamber, and the left side is a monochromator to disperse the #uorescence resulting from excitation by monochromatized synchrotron radiation focused into a gas cell perpendicular to this "gure. A #uorescence spectrum is recorded using a resistive anode-type position sensitive detector (PSD) behind a microchannel plate (MCP) coated with CsI.

oxygen. One is the state-to-state selective neutral dissociation, in which the n quantum number is conserved, O**(c&\nsp or ndp )POH(nsS)#O(P) .    

(25)

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Fig. 28. Two-dimensional #uorescence spectrum of excited O* atoms produced in the neutral dissociation of O** as  a function of both the excitation wavelength for 504k 460 nm and #uorescence wavelength for 804k 4140 nm  D (lower panel) [125]. The #uorescence yields presented by the gray rectangle plots increase from white to black linearly with 8 steps of the color. Horizontal line structures present the partial #uorescence yields of OI radiation lines. A nondispersed #uorescence spectrum for 1054k 4180 nm (upper panel) is shown together. D

The other is a vibrationally selective neutral dissociation. In process (25), the dissociation of the v"1 state of O** is much more enhanced than that of the v"0 state of O**. This is clear   evidence for a vibrationally resolved tunneling predissociation. Two-dimensional spectroscopy has also been applied to the dissociation dynamics of superexcited states of CO [131}133]. Fig. 29 is an example of the obtained spectra [132], from which new results have been obtained of the preferential dissociation of the vibrationally excited R (ns) and " R (n#1)s Rydberg states of CO into the fragments CI(2s2p(n!1)d/ns>L )#OI(2s2p >L ). " ( ( Fig. 29 also shows an interesting behavior that almost all the observed #uorescences are assigned to those emitted from fragment carbon atoms although those emitted from fragment oxygen atoms are energetically possible. It has been concluded that two-dimensional spectroscopy is a powerful method to study the dissociation dynamics at least of diatomic molecules in the superexcited states. Recently, twodimensional spectroscopy has been applied also to a triatomic molecule N O [130].  Fig. 30 shows a #uorescence spectrum of dissociation fragments produced from the neutral dissociation of N O in superexcited states at the excitation photon energy of 16.02 eV, where many  #uorescence lines are shown [130]. They are emitted by excited atomic and molecular fragments,

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Fig. 29. Two-dimensional #uorescence spectrum of neutral fragments produced in the photodissociation of CO as a function of the exciting photon energy for #uorescence wavelengths between 100 and 200 nm and for exciting photon energies between 18.5 and 29 eV [132]. The grey scale is linear. The sum of the observed #uorescence intensities, i.e., a VUV-#uorescence excitation function, is plotted on the left-hand side. Fig. 30. Fluorescence spectrum in the photodissociation of N O at a photon energy of 16.02 eV [130]. 

and some of them are assigned as shown in Fig. 30. The broad structure in the #uorescence wavelength of 150}200 nm is due to the molecular #uorescence of the N (a% ) fragment.   Fig. 31 shows an example of the two-dimensional #uorescence spectroscopy of N O as a func tion of both the excitation photon energy in the range between 14.28 and 36.35 eV and the #uorescence wavelength in the range between 80 and 200 nm, from which detailed information on the neutral dissociation of superexcited states of N O has been obtained [130]. Each rectangular  dot represents the yield of dispersed #uorescence in exponentially increasing order from light to dark with 20 degrees of gradation. Ionization potentials for some of the one-electron ionic states are also indicated in Fig. 31. The #uorescence spectrum shown in Fig. 30 corresponds to the horizontal cross-sectional view of the two-dimensional #uorescence excitation spectrum at the excitation photon energy of 16.02 eV. Diagonal structures in the spectrum are due to scattered primary light, which appear at higher-order positions of the secondary monochromator. The "nal dissociation products, dissociation limits, and correlation among neutral dissociation potentials of N O have been identi"ed. In most cases of the neutral dissociation of superexcited N O formed by   photoexcitation below 20 eV, it has been shown that the superexcited states undergo multistep predissociation. The neutral dissociation of the ndp Rydberg states converging to N O>(C&>)  has shown di!erent behavior from other Rydberg states. The ndp Rydberg states preferentially dissociate into N*(5dQL)#NO(X%). This has been explained by the dissociative character of the

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Fig. 31. Two-dimensional #uorescence excitation spectrum (2D-FES) is divided into two parts. One is a spectrum as a function of both the excitation photon energy in the range 14.28}36.35 eV and the #uorescence in the range 80}140 nm. The other is a spectrum as a function of both the excitation photon energy in the range 14.57}36.35 eV and the #uorescence wavelength in the range 140}200 nm. The #uorescence intensity in the 2D-FES increases exponentially from light to dark. The total #uorescence yield curve is also shown in the left portion of the "gure, of which intensity is on a linear scale. The positions of the #uorescence lines are indicated at the top of the "gure [130].

precursor Rydberg states based on the core ion model. The neutral dissociation into N (a% )#O   has also been shown to switch to a three-body neutral dissociation. Broad features of the enhanced #uorescence observed in the excitation photon energy range above 20 eV have been identi"ed clearly as due to multiply excited neutral states. 3.3.4. VUV-optical oscillator strength distributions of polyatomic molecules To understand the dynamics of superexcited states of molecules, in general, as pointed out already in the Introduction, it is necessary to measure absolute cross sections of photoabsorption, photoionization, and neutral dissociation processes as well as absolute photoionization quantum yields. Also as pointed out in the Introduction, these cross-section values have been measured for a variety of molecules in the wavelength region longer than the LiF cuto! at 105 nm, while those in

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the wavelength region shorter than the LiF cuto! at 105 nm have been very few. When the TKR sum rule is compared with the sum of the obtained p or f values for a molecule in the wavelength  H region longer than 105 nm which corresponds, roughly speaking, to about the "rst ionization potential of most molecules, it is found that the sum of the obtained f values corresponds to only H less than several percent of the value of Z. It is, therefore, concluded that the interaction of a photon with a molecule in the VUV-SX region, where synchrotron radiation is the most promising photon source as clearly shown in Fig. 1, is predominant over all the other wavelength regions. Optical oscillator strength distribution is, therefore, of fundamental importance in understanding the interaction of photons with molecules, providing important information about energy spectra for the formation of superexcited states. Instead of measuring these cross sections for a variety of complex molecules at random, these cross sections for molecules in several stereoisomer series have been systematically measured and compared with one another or with the sum rule [134}137]. The main purpose of such a systematic investigation is to see how df/dE changes with changing molecular structures. Since isomer molecules consist of the same kind and the same number of atoms, the df/dE of isomers is expected to have the following properties: (1) The sum of df/dE of an isomer over all the energy region is expected to be equal to that of another isomer, and also to the number of electrons in the molecule, according to the Thomas}Kuhn}Reiche sum rule. (2) The gross features of the df/dE of isomers are expected to be almost identical with each other in the wavelength region where innercore electrons are excited, because the molecular structure of isomers would have relatively little in#uence on the excitation of innercore electrons. Moreover, the value of df/dE in such a wavelength region would be almost equal to the sum of the df/dE values of the constituent atoms. With the above-mentioned expectation, the photoabsorption cross sections have been measured for isomers [134}137]; C H (cyclopropane and propylene), C H (1-butene, isobutene, cis-2    butene, and trans-2-butene), C H (cyclohexane, 1-hexene, and tetramethylethylene), C H O    

Fig. 32. Absorption cross sections of cyclopropane and propylene [134].

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(ethyl alcohol and dimethyl ether), and C H O (n-propyl alcohol, i-propyl alcohol, and ethyl   methyl ether) in the wavelength region from about 30 to 140 nm, and compared with one another and with the sum rule from the above viewpoints. Fig. 32 shows, as an example, the absorption cross-sections, i.e., the oscillator strength distributions of C H isomer molecules, cyclopropane   and propylene [134]. Similar cross-section data have also been obtained for the other isomer series listed above [135}137] giving the common new features of absorption cross sections or oscillatorstrength distributions summarized as follows [8}11]: (1) The p values show a maximum at about 70}80 nm (about 16}18 eV) for each molecule.  (2) In the wavelength region shorter than that at the maximum the p values are almost the same  among the isomer molecules, e.g., cyclopropane and propylene as shown in Fig. 32, and equal to the sum of the cross sections for the constituent atoms. (3) In the longer wavelength region, the cross sections have di!erent peaks and shoulders depending on the isomer, i.e., on its molecular structure as shown again in Fig. 32. The sum of the cross sections in this wavelength region is, however, almost equal among the isomer molecules, which means, therefore, that the total oscillator-strength distribution normalized to Z according to the total (TKR) sum rule can be divided into two regions, i.e., the longer and shorter wavelength regions, and that the sum of the oscillator strength in each region does not depend on the molecular structure of an isomer and constant among the isomer molecules; in other words, that the partial sum rule satis"es the oscillator-strength distribution in each region. These results will make an important contribution to chemical physics and physical chemistry particularly to motivate the development of new quantum chemistry [9,10], and will also make a helpful contribution to radiation research in estimating the energy deposition spectra of molecules in the interaction of ionizing radiation with molecules [2,8,11]. The results, e.g., those in Fig. 32, well satisfy the TKR sum rule as summarized in Table 3 [134]. The agreement is extremely good between the sum of the obtained oscillator strength values (partly including semiempirical ones [138}140] in the higher-energy region) and the number of electrons, Z. Table 3 also clearly shows that the sum of the oscillator-strength distributions in the energy region below the "rst ionization potential occupies only a few percent of the total and the distributions in the VUV-SX region are of great importance in understanding the ionization and excitation of molecules. An interesting comparison is presented in Fig. 33 of the oscillator strength distribution of ethyl alcohol Table 3 Sum of dipole oscillator strength distributions of C H [134]   Wavelength (nm)

Cyclopropane

Propylene

Below Ip Ip-105 105-35

0.746 0.715 11.774

0.507 0.666 12.176

35 ' Total Z

23.46

10.251 23.60 24

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Fig. 33. Absorption cross sections of ethyl alcohol: experiment (*) [135], theory by Platzman (2) [141].

between the result of recent experimental measurements [135] and that of Platzman's educated guess [141]. Gross features of the spectra agree well with each other, although several large peaks expected by Platzman are absent in the measured spectrum. 3.3.5. Photoionization quantum yields A photoionization quantum yield g, which is the absolute photoionization probability of an atom or a molecule on a single photoabsorption event, is the quantity of a considerable importance to evaluate photoabsorption processes above the "rst ionization potential, i.e., a competition, as described in the beginning of the present section, between direct ionization and excitation to the superexcited states opening to autoionization and dissociation. The measurement of g is, therefore, of great importance for further characterizing superexcited states. Several experimental e!orts have been devoted to the measurement of g. Serious discrepancies have existed, however, even for simple molecules, not only between photon-impact experiments and `simulateda ones by electron impact but also between photon-impact experiments themselves [8}11,34,80,81] and references cited in Ref. [34]. Problems have originated mainly from the lack of an intense cw light source and a suitable window material in the VUV region particularly in the wavelength region shorter than the LiF cuto! at 105 nm. Without a window for the entrance of a photon beam into an ionization chamber, there have been several signi"cant problems, such as e!usion of sample gases into the beam transportation region or a contribution from a di!racted photon beam in the higher order, making it di$cult to determine correct, absolute, and comprehensive values of g. In most cases, the g-values have been assumed to be unity in the energy region far above the "rst ionization potential. Recently, there were reports on a systematic observation of the photoionization quantum yields [136], as well as the photoabsorption cross sections, of C H , C H , C H , C H O, and C H O           isomers using a multiple-staged photoionization chamber [142}145] and a synchrotron radiation light source in the wavelength region from 105 nm up to their respective ionization potentials. This

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Fig. 34. Photoionization quantum yields of C H isomers [136].  

measurement has further been extended to the "rst attempt of measuring a photoionization quantum yield in the wavelength region of 54}92 nm using synchrotron radiation combined with thin metal foil window [49,146}151]. In this section, therefore, a survey is given of the recent progress. The values of g have been measured systematically in the wavelength region longer than the LiF cuto! at 105 nm for the molecules in several isomer series of which, as described in the preceding section, the photoabsorption cross sections have also been measured [136]. Examples of the results obtained are shown in Fig. 34 for cyclopropane and propylene. The g-curves for these two molecules are very di!erent from each other; the curve for cyclopropane increases almost monotonically and much more steeply than that for propylene, which increases and shows a step or a shoulder with decreasing wavelength. Other examples are also shown in Fig. 35. The positions of ionic states are also shown in Figs. 34 and 35 [152]. The obtained g-curves for these molecules, as well as for other molecules whose "gures are not given here, show common new features as functions of wavelength or photon energy; the energy di!erence between the "rst and the second ionization potentials correlates well with the shape of the g-curves. A larger energy di!erence corresponds to a longer step length. This result means that the g-value increases steeply in the wavelength or energy region just close to the ionization potential, and is explained by the conclusion, as presented in Section 2.4, that the most important part of the superexcited states is high-Rydberg states converging to each ion state. An attempt to elucidate the shape of the g-curves for complex organic molecules has also been made recently by assuming that a predominant decay of superexcited states is its dissociation into neutral fragments, in other words, that ions are formed only through direct ionization [153]. In the wavelength region shorter than the LiF cuto! at 105 nm, metal foil "lters have been employed as window materials for the incoming synchrotron radiation beam from a vacuum ultraviolet monochromator and used for comparative studies of ionization quantum yields for

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Fig. 35. Photoionization quantum yields of C H isomers [136].  

Fig. 36. Photoionization quantum yields of several molecules in the wavelength regions shorter and longer than the LiF cuto! at 105 nm [146].

some molecules with excitation spectra of optical emission from dissociation fragments [146}151]. The "lters prevent sample gas e!usion and eliminate higher order radiation. In the wavelength regions of 54}80 nm and 74}92 nm, Sn and In foils, respectively, of about 100 nm thickness with about 1% transmittance are used. Fig. 36 shows the ionization quantum yields for some molecules measured in the wavelength region of 54}92 nm together with those in the wavelength region longer than 105 nm [146]. No data are shown in the wavelength region between the two because there is practically no thin metal window which is convenient for the measurement of ionization quantum yields and because there exists a relatively large e!ect of higher order light in this wavelength region. The results may be summarized as follows. (1) g-values in the region above, but close to, the "rst ionization potential are much less than unity, which means that most molecules, at least the molecules shown in Figs. 34}36, are not easily

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ionized even when they have just su$cient amount of their internal energies. In this region, therefore, the neutral fragmentation of superexcited molecules is of great importance in the total decay channels as expressed by processes (9)}(14). (2) g-values do not reach unity even in the energy range more than about 10 eV above the "rst ionization potential. (3) g-values increase with increasing photon energy and reach unity in the absolute scale at the energy range above about 23 eV (or 54 nm). (4) g-curves show interesting structures. It is concluded from the results (1)}(4) that non-ionizing processes, such as the neutral fragmentation of superexcited molecules, have an important role in their decay channels. Since, as pointed out in Section 3.3.4, there exists an important part of the magnitude of the oscillator strength distributions of most molecules in the wavelength range of 60}100 nm in which the g-values are clearly and largely deviated from unity, it is concluded in general that molecules are not easily ionized but dissociated into neutral fragments even when they have energies much larger than their ionization thresholds. 3.3.6. Dissociation of superexcited states of polyatomic molecules } Comparative studies of ionization quantum yields with excitation spectra of optical emission from dissociation fragments Neutral fragments formed from the dissociation of superexcited molecules often have excess energies electronically, vibrationally, rotationally, and/or translationally excited, because the photon energies corresponding to the wavelengths in Fig. 36 are much higher than the bond dissociation energies to form fragments in their ground states. It is, therefore, of great interest to observe optical emissions from excited fragments as a function of wavelength, i.e., to obtain excitation spectra of optical emission and to compare with the structures in g-curves. In the following, examples of such a comparison are presented to clarify further in detail the dynamics of superexcited molecules. 3.3.6.1. Acetylene. The photoionization quantum yield of acetylene measured recently using synchrotron radiation combined with a metal thin foil window is shown in Fig. 37 [148] together with those measured previously using discharge lamps or electron beams as a virtual photon source [154}156]. In the energy range of 18}24 eV, the yields obtained using the virtual photon source are considerably larger than those obtained using synchrotron radiation as a real photon source, while in the lower energy range, agreement is relatively good in appearance between the yields obtained using the di!erent methods. The yields obtained using synchrotron radiation, represented by the solid curve, show clearly the structures with at least three minima, whereas those obtained using either the virtual photon source or discharge lamps do not show clearly the structures in the plot. Obvious deviations from unity in the ionization quantum yield of acetylene in Fig. 37 indicate the neutral dissociation of a superexcited acetylene molecule predominantly competing with autoionization. Fig. 38 shows the photoabsorption cross sections of acetylene [148], the structures of which are assigned to Rydberg series converging to each of the ionic states, (3p )\, (2p )\ and (2p )\    [157}159]. Combining the photoionization quantum yields and the photoabsorption cross sections, the cross sections for the total ionization (p ) and for non-ionizing decay processes (p )  are obtained and shown also in Fig. 38 [148].

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Fig. 37. Photoionization quantum yield of acetylene, : Metzger and Cook [154]; 夹: Person and Nicole [155]; #: Cooper et al. [156]; and !: Ukai et al. [148].

Excitation spectra of #uorescence from excited fragments produced in the dissociation of a superexcited acetylene molecule have been observed [148]. According to the emission spectra [160], as shown in Fig. 39, of excited fragments which may probably be produced from superexcited acetylene molecule, band-pass "lters have been chosen to disperse the total #uorescence. In Fig. 40, the excitation spectra of the dispersed #uorescence for some excited fragments are summarized and compared with the g-curve cited from Fig. 37 [148]. In comparison with the threshold wavelengths of the excitation spectra in Fig. 40 with the thresholds of related dissociation processes calculated from the dissociation energies based on the heats of formation and the electronic energies of excited fragments, the spectra (b)}(e) in Fig. 40 are interpreted as due to the following dissociation processes, respectively [148]: C H #hlPC (d% )#2H ,     C (C% )#2H ,   CH(A*)#CH (or CH(A*)) ,

(26) (27) (28)

H(2p)#C H (or C #H) . (29)   The "ve peaks as denoted by (1)}(5) in Fig. 40f correlate well with characteristic structures observed in the excitation spectra (b)}(e). By a comparison with the correlation further with the p and p curves, the precursor superexcited states for the dissociation processes (26)}(29) have been R G

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Fig. 38. Photoabsorption (p ), photoionization (p ), photodissociation cross sections (p ) (upper), and photoionization   quantum yield (lower) of C H [148].  

assigned [148] in detail to high Rydberg states and/or innervalence excited states which have been investigated theoretically [157}159]. The excitation spectrum in Fig. 40d as well as the corresponding dissociation (28) is of particular interest from chemical viewpoints because of a predominant breaking of the triple bond of

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Fig. 39. Fluorescence spectra from photoexcitation of acetylene using discharge lamps at (a) 83.4 nm, (b) 76.5 nm, and (c) 68.5 nm [160].

acetylene in a speci"c energy range, which reminds the present author of a theoretical investigation by Jesse and Platzman [161] indicating a predominant breaking of the double bond of ethylene in interpreting the hydrogen isotope e!ect on g. Meisels made a critical comment on this conclusion based on experimental results of the radiolysis of C H and C H -Ar systems [162].    

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Fig. 40. Fluorescence excitation spectra for (a) the total emission in 200}650 nm detection range, (b) C (d% Pa% ,    *t"0) (c) C (C% PA% , *t"1) (d) CH(A*PX%, 0P0) and (e) Lyman-a from H(2p) (121.6 nm) [148].   

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3.3.6.2. Silane. Dynamics studies of superexcited states of silanes and heavier group-IV compounds are of particular interest in comparison with those of hydrocarbons. Optical cross-section data on these compounds, however, have been very few in comparison with the data on hydrocarbons which are relatively more abundant as described in the preceding sections. Absolute values of the photoabsorption cross section, the photoionization cross section and the ionization quantum yield of SiH have been measured recently in the energy range of 13}40 eV  using synchrotron radiation (see Fig. 11) [49], a part of which is shown in Fig. 41. A broad peak is observed in the photoabsorption cross-section curve in Fig. 41 with the main peak maximum at 14.6 eV. This is observed also in electron energy-loss spectra [163] and assigned to the optically allowed transition of a 3a electron to an antibonding p*(t ) orbital. In the energy   region around this broad peak, the ionization quantum yield, as clearly shown in Fig. 41, deviated from unity, and the cross-section curve for the neutral fragmentation has a maximum. In the energy range just above this range in Fig. 41, a notable structure is observed in both absorption and ionization cross sections. Both of the cross sections are nearly equal; in other words, the ionization quantum yield is unity. This structure is assigned to the (3a )\(npt ) Rydberg state [164}166]   which decays predominantly through autoionization. Subtracting the photoionization cross sections from the photoabsorption cross sections, as shown in Fig. 41, one obtains the total cross sections for processes other than ionization, namely, the neutral fragmentation. An outstanding feature of the cross section is a peak centered at 14.6 eV. The peak position corresponds well to the maximum of the photoionization cross section and the minimum of the photoionization quantum yield. The cross section for the neutral fragmentation decreases with increasing photon energy to almost zero at 16 eV and rises up slightly again above 17 eV. In view of the rising-up behavior of the photoionization quantum yield above the ionization threshold, which is systematically observed for various molecules [136,138,147] as described in Section 3.3.5, the cross section for neutral fragmentation should have a considerable magnitude in the lower-energy region than the present lowest limit of the photon energy in Fig. 41. In summary, there are at least two kinds of superexcited states with di!erent characters in the energy range shown in Fig. 41. The superexcited state produced by the transition of 3a electron to  the optically allowed antibonding p*(t ) orbital, which is shown as the 14.6 eV broad band in both  g- and p -curves, decays through fast neutral-fragmentation processes competing with autoioniz ation, whereas the superexcited Rydberg states in the 16}17 eV region converging to the second ionization potential of silane decay mainly through autoionization. As a result, the neutral fragmentation cross section is inappreciable in this region. The neutral fragmentation in the energy region above 17 eV is ascribed to the dissociation of doubly excited states [49]. A similar investigation has been carried out also of Si H [149].   Other Si-containing molecules, SiF , SiCl , and Si(CH ) , which are also of great importance in    photoinduced or electron}collision-induced plasma-processing, have been also studied recently by measuring p , p , p , and g values in the VUV region, providing the important information on R G L electronic structures and dissociation dynamics of superexcited states of these molecules [150]. 3.3.6.3. Dimethyl ether. An interesting structure has been observed in the g-curve in the wavelength region of 54}92 nm. The optical emission at the wavelength of 115}200 nm, at which one expects in this experimental condition the observation of Lyman-a emission from fragment H*(n"2), is shown in Fig. 42 together with the ionization quantum yield for comparison [147].

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Fig. 41. Absorption cross sections (p ), photoionization cross sections (p ), photoionization quantum yields (g) and R G neutral-fragmentation cross sections (p ) of silane in the energy range between 13.6 and 22 eV [49]. 

Fig. 42. Photoionization quantum yield and Lyman-a excitation spectrum of dimethyl ether [147].

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A clear deviation of the g-value from unity with the three characteristic minima shows a superexcited dimethyl ether molecule decaying into dissociation and other nonionizing channels. The minima correspond well to the energies of ionic states with the vacancies of the p  and !& p orbitals or the MO correlating to the C(2s) orbital which are known from the HeII photo!electron spectra for dimethyl ether [167]. This correspondence has led to the conclusion that the superexcited states corresponding to the minima are high-Rydberg states converging to each of these ionic states; i.e., the (3b )\, (5a )\, and (1b )\ ionic states at around 80 nm and the (4a )\     and (2b )\ states at around 61 and 66 nm [147]. The large deviation of the g-value from unity,  which diminishes with the increase in the photon energy, is explained by the rate of autoionization in competition with dissociation and the state density of continuum correlating with the superexcited states. The rate of autoionization decreases inversely with the excess energy above the ionization potential. If the photon energy further exceeds other ionization limits, then, the new possible channels correlating with other ionization continua make the rate increase. A similar investigation has been carried out also of ethyl methyl ether [147]. 3.4. Other examples of synchrotron radiation studies on superexcited molecules A brief survey is given in this section of other examples of synchrotron radiation studies of superexcited states. Lee, Suto, and coworkers measured extensively the photoabsorption and #uorescence cross sections of various molecules in the excitation wavelength range of about 50}200 nm. The molecules studied are C H [160], SiH [50], Cl [168], H S [169], BF [170], BCl [171,172],        CF [173], CF X (X"H, Cl and Br) [174], SiF [175], GeH [176] and C H OH [177]. The "rst       two molecules, of which the dynamics of superexcited states is mainly discussed [50,160], are referred here already in preceding sections for comparison, while the other molecules are not described in this section because the experimental results of these molecules are not discussed mainly in terms of the dynamics of superexcited states [168}177]. Experiments by Lee, Suto, and coworkers have been focused mainly on the observation of new #uorescence spectra of free radicals or ions. The #uorescence excitation spectra of neutral fragments which they have obtained, however, are of great interest in understanding the dynamics of superexcited states of these molecules, as discussed already in preceding sections. When absolute ionization quantum yields are measured further for these molecules, a similar discussion will also be available on the dynamics of their superexcited states. Ibuki and coworkers also studied extensively the #uorescence spectra, lifetimes, and quenching processes of free radicals and ions produced from many molecules in the excitation wavelength range above about 105 nm using discharge lamps and partly synchrotron radiation. The molecules studied recently are CO [178], HCl [179], N O [180], CS [180], H S [181], CCl [182], CBrCl      [182], SiH [51], GeH [51], Si H [51], Si H [51], SiH Cl [183], SiHCl [183] and          (CH ) GeCl (n"0}2) [184]. Their experiments have been focused mainly on the assignment L \L of observed absorption and emission spectra to Rydberg and vibronic states, respectively. Such assignments have provided information helpful for understanding the dynamics of superexcited states of these molecules. Further measurements using synchrotron radiation are greatly needed of their ionization quantum yields and #uorescence excitation spectra in the wavelength range shorter than 105 nm.

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4. Related topics Some remarks are presented brie#y of the following three topics as related with the dynamics of superexcited molecules described in the preceding sections. 4.1. Superexcited states as reaction intermediates in various collision processes Important roles of superexcited states have been pointed out as reaction intermediates in a variety of collision processes [18,19]. The major collision processes concerned are: (1) photoexcitation of molecules (2) electron-impact excitation of molecules (3) electron attachment to molecules (4) electron}ion or ion}ion recombination (5) collisions of ions with molecules (6) collisions of excited atoms (or excited molecules) with atoms and molecules. Processes (1) and (2) have been discussed in detail in Sections 3 and 2, respectively. Processes (1), (4) and (6) are shown in Fig. 2. As for process (5) which includes both ion-impact excitation of molecules and ion}molecule reactions, no speci"c example seems to have been reported. However, a possibility of an important role of superexcited states is considered in a systematic investigation of ion}molecule collisions [1,4,185]. In process (3), electrons are captured by atoms or molecules to form negative ions. Recent understanding of the microscopic mechanism of electron attachment [186}189,192] is brie#y summarized in the following from the viewpoint of an important role of superexcited states in this mechanism. Electron attachment is classi"ed into two types; dissociative and non-dissociative processes as shown in the following reaction scheme: e\#AXPAX\HPA#X\

(30)

Pe\#AX PAX\#energy .

(31)

Interaction of low-energy electrons with molecules, AX, produces unstable negative ions, AX\H, with a cross section p or a rate constant k. The autodetachment of electrons from AX\H with a lifetime q may compete with the dissociation of AX\H or with the formation of a stable molecular negative ion, AX\, which requires the release of excess energies from AX\H. The lifetime q is related to the electron-energy width of the attachment resonance. The value of 1/q is the rate constant for the autodetachment process. In the presence of third-body molecules, AX\H is collisionally stabilized to form stable AX\. The branching ratios among the unimolecular processes of decaying AX\H depend on the interrelationship of the potential energy curves between AX and AX\, and also on electron energies. The relative importance of the collisional stabilization process in the overall decaying processes of electrons depends largely on these unimolecular processes, particularly on the lifetime q and on the number density and character of third-body molecules. Some environmental e!ects may thus be expected on the overall scheme of electron attachment processes. In addition to the determination of cross sections or rate constants for electron attachment or

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negative-ion formation and their electron energy dependencies, it has been of prime importance in electron attachment studies to clarify the attachment mechanism, not only the two-body mechanism but also the overall mechanism and how environmental conditions a!ect the mechanism. Since AX\ corresponds to a stable molecule in the above description of the electron attachment mechanism, as compared with the description of the mechanism of superexcitation in the preceding sections, AX\H should be in a superexcited state which autoionizes or autodetaches an electron to form AX in competing with dissociation to form A#X\. It is, therefore, interesting to compare theories of electron attachment with those of superexcitation. In electron}ion and ion}ion recombination processes, in particular dissociative ones, their cross-section values as well as their microscopic mechanism have been described in terms of the formation and decay dynamics of a neutral highly excited molecule as the product of recombination whose total energy is larger than its ground ion-state [18,19,102,190,191]. Recombination in the bulk gas phase is, in most cases, a three-body process and is described in terms of the collisional deactivation of a highly excited molecule [192]. In some collision processes, as summarized in process (6), between electronically excited atoms or molecules (A*) and stable atoms and molecules (B) in their ground states, collisional ionization occurs as follows: AH#BPA#B>#e\ PAB>#e\ .

(32) (33)

The excitation energy of A*, which is often a rare-gas excited atom or molecule, should be larger than either the ionization threshold energy of B or the appearance potential of AB>. Processes (32) and (33) are called, respectively, Penning ionization and associative ionization. According to theories of these processes [18,193}195], their microscopic mechanism is described in terms of the formation of a superexcited autoionizing state AB* as an intermediate. The experimentally obtained cross sections for collisions of He(2S) or He(2P) with atoms and molecules have recently been compared in detail with theoretical results based on this mechanism. For He(2S) collisions cross sections and their energy dependence are well explained by the mechanism of the formation of a quasimolecule AB* as a function of the internuclear distance between A* and B [194}196], while for He(2P) collisions they are also well explained by the formation of a superexcited state of B due to a dipole}dipole interaction between A* and B [194,195,197]. In these collision processes it is interesting to investigate the neutral fragmentation processes in addition to ionization processes (32) and (33). 4.2. Superexcited states in reaction kinetics An important role of superexcited states of molecules and dissociation fragments formed in chemical reactions, in particular, in the radiolysis of molecular compounds was "rst pointed out theoretically by Platzman [5,12], as described in the beginning of this paper. Based on the current status of the understanding of superexcited states, as described in the preceding sections, a similarly important role of superexcited states is also considered reasonably in other types of reaction systems such as various phenomena in ionized gases; electric discharge plasmas, particularly reactive plasmas, and collisions in the upper atmosphere and planetary space.

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After Platzman's theoretical indication of superexcited states in early 1960s [5,12], the present author and coworkers reported for the "rst time experimental results which were explicable in terms of an important role of superexcited states of molecules and formed dissociation fragments in the mechanism of "nal product formation in radiolysis [6,198}200]. Similar investigations which were extensively made both in radiation chemistry [6,201] and VUV-photochemistry [202,203] motivated electron- and photon-impact experiments described in the preceding sections. In the radiolysis of ole"ns [198}200], bimolecular hydrogen formation was observed with large yields, on which the e!ects of absorption dose, temperature and scavengers were examined. The most reasonable conclusion to be drawn from detailed discussion of various possibilities of mechanisms to explain these experimental results was obtained as follows: RH PRH PH #R .    PH#RH ,

(34) (35)

H#RH PH #RH , (36)   where RH are highly excited states which at least partly include superexcited states and H are hot  hydrogen atoms either translationally or electronically excited. The product hydrogen is formed by the so-called direct excitation to form RH followed by both molecular detachment (34) and  hydrogen abstraction (36) by H. The former is unimolecular hydrogen formation, and the latter bimolecular. The use of a perdeuterated compound RD as mixed with RH and the observation of   a relative yield of H , HD, and D could discriminate these two types of hydrogen formation from   each other. Mechanism (34)}(36) could reasonably explain all the experimental results. The observed hydrogen yield also agreed quantitatively with a theoretical estimate by the optical approximation [198}200]. In the radiolysis of alkanes a similar conclusion was also obtained [6]. Recently, it has been generally accepted that superexcited states of molecules have an important role not only in the reaction mechanism of radiolysis but also in the energy deposition and track structure in the interaction of ionizing radiation with matter [204]. They have an important role also in the interpretation of the magnitude of =-values [205,206]. 4.3. Superexcited states in the condensed phase The investigation of superexcited states in the condensed phase is closely related with that of ionization itself in the condensed phase. It is a still unresolved important problem, as described below, to understand the ionization potential of condensed matter or to discriminate ionized states from highly excited states in the condensed phase. It is of great importance also to investigate in detail geminate electron}ion pairs and high-Rydberg states in the condensed phase. It should be noted that a key experiment to clarify these problems, on which there have been very few reports, is the absolute measurement of ionization quantum yields in the condensed phase using synchrotron radiation [8,11]. In this section a survey is given of recent advances in such investigations. It has been pointed out that photoabsorption, photoionization and photodissociation crosssection data are greatly needed for molecules not only in the liquid or solid phase but also in high-density gases or cluster systems to substantiate superexcited states of molecules in the condensed phase [8,11]. Such cross-section data have been very few in comparison with those in the gas phase. In radiation chemistry, however, electron or ion transport and reactivities which are

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closely related with such cross-section data have been investigated extensively for a variety of molecules in the condensed phase [192,207}212]. Picosecond pulse radiolysis studies of geminate electron}ion recombination are helpful to understand speci"c features of ionization processes of molecules in the condensed phase [213,214]. Formation and decay processes of electronically excited states of molecules in the condensed phase have also been studied extensively [215}223]. Coincidence techniques should be applied more to measurements of ionization-excitation processes in the condensed phase [224]. Synchrotron radiation has been applied to the measurement of ionization threshold energy values of liquids [225,226] and also to the experiment for an attempt to give evidence for Rydberg states near the threshold in liquid alkanes [227]. Recent photoionization studies of doped supercritical #uids using synchrotron radiation are of considerable importance to substantiate ionization mechanisms of molecules in the condensed phase [228,229]. Recent advances have been remarkable also in laser photoionization studies of molecules in the condensed phase. Autoionization of highly excited states has been observed in the laser photoionization experiment of molecules in the condensed phase [230}233]. Photoionization quantum yields are measured as a function of excitation energy. Similar phenomena have also been observed in VUV single photon ionization of molecules in the condensed phase [234]. Superexcited states have an important role also in understanding of the magnitude of =-values in the condensed phase [235}237]. Such investigations have been made extensively of liquid argon, krypton and xenon systems. The e!ect of the addition of molecular impurities to liquid rare gases has been examined and explained in terms of the Penning ionization of additive molecules due to excited rare gas molecules [238]. Optical oscillator strength and related data have been measured extensively for a variety of organic and biological molecules in the condensed phase [239}241]. An important role of superexcited states has been considered also in the VUV photolysis of biological systems using synchrotron radiation [242].

5. Conclusions and future perspectives This review article has summarized recent progress in experimental studies of the interaction of vacuum ultraviolet photons with molecules, i.e., those of photoabsorption, photoionization, and photodissociation of molecules in the excitation photon energy range of 10}50 eV. A particular emphasis in this summary has been placed on current understanding of the spectroscopy and dynamics of molecules in the superexcited states which are produced not only in the interaction of photons in this energy range with molecules but also in electron}molecule collisions as a comparative interaction. Most of the observed molecular superexcited states are assigned to high Rydberg states which are vibrationally (or/and rotationally), doubly, or inner-core excited and converge to each of ion states. Non-Rydberg states are also observed. Dissociation into neutral fragments in competing with autoionization is of great importance in the observed decay of each of these state-assigned superexcited molecules. Some future problems needing more work from the viewpoints of the present summary are the following.

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(a) The dissociation of superexcited molecules into optically non-emissive fragments which is much more important than that into optically emissive ones should be measured by using laser-induced #uorescence or laser multiphoton ionization spectroscopy. (b) The yield spectra of the dissociation of superexcited molecules into neutral fragments should be measured with rotational-state resolved spectroscopy, from which the dissociation lifetimes of superexcited molecules are accurately obtained. (c) The coincident electron-energy-loss spectroscopy of the dissociation of superexcited states into neutral fragments should be applied more to other molecules than hydrogen and deuterium to obtain the information on the neutral dissociation of optically forbidden superexcited molecules. (d) New theoretical approaches should be developed to elucidate the dissociation dynamics of each of the state-assigned superexcited molecules as well as more simply the oscillator strength distributions of molecules in the present wavelength range. (e) Synchrotron radiation should be applied more systematically to molecules in dense gases and clusters to understand or to substantiate the e!ect of intermolecular potentials on gas-phase cross sections which are summarized in this review article. (f) More generally for better substantiation of speci"c ionization phenomena in the condensed phase as opposed to those in the gas phase, the absolute photoionization quantum yields in dense gases, clusters, liquids and solids as a key quantity for this purpose should be measured.

Acknowledgements The author wishes to thank Drs. M. Inokuti, F.J. de Heer, C.E. Brion, and H. Nakamura for helpful discussion and comments. He is indebted to Drs. N. Kouchi, M. Ukai, K. Kameta, T. Odagiri, H. Koizumi, S. Arai, A. Ehresmann, M. Kitajima, and S. Machida in his group for their excellent collaboration. The author's synchrotron radiation research described herein at the Photon Factory has been supported scienti"cally by Drs. K. Ito, K. Tanaka, T. Hayaishi, and A. Yagishita, and "nancially by the Ministry of Education, Science, Culture, and Sports, Japan.

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POLYMERS AS SOLIDS: A QUANTUM MECHANICAL TREATMENT

Janos J. LADIK Department of Theoretical Chemistry and Laboratory of the National Foundation for Cancer Research, Friedrich-Alexander University Erlangen-Nu( rnberg, Egerlandstrasse 3, D-91058 Erlangen, Germany

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 313 (1999) 171}235

Polymers as solids: a quantum mechanical treatment Janos J. Ladik* Department of Theoretical Chemistry and Laboratory of the National Foundation for Cancer Research, Friedrich-Alexander University Erlangen-Nu( rnberg, Egerlandstrasse 3, D-91058 Erlangen, Germany Received August 1998; editor: S.D. Peyerimho! Contents 1. Introduction 2. The ab initio Hartree}Fock crystal orbital method for periodic polymers 2.1. The case of simple translation 2.2. Calculation of 1D HF crystal orbitals in the case of a combined symmetry operation 2.3. The distance-dependence of the di!erent types of integrals 2.4. Four-component relativistic Hartree}Fock crystal orbital method 2.5. Selected ab initio Hartree}Fock band structures 3. The treatment of correlation in periodic polymers 3.1. Application of the Moeller}Plesset manybody perturbation theory for in"nite systems 3.2. The correction of the band structure of a periodic system on the basis of the electronic polaron models 4. The treatment of aperiodicity in quasi 1D polymers 4.1. The negative factor counting (NFC) method in its ab initio matrix-block form

174 177 177 180 182 185 191 197

198

201 208

4.2. The inclusion of correlation in the calculation of density of states of disordered chains 4.3. Selected examples for the calculation of the electronic structure of aperiodic chains both at the Hartree}Fock and at the correlation corrected level 4.4. Calculation of the hopping conductivities of proteins and nucleotide base stacks on the basis of random walk theory 5. Calculation of selected properties of polymers 5.1. Application of the intermediate exciton theory to the electronic spectra of di!erent polymers and to the high ¹  superconductor YBa Cu O    5.2. Calculation of the superconducting state of YBa Cu O    5.3. Calculation of the bulk modulus of polyethylene 6. Conclusions Acknowledgements References

208

* Tel.: 49-9131-8528831; fax: 49-9131-857736; e-mail: [email protected]. 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 8 ) 0 0 0 8 7 - 8

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Abstract Hartree}Fock and correlation corrected band structure calculation methods for periodic polymers are described together with many applications. The ionization potentials and gaps are in the second case in rather good agreement with experiment. Using the negative factor counting method the density of states, Anderson localization and hopping frequencies were computed for di!erent proteins and nucleotide base stacks. Using a random walk theory their hopping conductivities were also determined. The results agree well with those of amorphous glasses. Applying the intermediate exciton theory the spectra of di!erent polymers and of YBa Cu O (`123a) were computed again in good agreement with    experiment. For `123a the e!ective interaction (screened Coulomb#excitonic) substituted into the BCS gap equation gives for ¹ "87 K (exp: 92 K), D "19 meV (exp: 20 meV) and a d   gap symmetry with a small s admixture (like in 

 V \W experiments). These successful calculations suggest applications (1) to predict polymers with several optimal properties, (2) to extend them to 3D periodic solids.  1999 Elsevier Science B.V. All rights reserved. PACS: 71.20.Rv; 74.70.!b Keywords: Hartree}Fock and correlation corrected band structures of polymers; Hopping conductivities of proteins and DNA; Intermediate exciton theory with applications to polymers and YBa Cu O (&&123a); Calculation of ¹ , the gap     and its symmetry in &&123a.

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1. Introduction The de"nition of the word `polymera seemed sometime ago easy for the chemist. It was understood mostly as a 1-dimensional (1D), 2D or 3D system with the same covalently bound repeat units. In the meantime the importance of non-periodic polymers [like proteins or doped alternating trans-polyacetylene (PA)] has become clear. At the same time one speaks also about copolymers with two or more components when the di!erent units are arranged randomly (the copolymer is an aperiodic polymer), or the same unit is repeated a larger number of times (say 15 times) and another unit say 20 times. Such copolymers can be denoted in the binary case as [(A) , (B) ] if the (A) , (B) superunits are repeated again periodically (one speaks in this case also V W L V W about superlattices). On the other hand in most copolymers x and y are not constant, because one has a sequence (A) (B) (A) (B) 2(A) L(B) L . V W V W V W Finally nowadays in many cases not covalently bound periodic or aperiodic large systems are also called polymers. For such cases hydrogen-bonded systems, like F}H2F}H2F}H2, water chains

2D or 3D networks of H O molecules, or stacked systems can serve as examples. A tetra cyanoquinone (TCNQ) or tetrathiofulvalene (TTF) stack are examples of periodic 1D systems not bound by H-bonds (see Fig. 1). At the same time a nucleotide base stack in DNA (chemical formulae given later) is a non-periodic (four-component) 1D polymer in a general sense. Looking at all these di!erent cases it seems that one can obtain the best de"nition of a polymer if one uses the meaning of the Greek word polymer nujnkg1 which means many parts. We shall use in this paper the word `polymera in this broad sense. Polymers (in the above given general sense) have an ever increasing importance in many cases. They form plastics from which a large part of the bodies of cars, planes, household appliances and (especially in the United States) the houses themselves are built. They have also an important role in the construction of spaceships. Biopolymers like proteins, DNA, RNA, polysaccharides, parts of lipids play a crucial role in the living cell. Many polymers "nd important medical applications as the materials of di!erent protheses. Highly conducting polymers can serve as new types of batteries (here their light weight plays an important role). Polyparaphenylene (PPP) and polyparaphenylenevinylidene (PPPV) (Fig. 2) have been recently applied as light emitting diodes. The simple polymer tetra#uoroethylene (Te#on) [(}CF }CF }) ] is a very good insulator. Polymers like nylon, polyacrylnitril, etc. are important in   V the production of stockings, socks, shirts, dresses, etc.

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Fig. 1. The chemical formulas of tetracyanoquinone (TCNQ) (a) or TTF (tetratiofulvalene) (b). These planar molecules form alternating stacks in a 3D molecular crystal (c).

Fig. 2. (a) Polyparaphenylene, (b) Polyparaphenylenevinylidene.

Organic polymers will certainly play an important role in the near future as good materials for non-linear optics, for space research, in computer technology. 2D periodic organic polymers (possibly containing also heavier metal atoms) are still good candidates for future high-temperature (¹ ) superconductors (attempts to "nd 1D high ¹ superconductors have failed because dynamic   (time-dependent) thermodynamic charge #uctuations destroy the superconducting state in a quasi 1D system). The great advantage of polymers at di!erent applications over conventional solids is that they have a much larger number of freedoms. If we take the example of PPP (Fig. 2) and if we choose only 5 di!erent substitutents, but we allow besides single also all double, triple and quadruple substitutions, the number of possibilities is over 4000 [1]. If we increase the number of possible substituents to 6, there will be over 6000 possibilities. The most frequent task of practical polymer research is to "nd out among the members of a polymer family (the same ring or rings with given di!erent substituents) those few (4}5) which possess for a certain purpose usually 3}4 unrelated optimal properties. In the experimental

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polymer research out of say 4000 possibilities they choose (on the basis of experience and chemical intuition) 200}400 polymers, they synthesize them and measure the properties in which they are interested in. Finally on the basis of their results they decide which polymers they try to apply for the given purpose. This procedure is obviously very expensive and it may happen that the polymer family does not consist of those polymers which have optimal properties for the prescribed purpose. Further if the research has to "nd out polymers for a completely new purpose (which means several new optimal properties), the reduction of the number of polymers to be investigated (in our example from 4000 to a few hundred) may not work at all. A much less expensive and safe procedure is to calculate the electronic structure in a good approximation (which means a good quality Hartree}Fock (HF) band structure calculation and correction of the band structure for correlation) and calculate the desired properties of the chosen smaller pool of polymers (in the above example a few hundred polymers). On the basis of the theoretical results one can predict which 15}20 polymers should be synthesized and their properties measured to come up with the few polymers optimal for a given purpose. At the same time such a study could show, whether the chosen pool of polymers contains at all the required polymers, or it should be changed. It would also indicate whether the polymer family with which the investigation has been started is appropriate at all for a given (especially in the case of a completely new) purpose, or one has to restart the research with another polymer family (other basic ring system and/or other substitutents). Though such calculations require very much CPU time also on very fast vectorized computers with both a large core and peripheral memory, they certainly are orders of magnitudes cheaper than the conventional experimental approach. To underline the reliability of theoretical `tailoringa of polymers, one can refer to the large number of results in good agreement with experiment for very di!erent properties of quite di!erent polymers which we have obtained with the help of our correlation corrected band structures. These results include the ground state properties of (SN) , the relative vibrational V frequencies of di!erent polymers (both obtained already at a better basis HF level), the fundamental gap, ionization potential and electron a$nity, the exciton spectrum of different polymers, as well as the Young modulus of polyethylene (all obtained with the aid of correlation corrected band structures). Further we have obtained the frequency-dependent hopping conductivity of proteins (on the HF level) and of nucleotide base stacks (with correlation correction) to mention results on aperiodic systems which again agree well with experiment. (For references see [1] and further they will be given at the applications of di!erent methods in this paper.) Finally it should be mentioned that using the excitonic mechanism of high ¹ superconductivity  (which is the "rst-order term of the polarization propagator; details given later) starting from the correlation corrected 2D band structures of YBa Cu O [both for the CuO plane and the CuO     chain (including in both cases the so-called apical O-atoms in the BaO plane between them)] we have obtained very good agreement with experiment for the photoelectron spectrum [2] and exciton spectrum [3] in the normal state of this system. Further solving the BCS gap equation [4] we have again obtained excellent agreement with experiment for ¹ , the maximal superconducting  gap and its symmetry at ¹"0 [5]. It should be emphasized that this success was achieved the very "rst time without any adjustable parameters in a model Hamiltonian as it was predicted in a previous paper [6].

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2. The ab initio Hartree}Fock crystal orbital method for periodic polymers 2.1. The case of simple translation [7}9] Let us assume we have a 3D polymer or molecular crystal with m orbitals (on one or more atoms) in the unit cell and the number of cells in the direction of each crystal axis is equal: N "N "   N "2N#1. In such a case we can describe a delocalized crystal orbital (CO) of the system with  the help of a linear combination of atomic orbitals (LCAO) as K (1) Wp(r)" C( p) q sq . G G Q Q q Q Here p"( p , p , p ), q"(q , q , q ), where the integers p and q ( j"1, 2, 3) have the values       H H !N,20,2N, q is a summation over all the cells in the lattice and sq"s (r!Rq!r ) is the sth Q Q Q AO (belonging to the atom with position vector r ) in the cell with the position vector Q Rq"q a #q a #q a .       a , a , a , are the three basis vectors of the crystal, m the number of basis functions in the unit cell    and "nally the C(p) q 's are the LCAO coe$cients. G R One can form the expectation value of the Fock operator (in atomic units) of the crystal. + Z K 1 ? q # [2JK (p, i)!KK (p, i)] (2) FK "! D! " "r!R 2 q p ? ? G (D is the Laplace operator, M the number of nuclei in the unit cell, Z the charge of the ath nucleus ? and Rq its position vector in the cell Rq. Further in a closed-shell system nH"n/2, JK the Coulomb ? and KK the exchange operator), with the CO's (1) 1tp"FK "tp2 G G . (3) 1tp"tp2 G G In this way one obtains after a Ritz variational procedure the generalized matrix eigenvalue equation FC(p) "e(p) SC(p) . G G G The elements of the Fock and overlap hypermatrices F and S are de"ned, respectively, by

(4)

(5a) Fo q "1so"FK "sq2 Q_ R Q R (5b) So q "1so"sq2 Q_ R Q R (Here o denotes the reference cell). In these equations the subscripts mean basis functions and the superscripts are cell indices. One can easily see [7] that if we introduce Born}von Karman periodic boundary conditions, that is 2N"!1 2N!1"!2

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and so on, the hypermatrices F and S which have originally the structure



O

1

2

!1

O

1

2

2

2



2

2N

2

2

2N!1

2

2

!2N

2

2

!1

O

(6)

become cyclic hypermatrices. (Here the block O means interactions within the same cell, the submatrices !1"1 describe "rst neighbours' interactions and so on.) For a cyclic hypermatrix one can write down immediately the unitary transformation which blockdiagonalizes it

The unitary matrix U is given by its pth!qth block [7].






(8a)






(8b)

2pi pq 1 exp 1 Upq" 2N#1 (2N#1) or in the 1D case 1 2pi pq U" exp 1. N O (2N#1) 2N#1

One can see that Eqs. (8a) and (8b) would reduce to the unitary matrices which belong to the cyclic group, if we would have in Eq. (6) instead of subblocks only matrix elements (a cyclic matrix instead of a cyclic hypermatrix). One can easily prove that after the unitary transformation (7), the pth block of the blockdiagonal matrices obtained is





(9a)





(9b)

2pi pq F(q) 2N#1

F(p)" exp q

and S(p)" exp q

2pi pq S(q) . 2N#1

Here F(q) and S(q), respectively, are the submatrices of the original matrices F and S, respectively. Multiplying Eq. (4) from the left by U> and inserting UU>"1 on both sides we can write U>FUU>C( p) "e( p) U>SUU>C( p) , G G G or FD( p) "e(p) SD( p) ; G G G

D(p) "U>C( p) . G G

(10)

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179

Since F and S are block-diagonal matrices the latter equation can be decomposed to the simpler equation F( p)d( p) "e(p) S( p)d( p) . G G G where the matrix blocks F( p) and S( p) are only of the order m;m. If NPR we can introduce the continuous new variables

(11)

2pp 2pp 2pp  , k "   , k " k "  a (2N#1)  a (2N#1)  a (2N#1)    which change between !p/a and p/a ( j"1, 2, 3). In this way we can de"ne the vector H H k"k b #k b #k b .       Here b , b and b are basis vectors of the reciprocal space (by de"nition a b "2pd ).    G H GH With the help of the vector k we can rewrite Eqs. (9a), (9b) and (10) as

(12)

(13)

 F(k)" e kRqF(q) , q \

(14a)

S(k)" e kRqS(q) ,

(14b)

q

F(k)d(k) "e(k) S(k)d(k) . (15) G G G The solutions e(k) of Eq. (15) for di!erent i values provide the di!erent bands of the periodic system G and the di!erent k vectors (3D case) or k values (1D case) give the di!erent energy levels within a given band. To be able to solve Eq. (15) one has to eliminate the overlap matrix S(k) which can be done also in this case with the help of LoK wdin's symmetric orthogonalization procedure [10]. Having the di!erent vectors d(k) one can write Eq. (1) of the CO's in the form G K (16) tk(r)" e kRqd(k) sq . G GQ Q Q q that is the CO's are a linear combination of the Bloch functions (if m'1). The exponential factor in Eq. (16) comes from transformation (10) if one takes into account the form given by Eqs. (8a) and (8b) of the unitary matrix (U). Substituting into Eq. (2) in the Coulomb and exchange operators JK (k, i) and KK (k, i) the CO's (16), one obtains after a straightforward calculation [7}9] for the matrix elements (5a)






+ Z 1 ? sq Foq" so ! D! QR Q "r!Rq" R 2 q  ? ? K , P (q !q ) # ST   q q   \, ST








1 PK so(r )sq(r ) 1! @ sq(r )sq(r ) Q  S  r R  T  2 

.

(17)

 The quasi-momentum k can be considered as continuous also for large but not in"nite N if the level spacings within a band are smaller than k ¹ (k is the Boltzmann constant).

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Here r ""r !r ", the operator PK exchanges electrons one and two on its l.h.s. and the    @ generalized charge-bond order matrix element P (q !q ) is de"ned as [7}9] ST   1 L* p (q !q )" (18) 2d(k) d(k) e kRq\Rq dk ST   GS GT u G S (u is the volume of the "rst Brillouin zone.) Eqs. (5b), (17) and (18) together with Eqs. (14a), (14b) and (15) de"ne the ab initio HF CO method in its LCAO form. They form the counterpart of Roothaan's 1951 paper for molecules [11]. Because of the occurrence of the matrix elements P (q !q ) in Eq. (17) with their de"nition of ST   Eq. (18) the HF CO problem (as all HF problems) is non-linear. Therefore one has to start with a guess of the P(o) and P(q !q ) matrices, substitute its elements into Eq. (17) and this into   Eq. (14a). Solving Eq. (15) with the matrix F(k) obtained in this way one obtains the "rst set of eigenvectors d(k) . For this purpose one has to perform the calculation at 8}9 k values in the 1D G case and at least with 15}20 k vectors in the 2D case if one performs a numerical integration in Eq. (18) using Simpson's rule. The application of a Gaussian quadrature [12] decreases the number of necessary k-points to 3}4 in the 1D case [13]. With the new vectors d(k) one can calculate the new matrices P(q !q ), F(q) and F(k), G   respectively. Continuing this procedure until self-consistency (the eigenvalues, eigenvectors or P-matrix elements change at two subsequent interactions less than a prescribed criterion) one obtains an SCF (self-consistent "eld) solution of the problem. Our latest ab initio HF CO programs are in the 1D case fully vectorized, in a good deal parallelized, are easily portable to di!erent types of computers (they are written in FORTRAN 90) [14] and can handle linear chains with 300 contracted Gaussian basis functions in the unit cell (in the second neighbour's interactions approximation) [14]. One version of them uses di!erent cut-o! radii for di!erent types of integrals and applies a multipole expansion to sum up Coulomb interactions until in"nity [15] (see Section 2.3).



2.2. Calculation of 1D HF crystal orbitals in the case of a combined symmetry operation If we have in the 1D case instead of a simple translation a combined symmetry operation, the formalism developed in the previous point can still be applied with a little modi"cation. Let us consider a helix (one passes from one unit to the next by a translation s and a rotation a). Then we can write down the helix operator as SK (a, s)"DK (a)#s

(19)

 One should mention that if the "rst Brillouin zone possesses certain symmetries, one has to perform the calculation in k space only in its irreducible part. For instance in the simple 1D case, because of the relation c (!k)"!c (k) one has G G to perform the integration over k in Eq. (18) only between 0 and p/a and multiply this integral by 2 instead of integrating between !p/a and p/a.  Especially in the 2D and 3D cases one can start with a smaller number of k vectors and increase their number as self-consistency is approached.

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where the operator DK (a) performs a rotation around the helix axis by an angle a. We assume further that after n helix operations we obtain the large translation ¹K : SK L(a, s)"¹K

(20)

(in the case of DNA B helices n"10). By introducing again the Born}von Karman periodic boundary conditions we can write SK ,>"1) .

(21)

The Fock operator of the helix FK commutes with SK [SK , FK ]"0 .

(22)

Therefore we can classify the eigenfunction of FK according to the one-dimensional irreducible representations of the Abelian group G"+SK K; m"1, 2,2, 2N#1,. The kth representation is thus given by





2piqk f "exp OI 2N#1

(23)

and therefore the eigenvalue equation





2piqk SK OW "exp W "f W I OQ Q 2N#1 Q

(24)

is satis"ed. Here the CO W can be written down again in an LCAO form, but k is de"ned now on I the combined (helix) operation (F and S will be again cyclic hypermatrices and therefore can be blockdiagonalized, again with the help of an unitary transformation. Further if NPR instead of a discrete variable one can introduce in this case also the continuous variable k). For the generation of the eigenfunctions W of SK O it is advantageous to introduce the projection I operator OK [16] I ,> OK "(2N#1) f SK \O . (25) I OI O If we apply OK to an AO sO we can generate the Bloch orbital Q I tQ (r)"OK s (r!Rq!R ) I I Q Q ,> (26) "OK (r!RQq)"(2N#1)\ f SK \Os (r!RQ) (RQ"R #R ) . O O OI Q O Q I O The same procedure can be performed also if there are more than one AO's in the unit cell. For the application of Eq. (26) one can apply the well-known relation SK \Os (r!RQ)"s [SK O(r)!RQ] . Q Q O O Further one can write s [SK O(r)!RQ]"s [DK (qa)r#qs!Rs] O O Q Q

(27)

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that is s [SK O(r)!RQ]"s [DK (qa)r#qs!RQ] . O O Q Q One can apply here the identity [16]

(28)

DK (qa)r!DK (qa)SK \O(RQ)"DK (qa)r !DK (qa)[DK (!qa)RQ!qs]"DK (qa)(r)!RQ#qs . O O O On the basis of this one can write (29) SK \Os (r!RQ)"s +D(qa)[r!SK \O(RQ)], . Q O O Q Our result shows that by applying the helix operator to an AO (1) we have to repeat the helix operation on the position of the nucleus q times and (2) rotate the argument of the AO with the angle qa around the helical axis. With the help of Eq. (29) we can write down the CO's of a helical chain in an LCAO form as





, K 2pikq 1 (30) d(k) s +DK (qa)[r!SK \O(RQ)], . exp W(r)Q" O GQ Q G (2N#1) 2N#1 O\, Q The correspondingly modi"ed programs were applied to helical DNA B chains and to a-helical homopolypeptides. 2.3. The distance-dependence of the di+erent types of integrals In practical calculations we cannot perform the in"nite summations (14a) and (14b). Therefore we have to apply certain approximations. One has to notice that di!erent integrals decay in di!erent ways with the distance(s) between the unit cells. Therefore they have to be treated using di!erent cut-o! criteria. One has to be careful that by doing so one should not violate the translational symmetry and the electrical neutrality. One should mention that the Coulomb type two-electron integrals have a very long range and therefore after treating a part of them explicitly, one has to use a multipole expansion to take into account the remaining integrals (see next point). The matrix elements FOq,FOJ can be separated into three parts QR NO (31) FOJ"AO((N)#BOJ#COJ . NO NO NO NO The "rst term is the kinetic energy term, the exactly treated part of the electron-nuclear interaction and also the exactly considered two-electron Coulomb integrals,


 



OJ HL , !


PHL" PQ

PG dk dH (k)d (s)e kRH\RL) , FT FQ  8 F

(32)

(33a) (33b)

J.J. Ladik / Physics Reports 313 (1999) 171}235

 
 


 

Z ?
,

 

183

(33c)



1 H , sO(r )vJ(r ) s (r )sL(r ) N  O  r P  Q  

(33d)

(Here for the sake of later convenience at the handling of the groups of two-electron integrals, we have changed their notation from Eq. (17) to Eq. (33d).


  
 

  BOJ(N)" 2pHL NO PQ H \ L\ PQ

, , ! 2pHL PQ H \, L, PQ

OJ HL pq rs

OJ HL pq rs

 

!
!
(34)

The third term is the exchange part of the two-electron interaction:


 

OH JL   COJ" . PHL NO H PQ p r q s L \ \ PQ

(35)

The convergence properties of the lattice summation in Eq. (35) is determined by the charge density-bond order matrix P (the two-electron integrals show slower decay). According to the detailed investigations for metallic systems P converges very slowly but for insulators it converges rather fast [17}19]. The long-range terms gathered in B can be treated using a multipole expansion method. This was formulated for quasi 1D polymers with helical symmetry by the Namur Group [20,21]. According to their results   B-( (N)" ;-(QJ(N)a\I>J> . NO NO I J

(36)

Here a is the elementary translation of the quasi 1D polymer and 2(1)J>K =(k#l#1)D (N, m, a)M-( MJK . ;-(IJ(N)" I>J> NO NO (k#"m")!(l#"m")! K

(37)

M-(IK and MIK are the mth components of the 2Ith pole moment of the charge distribution NO associated to the orbital product s-s(, and the total (electron#nuclear) charges, respectively, N O M-(IK"1s-(r)"rIpK(cos (0))e Ku"s((r) , NO N O I

(38)

, MJK" Z RJ pK(cos (H ))e KU?# 2p-( 1s-(r)"rIpK(cos (0))e Ku"s((r)2 . ? ? J ? NO N O J ? (\, NO

(39)

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Here R , H , U are the spherical coordinates of the position vector of the ath nucleus and r, 0, u are ? ? ? those of the position vector of the electron, respectively. Further



if k#l#1 odd ,

= " if k#l#1 even and one has right-handed helix , I>J> if k#l#1 even and the helix is left handed , , f (kma) , (40) D (N, m, a)"f(k#l#1, ma)! I>J> hI>J> F  f (kma) . (41) f(k#l#1, ma)" hI>J> F Further the function f is a cosine function if k#l#1 odd and a sine function if k#l#1 even, a is the rotation angle. f reduces to Riemann zeta if a"0. For the calculation of the f function Mintmire [22] suggested a series expansion, H H #2 . f (H)"!ln H# #  24 2880 Using the relation f (h)"(!1)Jf (h), he applied this series expansion to the 40th order.  J\ Explicitly integrating each term to obtain f for l'1 yields a relative precision of +10\ for J 04h4p. Even in a 1D HF CO calculation the number of two-electron integrals


  OJ H¸ pq rs

can be enormous even for a polymer with a small unit cell (in case of a larger number N of cells explicitly taken into account). Therefore it is very important to reduce the number of integrals to be calculated as much as possible. For this purpose one has to use the permutational symmetry of the two-electron integrals


 
 
  OJ H¸

OJ ¸H

"

pq rs

H¸ OJ

"

pq sr

"2

rs

pq

and to "nd a fast method which predicts the negligible integrals without calculating the whole list. In this way we have to calculate only the selected integrals and so save storage place and CPU time. To reach this goal one can apply the method which is also implemented in the CRYSTAL code [23]. To understand this method one can start with a contracted Gaussian basis set, in which the basis functions are built up as a "xed linear combination of primitive Gaussians with di!erent exponents u , I s " l g (u ) . N NI I I I

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185

We have to select the primitive Gaussian with the smallest exponent (the most di!use function) and have to use it as a representative of the basis function s (this is called adjoint Gaussian by Pisani). N This means that only one primitive Gaussian is selected for every basis function. We shall use this function for selecting the two-electron integrals to be calculated. An integral


  OJ H¸ pq rs

can be neglected if the overlap integral S-("1g-"g(2 or the overlap integral NO N O S&*"1g& " g*2 PQ P Q are under a certain threshold (overlap threshold). Here the functions gM , etc. are the above-de"ned N adjoint Gaussians. The overlap integral of the functions g-( and g&*1g-("g&*2 are used to control NO PQ NO PQ the exactly treated zone of the Coulomb interaction. g-( is the product of the primitive Gaussians NO g- and g( and g&* is de"ned similarly. A Coulomb integral is in the exactly calculated zone if the N O PQ overlap integral of g-( and g&* is larger than a given threshold (Coulomb threshold). The highest NO PQ H appearing among these integral gives the border of the exactly treated zone. The contribution of the remaining part of the Coulomb integrals is approximated using the method described previously. The two-electron integrals are grouped according to the cell indices (OJHL). A complete group of integrals can be neglected if (using the above described estimation procedure) all the members of the group are smaller than at least one of the two thresholds. From the remaining integral groups only those which cannot be obtained from another group of integrals by permutation symmetry are calculated. In a certain group of remaining integrals only those integrals have to be calculated for which both S-( and S&* are larger than the overlap threshold (see above). Those pq index pairs for NO PQ which S-( are larger than the threshold are stored and used during the integral calculation. NO 2.4. Four-component relativistic Hartree}Fock crystal orbital method Many solids or polymers consist completely or partially of heavy atoms. To treat them (especially their inner electrons) correctly one needs a relativistic formalism. As a "rst step for this, one has to construct relativistic HF crystal orbitals which have four components. To derive the Dirac}Hartree}Fock (DHF) equations for an atom, molecule or periodic solid for n electrons (or n electrons in the unit cell), one can perform the variation (as in the non-relativistic case) of the quantity E"E# e 1W "W 2; dE"0 . (42) GH G H GH In Eq. (42) the total energy E (if one takes into account only Coulomb interactions) can be written as E"1U "HK "U 2; U "AK [W (1)W (2)2W (n)] .       L

(43)

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Here U is a Slater determinant expressed in terms of the four-component W (i) relativistic  G one-electron orbitals:




W* G W* W" G G W1 G W1 G

(44a)

and the relativistic Hamiltonian can be written as 1 + HK " HK (i)# . "  r G GH GH

(44b)

where HK is the Dirac operator which contains the 4;4 Dirac matrices. " The radial parts of the two large P*(r) and two small Q1(r) components G G




P*(r) u (r)" G G Q1(r) G

(45)

have to obey the following relations [24}27]: (1) The radial density





o (r)dr" ["P*(r)"#"Q1(r)"] dr""nite if r(R . G G G

(46a)

(2) The expectation value of the scalar potential <(r) must be "nite,



o (r)<(r) dr""nite . G

(46b)

(3) If we expand P*(r) and Q1(r) in the following way: G G P*(r)"n (r )p #2#n (r)p , G    , ,

(46c)

Q1(r)"0 (r)t #2#0 t ,   ,Y ,Y G

(46d)

it is required that N"N and the pairs of the corresponding functions n (r)!0 (r) have to match to G G ful"ll the boundary conditions at r"0. The simplest way to obtain this pairing is to construct the  A four-component relativistic AO always has only two di!erent radial functions (large and small components). The four components are then constructed with the aid of di!erent combinations of the two radial functions with the angular dependent parts of the wave functions. This is already the case of the 1s state of the H-atom [28]. 

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small component functions 0 (r) with the help of the relation [24}27] G d K 0 (r)" # n (r) (i"1, 2,2, N) G dr r G





(47) (K"!( j#) or j#) .   Eq. (47) is called the kinetic balance condition [24,25]. If one uses Gaussians as basis functions with an integer exponent l, s%(r)"rJ exp[!jr] and a "nite nucleus with a distributed charge, the above given 3 criterions usually (but not necessarily) lead to meaningful HF results (one does not obtain deeper lying states than the HF limit). Formally one has to apply the projector PK " "u"21u"2 to project out the positive energy solutions u from "u>2" c "u 2# . . . . . N . G c "u 2 L L L (48a) PK FK PK "u>2"e>"u>2 . G G G Returning to Eq. (42) the second term of its r.h.s. contains the relativistic orthonormality conditions and the e 's are Lagrange multipliers. After performing the variational procedure in the GH standard way one obtains the relativistic HF (DHF) equations





L (48b) FK  W " HK "# (JK !KK ) W "e W . H H G G G G H One can solve these equations numerically (in the case of atoms) [29], or applying Gaussian basis sets (their generation see below) for molecules and solids in an iterative way (the Coulomb operators JK and exchange operators KK contain, in complete analogy to the non-relativistic case, the unknown spinors W ). G The DHF equations have been derived in an LCAO form before [30]. Therefore here only a brief summary of their derivation will be given. As in the non-relativistic case let us assume we have a crystal of 2N#1 unit cells in each direction and m basis functions in the unit cell. We assume further that we have four di!erent basis sets sR (t"1, 2, 3, 4). In this case we can write for a relativistic crystal orbital (CO) in an LCAO form






sq Cq G Q Q sq , K C q Q G Q . (49) W!-" G sq Cq q  \, Q Q G Q Cq sq Q G Q Here the shorthand notation q has the same meaning as in the non-relativistic case (see Section 2.1) and the basis functions have to be understood in the same way as in Section 2.1, only

 Due to the kinetic balance we have basically one independent radial function (though for Q1(r) one usually needs G additional basis functions) and the angular dependence of the functions is "xed (at least in the case of the H-atom). However, to make it possible to consider also basis functions not centred on the nuclei (functions centred on the middle point of a bond, #oating Gaussians, etc.) we prefer to work with four di!erent basis sets and four di!erent sets of coe$cients which gives more variational freedom.

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they have the additional label t (t"1, 2, 3, 4). We can rewrite now the DHF equations for a crystal as





K , (JK ( p) !KK ( p) ) W!-"e W!- . FK  W!-" HK # HY HY " p G G G G Y\, HY The scalar potential of the crystal (as in the non-relativistic case) is again

(50)

, Z + ? . (51) <" p  "r!R p " ? ? Y\, (M is again the number of nuclei in the unit cell and Z the nuclear charge of the ath nucleus). If no ? external "eld is present, we can write Eq. (50) as





c  L R #<#a m c# [JK ( p)!KK ( p)] W!-(q) "e(q) W!-(q) . (52) a  M G G G H HY HY i RX p H H Y HY Here the a 's ( j"1, 2, 3, 4) are the 4;4 Dirac matrices. In the JK (p) Coulomb and KK (p) exchange H HY HY operators for later convenience we have separated the band indices j and level indices q. To make clear the meaning of these operators we write in detail in the case of the exchange operator



KK ( pG, r ) W(q, r ) " (W( p, r ) W( p, r ) W( p, r ) W( p, r ) )  HY  HY  HY  HY  HY G




W(q, r ) G W(q, r ) 1  G W(p, r ) . ; (53)  HY r W(q, r )  G W(q, r ) G Introducing the LCAO forms of the four functions WR(p) (t"1, 2, 3, 4) (see Eq. (49)) one arrives G in the usual way [11] to the matrix equation

 






C(q) C(q) G G C(q) C(q) G "e(q) S G . (54) F G C(q) C(q) G G C(q) C(q) G G It is easy to show that the generalized overlap matrix S has the block-diagonal form [30]. S

S "

S

S

S

.

(55)

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189

The block S for instance contains only overlap integrals between basis functions belonging to the second basis set (S)q

"1sqY"sq2 . YQYOQ I Q

One should not forget that the dimension of the matrices SRR is M;M; M"m(2N#1). To express the 1.h.s. of Eq. (54) one has to perform the matrix multiplications in Eq. (52), substitute the LCAO forms of the functions WR(q) (t"1, 2, 3, 4); multiply from the left by the G di!erent sqYRY basis functions and "nally execute the integration according to the space variables QY (like in the non-relativistic case [11]). One thus obtains F as the sum of hypermatrices of the order 4M;4M. They correspond to the di!erent derivatives in HK (the "rst three hypermatrices), " the fourth one is just m cS, while the "fth one contains < (Eq. (51)) and the sums  p j [JK ( p) !KK ( p) ]. For instance Y Y HY HY

  

R c [FRYR]q q " sRYq sRq  YQY_ Q i QY Ry Q

.

One can write out the hypermatrix Eq. (54) in the form of four matrix equations [30] which all contain M;M matrices F RRY (l"1, 2,2, 5). It is easy to show that these matrices become cyclic J hypermatrices if one takes into account the translational symmetry of the crystal and introduces periodic boundary conditions (see Section 2.1). So one can apply again a unitary transformation to all the terms of the four matrix equations which will block-diagonalize all these matrices (see again Section 2.1). So we can write U>FRYRUURCR"FRYR " DR J J

(56)

where the unitary matrix U was de"ned in Eq. (8a). Since we have now only block-diagonal matrices in the four matrix equations, we can separate again these equations according to their blocks. Further if NPR, we can introduce again instead of the discrete variables p "!N,!N#1,2,0,2N the continuous variables G k "2pp /a(2N#1) (see once more Section 2.1). G G In this way we could arrive at the relativistic Roothaan}Hartree}Fock equations of the crystal. The equations are the following: [F(k)!F(k)]d(k) #F(k)d(k) #[F(k)#F(k)]d(k)   G  G   G "e (k) S(k)d(k) , G G

(57a)

[F(k)#F(k)]d(k) !F(k) dk #[F(k)#F(k)]d(k)   G  G   G G "e(k) S(k)d(k) , G G

(57b)

[F(k)!F(k)]d(k) #F(k)d(k) #[F(k)!F(k)]d(k)   G G   G "e(k) S(k)d(k) , G G

(57c)

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[F(k)!F(k)]d(k) !F(k)d(k) #[F(k)!F(k)]d(k)   G  G  G "e(k) S(k)d(k) . G G

(57d)

All of them have the form of a Fourier transform [7] FRYR(k)" FRYR(q)e kRq , J q

(58a)

SRR(k)" SRR(q)e kRq .

(58b)

q

To obtain the form of the matrices FRYR)(q), we have to substitute in Eq. (53) and in the correspond ing equation de"ning JK ( p) , the LCAO form of the CO's WR( p) . A straightforward derivation HY HY after de"ning the relativistic generalized charge-bond order matrix elements



1 L p(q !q )RY" d(k)RYHd(k)RY e kRq\Rqdk HYS HT   ST u H S (u is the volume of the "rst Brillouin zone) leads to the expression

(59)

(FRR(q)) "1soR"<"sqR2  QQY Q QY









 K 1 ) sqR(1)sqRY(2) # p(q !q )RY soR(1)sqRY(2) (1!PK   ST Q S @ QY T r q q   ST RY 

. (60)

The other four FRYR matrices were de"ned before through their elements. J Instead of solving the four coupled matrix Eqs. (57a), (57b), (57c) and (57d), we can rewrite them as a hypermatrix equation. To be able to do this we can use the relation




d(k) G d(k) G "a dI (k) aJ (s)"   G d(k) G d(k) G

(61)

and similar relations for the other three hypervectors dI , dI  and dI : dI "(1/i)a dI , dI "a dI    and dI "a dI .  With the help of them, we can put together the matrix Eqs. (57a), (57b), (57c) and (57d) into the generalized eigenvalue equation

F (k)dI (k) "e(k) G G




S(k) S(k) S(k) S(k)


 d(k) G d(k) G d(k) G d(k) G

(62)

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191

with




F (k)"

F(k)#F(k)   O F(k)  F(k)#F(k)  

F(k)  F(k)#F(k)   F(k)!F(k)   O

O F(k)#F(k)   F(k)!F(k)   !F(k) 



F(k)!F(k)   !F(k)  . O F(k)!F(k)   (63)

It is simple to demonstrate with the aid of the relation F"!(F)"(F)>    and of similar ones in the cases of the other o!-diagonal matrix blocks, all of which can be proven through integration by parts of their elements. For instance

  

  

RsqRY RsoR soR QY "! sqRY Q . Q Rs QY Rs G G Therefore F is a Hermitian matrix. One will obtain therefore after a LoK wdin symmetric orthogonalization procedure [10] applied for a crystal [7]




b(k) G b(k) G (64) S\(k)F (k)S\(k)dI (k) "FI  (k)b(k) "e(k) G G G b(k) G b(k) G real (relativistic) one-electron energies e(k) . In this way Eqs. (62) and (63), together with the above G given de"nitions of the matrices occurring in them fully de"ne the relativistic Hartree}Fock} Roothaan (relativistic ab initio SCF LCAO) equations with di!erent coe$cients for all the functions belonging to the four di!erent basis sets of a crystal. A four-component spinor DHF calculation has been performed for a zig-zag Se chain (our program does not work yet for helix operations which is the true conformation of an Se chain). We have used for the calculation a (15s13p7d) Gaussian basis which we have generated in a complicated way starting by "tting the numerical Desclaux results [29] by Gaussians (for details, see Hu et al. in Ref. [30]). 2.5. Selected ab initio Hartree}Fock band structures 2.5.1. Polyethylene and some of its halogenated derivatives Polyethylene (PE) is produced in large amounts in the plastic industry. Therefore the theoretical study of its electronic structure and properties (especially its mechanical properties [discussed later]) is of great importance. Using di!erent medium level basis sets (6-31GHH [31] or Clementi's doublef basis with polarization functions [32]) di!erent authors have obtained valence band widths of 6.2 eV [33] and 5.5 ev [34], while the corresponding values for the conduction band are 4.3 and 3.8 eV, respectively. The fundamental gap is 13.4 eV in the "rst case and 13.9 eV in the second one. The experimental value (from photoelectron spectra) of the valence bandwidth is 6.0 eV [35] (in quite good agreement with

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the theoretical results), while the experimental fundamental gap is considerably smaller (8.8 eV [35]) than the theoretical value (as it is always the case with HF band structure calculations). This discrepancy can be, however, substantially decreased, if one takes into account correlation corrections (see Section 3). The HF ionization potential is 10.2 or 9.7 eV, while the corresponding experimental value is (depending on the di!erent measurements) between 7.6 and 8.8 eV [35]. Since Koopmans' [36] theorem is only approximately valid, the deviation of the experimental and theoretical values is not especially signi"cant. There is a more serious disagreement between the theoretical values (3.2 eV [33] and 2.8 eV [34], respectively) and the experimental ones (between 0.6 and 1.2 eV [35]) in the case of electron a$nity showing that Koopmans' theorem is still less valid for the conduction band, than for the valence one. One should observe that the rather large conduction and valence bandwidths indicate large electron or hole mobilities, respectively. Of course because of the large gap it seems hopeless to dope PE itself to bring free charge carriers into these bands. On the other hand these bandwidths indicate that if one would have substituted PE (for instance if they would be substituted by halogens, the whole band structure would move downwards and thus the halogenated PE's could be doped by electron donors) it does not seem hopeless to increase the conductivity of these polymers. Detailed HF calculations were performed also on all #uorinated PE's. Here we are going to discuss only the results for polytetra#uoroethylene (Te#on) which has again a large industrial importance. Though the gap obtained by the 6-13GHH [31] and by Clementis' doublef# polarization functions [32] basis (17.3 and 17.10 eV, respectively) are larger by about 3.5 eV than in the case of PE and the bandwidths hardly change, the lower limit of the conduction band is shifted in the case of the second basis set from 2.8 to 2.4 eV. In a subsequent calculation using another doublef basis of Clementi [37] which applies another contraction scheme the lower limit of the conduction band decreases from 4.6 eV (PE) to 1.1 eV (Te#on) [38]. Since correlation e!ects usually shift further down the conduction band (see Section 3) one may expect that Te#on could be doped by electron donors increasing in this way its conductivity (see below). For further details about di!erent halogenated PE chains see [39]. 2.5.2. Alternating trans polyacethylene (PA) It is well known that the conductivity of PA can be changed by doping it with electron donors or acceptors of various orders of magnitude (from p+10\ )\ cm to p+10 )\ cm) [40] changing it from an insulator to a good conductor. The electron acceptors usually applied are bromine, iodine and AsF and as electron donors lithium or sodium are applied [40].  The electronic structure of PA was calculated by di!erent authors [41]. Suhai [42] using di!erent basis sets has calculated the di!erence of the single and double bond distances, the total energies and the gap. In his calculations he has used up to ten neighbours' interactions and has optimized the geometry in the case of each basis. In his calculations he has used his method to apply di!erent cut-o! radii for di!erent types of integrals [43], but did not sum up the Coulomb interaction until an in"nite number of neighbours using a multipole expansion. His most important results are collected in Table 1. Looking at Table 1 one can see that the investigated quantities are strongly dependent on the basis set applied. According to the details [42] bandwidths (values between 6 and 10 eV) and R }R become constant at the better basis sets already after 5}7 neighbours' interactions.    
 This is, however, not the case for the total energy per unit cell [42].

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Table 1 Single-double bond length di!erence, the total energy per C H unit and the HF gap in the case of alternating trans  polyacethylene [42}44] Basis set

R !R (A> )    


E (a.u) !&

Gap (eV)

STO-3G 4-31G 6-31G 6-31GH 6-31GHH


0.16 0.11 0.11 0.10 0.10

!75.947 !76.777 !76.859 !76.887 !76.893

8.28 4.93 4.43

A set of d functions of the C-atoms.
A set of d functions on the C- and a set of p functions on the H-atoms.

2.5.3. The energy band structu re of (SN) V

is a highly conducting polymer (without doping), because its unit cell has 3 p electrons and therefore it possesses a half-"lled conduction band. Using a doublef basis [45] we have obtained a width of 4.12 eV for the half-"lled band [46]. The e!ective mass at the Fermi (E ) level was 1.7 m (the experimental value is 2m deduced from the $   analysis of polarized re#ection spectra of single crystals of (SN) [47]) and for the density of states V at E we have found o(E )"0.14/(eV.spin.molecule). For the latter quantity the experimental value $ $ (obtained from the contribution of the linear temperature-dependent term to the speci"c heat) is 0.18/(eV.spin.molecule) [48]. Further the theoretical value obtained from Mulliken's population analysis [49] for the charge transfer in the ground state from the S atom to the N atom is 0.40e, while from X-ray photoemission (XPS) spectra 0.3}0.4e extra charge was found on the N atoms [50]. All these results show that if a better basis set is used, already a HF calculation gives reasonable results for the ground state of a polymer (at its equilibrium geometry) in agreement with the general experience for molecules. On the other hand when only a minimal basis was used, the agreement with experiment was quite poor (mH"0.72 m , o(E )"0,06/(eV.spin.molecule)) [51].   $ 2.5.4. The band structure of polyparaphenylene (PPP) Detailed calculations have been performed for the ab initio HF band structure of polyparaphenylene (PPP) (see Fig. 2) [52]. Clementi's doublef basis set [32] with polarization functions was used and a geometry for the rings taken from the middle part of the terphenyl molecule determined by X-ray di!raction [53]. For the torsion angle 22.73 was applied which originates  Contrary to alternating trans PA in (SN) there is no Peierls instability (the occurrence of a gap in the middle of the V half-"lled band). This is most probably due to the di!erent interactions in the 3D crystal structure of the two di!erent polymers.

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again from X-ray di!raction of PPP [54]. The calculation was performed in the second neighbours' interaction approximation with 25 di!erent k-points in the irreducible part of the "rst Brillouin zone. In this way for the conduction bandwidth 4.87 eV was obtained and the lower limit of this band lies at 0.55 eV. The calculated width of the valence band is 4.95 eV and its upper limit lies at !7.96 eV. This means that the fundamental gap is 8.51 eV which is again far too large (as usual at HF calculations). It should be mentioned that performing subsequently correlation corrections for the band structure the gap decreases to 4.79 eV (see Section 3). The experimental ionization potential of PPP is 5.6}5.7 eV [55] which is &2.3 eV smaller than our &8.0 eV value. This discrepancy will be decreased again by about 1 eV if we take into account correlation corrections. This falls into the range of error (&1 eV) of theoretical ionization potentials estimated on the basis of Koopmans' theorem [36]. No experimental data are available for the electron a$nity of PPP. The comparison of the theoretical gap value with the experimental one will be discussed later, after correlation corrections have been introduced to the band structure. 2.5.5. Calculation of a cytosine (C) stack Since DNA plays a central role in the living cell, we have investigated the band structures of the di!erent nucleotide base stacks, both at the HF and at the correlation corrected level [56,57], as well as the band structures of the di!erent periodic polynucleotides (base#sugar#phosphate) at the HF level [58]. The calculations were extended also to non-periodic base sequences [59] calculating also the hopping conductivity through the base stacks both at the HF and correlation corrected level [60] (see Section 4). The HF periodic polynucleotide calculations [58] have shown that in a very good approximation their band structure is a superposition of that of the sugar}phosphate chain and of the nucleotide base stacks, respectively. Since according to recent biophysical experiments there is a rather strong hole conductivity even across aperiodic base stacks [61], one could concentrate on the calculation of the base stacks. As a "rst step one can perform periodic base stacks band structure calculations both at the HF and at the correlation corrected level. This has been done, using a Clementi's doublef basis [36] for all the four base stacks [57]. Since, however, a doublef basis is not good enough to obtain results which one can compare with experiment even after correlation corrections, in the case of a cytosine (C) stack we have performed a detailed investigation using a better basis set [57]. (For the chemical formula of a cytosine molecule see Fig. 3). For the C stack we have used the geometry of DNA B [62] with a stacking distance of 3.36 A> and a rotation of 363 (there are 10 nucleotide bases in a turn of the DNA helix). Instead of using only a doublef basis with the AO's centred on the nuclei of the bases we have put at the half stacking distance (1.68 A> ) rotated by 36/2"183 with respect to the neighbouring two real cytosine molecules a `phantoma molecule on which we have put at the position of the nuclei additionally two sets of p-functions (doublef) on the C, O and N atoms. It should be emphasized that neither the nuclei nor the electrons of this additional `phantoma C molecule were taken into account, it served only for the positioning of the additional sets of p-functions (in a stacked system with a stacking distance of &3.4 A> there is a large un"lled space. One could use either a triplef basis with a third set of di!use p-functions, but one believes that the described procedure with the phantom molecule further improves the results).

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195

Fig. 3. The chemical formula of a cytosine molecule.

The dependence of the band structure of the C stack (with the doublef basis) on the number of neighbours and number of k-points was also investigated at both the HF and correlation corrected level. It was found that the results hardly change going from second to third neighbours and already 7 k-points in the half of the Brillouin zone have given practically the same results as a larger number of k-points. In Table 2 we give the HF band structure of a cytosine molecule using 2nd neighbours' interactions, using a doublef basis centred (a) at the C molecules and (b) additionally also with two sets of p-functions at the phantom C molecules. One can see from the table that the introduction of the two sets of additional p-functions does not change the results very strongly. This is, however, not the case after the correlation correction of the band structure, when the gap decreases to 6.60 eV (which is quite close to the estimated experimental gap) and generally the band structure using the improved basis is quite di!erent than in the case of the simple doublef basis set (for details see [57] and Section 3). 2.5.6. The HF band structures of the homopolypeptides polyglycine and polyalanine Besides DNA the di!erent protein molecules are the most important biopolymer constituents of the cell. They act as catalysts (enzymes) making possible biochemical reactions at body temperature which otherwise could not take place. The structural proteins are the building stones of the larger cell constituents [63], they form hormones, etc. Though proteins which consist of 20 di!erent amino acids in a non-periodic way, the investigation of their electronic structure and the charge or/and energy transport processes taking place in them has to start again with the calculation of the band structures of periodic (homo) polypeptides. This takes place "rst at the HF level and afterwards the resulting band structures can be corrected again for correlation. Knowing their amino acid sequence or assuming a random sequence in them,

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the investigation of the charge transport in them can be investigated besides other mechanisms by calculating the hopping conductivity along their aperiodic main chains [63]. For the two simplest homopolypeptides polyglycine and polyalanine (see Fig. 4) a 6-31GH (doublef plus a set of d-functions on the non-hydrogen atoms [64]) calculations were performed [65] in the a-helical con"guration. Since the unit cell has a dipole moment, "fth neighbours' interactions had to be applied to obtain meaningful HF results. Applying a careful balance between the contribution of the one- and two-electron terms [66] the HF solutions were numerically stable. After that the resulting band structures were corrected again for correlation (see Section 3). From these corrected band structures the exciton spectra of these two simple homopolypeptides were computed (the method used see below) [65]. The results obtained were in good agreement with experiment. In another series of HF calculations using Clementi's minimal basis and doublef basis set HF computations were executed for polyglycine and minimal basis HF calculations for polyalanine and for other six homopolypeptides in the parallel b pleated sheet conformation. One has found [67] that the position of the conduction and valence bands and their widths and the gap depends strongly on the side chains. (The bandwidths vary between 0.02 and 1.4 eV.) In the case of polyglycine the bandwidths were substantially larger in the case of the doublef basis than those obtained with the minimal basis. The gap values also change strongly with the side chains (values Table 2 The Hartree}Fock band structure of a cytosine stack with (a) a doublef basis and (b) putting two sets of additional p-functions at the &&phantoma C molecules (in eV s) Basis

Conduction band

Valence band

Gap

max. min. width max. min. width

(a) Doublef

(b) Doublef#2 set of p-functions at the phantom molecules [57]

3.28 2.81 0.43 !8.74 !8.86 0.12 11.55

3.56 3.22 0.34 !7.98 !8.19 0.21 11.20

Fig. 4. The chemical formula of (a) polyglycine and (b) polyalanine. In the other 18 homopolypeptides 18 from CH  di!erent side chains occur.

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197

between 15.8 and 10.8 eV) usually decreasing with increasing side chain lengths. As usual they are much larger than the values which one can estimate as their real values. (No experimental values for the gap of the homopolypeptides are available.) Starting from these band structures calculations are in progress to correct them for correlation. On the basis of previous experience one can expect that the correlation corrections will substantially decrease these gaps and bring them closer to their estimated real values (7.8}8.5 eV) [68].

3. The treatment of correlation in periodic polymers As it is well known from theoretical molecular physics (quantum chemistry) to obtain reasonable results for most molecular properties generally one needs to take into account electron correlation. The correlation energy is the di!erence of the eigenenergy of the SchroK dinger equation in a given state (which can be estimated with the help of the experimental ionization potentials of all the electrons in the given state and of the relativistic energy corrections) and of the energy of the HF limit. If the investigated system is large or in"nite, the standard method of quantum chemistry to treat correlation, the con"guration interaction method cannot be applied in any approximate form, because then this method would not be size extensive (the correlation energy per electron would go to zero if the number of electrons goes to in"nity [69]). Since we cannot work with an in"nite number of con"gurations we have to apply to long or in"nite polymers (or more generally to solids) a size extensive method. Such methods are for instance the many body perturbation theory [70] and the coupled cluster theory [71]. One can perform the correlation calculation in both cases with the help of the delocalized HF CO's, but for 1D polymers the application of localized Wannier functions (they are well localizable in the 1D case) are more advantageous (can save CPU time). In the 2D case it did not succeed until now to localize well the Wannier functions, so in these cases the original HF CO's were applied. The Wannier function belonging to the nth band and localized mostly in a cell characterized by the position vector R can be obtained as the Fourier transform of the CO's of this band [72]. G w (r!R )"N\ Wk(r) exp(!ikR ) . L G L G k

(65)

Here N is again the number of unit cells, and Wk is the HF CO which can be written again as the L linear combination of Bloch orbitals (16). Substituting this into Eq. (65) one can express the Wannier function (65) after some manipulation in the form w (r!R )" dRq\RGsq LQ Q L G Q O

(66)

dRq\RG"N\ d (k)e kRq\RG . LQ LQ I

(67)

with

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In Eqs. (65) and (67), respectively, the summation over k has to be extended to the whole "rst Brillouin zone. This means that one cannot construct Wannier functions for partially "lled bands (metallic solids or polymers). 3.1. Application of the Moeller}Plesset many-body perturbation theory for in,nite systems The simplest and therefore most frequently applied many-body perturbation theory (MBPT) for the treatment of correlation goes back to the fundamental work of Moeller and Plesset (MP) [70]. In this form of MBPT the unperturbed Hamiltonian is the sum of Fock operators m

HK " FK (i) .  G This operator has the eigenvalue equation for the ground state

(68)

HK U "E U , (69)     where U is a Slater determinant  U "AK [W (1)2W (n)]"U (70a)   L &$ constructed from the HF orbitals. In the case of polymers (or solids) one has to write into Eq. (70a) instead of HF molecular spin orbitals (MO's) HF crystal spin orbitals (CO's) and n now means the number of electrons in the whole large or in"nite periodic solid (polymer). Further n"NnJ , where nJ is the number of electrons in the unit cell,





U "AK (Wk (r )2WkJ (r J )) .   L L  k

(70b)

In the theory of MP the total energy of the ground state is the sum of the HF one-electron energies, LJ (71) E " ek . G  k G From Eqs. (70b) and (71) we can easily write down the many-electron wave functions and energies of the singly-, doubly-, etc. excited states: U"U (IPA), ' &$




IPA U "U , '( &$ JPB

(72a)

L (72b) E" e #e , E " e #e #e . ' +  '( +  + + +$ +$'( In this compact notation the combined indices always mean a certain band index a k-vector, for instance I"i, k . I, J refer to occupied states and A, B to unoccupied ones in the ground state. G Applying the Raleigh}SchroK dinger perturbation theory one can write the total Hamiltonian as HK #HK #j
(73)

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where the perturbation operator
(76a) (76b)

1 U "
(77)

E "1U "HK "U 2"1U "H #








 1 W (1)W (2) (1!PK ) W (1)W (2)" ' ( @ r   . (79) E "  e #e !e !e ' (  '(  Here it was taken into account that singly excited states, because of Eq. (75) do not contribute to E and the excitation energy E !E reduces in the MP partitioning of the Hamiltonian to the   '( di!erences of one-electron-energies (see Eq. (72b)).

 We have truncated the perturbation expansions (76a) and (77b), respectively, after the second-order term, because, contrary to molecular calculations, there are very few MP3 and MP4 calculations of polymers, because they are very cumbersome. For these cases we refer to the literature [73,74].

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If we have more than one basis function in the unit cell (m'1) our CO's are a linear combination of Bloch orbitals (see Eq. (16)). Therefore after a HF band structure calculation we also have to perform in this case a four index transformation. This is, also in the 1D case, rather cumbersome even if we take into account the periodicity of the system (for the details see [75]). It was found [75] that in the case of non-metallic systems one saves substantial CPU time if one applies instead of the CO's W , etc. Wannier functions in their form given by Eqs. (66) and (67),  respectively. In the second order it is possible to write E as the sum of pair correlation energies  E " e  '( '(



(80a)







 1 W (1)W (2) (1!PK ) W (1)W (2)" ' ( @ r   . (80b) e " '( e #e !e !e ' (   Instead of calculating the integrals occurring in the numerators of the e 's with the help of the HF '( CO's; one can compute them using integrals expressed by Wannier functions:



 



1 w (r )w (r !Rh) w (r !Rh )w (r !Rh ) . (81) Y @  YY G  H  ?  r  One can perform the four-index transformation stepwise (at a step only one Wannier function is computed from the AO's [76]). Afterwards one has to take into account the k dependence of the Wannier functions (see Eqs. (66) and (67)). It is also necessary to transform back the Fourier transforms (65) into CO's to obtain the integrals occurring in the numerator of Eq. (80b). This can be done easily by dividing the BZ into a "ne mesh in k. The advantage of this procedure is that one has to calculate the integrals (81) only once (they are k-independent) and the subsequent numerical integration according to k is already independent of the AO indices and therefore it is already quite rapid. In the case of a metallic chain the "rst BZ was divided into several parts. In each one the coe$cients d(k ) were expressed as a simple trignometric expansion G LQ LG (82) d (k )" AR (i)cos[tk a] . LQ G LQ G R Here n is the number of terms in the expansion in the ith region of the BZ. One can determine the G constants AR (i) on the basis of the coe$cients d (k ) obtained from the HF calculation. In LQ LQ G Table 3 for a metallic hydrogen chain the total and cohesion energies and the H}H distance belonging to the energy minimum are shown for "ve di!erent basis sets both at the HF and HF!MP2 level. In the course of the calculations 60 neighbours have been taken into account. Looking at the table one can see that (1) both at the HF and at the HF#MP2 level the total energy and the cohesion energy per unit cell strongly depends on the quality of the basis, (2) the correlation energy, (E0  !E0 ) is still more strongly basis set dependent (it is nearly &$>+. &$ doubled going from the minimal basis set to the 3s1p basis) and (3) there is a small increase in the lattice distance due to the correlation.

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Table 3 The total and cohesion energies per unit cell and the equilibrium H}H distance for an in"nite metallic H chain at the HF and HF#MP2 level using "ve di!erent basis sets [75] Basis set

E /N &$

E&$ ,

E &$>+.,

E&$>+. ,

R
&$


R &$>+.

4s/1s 4s/2s 6s/3s

!14.232 !14.525 !14.543

0.669 0.939 0.940

!14.583 !14.961 !14.962

1.020 1.375 1.358

1.782 1.839 1.843

1.810 1.866 1.852

6s, 2s/3s, 1s 6s, 1p/3s, 1p

!14.580 !14.599

0.977 0.993

!15.090 !15.223

1.487 1.618

1.826 1.834

1.840 1.844

All energy values in eV.
Interatomic distances in a.u. s.  The line over the notation of the basis functions means that it is centred at the middle of the H}H distance.

The correlation energy calculated with our best basis (!0.63 eV/el) is &0.3 eV smaller, than the correct value of !0.9 eV/electron (estimated on the basis of calculation of the H molecule [77]).  This discrepancy can be attributed "rst of all to the missing part of the correlation. The total energy per unit cell was calculated also for trans-alternating polyacetylene [78]. In this case, since the undoped system has a gap, the MP2 calculation was performed with the help of Wannier functions as described above. With the best applied Gaussian basis set, 6-31GHH(doublef# a set of d-functions on the C atoms and a set of p-functions on the H-atoms [68]) and taking into account 7th neighbours' interactions, the total energy per C H unit has been   found in the HF#MP2 approximation to be !77.168 a.u. which is about 7.5 eV larger negative value than the corresponding HF one. One should compare this with the value of &!10 eV which was estimated for an acetylene molecule [79]. This means that already an MP2 calculation with a medium quality basis set provides 70}75% of the correlation energy which is not so in the case of molecules. The main reason for this is that by taking into account not only the unit cell, but also a number of its neighbours, one has a basis set superposition e!ect which improves every applied incomplete basis (this would cause an error only if we would calculate energy di!erences, as in the case of cohesion energies or interaction energies). One should point out further that according to the detailed results [78] the bond alternation *R (R !R ) decreased going from a minimal basis (STO-3G) HF calculation to an    
 HF#MP2 calculation with the 6-31GHH basis from 0.156 to 0.084 A> . One would expect that using a still better basis and going beyond MP2 (calculating also the MP3 and MP4 contributions) the bond alternation would still decrease somewhat further. 3.2. The correction of the band structure of a periodic system on the basis of the electronic polaron model Toyozawa [80] has introduced the concept of the electronic polaron (an excess electron surrounded by a cloud of virtual longitudinal excitons). This idea was further developed by Kunz et al. [81}83] taking an electron or a hole in a HF state as naked electron (hole) which is dressed by

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surrounding virtual excitons. In this way one can formulate an electronic polaron in an ab initio form which provides instead of HF correlation corrected (quasi particle; QP) one-electron states. In analogy to Koopmans' theorem [36] (82a) ekA(HF)"EkA(HF),>!E,(HF)"AkA , A A A ekT(HF)"E,(HF)!EkT(HF),\"!IkT . (82b) T T T Here E,(HF) is the HF total energy of the ground state and containing N electrons, EkA(HF),> A and EkT(HF),\, respectively, are those of the state where there is an extra electron in the T conduction band c with a quasi momentum k , or one electron is missing (there is a positive hole) in A the valence band v with the quasi momentum k , respectively. Finally, AkA is the electron a$nity A T belonging to the state c, k and IIT the ionization potential of the state v, k , respectively. T T A One can generalize Koopmans' theorem by writing in Eq. (82) everywhere instead of HF correlation corrected (QP) total energies [42] (83a) ekA(QP)"EkA(QP),>!E,(QP) , A A ekT(QP)"E,(QP)!EkT(QP),\. (83b) T T Here the QP energies are the exact total energies. As an approximation we can write for instance for the ground-state total energy E,(HF)#E,(MP2) (Moeller Plesset 2nd order). Applying this approximation and Eq. (82) one obtains (84a) ekA(QP)"ekA(HF)#EkA (MP2)!E (MP2), , A A A  (84b) ekT(QP)"ekT(HF)#E (MP2),!EkT (MP2),\ . T T  T As we have seen before the Moeller}Plesset correlation correction in the second order can be expressed as the sum of pair correlation energies (see Eq. (80b)). Writing the di!erences of E for the  (N#1)!N particles systems and for the N and N!1 particles systems one obtains after some algebra [42,84] the expressions ,> ,> e (QP)"e (HF)# (e)# (h) (C"c, k ) , (85a) ! ! A ! ! , , e (QP)"e (HF)# (e)# (h) (<"v, k ) . (85b) 4 4 T 4 4 Here the electronic and hole contributions to the self-energies [85] ,> and ,, respectively, ! 4 are de"ned as ,> " e,>!" q , '! '! !C ' ' $! $! e,>!" '! $! $!





U (1)U (2) (1!PK ) ' ! @ r

(86a) 1





U (1)U (2) 

 e #e !e !e ' ! 

 (87a)

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or if one takes into account also relaxation due to the presence of the extra electron [86] (2!" '! '! e #e !e !e ' !  $! $!

(87b)

In Eqs. (87a) and (87b) the quantities < are the usual two electron integrals '! 1 < " U (1)U (2) U (1)U (2) (87c) '! ' ! r   expressed in terms of HF CO's. For their calculation it is again advantageous to perform the necessary four index transformation in two steps. First one transforms the AO's to Wannier functions and after that transforms back only that Wannier function to a CO (in our example one has to transform back only u to U ) which belongs to the one-electron energy ekA which we want to A ! ! correct for correlation (see also Section 3.2). In the case of a partially "lled conduction band one may obtain poles in the denominator if both excitations occur with zero energy. Therefore we have to write instead of Eq. (87b)



 







  < (2!"lim '! '! e #e !e !e #ig !  E $! $! ' If one has in the real part of Eq. (87c) a pole, one can use the principal value theorem of complex analysis [87]. Returning to the self-energies in Eqs. (85a) and (85b), we can write on the basis of the detailed derivation [42,84], ,> (h)"! q! . '( ! ' ( $! Further

(88)

,\ (e)" e," q , (89) '4 '4 4 '$4 '$4  ,\ (h)"! q4 . (90) '( 4 '$4 ($4  Investigating in detail the expressions (84a)}(90) one "nds [42,84] that the lower limit of the conduction band is shifted downwards and the upper limit of the valence band is shifted upwards. This results in a reduction of the fundamental gap. The larger the decrease of the gap, the better the applied basis set. Further one can show that the upper part of the conduction band part is shifted more strongly downwards than its lower part resulting in a band narrowing. We can also see that the same happens with the upward shifts of the di!erent parts of the valence band resulting also in a narrowing of this band [42,84]. One should emphasize, however, that these band narrowings do not necessarily occur in all cases. The symmetry of the wave functions may be di!erent for di!erent empty bands and it can also happen that the higher lying empty bands have larger correlation contributions. These two

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e!ects in some cases can even result in a broadening of the bands. Further one should mention that if the width of the HF conduction band is larger than the HF gap, the scheme based on the MP2 approximation breaks down for the upper part of the conduction band. For the same reason also higher-lying empty bands cannot be corrected by this scheme. In such cases either one has to go to higher orders, or use in the MBPT another partitioning scheme. A further possibility is to use instead of MBPT the Coupled Cluster Approach [71] which in the ¹K ¹K approximation (the   operator ¹K includes all double excitations) permits also the partitioning of the correlation energy  into pair correlation function. On the other hand our attempt to use this theory even with very well localized Wannier functions for PA demanded extremely large CPU times [88] which shows that at least for the time being this theory is not well applicable to extended systems. One can calculate the QP energies in a somewhat better approximation applying (following the suggestion of Liegener [73]) the inverse Dyson equation in its diagonal approximation [89]. One can start from the one-particle Dyson equation in its matrix form with already Fourier transformed Green's matrices. G"G #G R G ,   1 1 , G" . G "  x!e&$ x!e&$!R

(91)

Here G is the Green's matrix of the unperturbed (uncorrelated) system, G the one describing the  correlated system and R is the self-energy matrix. Further x is the diagonal matrix containing the quasi particle energies and the diagonal matrix e&$ contains the HF energies. Taking only the diagonal part of R and multiplying Eq. (91) from the left by G\ and from right  by G\ one obtains the inverse Dyson equation G\"G\#R . (92)  Applying for this equation the diagonal approximation (which at least at the calculation of excited and ionized states of molecules did not cause a larger error [90]) one can write (G\) "(G\) #(R) .  '' '' '' Since (G\) at a pole is 0 (u !e&$!(R) "0) one obtains '' ' ' '' "0 . u "e&$#[R(u)] ' ' SS' ''

(93)

(94)

For [R(u )] one can again apply any size extensive method if one would like to apply Eq. (94) for ' '' the correlation correction of the energy bands of an extended periodic system (for the application of non-periodic systems see Section 4). As the simplest case we shall substitute for R again the MP2 '' expression in the form which corresponds to Eq. (87d).



< (2


(95)

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The integral < is de"ned again by Eq. (87c) and can be calculated in a way as described '( above. One should notice that Eq. (94) is non-linear and therefore can have even a number of real solutions u . According to general experience if the periodic system which we investigate has a gap ' the pole strength de"ned by




P " 1! '

 

RR(u) '' Ru

\

(96) SS' always has a main value P 50.6. In this way the physically relevant solution for every I can be ' easily found. A vectorized program to solve Eq. (94) was written which has used a graphical method. Afterwards it was found that the program is much faster (by a factor of 240) if instead of the graphical method an iterative procedure was applied [90] (in the latter case to solve Eq. (94) only a single loop was needed which is very advantageous for the vectorization). In the case of a partially "lled conduction band (metal) there are no main values of u !s (there ' are many P values of the order 0.1}0.2) and therefore no physical solutions can be found. The only ' possibility in this case is to put for u e&$ and do not perform any iterations, i.e. in this case one has ' ' to apply instead of Eq. (95) the one which corresponds to Eq. (87d). (In other words if one cannot perform iterations the two formalisms become equivalent.) At the same time in this case one cannot work with Wannier functions (see above). Finally if one has poles in the expression of Eq. (95) one has to apply again the principal value theorem [87]. 3.2.1. Selected examples for quasi particle band structures 3.2.1.1. Alternating trans polyacetylene. In Table 4 QP (correlation corrected) conduction and valence band limits of alternating trans PA are presented (using di!erent Gaussian basis sets) which determine the fundamental gap. As mentioned before, ten neighbours' interactions were applied

Table 4 The quantities determining the fundamental gap of alternating trans PA : HF one-particle energies, electron and hole self-energies and quasi particles energies (all quantities in eV s) [42,84] Quantity

STO-3G

6-31G

6-31GHH

e (HF) A  ,>(e) A  ,>(h) A  e (QP) A  e (HF) T  , (e) T  , (h) T  e (QP) T  Gap (HF) Gap (QP)

3.719 !0.452 0.112 3.379 !4.563 !0.312 0.583 !4.293 8.282 !7.672

!0.806 !0.623 0.193 !1.236 !5.732 !0.436 0.896 !5.361 4.926 4.125

!1.322 !0.938 0.274 !1.996 !5.749 !0.274 1.497 !4.976 4.427 2.980

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together with Suhai's cut-o! criteria [41] and the band distances were optimized both at the HF and QP levels at each basis set. One can see that (1) the band limits come closer in the HF case if one uses better basis sets and therefore the HF gap decreases from 8.28 to 4.43 eV, (2) the larger the e!ect of correlation the better the basis. In the STO-3G case the gap decreases from 8.28 eV to 7.07 eV, while with the 6-31GHH basis it changes from 4.43 eV to 2.98 eV. The same is true also for the absolute values of the self-energies. Finally according to the detailed results [42] both QP bands are narrower than the original ones. In the case of the 6-31GHH basis one can estimate the valence-shell correlation taken into account to be 75%. One can estimate the gap value at 100% valence electron correlation to be 2.5 eV [42]. On the other hand the "rst peak in the absorption spectrum of PA is at 2.0 eV [91] which is probably smaller than the gap. To obtain about 100% valence correlation one would need a still better basis set and at least an MP3#MP4 calculation. In later papers Suhai used a much better triplef (5s3p3d2f for the C atoms and 4s2p1d for the H atoms) basis and added MP3 and MP4 correction to the MP2 one [74]. In this way he obtained !38.716 a.u. for the total energy per CH unit which is by 0.032 a.u + 0.9 eV larger negative value than this MP2 result. However, since no MP3 calculation was performed (which usually gives a positive contribution) one does not know, how large is the total energy per CH unit if MP3 is also taken into account. Further it should be mentioned that no attempt was made to calculate the gap in this high level of approximation. In a somewhat earlier calculation [92] he has determined the gap for PA in the MP2 approximation using for the self-energies the expression (87a) (no relaxation) and applying Dunning's basis sets [93]. In this way he has obtained after a geometry optimization instead of 2.98 eV 2.75 eV for the gap. In a still more recent calculation of Sun and Bartlett [94] applying the 6-31GHH basis for a HF#MP2 calculation of PA the correct expressions (87b) for the self-energies were used. They have also applied the multipole expansion for the long-range part of the Coulomb interaction, but did not use di!erent cut-o! radii for di!erent integrals at the part of the chain where the integrals were taken into account explicitly. They have applied 21 neighbours and 21 k-points in the HF and QP calculation and have obtained a gap value of 4.03 eV. The reason for this discrepancy is unclear. If one would like to obtain an etalon for the calculation of the gap one should have (1) a very good basis, (2) should include at least MP2#MP3#MP4 for the self-energies appearing in the inverse Dyson equation, using expression corresponding to Eq. (87b), (3) further di!erent cut-o! radii for di!erent types of integrals should be applied, (4) for the long-range part of the Coulomb interaction a multipole expansion should be used, (5) the number of kpoints and number of neighbours should be large enough to reach saturation, (6) in addition at the last stage of the calculation a full geometry optimization should be performed, (7) "nally there should be at least an estimation performed for the phonon polaron e!ect on the gap. In the author's opinion the calculations [78,92,94] are the most sophisticated ones, but none of them ful"lls all the above-mentioned criteria. Of course the execution of such a calculation even for PA requires much CPU time, but at least in this prominent case it would be worthwhile to do it. If one applies Koopmans' theorem again the theoretical (QP) ionization potential value of 5.36 eV [94] is in rather good agreement with the experimental one (4.6 eV [91]). Unfortunately, there is no experimental value known for the electron-a$nity of PA.

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3.3.2.2. Summary of the results of QP band structure calculations for other 1D periodic polymers. There are QP band structure calculations for polyethylene (PE) and polytetra#uoroethylene (Te#on) [95]. In the PE case a 6-31GHH [33] and Clementi's doublef basis, respectively, was applied. In the "rst case for the self-energies the expression (87a) (no relaxation was used), while in the second case formula (87b) was applied putting it in the inverse Dyson equation and performing the iterations. In both cases a geometry optimization was executed. In the "rst case a gap value of 10.3 eV was obtained [33]. In the second case [95] the gap (in the "fth neighbours' approximation) due to the poorer basis set increased to 11.6 eV. The value of e (QP) is 3.4 eV while e (Q) lies !  4  at !8.2 eV (which agrees quite well with the experimental values (7.6}8.8 eV [35]). From the large gap and high lying e (QP) and low lying e (QP) it is clear that there is no hope of doping !  4  PE and making it in this way better conducting though the widths of the valence and conduction bands, respectively, is quite large (6.5 and 5.1 eV, respectively). One has to admit that "rst of all due to the not good enough basis sets and probably also due to not enough neighbours the theoretical gap value in the second calculation [95] (11.6 eV) is still substantially larger than the experimental value of 8.8 eV [35]. In the case of Te#on the QP band structure [95] is similar to that of PE. However, due to the four negative F-atoms, the whole band structure is strongly shifted downwards. The lower limit of the conduction band e (QP)"!5.5 eV (at room temperature stable conformation of it). !  This means that with the help of appropriate chemical donor doping (or with the aid of electrochemical doping) one could bring electrons into its conduction band and in this way increase its conductivity. In the case of polyparaphenylene (PPP) it was also a QP band structure calculation performed [52] using Clementi's doublef basis supplemented by a set of d functions on the C-atoms. The geometry of PPP was the same as in the HF case [53,54] taken from X-ray di!raction data (the middle part of the terphenyl molecule [53] and from the X-ray di!raction investigation of PPP [54] for the determination of the torsion angle of 22.73). The computation was performed again in the second neighbours' interactions approximation, but using instead of 25 (HF) only 7k-points in the QP calculation. We have obtained for e (QP)"1.26 eV and for e (QP)"!7.10 eV !  4  resulting in a gap of 4.8 eV. The theoretical ionization potential e (QP)"!I and the gap are 4  4 still too large as compared to the experimental values (5.7 eV [55] and 2.7 eV [96], respectively). In this connection it should be mentioned that besides the shortcomings of the theoretical calculation the discrepancy of &1.4 eV in the ionization potential is in the range of the usual error if one applies Koopmans' theorem for the determination of the ionization potential. The discrepancy of over 2 eV in the gap is, at least partially, due to the fact that the experimental value was estimated on the basis of the UV absorption band. If there is an exciton band in the gap (which is most probably the case) the UV spectrum gives a substantially smaller value for the gap, as it is in reality. Only the measurement of the inverse photoelectron spectrum of PPP (which is not available could give a realistic value for the gap). In the case of a cytosine (C) stack (using the experimental geometry of DNA B [62], again second neighbours' interactions, 25k points at the HF and 7 at the QP level) we have obtained using  The measurement of the gap (which could be done if a small number of extra electrons would be injected into the conduction band and then ionizing some of them out of this band) and subtracting the experimental ionization potential from the electron a$nity.

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Clementi's doublef basis from the inverse Dyson equation with an MP2 self-energy (Eq. (87b)) for e (QP)"2.81 eV, for e (QP)"!7.20 and a gap of 8.66 eV [57]. These values are some!  4  what (especially the gap) di!erent from the HF ones, but obviously "rst of all due to the rather poor basis they are still not satisfactory. On the other hand if one introduces additionally two sets of p-functions on the `phantoma molecules at the middle of the stacking distance (see Section 2.5.5), the results become much better: e (QP)"1.13 eV, e (QP)"!5.47 eV and correspond!  4  ingly the gap is 6.60 eV. This value is rather close to the one of &5.5 eV which can be estimated on the basis of the UV spectrum of [68] (for which the exciton spectrum was calculated; see Section 5). The QP band structure calculations have been extended also to the two simplest homopolypeptides in their a-helical con"guration [97]: polyglycine and polyalanine using a 6-31GH basis and "fth neighbours' interactions [65]. In this calculation for the self-energies the expression (87a) (no relaxation) was applied for the self-energies. These calculations resulted in a gap of 9.81 eV for polyglycine and 9.52 eV for polyalanine (the corresponding HF values are 13.12 and 12.78 eV, respectively, which means a decrease of &3 eV): These theoretical gap values are smaller than the estimated experimental values by &1.4 eV [68]. One should "nally mention that several 2D QP band structure calculations have been performed using again the inverse Dyson equation in its diagonal approximation with the 2D analogue of the MP2 self-energy (87b) (relaxation included). Since in the 2D case the localization of the Wannier functions remained poor after all attempts, the two-electron integrals were calculated by going directly from AO's to CO's. As an example we should mention here the cases of a 2D polyglycine network (polyglycine chains with H-bonds between them) [98] and of boron-nitride [99]. For the details we refer to the original papers.

4. The treatment of aperiodicity in quasi 1D polymers Many polymers, like the biopolymers DNA, RNA and proteins, as well as doped (conducting) polymers are aperiodic. Partially or completely aperiodic are also di!erent copolymers which are frequently used in the plastics industry. The standard method of solid state physics to treat disordered (aperiodic) systems is the coherent potential approximation (CPA). Since this method works "rst of all for 3D systems and one is interested in the framework of this paper primarily in 1D systems, we do not discuss this method here, but give some key references: [100}102]. At the same time only for quasi 1D systems there is a method [the negative factor counting (NFC) method] which is de"nitely superior to the CPA method for aperiodic chains. The CPA method was applied to a mixed (SN) (SNH) chain (for details see [103]). V V 4.1. The negative factor counting (NFC) method in its ab initio matrix-block form The negative factor method which is based on Dean's negative eigenvalue theorem [104], provides the distribution of eigenvalues (density of states; DOS). The method was originally applied to vibrational spectra of disordered 3D systems [104]. For one-electron energies it was "rst applied by Pumpernick et al. [105] for a bundle of short disordered alloy chains and by Seel [105] to long disordered polypeptide chains [106] in both cases in the tight binding approximation (giving the DODS coming from one-band of the components).

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209

One can formulate, however, the method also in an ab initio form [107,108] having an arbitrary number of orbitals per site (in this way one obtains the total DOS of the disordered chain). One can write for an arbitrary long disordered chain with an arbitrary number of di!erent units in the "rst neighbours' interactions approximation a triblock-diagonal secular determinant "M(j)"""F!jS"



F !jS   F !jS   0

F !jS   F !jS   F !jS  

"

F !jS   F !jS  

F !jS  

\



"0 .

\ F ,,

!jS ,,

(97)

Here the F 's are the diagonal and the F 's the o!-diagonal blocks of the Fock matrix, respectively GG GH (belonging to the ith unit and to its interaction with the jth unit, further N is the number of units in the chain). Since we are dealing with di!erent type of units, generally F  OF  . One could solve the GG GG eigenvalue problem corresponding to Eq. (97) directly, but one would need much CPU time. For a protein for instance consisting of 200 units (with 20 di!erent-type of amino acid residues) and with only 50 basis functions per unit (in the case of a minimal basis) the dimension of the matrix is 10. There may be, however, many such problems, the systems may be essentially larger and one should use also a better basis set (which is important to treat also the correlation problem of the disordered chain; see below). Therefore the calculations could become quickly unmanageable. To write down the whole F and S matrices for a long chain can become in itself a formidable problem. Therefore we have introduced the approximation that we construct them from overlapping dimers [108]. For instance in the case of an ABBCDADB: 2 etc. chain one calculates the (SCF) Fock and overlap matrices only of the dimers AB, BB, BC, CD, DA, AD, DB, etc. In this procedure we have taken for the diagonal Fock matrix blocks their algebraic mean value (for instance F "[F#F], where F is the SCF Fock matrix block of the dimer BC and !! !! !!  !! F that of the dimer CD, respectively. In another paper [109] it was also shown for a shorter !! chain with smaller units that the SCF Fock matrix generated from a direct calculation and the one constructed from overlapping dimer calculations are hardly di!erent. Finally one should point out that one can apply also further neighbours' interactions in Eq. (97) (in this case the matrices F and GH S will not be quadratic matrices, but rectangular ones with n rows (as in F but with m"m #m GH GG   columns, where m is the number of columns in the "rst and m in the second neighbour).  

 For this one has to diagonalize the matrices




F  F 

F  F



etc.

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We can rewrite the matrix M(j) in the form M(j)"F!jS"S[S\FS\!j1]S"S[FI !j1]S

(98a)

or in determinantal form det M(j)"det Sdet(FI !j1)det S "det S det S det(FI !j1)"det S det(FI !j1) .

(98b)

The determinant "F!jS " can be brought to a diblockdiagonal form with the help of a successive Gaussian elimination [110]. In this way one obtains for the diagonal blocks (as it could be shown easily) the recursion formula U (j)"F !jS !(F !jS )U\ (F !jS ), G G>G GG GG> GG> G\ GG> GG> U (j)"F !jS .    Therefore we can write











(99)



L L JG , , “ (j !j) " “ “ u (j) n" l . (100) det(F!jS)" “ s G H GI G G H G I G Here the s 's are the eigenvalues of S and the j 's are the roots of the generalized eigenvalue G H equation Fc "j Sc . H H H Further N is again the number of units, l the dimension (number of basis functions) of the ith unit G and u (j) denotes the kth eigenvalue of the matrix block U (j). The latter matrix block can be easily GI G diagonalized for a given trial value of j. From Eq. (100) it follows that for a given j the number of negative u (j) factors has to be GI equal to the number of eigenvalues j which are smaller than the chosen j. Taking another trial H j value one can determine the number of negative u (j)'s again. The di!erence of the negative GI u factors gives immediately the number of eigenvalues j which lie between j and j: GI H j'j 'j . (101) H Taking a "ne grid in the trial j's and scanning the whole eigenvalue spectrum one obtains immediately the total DOS. To test this method we have calculated the DOS of the valence and conduction bands of the periodic homopolypeptide polyglycine. The band limits obtained form a direct crystal orbital calculation of this system agree very well with the limits of the DOS histograms for chains between 1000 and 2000 units (histogram interval 0.002 a.u.) for the corresponding regions [111]. This example shows the accuracy of the matrix block NFC method. Unfortunately for topological regions it cannot be applied for 2D- and 3D aperiodic systems (in the 2D case we have 5- and in the 3D case 7 block-diagonal matrices). Using the Givens}Householder method for the diagonalization of matrices of order 100 and 200 (describing a 2D disordered system with 10;10 and 20;20 sites, respectively), we have found that to bring the occurring "ve diagonal matrices into a tridiagonal form it took about one-third of the complete diagonalization time [112]. Finally one should

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point out that the NFC method recently was further extended [113] to the case of disordered chains with cross links (like S}S bonds between di!erent parts of a folded protein chain). It should be mentioned further that having the DOS of any disordered chain one can easily generate to any given energy level the corresponding eigenvector using the inverse iteration technique. Since this is again a standard method written down in many books [114] we do not describe it here again. One should mention, however, by analysing the eigenvectors calculated in this way one can investigate the di!erent one-electron energy levels for Anderson localization and "nd out the mobility edges in the di!erent energy level ensembles (quasi bands) [115]. Having the energy levels and the wave functions one can calculate the primary jump rates (hopping frequencies) due to the interaction of electrons with acoustic phonons in a disordered chain. As a "rst approximation one can use the very simple formula given by Mott and Davis [115,116] for the hopping probability from site A and level i to site B and level j:






 (102) cc 1s"s 2 exp(*E "k ¹) . = "l GH GP HQ P Q G H  PZ QZ In the expression of = (which is a generalization of the original expression given in Refs.  [115,116] to the case if one has an arbitrary number of orbitals at sites A and B, respectively), l is the acoustic phonon frequency, the c's are the LCAO coe$cients of the localized orbitals   W and W  , respectively, and the (s"s )'s are the overlap integrals between atomic orbitals G H P Q (AO's) centred on site A and B, respectively. Finally *E in the Boltzmann factor is the energy GH di!erence between e and e . G H One should point out that instead of Eq. (102) it would be much more correct to take into account the interaction of the localized electrons with the also localized phonons. In this way one would obtain an expression [119]  = "1W "HK } "u2 cc  ) 1s"s "2 GP HQ P Q G H G   G PZ QZ e\ #GHI 2 if '*E '0 GH ;1u "HK  } "W  2 ) . (103) H   H 1 if (*E (0 GH Here the operators HK } are the electron}phonons interaction operators and u and G   u  are the wave functions of the localized phonons at the two sites. It should be mentioned, H however, that until now no hopping conductivity calculations on any disordered chain has been performed using the more complicated expression (103).








4.2. The inclusion of correlation in the calculation of density of states of disordered chains We can easily substitute instead of the matrix blocks F and F into the determinant (97) also GG GH such matrix blocks of dimers which take into account also the major part of the valence correlation. For this we can write for a dimer the more complicated expression given by Liegener [117] and by Liegener et al. [118]. Modifying their expression we can write






FI FI GH "F FI " GG #S U R U>S .         FI  FI GH HH

(104)

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Here F and S have the same de"nition as before, the unitary matrix U is formed from the     solutions u of the generalized eigenvalue equation G F u "e S u   G G   G and R is the self-energy matrix. We can apply for the diagonal elements of the self-energy matrix. R(u ), in the same way as in the G case of periodic polymers, in the MP2 approximation < (2


 

(106)



1 u (1)u (2) (107a) < " u (1)u (2) G H r ? @ GH?@  O u (r)" c s (r ) . (107b) G GP P  P In Eq. (106) the summation runs over all the g basis functions of the dimer. One should observe that Eqs. (105)}(107b) are the molecular counterparts of Eqs. (87a), (87b), (87c), (87d), (94) and (95), only one has to write the MO indices i etc. instead of the combined indices I(i, k ) of the CO's. G Having calculated [R(u )] in an iterative way with the help of the inverse Dyson Eq. (105), G GG one can substitute the values obtained as the diagonal elements of R into Eq. (104) (neglecting its o!-diagonal elements). In this way one can construct FI . Having these block-matrices for all   overlapping dimers, they can be substituted into Eq. (97) and the NFC procedure can be performed again with the help of Eqs. (99) and (100). In this way we can obtain correlation corrected DOS (based on the ab initio form of the electronic polaron model in the MP2 approximation) also for aperiodic polymers. 4.3. Selected examples for the calculation of the electronic structure of aperiodic chains both at the Hartree}Fock and at the correlation corrected level The "rst application of the NFC method in its matrix block form was performed on an aperiodic 4-component polypeptide chain. The chain contained the amino acid residues glycine (gly.), serine (ser.), cysteine (cys.) and asparagine (asn.) (Fig. 5) in a 1: 1: 1: 1 (25%) composition, had a length of

 If we multiply Eq. (104) from the left by U> and from the right with U, take into account the normalization condition U>SU"1, the relation U>F U"e&$ and apply again the diagonal approximation, we recover the inverse Dyson     equation in its diagonal approximation u "e&$# . (105) G G GG  In the case of close-shell dimers we can always "nd a dominant pole strength P (see Eq. (96)), which is larger than G 0.6. In this way we can always "nd from the solutions of the non-linear Eq. (105) the physically relevant ones for all the u 's of a given dimer, and we apply the same procedure to every dimer which occurs in the non-periodic chain. G

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213

Fig. 5. The chemical formulae of (a) serine, (b) cysteine, and (c) asparagine units in a polypeptide chain.

300 units and its random sequence has been generated by a Monte Carlo program. The results obtained for the DOS show a very strong broadening of both the conduction and valence band regions of the aperiodic chain (in the valence band region there are allowed states between !15 and !11 eV and in the conduction band region there are allowed energy intervals between 3 and 4, and between 5 and 7.5 eV). On the other hand if we arrange the four components in a periodic way (gly. ser. cys. asn. gly. ser. cys. asn., etc.) one obtains extremely narrow, d-function type peaks in the DOS histogram. Fig. 6 shows the DOS of the valence bands region: (a) of the above described aperiodic 4-components polypeptide and (b) its DOS spectrum if the 4-components are arranged in a periodic way. The reason of the extreme large broadenings of the allowed states in the histogram is that while in 4-component ABCD ABCD 2 chain the number of "rst neighbours' interactions say for C is only two, BCD, in the random sequence there are 16 di!erent such interactions A B



A B

C C C D

.

D

One has found similar DOS broadenings of the valence and conduction bands region of 6- [121] and 8-component [122] aperiodic polypeptide chains and in collagen [123]. Finally the same kind of broadenings occurred in a somewhat lesser amount due to the weaker coupling between the units in disordered (4-component) DNA stacks [124]. If the four nucleotide bases were arranged in a periodic way, say AGCT AGCT 2 the DOS histograms have shown again only extremely narrow d-function type peaks both in the valence and conduction band regions [124]. In the case of the 4-component (gly., ser., cys., asn.) disordered polypeptide chain we have determined for the 25 "rst un"lled levels also the corresponding eigenvectors using the inverse  In a nucleoprotein (DNA and protein complex, the protein part, as for instance nucleohistone, contains positively charged groups. Therefore one can expect a strong electron transfer between the negatively charged PO\ groups of DNA  and the positively charged groups of the protein which will populate the low lying empty levels of it. This was the reason to perform the calculations on the "rst 25 lowest lying empty levels of the disordered polypeptide chain.

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Fig. 6. The DOS distribution (a) of the valence band region of a 300 units long 1 : 1 : 1 : 1 gly., ser., cys. asn. polypeptide chain having a random sequence. The histogram interval is 0.05 eV and N is the number of states in this interval. (b) the DODS curve for the same chain but with a periodic sequence (gly. ser. cys. asn) . V

iteration method [120]. We have found that the wave functions corresponding to these levels are always very well localized on a single amino acid residue which are not following each other in the chain according to the increasing energy levels (as it is also not expected on the basis of the theory of disordered systems). The detailed results of this Anderson localization of the wave function can be found in [120]. As the next step we have applied Eq. (102) for the calculation of the hopping frequencies (primary jump rates) in the same system. For the frequency of the acoustic phonons (vibrations changing the distances and relative orientations of the side chains) occurring in this equation we have chosen the value of 10 s\ which is characteristic for these kind of phonons [125]. The energy levels and wave functions appearing in Eq. (102) were taken from our previous calculation of the 4-component polypeptide [120]. We have found that in the case of hoppings between "rst neighbours the = values are (at G H body temperature, ¹"310 K), between 5;10 and 7;10 s\ (the hopping frequencies are not normalized to one) and between second neighbours they decrease to 10 s\ [120]. If the hopping occurs between di!erent levels at the same site (amino acid residue) the primary jump rates increase to 7}9;10 s\. It should be pointed out, however, that from the point of view of hopping conductivities along the main chains of proteins hopping frequencies at the same site are irrelevant.

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215

One should mention further that the described calculations have used a minimal basis for the electronic wave function. If one would apply a better quality basis and would take into account also correlation e!ects the average level spacing and with it the *E values would decrease and GH therefore both the Boltzmann factors and with it the hopping frequencies would increase (according to our estimate by about an order of magnitude). On the other hand since no calculations are available for the case, if one would have used the more correct Eq. (103) instead of Eq. (102), one does not know what e!ect this change would have on the hopping frequencies. 4.4. Calculation of the hopping conductivities of proteins and nucleotide base stacks on the basis of random walk theory Szent-GyoK rgyi [126] had already assumed in 1941 that there is electronic conduction along the main chain of proteins. He came to this idea on the basis of his studies on oxygen metabolism in the living cell (Szent-GyoK rgyi}Krebs cycle [127]) in which electrons are transferred through a series of proteins to the "nal acceptors, the O molecules.  Szent-GyoK rgy's idea has been mostly rejected by the scienti"c community because (1) of the large gap (7}8 eV) and (2) because polypeptide chains form disordered systems. It was rather assumed that electron transfer in proteins happens via hopping and tunnelling between di!erent segments of a folded protein or with the help of the motion of electrons coupled to protons and ions in hydrated proteins. Though measurements performed on proteins and DNA have shown a weak semiconductivity [128}136] also in the dark, which was probably due to electron transport, it was not clear whether the conductivity originated from the protein chains themselves or are due to impurities. There is a larger number of theoretical calculations on the conductivity in proteins and DNA. For the references one should consult [137] in which the "rst hopping conductivity calculation on a native protein (pig insulin) is described. Nowadays there exists the theoretical and computational background to calculate such a conductivity in proteins or in an aperiodic DNA stack which makes it possible to overcome the di$culty (2) mentioned above. Concerning the counterargument (1) (too large gaps) one should take into account that in vivo there is always a possibility of charge transfer between di!erent macromolecules or between macromolecules and molecules. Therefore one does not have to excite an electron from the valence band region to the conduction band region to obtain free charge carriers. In the previous points we have seen that one can calculate relatively easily the DOS, the Anderson localization and the hopping frequencies between di!erent amino acid residues in a disordered protein chain. Having the latter quantities one can use them as input into the random walk theory of Lax and coworkers [138] (somewhat generalized to have an arbitrary number of orbitals on each site [137]). There is a more general formulation of the random walk theory [139] than the one due to Lax and coworkers [138], but this could be applied, only if such special restrictions would be introduced in the theory which are not compatible with the ab inito formalism applied for these complicated biopolymers. The random walk theory of Lax and coworkers uses as the starting point the master equation RP[(n, j), t"(n , i ), 0]   "!C P[(n, j), t"(n , i ), 0]# h P[(n, j), t"(n , i ), 0] L H   LYHYLH   Rt LYHY$LH (108)

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where C " h . (109) LH LHLYHY LYHY$LH Here P[(n, j), t"(n , i ), 0] is the probability to "nd a charge carrier at energy level j of site n at time t,   if it was at level i of site n at time 0 and the h are the hopping frequencies de"ned before.   LHLYHY Eq. (108) ensures the conservation of charges. The AC conductivity p(u) is given by the generalized Einstein relation n e p(u)" J D(u) . k ¹

(110)

Here n is the number of charge carriers in the volume of the protein molecule. The di!usion J constant of the electrons (holes) D(u) can be written with the help of the Laplace transform of PI [(n, j), t"(n , i ), 0],    (111) PI "[(n, j), iu"(n , i )]" e\ SRP[(n, j), t"(n , i ), 0] ,      u (112) D(u)"! [(n, j)!(n , i )]PI [(n, j), iu"(n , i )];f [E  ] .     L G 2d LG L H Here d is the dimensionality of the systems and the functions f [E ] are the equilibrium distribuLH tion functions of the localized carriers. They have to obey the equation



h

f "h f . LYHYL\H LYHY LHLYHY LH Finally E is the jth energy level of the charge carrier at site n. LH Further it can be shown [138] that the Laplace transform PI [ ] ful"lls the relation

(113)

PI [(n, j), iu"(n , i )]! h ;PI [(n, j), iu"(n , i )]"d  .   LYHYLH   LHL G LYHYLH This leads to a formal solution for PI [ ] [138]

(114)

PI [(n, j), iu"(n , i )]"[(iu)1!H ]\   .   LHL G Here the elements of the hopping matrix H are given as

(115)

(iu#C

LH

H "!C , LHLH LH H "h (n, jOn, j) . LHLYHY LYHYLH Eq. (116b) shows that the matrix H is non-symmetric.

(116a) (116b)

 In the actual calculations since our wave functions are localized on an amino acid residue (see above) which contains several atoms, one had to de"ne a centre of the site on the basis of its charge distribution and assume that the hopping takes place between these di!erent localization centres. To simplify the notation we did not introduce them in the de"nitions of the hopping frequencies and the functions P or PI and f, respectively.

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217

The functions f can be obtained as the components of the normalized right eigenvector of LG H belonging to the non-degenerate eigenvalue O [138]. This eigenvector can be written as

  f (E )  $ $

.

(117)

f (E ) K Here E ,2, E are those energy levels E which were applied in the construction of the matrix  L LH H (their indices are reordered as 1,2, m). To perform an actual calculation for a disordered chain one has to know of course the sequence and the conformation of the whole macromolecule. One can take d"1 and can calculate the volume of the macromolecule (protein). From the latter the number density n can be computed. J After this preparation one can form from Eqs. (116a) and (116b) the hopping matrix H (if the hopping frequencies are known). With the help of H using Eq. (115) one can determine the Laplace transforms PI . Finally solving the eigenvalue equation of H one obtains from its right eigenvector belonging to the zero eigenvalue the equilibrium distribution functions f [E  ]. Substituting all L G these quantities into Eq. (112) the di!usion constant (112) can be computed. Finally putting this into the Einstein relation (110) one obtains the hopping conductivity (110). Schematically:

As a "rst application of the above-described random walk theory, after determining the DOS and hopping frequencies [140], the frequency-dependent complex conductivity "p(u)" of native pig insulin (in its active form) has been calculated [137]. This hormone has 51 amino acid residues and 3 disulphur bridges. Its sequence and its detailed stereostructure is known. The obtained p(u) (in )\ cm\) versus u(s\) curve show the following features: the real part of p (u) and its absolute value "p(u)" show a very similar u-dependence (see Figs. 4 and 6 of [141]). 0 They have both their saturation value of "p(u)""10\ )\ cm\ at u+10 s\ (it should be mentioned that conductivity measurements still can be performed at such large frequencies with the help of microwave techniques). The reason for this is that the imaginary part of p, p (u) goes to ' zero as uPR [141]. lim p (u)P0 ' S which can be easily shown if one takes the reciprocal of the impedance with zero inductivity, ¸"0 (¸"0 in disordered systems). Since there are no AC conductivity measurements on proteins one can compare these results only with the hopping conductivities of amorphous glasses like the chalcenogides (which can be found for instance in Fig 7.15 of [115]) in the frequency range 10}10 s\. One sees that "p(u)" in

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this frequency range lies between those of Te AsSi and As Se and has the same order of    magnitude in most of the frequency ranges as Te As S Ge . (The same is true for p(u) of hen     egg white lysozyme). Since these calculations have applied a minimal basis, one can raise the question as to what would happen if one would apply a better basis set and also take into account correlation e!ects. Since p(u) is strongly dependent on the hopping frequencies and their magnitude is "rst only in#uenced by the Boltzman factor e\ #GHI 2, we can conclude that better quality calculations would provide still larger p(u) values because in the case of such calculations the level distances } according to our experience } decrease (in the case of an aperiodic nucleotide base stack such calculations, using the formalism described in Section 4.2 have been performed [60]). This means that our calculated p(u) curves give lower bounds to the real conductivity values at a given frequency (according to our estimates they would be 1}1.5 order of magnitude larger than the values obtained in these "rst calculations). We have performed the same kind of random walk calculations for the inactive form of pig insulin [142]. We have found that in this case the saturation value of "p(u)" is in the high frequency range, from u"10 s\ (which corresponds to the time period of an elementary step in biochemical reactions) about 2 orders of magnitude smaller (10\ )\ cm\). In this case the sequence remains the same but the conformation of the protein is di!erent. We have calculated as the next step the hopping conductivity of the enzyme hen egg white lysozyme [141] (129 amino acid residues, and again 3 S}S bridges). In this case in the active form of the enzyme "p(u)" reached its saturation value 10\ )\ cm\ at &10 s\ [141]. p (u)P0  again at high frequencies and therefore p (u)+"p(u)" at higher frequencies. The inactive form of 0 lysozyme shows a saturation already at 10 s\ with a "p(u)" value of 10\ )\ cm\ [143]. This again proves that the conformation and not the sequence of the amino acid residues is the decisive factor determining "p(u)". This statement is especially relevant in the case of the active sites of enzymes which usually change their conformation (especially in the presence of reactants) quite easily [143]. These considerations point in the direction that in the case of enzymatic reactions charge transport via hopping conductivity is } according to all probability } quite important. In another investigation the p(u) of subtilisin both in its native and inhibited form [143] was studied. Native subtilisin shows a "p(u)" saturation value of 10\ )\ cm\ at u"10 s\, while in its inhibited form (which has obviously another conformation) has a larger "p(u )""10\ )\ cm\ value again at u"10 s\.  From all these results we can conclude that proteins have a signi"cant electronic hopping conductivity (in a DNA-protein complex the negatively charged DNA molecules give over electrons to the proteins which possess some positively charged side chains), if a segment of a folded protein closes a small angle with the local e!ective electric "eld. Of course to "nd out the rate of electron transfer through a protein one had to take into account also the other electron transport channels. (1) electronic motion coupled to the motion to protons and ions in hydrated proteins, (2) hopping between di!erent segments of folded proteins, (3) multi-channel tunnelling between more distant segments. Since all these mechanisms occur in many channels, to obtain the overall electron

 In these calculations the "rst 50 un"lled levels have been taken into account.

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219

transfer through a protein an appropriately modi"ed form of the Feynman path integral formalism could be applied. Since, especially in the case of tunnelling, there are very many channels one could select those paths which have about the same probability and one could bundle them together to an e!ective path multiplying it with an appropriate weight factor (for this one has to know the potential hypersurface and the number of channels which belong to the e!ective path). Such a theory seems rather complicated, but most probably no simple formalism exists which would provide electron (or hole) transfer rates through di!erent proteins and between them. Finally we have applied the random walk theory also for aperiodic nucleotide base stacks [60] both for single and double stranded DNA. Since in the case of DNA the charge carriers are most probably holes (which was con"rmed also by the chemical experiments of the Group of Barton [61]) we have used the 200 highest "lled levels in the double stack and the highest 100 "lled ones for the single stacks for our calculations. As sequences we have used (1) a part of an oncogene [144] or (2) generated a random sequence with the help of a Monte Carlo program. For the geometry we have applied the re"ned data of Dickerson's Group on DNA B [62]. The problem of charge transport in DNA is on the one hand simpler than in proteins, because one has instead of 20 only 4 components. On the other hand in the case of DNA the role of the sugar}phosphate groups and of the water-ion surroundings has to be taken into account. Since the phosphate groups have !1.2e charge [145], the sugars �.6e and the nucleotide bases !0.2e [146] the e!ects of these di!erent charges (including those of the counterions) mostly screen out each other. Further HF calculations of periodic polynucleotides } as mentioned before [58] } have shown that the band structures of these systems are in a quite good approximation equal to the superposition of the band structures of the base stacks and that of the sugar}phosphate chain. For these reasons one would not expect a larger e!ect of the sugar}phosphate chain on the hopping conductivity through the aperiodic base stacks. An early HF calculation on the e!ect of water clusters [146] on the band structure of a cytosine (C) base stack [147] has shown that the water environment has hardly in#uenced the band structure of the C stack. Therefore one would not expect that the results of a hopping conductivity calculation on an aperiodic base stack would be very di!erent from a full DNA calculation (which nowadays still hardly could be performed). The results of the base stack calculations [148] show that "p(u)" is still larger, than in the case of proteins. For a double stack of 100 base pairs (200 HOMO levels) or for a single stack of 200 bases (again 200 HOMO levels) "p(u)" is 5;10\ )\ cm\ at u"5;10 s\. The detailed analysis of the results [148] shows that the "p(u)" values in these cases are basically controlled by the Boltzmann factors. This can be seen at a single base stack with 100 levels (larger level spacings) in which case "p(u)" is only 10\ )\ cm\ at its saturation at u"5;10 s\. It should be mentioned that there is an AC measurement by smaller frequencies for DNA [130}132]. Since, however, the samples were not puri"ed and characterized well enough, we prefer not to compare our results with the measured ones (at some u values the agreement is quite good, for other ones, however, rather poor.) Finally it should be mentioned that doublef#MP2 calculations are in progress for all the 16 DNA base dimers from which FI (see Section 4.2) can be constructed for every aperiodic single DNA stack. We expect that the results of these calculations will give substantially increased "p(u)" values, but it is yet premature to comment on the preliminary results [148].

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5. Calculation of selected properties of polymers 5.1. Application of the intermediate exciton theory to the electronic spectra of diwerent polymers and to the high T superconductor >Ba Cu O     5.1.1. The intermediate exciton theory in its ab initio form In a quasi-1D in"nite chain the HF singlet excitation energy *E "e&$!e&$!I #2K (118) G? ? G G? G? reduces to the di!erence of the one-electron energies, because the single Coulomb and exchange integrals I and K disappear in an in"nite system as [149] G? G? ln N P0 (119) lim I " lim K " lim G? G? N , , , where N is again the number of unit cells. Eq. (118) with Eq. (119) would give unrealistically large excitation energies. In in"nite or large systems possessing a band structure another physical phenomenon has to be taken into account, namely the attractive interaction between the excited electron and the remaining positive hole in the "lled band from which the excitation occurred. This electron}hole pair is called an exciton. Generally the excited electron is not localized to the same cell from which its excitation occurred, but it is spread out in both directions to 4}5 neighbouring cells (in the case of a chain). If this happens one speaks about an intermediate exciton [150]. If the distance between the electron and the hole is much larger one has a Wannier exciton [151] and if the electron}hole pair is localized in the same cell one has a Fraenkel exciton [151]. The latter happens if the units are weakly coupled, while a Wannier exciton occurs if the dielectric screening is large (the binding energy of the exciton is small). The mathematical formulation of the intermediate exciton theory will be only sketched here and we refer for details to Refs. [152,153]: "rst one can create an electron}hole pair from a completely "lled band (120) "U2"2dK dK 2dK ,"02 J J J with a separation R 1 "W 2"aL > dK >"U) . (121) J>QJ J>Q J Here aL > creates an electron in the cell l#s in the conduction band c (aL > ,aL > ) and a hole in J>Q J>Q J>QA the valence band in the cell l [the operator dK annihilates a hole (creates an electron) in the valence J band in the cell l, dK ,aL > ]. As next step one can construct stationary states with quasi-momentum J JT K in a Bloch-function form "W K2"N\ exp [iKR ]aL > dK >"U2 . Q J J>Q J J As "nal step one allows also the distance R to change which gives Q "WK2" X K"W K2 . Q Q Q

(122)

(123)

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221

The wave functions "W 2 (and the energy dispersion of the exciton band) can be obtained from Q) the SchroK dinger equation HK WK"EKWK

(124)

where HK "E #HK #HK #H , &$ C F CF E "1U"HK "U2 . &$

(125) (126a)

Here H " "aL > aL 1w (x!l )"FK "w (x!l )2 , C J J A  A  JJ

(126b)

HK " dK > dK 1w (x!l )"FK "w (x!l )2 , F J J T  T  JJ

(126c)

(127) HK } " aL > aL dK > dK (1wJwJ"wJwJ2!exch) . A T T A CF J J J J     JJJJ In these expressions (in which a two-particle and two-hole operators as well as vertex corrections were neglected) x stands for r, p, p being the spin variable. Eqs. (124)}(126a), (126b), (126c) and (127) were derived from the second quantized form of the total Hamiltonian of the system, where the "eld operators WK >(x) and WK (x) were expanded with the help of the Wannier functions wJQ, etc. ! (for details see Refs. [152, 153]). This is advantageous in the quasi-1D case, when the Wannier functions can be rather well localized. If one puts E ,0 as the zero point of the energy scale, the resolvent method can be applied to &$ Eq. (124) which gives WK"[EK!(HK #HK )\]HK W . C F CF ) Further after some manipulation [152,153] one obtains the system of equations

(128)

X K" 1W K"[E !(HK #HK )\]"W K21W K"HK "W K2X K (129) P P ) C F Q Q CF R R Q R for the determination of the unknown coe$cients X K. Using Eq. (127) one obtains for the matrix P elements 1W K"HK "W K2"! exp[!iuKR ][1wQ>Sw"wS ) wR 2!exch],<TA(R , R , K ) . S A T T A Q R Q CF R S

(130)

Substituting this and the matrix elements of HK and HK (expressed again with the help of Bloch C F functions [152,153] into Eq. (129) one obtains "nally the system of equations XTA " GTA(R , R , E )<TA(R , R , K) . PK P Q ) Q R Q R

(131)

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Here the Green's matrix element GTA is exp[ik(R !R )] P Q . GTA(R , R , EK)"N\ P Q k EK!(e k!e k K) A T \ The exciton band energies EK can be determined from the condition D""GV!1""0 .

(132)

(133)

. Having them one can obtain from Eq. (131) the coe$cients XTA PK One has to point out, however, that according to our experience one obtains a reasonable exciton spectrum from this formalism only, if one substitutes in the denominator of the Green's matrix elements as one-electron energies e and e k K not HF energies, but correlation corrected AI T \ QP energies. 5.1.2. The excitonic spectra of some polymers and of YBa Cu O    Using the above-described formalism the excitonic spectra of di!erent polymers were calculated. In Table 5 the results for singlet exciton bands of these polymers are shown. In this table we show besides the QP gap the edges of the "rst singlet exciton bands at (K"0 and K"p/a, respectively). The exciton bands have widths between 0.3 and 3.5 eV. They (with the exception of the cytosine stack, where only a too poor doublef basis could be applied) show rather good agreement with the electronic spectra of these systems. The used basis sets and the references to the original theoretical papers and to those containing the experimental spectra are given below the table. One should notice the rather large excitonic binding energies (the di!erences of the QP gaps and the lower edges of the "rst singlet exciton bands (at K"0)). They again show that it is incorrect to equal to the energy of the "rst u.v. peak with the fundamental gap as it is done frequently in the literature. For data for the second singlet exciton band and the "rst triplet band (which are not in all cases available), we refer to the original papers. One should also observe that it is necessary to take into account 5}6 neighbours (in both directions from the reference cell) and at least 2 neighbours in the cases of the cytosine stack to obtain a saturated exciton spectrum. In the case of a Fraenkel exciton (n"0) one obtains only about 30% of the binding energy of the exciton in the cases of the two polydiacetylenes, only 20% for polyethylene, 80% for the C stack, and 70}80% in the cases of polyglycine and polyalanine. Finally one should mention that in the case of polydiacetylenes PTS has the chemical formula ("C!C,C!C") V " " R"!CH !SO !C H !CH      R R

(134)

while TCDU stands for (!C"C"C"C!) " " V R"!(CH ) OCONHC H    R R

(135)

In the actual calculations [154] these side chains were substituted by an H-atom.  To obtain singlet bands one has to put a factor !2 before the exchange term in Eq. (130) and one should use in it instead of Wannier spin orbitals only Wannier functions with spatial dependence [in the triplet case the exchange term is zero in Eq. (130)].

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Table 5 The "rst singlet exciton band of some polymers calculated with the aid of intermediate exciton theory Polymer

QP Gap (eV)

Edge(s) of the "rst singlet exciton band (eV)

Experimental excitation energy (eV)

Number of neighbours (n)

Exciton binding energy (eV)

Polyethylene

10.3

8.7 (K"0)

7.6}8.8


6

1.6

Polydiacetylene PTS

3.7

1.8 (K"0) 4.7 (K"p/a)

2.0

5

1.9

Polydiacetylene TCDU

3.2

1.4 (K"0) 4.9 (K"p/a)

1.9

5

1.8

Cytosine stack

8.0

4.7 (K"0) 5.3 (K"p/a)

4.5

2

3.3

Polyglycine (a helix)

9.8

6.2 (K"0) 6.6 (K"p/a)

6.0}6.5

5

3.86

6.1 (K"0) 6.4 (K"p/a)

6.0}7.5

5

3.4

Polyalanine (a helix)

9.5

6-31GHH basis, Ref. [154].
See Ref. [155]. 6-31GHH basis Ref. [152]. Ref. [156]. Doublef basis, Ref. [157]. The "rst excitation energy of the monomer is 4.3 eV [158]. The discrepancy is due to the not good enough basis which has been applied. 6-31GH basis, Ref. [159]. Ref. [160]. Ref. [161].

To compute the "rst single exciton band of the high ¹ superconductor YBa Cu O (`123a) "rst     a 2D HF band structure calculation of the CuO plane and CuO chain (taking into account in both  cases also the apical O-atom in the BaO plane which lies between the two above-mentioned planes) was performed in a mutually consistent way. This means that the output charges obtained after an ab initio SCF band structure calculation for instance of the plane were used as input charges for the chain and vice versa, until mutual consistency was reached. The calculated two in"nite 2D systems were imbedded in an in"nite 3D Madelung potential in which the Cu and O charges were changed during the iterations to reach self-consistency and mutual consistency, respectively, in the explicitly calculated systems. The charges in the in"nite 3D crystal of `123a were summed up with the help of Ewald's [162] method in each iteration (for further details and the mathematical formulation see [163]). The HF band structures obtained in this way were corrected both in the plane and chain cases for correlation using the inverse Dyson equation in its diagonal approximation using as self-energy the MP2 expression with orbital relaxation (see Eqs. (98a) and (98b) in Section 3.3.9).

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The "lled parts of the correlation corrected band structures obtained in this way agree well with the photoelectron spectra of `123a [164]. In these band structures we have found (in agreement with the photoelectron spectra) 4 relatively low lying (between 8.5 and 10.5 eV below the Fermi level) very narrow, essentially Cu 3d bands. These are the only "lled bands in which the electrons can be strongly polarized [because they are quite well localized due to the narrowness of these bands (a few tenth of an eV)]. At the same time on the basis of their positions one can expect that the transitions from them to the partially "lled conduction band will have a large probability (intensity). Therefore we have used these transitions to calculate the excitonic spectra of `123a [165]. Since in 2D systems we cannot well localize the Wannier functions, we have performed the whole calculation in k space using crystal orbitals also in Eqs. (126b), (126c), (127) and in Eq. (130) [165]. In this way we have obtained "rst excitation energies (excitations to the lower edge of the four "rst singlet exciton bands arising from excitations from the 4 narrow Cu 3d bands to the conduction band just above Fermi level). The obtained exciton energies Eq lie between 4.4 and 5.7 eV (in the CuO plane case) which agree very well with the energy range of the "rst u.v.  absorption peak of `123a which lies between 4.5 and 5.5 eV [166]. One should further mention that by taking the di!erence between QP electron energy di!erences in these transitions and the obtained exciton energies as binding energies of the excitons, their range is 3.4}4.4 eV. We have also calculated exciton energies of the plane containing the Cu}O chains. We have obtained for the exciton energies larger values (6.4}7.7 eV) which are again in good agreement with the experimental data of 7.0}8.0 eV determined by re#exion spectra measurements of `123a [167]. In the case of the chains the exciton binding energy is about 3.5 eV for all the four transitions of the four Cu 3d bands. 5.2. Calculation of the superconducting state of YBa Cu O    Having the HF CO's and the exciton spectra of the CuO plane and of the plane containing the  Cu}O chains, one can calculate the e!ective interaction potential between the two electrons of a Cooper pair. For this one can write < "  A \  


 

(136)



(k "!k ) .  

(137)

 In the mechanism which we shall assume in the next point to calculate the most important characteristic quantities of the superconducting state of `123a, the polarization of the electrons in narrow "lled bands plays a key role. This polarization can be described in the "rst order with the help of excitons (for details and references see the next section).

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225

Applying the simple Thomas Ferni (TF) dielectric function which is equivalent to the Linhardt function [169] in the RPA [170] approximation: e(q , XP0)"1#(k /q) (138a)  2$ k "6nn e/E (138b) 2$ C $ where n is the average electron density in the quasi 2D unit cell and the Fermi energy E is C $ measured from the bottom of the lowest lying correlated Cu 1s band (!299.8 a.u.). We have obtained for k in the case of the CuO plane 1.479;10 and for the plane with the CO chains 2$  7.39;10\. One should emphasize that Eqs. (138a) and (138b) provide an e!ective screening only at small "q" values. In this way only in a part of the k space will < be negative. To improve this   situation we have started in international cooperation the calculations of the dielectric function from the correlation corrected band structure of the CuO plane using the HF CO's Wk and their  matrix elements in the numerator of Eq. (137). For the calculation of the attractive interaction we have applied the so-called `two-bandsa model which assumes that in the system there is (are) one or more very narrow "lled bands and a broad partially "lled conduction band. The motion of the delocalized electrons in the conduction band polarizes the localized electrons in the "lled narrow bands which gives rise to an attractive interaction. If the attractive potential (which can be described by a polarization propagator) is in absolute value larger than the screened Coulomb interaction "< "'
(139b)

 This kind of research had to be given up, because it has soon become clear that due to dynamic charge #uctuations, no superconductivity can exist in quasi 1D systems.

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with  A (q)"! 2 G BH IBH (i"1, 2) .





 





1  (r )

(r )

kG$q(r ) kBH q(r ) !exch AkG  BHkBH  r A  A !  

uBqH! 8 (140)

The matrix element (140) describes the interaction of a mobile electron in the conduction band (with quasi momentum k (i"1, 2)) with a second 3 d electron with momentum k H which gives G H B over (or takes away) after the excitation an exciton momentum q to (from) the "rst electron. Since this procedure can take place with both delocalized electrons we have two A (i"1, 2) terms which G corresponds to the two vertex points. The detailed calculations show [168] that near to the C point (k"0, 0), the > point (k"(0, p)) and X point (k"p, 0) of the rectangular "rst Brillouin zone at small "q" values between q"(0,0) and q"(0, p/10) the e!ective potential q is really negative (its absolute value decreases with  increasing q ). W Using the approximation that we take into account only the negative part or the e!ective potential q and put zero for its positive parts (as was done also in the original BCS paper  [177] and in Schrie!er's book [178]; for a more detailed argumentation see [168]), we can substitute it into the gap equation [179]






< (k , k)Dk (¹) [(e/. 1 k !e/. k )#"Dk (¹)"]  Y Y Y   Dk(¹)"! tanh 2k ¹ 2 k [(e/.!e/. k )#"Dk (¹)"]  Y IY Y (k"k #q) . (141)  Substituting into Eq. (141) Dk(¹ )"0, a careful numerical procedure resulted in ¹ "87 K   [168], while the experimental value is 92 K [180]. As a next step we have solved the gap equation for ¹"0. The iterative numerical procedure has provided for the maximal superconducting gap, D at the k vector (0, p) the value

   D "18.84 meV [168], while the experimental value is &20 meV [181]. With these values of

 ¹ and D one obtains for the ratio 

 2D /k ¹ "5.03 (142)

 @  with complete agreement of the experimental value of &5 [181]. Finally we have investigated the symmetry of the gap as a function of k , D(k ). The detailed   results show a dominant d   symmetry (at least for the positive part of D(k ), we could not obtain V \W  their negative parts because we have taken into consideration only negative < (k , k ) values with    a small s wave admixture [168]. These results are again in complete agreement with the recent experimental results of SQUID Interferometry and Single Junction Modulation measurements [182]. One should mention that in the planes containing the CuO chains in agreement with experiment no superconducting state was found from the gap Eq. (141). One should emphasize that though in our calculations no phonons were taken into account, the second-order vertex corrections were neglected, and last but not least only small "q" could be taken into account because of the screening by the Thomas}Fermi dielectric function (which was also the

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cause for neglecting the positive parts of < ), we have obtained astonishingly good agreement with  experiment for 5 characteristics of the high ¹ superconductor `123a. (In the CuO plane ¹ , D ,     dominant d   symmetry with a small s admixture of the function D(k ); no superconductivity in  V \W the chains). Therefore one would hope that more re"ned calculations which take into account all the neglected e!ects would not change very much the results. By executing the program described in a previous publication [183] ((1) HF- and (2) correlation corrected band structure, (3) ab initio calculation of the excitonic spectra using the intermediate exciton theory, (4) calculation of < and (5) solving the gap equation) we have obtained many  results in very good agreement with experiment for both the normal and superconducting state of YBa Cu O .    As a next step, as was already mentioned above, a better dielectric function should be generated from the existing data which will give a better screening also at larger "q" values. In this way we shall not be forced to neglect the positive parts of < and can also calculate the negative domains of the  order parameter *(k ). Further it would be good to repeat the whole calculation in a spin polarized  way to see whether spin e!ects play a role in the superconductivity of `123a (To obtain, however; also time-dependent spin #uctuations it will not be enough to solve the BCS gap equation, but one has to apply Eliashberg's equation [184]). Further it will be necessary to repeat the calculations on at least one (but possibly two) other high ¹ cuprates to see whether the `two bandsa model is valid also in these cases. If this would be  so, one could begin to design new superconducting materials with which it would be easier to work for practical applications than with the known high ¹ ceramic materials.  5.3. Calculation of the bulk modulus of polyethylene As a further example we have performed the calculation of the bulk (Young) modulus of polyethylene (PE) [154,185] B"(1/A)(RE/Ra)  ??

(143)

(A is the average cross-section perpendicular to the long axis of the chain, a is the lattice constant  at equilibrium) with the help of the correlation corrected total energy per unit cell, E. To obtain the second derivative of E according to a, by larger relative elongations (e"da/a 50.05) the band  structure and with it E had to be calculated at di!erent lattice constants a, the E(a) function obtained in this way numerically and RE/Ra had to be determined by a polynomial "t of the E(a) values not far from a [186].  The theoretical bulk modulus obtained in this way using a 6-31G basis and MP2 for correlation is 303 GPa for a single chain. (The less sophisticated methods give larger values). Experimentally the B values found for a single PE chain vary between 235 and 340 GPa depending on the method of measurement. Thus X-ray di!raction gives values 235}255 GPa [187], inelastic neutron scattering 239 GPa [188], Raman spectroscopy 258}290 GPa [189]. One can see that the later X-ray and Raman spectra values are between 255 and 290 GPa (only the rather early neutron scattering value of 239 GPa is smaller) which are not far from our perhaps somewhat too large theoretical value of 303 GPa.

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6. Conclusions As it was discussed in the Introduction, polymers play a very important role in modern techniques and it can be expected that their role will increase further. Since they have substantially more degrees of freedoms for changes (substitutions, etc.) than usual solids they can and will be used for purposes for which ordinary solids are not suitable. In Section 2 (ab initio Hartree}Fock crystal orbital theory), 3 (treatment of correlation in polymers) and 4 (treatment of aperiodic polymers both at the HF and correlation corrected levels) we have systematically built up quantum theory polymers as complicated solids. The only important part of polymer theory which was not discussed is the theory of interaction between polymeric chains. This has happened for space reasons. (The interested reader could consult Chapter 6 of [39] and the references therein.) In the three mentioned sections, examples of band structure and DOS calculations, respectively, were included. For these examples, whenever it was possible, comparisons with experiment were performed. Thus in Section 2 the experimentally measured most important properties of the electronic structure of (SN) are compared and found to be in good agreement with our theoretical V results. In Section 3 the experimental ionization potentials (photoelectron spectra) and gaps (estimated on the basis of the u.v. spectra) were compared with our theoretical "ndings for di!erent polymers. Though in hardly any case direct (inverse photoelectron spectra) measurements are available, the gap values estimated on the basis of the u.v. spectra are in reasonable agreement with our results if a good enough basis set was used for the calculation. On the other hand the experimental ionization potentials are in good agreement with our theoretical results in the case of QP (correlation corrected) band structures. In the case of di!erent native (and therefore aperiodic) protein chains like pig insulin, hen white egg, lysozyme, subtilisin, the calculated frequency-dependent hopping conductivities (p(u)) are in good agreement with those measured for di!erent chalcenogides (well conducting amorphous glasses) in the frequency ranges 10}10 s\, where the measurements were performed (there are no AC conductivity measurements of proteins). In Section 5 after shortly describing the development of the ab initio form of the intermediate exciton theory, in di!erent examples the characteristics of the "rst singlet exciton bands of di!erent polymers are shown and found again in good agreement with visible or u.v. spectra measurements. The same is true for their "rst triplet bands and second singlet bands (if the experimental data are known, or the 2nd singlet band exists, respectively, for details see Chapter 8 of [39]). We have also found the calculated exciton spectrum of the high ¹ superconductor YBa Cu O (`123a) in very     good agreement with its experimental absorption and re#exion spectrum, respectively. In the case of `123a as a next step we have computed on the basis of the QP band structures of its subunits (CuO plane with one apical oxygen or the plane consisting of the CuO chains with the  two neighbouring apical oxygens) and the HF crystal orbitals of these subunits the repulsive Coulomb and the attractive excitonic interaction terms (which occur in the "rst order if one applies a polarization propagator) between two electrons in its conduction band. We could apply only a Thomas Fermi dielectric function of the screening of the Coulomb interaction and have neglected the positive parts of the e!ective potential (as was done in the original BCS theory). In this way we restricted ourselves only to small q (the quasi momentum of the exciton). To improve our approximate way to compute < , we have started in international cooperation the calculation 

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of a dielectric function (or better tensor) using the quasi particle (QP) band structures, the already computed interaction matrix elements and HF CO's. Despite the approximations applied after substituting the negative parts of our e!ective potential into the BCS gap equation we have obtained for ¹ "87 K(exp. 92 K), for the superconducting gap  D "19 meV(exp. 20 meV) and for the symmetry of the D(k ) function d   with a small

  V \W admixture of s symmetry (the latter results are also in excellent agreement with the experimental conclusions reached in the last few years). Finally also the bulk modulus of a single chain of polyethylene was computed. Its value (302 GPa) is again in very good agreement with the experimental data. Besides the polymeric properties described in the text we have performed (after the necessary theoretical developments) extended ab initio calculations for the non-linear optical properties of polymers (polarizability and "rst-order hyperpolarizability tensor elements). Since no such experimental data are available for polymeric chains, we did not include (besides space reasons) these investigations in this review paper. For the same reasons we have omitted the numerous soliton calculations on polymers ("rst of all the works of W. FoK rner). The quantum theory is only partially or not yet worked out for many other polymeric properties (theory of vibrational spectra including correlation e!ects, of charge and heat transport, magnetic, most of the mechanical and of rheological properties, NMR spectra, phase transitions etc.). Having a good theoretical description (including the e!ects of correlation) of the electronic structure of a polymer, it seems comparatively easy to develop the theories and the corresponding codes also for these polymeric properties (for macrocosmic properties one has to apply of course also statistical mechanics). On the other hand, on the basis of the already performed numerous polymeric property calculations on very di!erent properties described in this work, one can see that polymer theory has reached a level of sophistication which makes it possible to calculate nearlyany polymeric property of any polymer, if its chemical structure and possibly its geometry (geometry optimizations would of course increase the necessary CPU times by several orders of magnitude) is known. Therefore we hope so that we have proven our claim described in the Introduction that with the help of the present-day sophisticated polymer theory one can predict theoretically from a large number of members of a polymer family those few ones which possess 3}5 di!erent (unrelated) optimal properties for a given purpose. Finally on the basis of the example of the high ¹ superconductor YBa Cu O one can say     that the methods developed for quasi 1D periodic polymers could be successfully applied also to di!erent 2D and 3D solids with periodic symmetry. This would provide an alternative to the density functional methods. One visualizes that in the future a successful synthesis of the classical quantum chemical approach (HF#treatment of correlation) and a further re"ned form of the density functional methods (which can treat well also virtual energy levels or bands, respectively, as well as excited states) will emerge both for molecules and solids (including polymers).

 It may happen that for a certain special polymeric property the theory fails or is very di$cult. Of course such examples can also be found in molecular physics (e.g. the electric "eld gradient, at the nuclei of a large molecule).

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Acknowledgements The author would like to express his gratitude to many of his colleagues both in Erlangen and elsewhere both on a junior and senior level. Without their help the theory and results described in this review article would not have been achieved. First of all the names of Professors A.K. Bakhshi, F. Beleznay, R.S. Chen, J. Cizek, E. Clementi, T.C. Collins, G. Del Re, W. FoK rner, F. Martino, P. Otto, M. Seel, S. Suhai, V. van Doren, Y.-J. Ye and A. Zawadowski as well as of Drs. G. BiczoH , F. BogaH r, R. Day, D. Dudis, D. Ho!mann, C.L. Palmer, J. Pipek, C.-M. Liegener, P. Saalfrank, W. Utz and D. Wechsler among the many others should be mentioned. He is very grateful to the late Professors A. Szent-GyoK rgyi, Nobel Laureate, C.A. Coulson, E. Krokowski, K. Laki, U. Poppe and G. Schay as well as to Professors E. HaK usser, Former President of the German Patent O$ce, G.L. Hofacker, F. Ho!mann and P.O. LoK wdin for their long-lasting inspiration, encouragement, support and their critical comments. He is further very much indebted to Former President of the Friedrich-Alexander-University Erlangen-NuK rnberg Professor N. Fiebiger and Former Chancellor of the same University Dr. h.c. K. KoK hler for their interest and support. He would like to express his deep gratitude to the leaders of the National Foundation of Cancer Research, to the late Dr. F. Salisbury, to Mrs. T. Salisbury and Mr. F. Salisbury, Jr., for their interest and for their over 20 yr lasting substantial "nancial support without which most of the calculations would not have been possible. He is further very much indebted to the `Deutsche Forschungsgemeinschaft (DFG), to the Alexander-von-Humboldt Foundation, The Volkwagen Foundation, The German Academic Exchange (`DAADa) and to the `Fonds der chemischen Industriea, to the US Air Force O$ce of Scienti"c Research and the NATO Scienti"c A!airs Division whose substantial support made it possible to "nance a larger number of coworkers and graduate students as well as to invite numerous Visiting Professors and Guest Scientists from abroad. Last but not least he would like to express his special thanks to the Regional Computing Center of this University and especially to its Director Dr. F. Wolf for his much help and patience during the many years in which he has provided as much free computer time as possible and also for his help to obtain much computer time on the di!erent computers of the Leibnitz Computing Center of the Bavarian Academy of Sciences. Finally he is very much indebted to Siemens A.G. (Dr. F. Kuhrt) and Kraftwerk Union as well IBM Germany and IBM USA (Poughkeepsie and Kingston, N.Y.) who also provided substantial amounts of free CPU time for our di!erent large-scale calculations.

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EARTHQUAKES: FROM CHEMICAL ALTERATION TO MECHANICAL RUPTURE

DIDIER SORNETTE 
Department of Earth and Space Science and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, USA
Laboratoire de Physique de la Matie` re CondenseH e CNRS and UniversiteH des Sciences, B.P. 70, Parc Valrose 06108, Nice Cedex 2, France

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 313 (1999) 237}291

Earthquakes: from chemical alteration to mechanical rupture Didier Sornette 
* Department of Earth and Space Science and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095, USA
Laboratoire de Physique de la Matie% re Condense& e, CNRS and Universite& des Sciences, B.P. 70, Parc Valrose 06108, Nice Cedex 2, France Received July 1998; editor: I. Procaccia Contents 1. Introduction 2. Earthquakes: where are the challenges? 2.1. The N-body problem 2.2. Self-organized criticality 2.3. Application of statistical physics concepts 2.4. The one-body problem 3. The mechanical view 3.1. Seismology versus rock mechanics 3.2. Friction 3.3. The standard model: elastic rebound 4. The strain paradox 5. The stress paradox friction 5.1. Statement of the paradox 5.2. Proposed resolutions 6. The heat #ux paradox 6.1. Statement of the paradox 6.2. Proposed resolutions 7. Water and rocks: metamorphism and faulting 7.1. The ubiquity and importance of water 7.2. E!ect of water on minerals: hydrothermal alteration under stress 7.3. Role of water in rock metamorphism and relation to earthquakes 8. Structure and polymorphism of crustal minerals

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9. The mechanical rupture: possible role of high frequency acoustic vibrations 10. Seismic radiation 10.1. The di!erent regimes 10.2. Earthquake dynamical rupture 11. Other observations and questions 11.1. Laboratory experiments 11.2. The longer the recurrence time, the larger the stress drop 11.3. Shear heating 11.4. Seismic P-wave nucleation phase 11.5. Earthquakes occur preferentially in strong rocks 11.6. Earth tides and seismicity remotely triggered at long distances 11.7. Non-double couple components 11.8. Creeping sections and slow earthquakes 11.9. Deep earthquakes 11.10. Precursory phenomena 12. Concluding remarks 13. Note added in proof Acknowledgements References

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* Tel.: #1 310 825 2863; fax: #1 310 206 3051; e-mail: [email protected]. 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 8 ) 0 0 0 8 8 - X

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Abstract In the standard rebound theory of earthquakes, elastic deformation energy is progressively stored in the crust until a threshold is reached at which it is suddenly released in an earthquake. We review three important paradoxes, the strain paradox, the stress paradox and the heat #ow paradox, that are di$cult to account for in this picture, either individually or when taken together. Resolutions of these paradoxes usually call for additional assumptions on the nature of the rupture process (such as novel modes of deformations and ruptures) prior to and/or during an earthquake, on the nature of the fault and on the e!ect of trapped #uids within the crust at seismogenic depths. We review the evidence for the essential importance of water and its interaction with the modes of deformations. Water is usually seen to have mainly the mechanical e!ect of decreasing the normal lithostatic stress in the fault core on one hand and to weaken rock materials via hydrolytic weakening and stress corrosion on the other hand. We also review the evidences that water plays a major role in the alteration of minerals subjected to "nite strains into other structures in out-of-equilibrium conditions. This suggests novel exciting routes to understand what is an earthquake, that requires to develop a truly multidisciplinary approach involving mineral chemistry, geology, rupture mechanics and statistical physics.  1999 Elsevier Science B.V. All rights reserved. PACS: 91.30.Dk; 91.45.Vz; 05.70.Ln

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1. Introduction A seismological map of the Earth reveals that the epicenters of earthquakes are for the most part located on long arcing strips } the seismological belts that encircle the planet. These seismological belts mark the Plate Tectonic boundaries of convergence, divergence and transform that de"ne the lithosphere plates of the Earth. Much of the active seismological belts lie close to or are underneath densely populated areas of human habitation. The fact that large populations are often found in tectonically active, earthquake prone areas may be due to the favorable environment they provided for the development of prehistoric human populations (Bailey et al., 1993; King et al., 1994). Furthermore, an important factor in the development of ancient civilizations (Knopo! and Sornette, 1995), i.e. the development of large communal populations, may have been the geographical isolation, induced by tectonic processes of mountain building that presented natural defenses. Examples of such ancient civilizations are to be found in Anatolia, China, Greece, India, Japan, Java, Mexico, Persia, Peru and Rome. The absence of tectonic defences in the plains promoted di!usion by marauders and hence the incubation processes necessary to develop large populations could not emerge. The growth of these populations has continued to this day. It was much later, in the Middle Ages, that global trade emerged as an even more potent force that allowed for the development of large populations in the plains, for example, of northern and western Europe. And the exploration and development of North America took place because of these more potent forces as well. An additional factor is that earthquake prone areas are geologically active and thus rich in mineral and oil reservoirs, as well as in the quality of the soil for agriculture exploitation. It has been estimated that the economic balance between property losses incurred during earthquakes and gains from their indirect economic bene"ts is largely positive. Thus, people continue to live in large metropolitan areas such as Tokyo, San Francisco, Los Angeles, Santiago, and Rome to name a few, which are in danger of destruction by earthquakes. Statistics compiled by insurance companies such as Lloyds of London, rate earthquakes "rst among natural disasters in terms of severity of damage and loss of life. The last destructive Japanese earthquake at Kobe of magnitude 6.9 that occurred on January 17, 1995, led to about $ 200 billions of damage in a country that is a leader in engineering construction safety rules and hazard reduction e!orts. Estimates by Japanese banks and insurances put the damage for the next great Tokyo earthquake at more than a trillion dollar, with important economic repercussions worldwide in the following years. Recent careful analysis of the seismic energy-frequency Gutenberg}Richter relation using an independent regionalization of global seismicity, the Harvard centroid-moment tensor catalog data and a combination of observed rates and tectonic deformation rates shows that all regions, except mid-ocean ridges, are susceptible to the same maximum earthquake magnitude around 8.5 (Kagan, 1997). According to this result, there are almost no  The last great Tokyo earthquake occurred in 1923 and completely destroyed the city. The estimate recurrence time is about 70 years plus or minus a large uncertainty.  This view is controversial in comparison to the standard way of estimating the largest possible earthquake, based on the maximum fault length, which thus leads in general to much smaller estimations. However, the limitations of this `olda approach are more and more recognized because most recent damaging earthquakes occurred on unknown faults or activated several disconnected faults.

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regions that are safe from great earthquakes on the long term. The only di!erence is the typical time scales, tens of years in Chile, a hundred or a few hundreds of years in Japan and California to tens of thousands of years in so-called non-seismic regions. These time scales are simply controlled by the deformation rates, i.e. the rate with which faults are reloaded, which go from several centimeters per year to a fraction of millimeter per year in di!erent parts of the world. These time scales separating major earthquakes are larger than the historical development time of our industrial society leading to a global interconnected economy. We can thus expect surprising impacts that have never been experienced in the past. Since the inception of seismology, there has been a great urgency upon the search for reliable precursor phenomena of earthquakes and their generative processes (Gokhberg et al., 1995). It is widely accepted that the extensive e!orts of the last thirty years to "nd reliable earthquake prediction methods have largely failed (Evans, 1997; Geller et al., 1997; Scholz, 1997; Snieder and van Eck, 1997; Wyss et al., 1997). This has even led some to assert that earthquakes are unpredictable (Geller et al., 1997). More researchers agree now that it is necessary to improve our fundamental understanding because `black boxa prediction schemes do not work reliably. This review attempts to bring up to the physicists a #avor of the crisis (Jordan, 1997; Kagan, 1998) and the challenges felt by the earthquake science community.

2. Earthquakes: where are the challenges? 2.1. The N-body problem The most common view among physicists interested in earthquake modelling is that the real challenge lies in its `N-bodya nature. This has led to a #urry of activities starting in the late eighties with hundreds of publications on earthquakes in physical as well as geophysical journals since. To simplify to the extreme, the one-earthquake or one-fault problem is usually thought to be relatively well-understood (even if an `isolateda fault has never been observed!) and the excitement (for the physicists) emerges when coupling (via long-range elasticity) many faults presenting highly nonlinear responses (threshold dynamics) in the presence of quenched (and also annealed) heterogeneity (to account for inhomogeneous rocks and fault geometries). 2.2. Self-organized criticality This is the vision that led to the proposed analogy between earthquake organization and self-organized criticality (Sornette and Sornette, 1989; Bak and Tang, 1989; Sornette et al., 1990a,b; Ito and Matsuzaki, 1990; Scholz, 1991; Sornette, 1991) (see for reviews Main, 1996; Grasso and Sornette, 1998) and the recognition that earthquake dynamics o!er for the solid earth a complexity and richness (and di$culties!) equivalent or even greater to that of hydrodynamic turbulence (Kagan, 1994). Self-organized criticality provides a conceptual framework for understanding the Gutenberg}Richter size distribution of earthquakes, the power law fault length distribution, the fractal geometry of sets of earthquake epicenters and of fault patterns among others (Sornette, 1991). It has recently been exploited to advance our understanding on the crust organization and

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about the rich and subtle properties found in tectonics and seismology, such as induced seismicity by water impoundment, mineral and gas extraction (Grasso and Sornette, 1998). In a nutshell, SOC refers to the spontaneous organization of a system driven from outside in a globally dynamical statistical stationary state, which is characterized by self-similar distributions of event sizes and fractal geometrical properties. SOC applies to the class of phenomena occurring in continuously driven out-of-equilibrium systems made of many interactive components, which possess the following fundamental properties: (1) a highly non-linear behavior, (2) a very slow driving rate, (3) a globally stationary regime, characterized by stationary statistical properties, and (4) power distributions of event sizes and fractal geometrical properties. The crust obeys these four conditions (Sornette, 1991). 1. The threshold response can be associated with the rupture instability leading to the sudden unlocking of the fault. 2. The slow driving rate is that of the slow tectonic deformations thought to be exerted at the borders of a given tectonic plate by the neighboring plates and at its base by the underlying lower crust and mantle. The large separation of time scales between the driving tectonic velocity (cm/yr) and the velocity of slip (m/s) makes the crust problem maybe the best natural example of self-organized criticality, in this sense. 3. The stationarity condition ensures that the system is not in a transient phase, and distinguished the long-term organization of faulting in the crust from, for instance, irreversible rupture of a sample in the laboratory. 4. The power laws and fractal properties re#ect the notion of scale invariance, namely measurements at one scale are related to measurements at another scale by a normalization involving a power of the ratio of the two scales. These properties are important and interesting because they characterize systems with many relevant scales and long-range interactions as probably exist in the crust. A remarkable feature of the SOC state is in the way it emerges. The spatial correlations between di!erent parts of the system do not appear as a result of a progressive di!usion from a nucleus but from the repetitive action of rupture earthquake cascades. In other words, within the SOC hypothesis, di!erent portions of the crust become correlated at long distances by the action of earthquakes and their relaxations which `transporta the #uctuations of the alteration and stress "elds in the di!erent parts of the crust many times back and forth to "nally organize the system. This physical picture is substantiated by various numerical and analytical studies of simpli"ed models of the crust (Bak and Tang, 1989; Zhang, 1989; Chen et al., 1991; Cowie et al., 1993, 1994). The crust is heterogeneous at many scales from the smaller atomic scale, with the presence of dislocations, impurities and grain boundaries to the scale of tectonic plate boundaries. Its structure can be approximatively described as fractal, probably better by multifractal or even better as hierarchical (Ouillon et al., 1996). Water is probably found everywhere within the crust, however with a very large heterogeneity. We thus envision the crust containing a 3D hierarchical network of faults and discontinuities with water "lling them partially in varying amounts. As a consequence and in the presence of the heterogeneous strain "eld, rock alteration varies from place to place. Integrated over time, a substantial fraction of the crust is close to rupture instability. However, at a given instant of time, only a small subset of the crust is altered, su$ciently stressed and riped enough to destabilize into a rupture earthquake. Together with the localization of seismicity on

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faults, this leads to the conclusion that a signi"cant fraction of the crust is susceptible to rupture, while presently being quiescient. The quantitative determination of the susceptible fraction is dependent on the speci"city of the model (Zhang, 1989; Pietronero et al., 1991; Sornette et al., 1994) and cannot thus be ascertained with precision for the crust. What is important however in that the susceptible part of the crust can be activated with relatively small perturbations or by modi"cation of the overall driving conditions. This remark leads to a straighforward interpretation of induced seismicity by human activity (Grasso and Sornette, 1998). 2.3. Application of statistical physics concepts Other modern physical concepts has been shown to be relevant to earthquakes. For instance, faults result from growth processes (Sornette et al., 1990a,b) and, in some limited cases, converge to geometrical structures that can be mapped onto optimal random manifolds (Miltenberger et al., 1993; Sornette et al., 1994a). Nucleation and spinodal decomposition has been suggested to describe an incipient earthquake (Rundle and Klein, 1995; Klein et al., 1997; Dieterich, 1992). The presence of quenched heterogeneity coupled with dynamics is also thought to play a fundamental role (Miltenberger et al., 1993; Sornette et al., 1994b; Knopo!, 1996a; Fisher et al., 1997) and leads to complex spatio-temporal behavior. Long-range elasticity can induce `frustrationa in the motion of adjacent tectonic blocks (Sornette et al., 1994a; Sornette and Vanneste, 1996) and lead to a hierarchical organization of faults and earthquakes in space and time (Alekseevskaya et al., 1977; Scholz, 1990; Ouillon et al., 1996), suggested to be qualitatively similar to spinglasses (Sornette et al., 1994b). The tensorial nature of crust motions leads to kinematic compatibility and geometric compatibility requirements (Gabrielov et al., 1996). Kinematic compatibility (also known as St-Venant compatibility) stems from the fact that the displacement must be uni-valued along a closed contour. Geometric compatibility measures the tendancy for tectonic blocks to create or anihilate surface, by the activation of so-called thrust and normal faulting that relax the motion along the vertical direction perpendicular to the tectonic plates (Gabrielov et al., 1996). Recent e!orts have been undergoing at the frontier between seismology and statistical physics to understand the origin of the observed complexity of the spatio-temporal patterns of earthquakes. An interesting debate has been whether space-time complexity can occur on an homogeneous fault solely from the nonlinear dynamics (Shaw, 1993, 1995, 1997; Cochard and Madariaga, 1994, 1996) associated to the slip and velocity dependent friction law (Dieterich and Kilgore, 1994; Dieterich, 1987, 1992, 1993). Or, is the presence of quenched heterogeneity necessary (Rice, 1993; Ben-Zion and Rice, 1993, 1995; Knopo!, 1996b)? It is now understood that complexity can emerge purely from the nonlinear laws but heterogeneity is probably the most important factor dominating the multi-scale complex nature of earthquakes and faulting (Ouillon et al., 1996). It is also known to control the appearance of self-organized critical behavior in a class of models relevant to the crust (Sornette et al., 1995; Shnirman and Blanter, 1998). 2.4. The one-body problem More than six years ago, a "rst draft of this review was focused on statistical models of rupture in heterogeneous media and their self-organizing principles, with emphasis on cooperative behavior in order to account for the many observed power law behaviors (see, for instance, Kagan, 1994;

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Knopo!, 1996a, 1997). Carlson et al. (1994), Rundle and Klein (1995) and Langer et al. (1996) have written reviews in this spirit, however mostly specialized to spring-block models. In the last few years, my point of view has evolved dramatically. I am still convinced that the dynamics of earthquakes and their space-time structure provide a nice example of self-organizing nonlinear spatio-temporal systems and that this deserves investigations. I am con"dent that the physical community does not need encouragement in this area. However, I have come to realize that maybe the biggest challenge of all resides in the so-called `one-bodya problem, i.e. in the understanding of the single earthquake `cyclea. This came while I was trying to synthetize many geological observations and experimental data. It progressively dawned on me that even the supposedly `simplea one-earthquake process is not at all understood and that this poses many fundamental questions. This review is focused at the basic facts that poses di$culties for their interpretation and for making them compatible with the standard pictures. The review is thus organized around the three main paradoxes, the strain, stress and heat #ow paradoxes. The emphasis is on geological, seismological, mechanical and chemical data and concepts rather than on statistical physics. The challenge is to attempt to synthetize and account for the large body of phenomenology existing in these diverse disciplines that are rarely put together. The style of work is similar to that of a detective slowly and patiently assembling clues to identify the `shooting guna: what is really an earthquake? It is hoped that this will help realize that the one-earthquake problem is amazingly diverse and multidisciplinary in essence and that mechanical and chemical processes are intimately coupled in earthquake processes. The main theme developed here is that it is not possible to get a reasonable description of an earthquake if one forgets about the chemical processes occurring at the small scale. In contrast to the homogenization approaches that lead to treatments of earthquakes solely based on macroscopic descriptions, here we propose that the preparation of an earthquake and its triggering start at the atomic level and that keeping such a (multi-scale) description is necessary for understanding what an earthquake is. This may be an example where di!erent levels of the hierarchical structure of science meet and interfere, in contrast to the more usual bottom-up structure (Anderson, 1972).

3. The mechanical view The fault of the geologist is not the same object as the fault of the rock mechanician which is itself a di!erent structure than the fault of the seismologist. For the geologist, complex cataclistic deformations are ubiquitous in fault zones. For the rock mechanician, gouge and localized deformation are the key elements. For the seismologist, the earthquake source as one (or several) double-couple set(s) of forces (see Section 10 for a de"nition of a double-couple) on well-de"ned rupture planes is the fundamental quantity. We shall try to synthetize these di!erent view points and, in this way, gain a better understanding of the physical mechanism of earthquakes.

 The gouge is the fragmented core of a fault.

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3.1. Seismology versus rock mechanics It is a common wisdom that an earthquake is primarily a mechanical process which appears as a genuire rupture (or a reactivation of a previous rupture) of the crust, and the earth behaves as an elastic body during the short time span of the phenomena (Aki, 1988). Such model is very good at calculating the static strain "elds that are created by large dislocations at fault planes. They are also very good at calculating the propagating compressional and shear waves that are important dynamical consequences of earthquakes. It is recognized that some complication occurs in the earthquake source region before and during the earthquake but it is encapsulated in the purely mechanical formulation in terms of the double-couple source mechanism (Burridge and Knopo!, 1964). At the simplest level of description, elastic energy is progressively stored until a static friction threshold is reached at which the fault slips according to dynamical friction, creating the equivalent of an extended dislocation. This description is evidenced by the many super"cial observations of fresh surface rupture traces, fresh scarps and fresh displacements seen at the surface of the earth after large earthquakes. Fractures are also widely observed in surface rocks of all types. When trying to understand the mechanism underlying an earthquake, it thus seems natural to think that some sort of rupture is the cause of the earthquake. The standard earthquake picture is thus based on the idea that pre-existing cracks can grow progressively by subcritical growth, leading ultimately to a catastrophic failure propagating at a fraction of the shear wave speed. A similar and related mechanism is to invoke friction of pre-existing faults, which goes from very slow stable growth to unstable, depending on the stress loading and other water content. The present understanding is vividly summarized by Scholz (1992a), who contrasts the two main paradigms of the earthquake mechanism. On the one hand, seismologists tend to view an earthquake as a seismic source characterizing a fracture phenomenon, i.e. the rupture of a strong part of the crust. This view is in#uenced by the seismological observations which put the emphasis on the generation of elastic waves and their propagation to the recording stations. On the other hand, rock mechanicians consider an earthquake as a stick-slip event controlled by the friction properties of the fault, i.e. the destabilization of a weak part of the crust. This formulation has been shaped by numerous laboratory experiments under a variety of pressure and temperature conditions (which however reproduce only imperfectly the conditions prevailing in the crust). Scholz (1992a) sees the transition from the "rst paradigm to the second one as ineluctable: the seismological paradigm uses concepts such as asperities and barriers which remain loosely de"ned (see however for instance (Aki, 1979)); in contrast, the stick-slip friction mechanism has developed quantitatively both with laboratory experiments and theoretical formulations. 3.2. Friction As a scienti"c "eld, solid friction has a long history, probably starting in the western world about 500 years ago with Leonardo di Vinci, continuing with the empirical Amontons' laws two centuries  A dislocation is a planar fault domain over which a slip discontinuity is imposed.  An admissible earthquake must be such that the medium is in mechanical equilibrium after the introduction of this event. Thus, the total force and the total torque induced by the earthquake must vanish, hence the term `double-couplea.

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later and Coulomb's investigations of the in#uence of slipping velocity on friction in the XVIII century. Only three decades ago was it recognized that friction has probably a fundamental role in the mechanics of earthquakes (Brace and Byerlee, 1966). Rock mechanicians consider an earthquake as a stick-slip event controlled by the friction properties of the fault, i.e. the destabilization of a weak part of the crust. Numerous laboratory experiments have been carried out to identify the parameters that control solid friction and its stick-slip behavior (Persson and Tosatti, 1996). The most signi"cant variables appear to be the mineralogy, the porosity, the thickness of the gouge, the e!ective pressure, the temperature and the water content (Byerlee et al., 1968; Brace, 1972; Beeman et al., 1988; Gu and Wong, 1991; Johansen et al., 1993; Streit, 1997). Low velocity experiments have established that solid friction is a function of both the velocity of sliding and of one or several state parameters, roughly quantifying the true surface of contact (Brace, 1972; Dieterich, 1972, 1978, 1979, 1992; Ruina, 1983; Cox, 1990; Beeler et al., 1994, 1996; Baumberger and Gauthier, 1996; Scholz, 1998). These Ruina}Dieterich laws constitute the basic ingredients in most models and numerical elastodynamic calculations that attempt to understand the characteristics of earthquake sources. A well-known and serious limitation of these calculations based on laboratory friction experiments is that the friction laws have been determined using sliding velocities no more than about 1 cm/s, i.e. orders of magnitude below the sliding velocity of meters or tens of meters believed to occur during an earthquake. Is it correct then to extrapolate these laws, and especially the velocity weakening dependence, to large velocities? This question is all the more relevant when one examines the underlying physical mechanisms of the friction laws. At law velocity, hysteretic elastic and plastic deformations at the scale of roughness asperities seem to play a dominant role (Bowden and Tabor, 1954; Sokolo!, 1984; Jensen et al., 1993; Dieterich and Kilgore, 1994; Caroli and Nozie`res, 1996; Tanguy and Nozie`res, 1996; Tanguy and Roux, 1997; Caroli and Velicky, 1997; Bocquet and Jensen, 1997). At large velocities, di!erent mechanisms come into play. Collisions between asperities and transfer of momentum between the directions parallel and perpendicular to the motion may become an important mechanism (Lomnitz, 1991; Pisarenko and Mora, 1994). This regime is probably relevant for the resolution of the apparent low-friction problem stemming from the heat #ow paradox and the low strength of the San Andreas fault as we discuss below. Recently, Tsutsumi and Shimamoto (1996, 1997) have reached a completely novel regime, by performing friction measurements on rotating cylindrical samples at velocities up to 1.8 m/s and for slips of several tens of meters. The results cannot be taken completely for granted due to several experimental problems but seem to indicate the existence of a change of regime from velocity weakening to velocity strengthening and then again to velocity weakening at the largest velocities. This last regime seems to be associated to the melting of a very thin layer. It is quite possible that di!erent physical mechanisms might lead to a change of regime in the velocity and slip dependence of the solid friction law at large slip velocities. One thus needs to explore the high velocity regime in a variety of conditions, as has been done for the low velocity regime. In this goal, recent realistic 3D numerical simulations of elastic bodies sliding on top of each other in a regime of velocities of meters to tens of meters per second have been performed that allows one to probe more intimately the response of the bodies and the nature of the friction, that any experiment could. A velocity strengthening is found with a doubling of the friction coe$cient when the velocity increases from 1 to 10 m/s, which re#ects the increasing strength of vibrational damping (Maveyraud et al., 1998).

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3.3. The standard model: elastic rebound Geodetic measurements of crustal deformations starting in 1850 in California and conducted after the 1906 San Francisco earthquake indicated that signi"cant horizontal displacements parallel to the ruptured San Andreas fault had occurred both before and after the earthquake. In particular, Reid noticed that distant points of opposite sides of the fault had moved 3.2 m over the 50-year period prior to 1906. This led him to the famous elastic rebound theory for the cause of earthquakes (Bolt, 1993). The theory is best explained using a simple diagram shown in Fig. 1, representing a locked fault within an elastic medium immediately after the last earthquake (Turcotte and Schubert, 1982). A relative uniform tectonic velocity u is applied at a distance b from each side of the fault and the  shear strain e increases with time according to u t e"  . 2b

(1)

The stress increases correspondingly uniformly as u t p"p #G  ,  2b

(2)

where p is the so-called dynamical frictional stress that is operative on the fault at the end of  faulting and G is the shear modulus. The locked fault can transmit any shear stress less than its assumed static frictional stress q . When this stress is reached at the time  2b t*" (p !p ) , (3)  Gu  

Fig. 1.

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slip occurs on the fault in a few seconds so that the edges of the plates can be assumed to be stationary during this time. The accumulated shear strain e*"u t*/2b is recovered by the plates by  a springing back or rebounding on each side of the fracture, thus the term elastic rebound. The resulting displacement on the fault is d"2e*2b. After the earthquake, the fault locks and the cycle repeats.

4. The strain paradox From the observation of the fault slip d of the earthquake, the elastic rebound theory of the previous section with Eq. (2) predicts the order of magnitude of the width b over which the shear strain develops progressively across the fault prior to the earthquake: G d. (4) 2b" (p !p )   We consider a typical displacement d+5 m on the fault during a large earthquake. The shear elastic modulus is of the order of 10 GPa. The stress drop p !p on the fault during the large   earthquake lie in the range 1}10 MPa. Eq. (4) leads to the estimate 2b"5}50 km.

(5)

Note that b is all the smaller, the larger is the stress drop. The order of magnitude of the result (5) is robust with respect to the many ways with which one can re"ne the model. A standard model (Turcotte and Schubert, 1982) (p. 371) consists of two lithospheric plates of thickness ¹ sliding past each other and in contact at the position of the fault. Initially, there is not stress on the fault which is locked to a depth a. This depth represents the thickness of the brittle seismogenic crust. Below, the material is ductile and #ows continuously without locking. The plates are then displaced progressively but no displacement occurs on the locked part of the fault. Assuming that the displacement rate is imposed at a large distance from the fault, a simple elastic calculation (Turcotte and Schubert, 1982) (p. 373) shows that the shear strain is concentrated in a narrow zone of width b+a around the fault. For a typical thickness a+15 km for the brittle crust, we thus get a con"rmation of the order of magnitude estimate (5). Modern geodetic measurements are su$ciently precise to test for the existence of strain localization. The surprise is that there is no geodetic evidence of the existence of a concentration of shear deformation at the scale of 1}10 km around faults in many situations, even if others exhibit it (Pearson et al., 1995). The geodetic displacement pro"les obtained across section of faults that have not ruptured in the last decades give an essentially uniform strain over distances of 150 km or more from the fault (Walcott et al., 1978, 1979; Shen et al., 1996; Snay et al., 1996). This leads to a problem. Indeed, if we put 2b+150 km in Eq. (2) while keeping the plate tectonic displacement

 Basalt and granite have a large modulus 30}50 GPa. However, the crust is heterogeneous with defects and cracks are many di!erent scales. As a consequence, the modulus determined in the laboratory from homogeneous samples overestimates signi"cantly the values in the "eld. Due to the large heterogeneity, it is di$cult to give a meaningful and precise value for the modulus. Our calculation must rather be taken as an order-of-magnitude estimate.

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of the same order d+5 m, we get a reduction of strain by a factor 10}100. This leads to a much smaller stress drop at rupture than usually measured for large earthquakes. Taking G+10 GPa leads to p !p +0.1 MPa (1 bars) (see below the discussion in Section 5). We note that there is   evidence that some small earthquakes can exhibit very small stress drop of the order of 0.1 MPa (Scholz, 1990) (p. 183, Fig. 4.10), but this is by no means the case for the population of large earthquakes. How a large earthquake can develop and pass over barriers with a small stress drop is another unresolved di$cult question in the standard conceptual framework. On the other hand, strain concentrations are observed very neatly after a large earthquake (Shen et al., 1994; Massonnet et al., 1994, 1996). The observed postseismic relaxation, with concentration in the vicinity of the fault, is attributed either to viscoelastic relaxation in the volume of the crust and upper mantle or to afterslip or continued slip on the fault planes. There are no particular di$culties to account for these observations within the current understanding of the crust rheology. We mention another phenomenon occurring at much smaller scales: horizontal fault slippage has been detected in several faults around the world (Bolt, 1993) (p. 92}93), including the San Andreas fault system, the north Anatolian fault in Turkey and along the Jordan Valley rift in Israel. The slip is usually highly localized on a width of one meter or less and has an amplitude of a few millimeter and a duration of a few minutes to a few days. The standard interpretation is to invoke the weak strength of the gouge material present within faults in the upper layers of the crust which cannot resist the tectonic applied shear. If the shear is applied more at the bottom of the crust, as in several model calculations, it is hard to argue that the upper gouge would deform while the lower stronger layer is stuck. We have seen that the upper layers respond and follow the after-earthquake continuous deformation. The observed localized slip at the upper surface should thus re#ect a process occurring in the stronger rock materials, namely very localized deformations. It is still an open question whether the faults having measurable creep could carry very large earthquakes (magnitude 5 earthquakes are been measured on these faults).

5. The stress paradox friction 5.1. Statement of the paradox In the laboratory, frictional sliding is observed to occur at lower stresses than cracking. This is the usual proposed rational for considering that earthquakes are better described as slip events than as rupture processes. There is a large body of literature documenting the maximum shear stress to initiate sliding as a function of normal stress for a variety of rock types. The best linear "t de"nes a maximum coe$cient of static friction f "0.85 (Byerlee, 1977), with a range maybe between 0.6 to 1 approxim ately. The range has been con"rmed by stress inversion of small faults slip data in situ with result 0.6$0.4 (Sibson, 1994). The stress inversion methods follow the assumption that slip along a fault

 Heuze (1983) reviews a large body of the literature until that time on the various properties of granitic rocks.

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occurs in the direction of the maximum resolved shear stress. These methods calculate the stress axes which minimize the angular deviation between the observed slip axis along the fault and the axis of the maximum resolved shear stress determined from the general stress. There is big discrepancy between this value of friction measured for rocks in the laboratory in situ and the corresponding stress threshold on the one hand and the stress that is available in nature to trigger an earthquake on the other hand. Indeed, assuming a Mohr}Coulomb relation, for a normal stress of the order of the lithostatic pressure, say 300 MPa at 10 km depth, the shear stress at threshold is thus of order 200 MPa. Using a shear modulus of 10 GPa, this yields a shear strain at threshold of e+3;10\. This is an enormous value that would demand a very large localization. Consider a large earthquake with rupture slip of u+5 m, then the localization width must be u/e+1.7 km. As already discussed, this is not usually observed. In addition, the stress of the San Andreas fault zone deduced from a variety of techniques is found to be low and directed at a large angle close to perpendicular to the fault (Zoback et al., 1987). This is in contradiction with the frictional sliding model. A more detailed tensorial analysis assuming hydrostatic pore pressures gives the depth-averaged shear strength of faults in the brittle continental crust under a typical continental geotherm and strain rate of about 35 MPa (350 bars) in normal faulting, 150 MPa in thrust faulting and 60 MPa in strike-slip faulting (Hickman, 1991). Computer simulations modeling fault networks in California with thin-plate "nite elements, when tested against measured slip rates on faults, against principal stress directions and rates of geodetic baselines, suggest that the time-average friction coe$cient of major faults is only 0.17}0.25, i.e. only 20}30% of the value that is assumed for the friction of intervening blocks (Bird, 1989; Bird and Kong, 1994). In the previous section, we have discussed the paradox that, in many situations, strain localization is not observed. Strain localization could be naively argued to be necessary to increase the stress level necessary to trigger earthquakes. The problem with this argument is that, if the gouge zone is weak, it will indeed accumulate and localize the strain but the stress will still be small due to the weak elastic modulus. On the other hand, if the fault gouge is not so weak, the strain will not be localized and thus the stress will also be weak due to the small strain, as the deformation is spread over a large domain. Let us make this quantitative by using a simple 1D model. Consider two shear springs associated in series of shear elastic modulus k and k . One extremity of the springs is "xed   and the other one is displaced by the amount u. The force k k F"   u k #k   is the same in each spring and their respective displacement is

(6)

k k  u,  u and u " (7) u "  k #k  k #k     which add up to the total displacement u. Note that if the weak zone (represented by spring 1) is such that k k , then u +u, i.e. essentially all the deformation is localized within the weak layer.     This must be compared to the average stress drop for large and great earthquakes which varies between 1 and a few 10 MPa (Kanamori, 1994).

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However, the force is weaker and weaker as k decreases since F+k u. In the other limit   where k Pk (no weakened zone), strain localization disappears (u "u "u/2) and the force     (k /2)u is of the same order.  Let us translate this spring model by considering that the weak spring has a size b corresponding to the width of the gouge zone and the strong spring has the width ¸ of a tectonic plate. The previous calculation gives the stress p within the weak zone  u k  . (8) p "  1#(¸/b)(k /k ) b   Let us take ¸/b+100 km/1 km. If k /k +1, we get u +u/100 (no strain localization) and    p +k (u/¸)+10(5/10)"0.5 MPa. On the other hand, if the gouge is soft, i.e. k +k /100, then     u +u/2 (strain localization) but p "k (u/b)+k (u/¸), which gives the same order of magnitude:     the enhancement of the strain is exactly compensated by the softening of the gouge. We thus see that the largest stress drops that are measured (for instance 30 MPa for Landers earthquake of June 18, 1992, M +7.3) cannot be accounted for by a localization of the strain within a soft gouge. 1 This simple calculation shows the di$culty in having both strain localization and signi"cant stress build-up by invoking solely a mechanical deformation process. We could have guessed this result from the start from stress conservation. A set of stress-"eld indicators, including borehole breakouts, earthquake focal mechanisms and hydraulic fractures, suggests that many (but not all) active faults are sliding in response to very low levels of shear stress (Zoback, 1992a,b). For instance, the maximum horizontal principal stress, as measured by borehole breakouts and hydraulic fractures, is nearly perpendicular to the San Andreas fault in Central California. Within the standard mechanical model, this implies that the San Andreas fault is sliding at a very low shear stress. It is very important to stress that this conclusion is reached by using the usual solid friction law and is thus model-dependent. Within alternative scenarios, this conclusion is no more necessary. Also noteworthy is the paradox of large overthrusts (Brune et al., 1993), namely that thrust faults exhibit an orientation too close to the horizontal to obey the usual friction law, thus also requiring an anomalously small resistance to friction. Such low-angle faulting is observed in many places and is in contradiction with Anderson's classi"cation of faults (Anderson, 1951) and with the friction theory usually used for the other modes of faulting. The conclusion on this weakness of faults is contrary to traditional views of fault strength based upon laboratory experiments and creates a serious problem as one cannot rely directly on the knowledge accumulated in the laboratory. The situation is made even more confusing when one refers to the occasional observations of very high stress drops (30 to 200 MPa) in moderate earthquakes (Kanamori, 1994). This would indicate that the stress drop (and therefore the absolute stress level) can be very large over a scale of a few kilometers. But then we should see large strain and anomalous heat #ux due to frictional heating (see below).

 Thrust faulting refers to a mode of sliding during an earthquake in which one block moves upward with respect to the other block on the other side of the fault, so as to accommodate a large scale shortening (or compressive) deformation.

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5.2. Proposed resolutions To account for these puzzles, many suggestions have been made. Let us mention in particular that Chester (1995) has proposed a multi-mechanism friction model including cataclitic #ow, localized slip and solution transfer assisted friction in order to describe the mechanical behavior of the transitional regime at mid-crustal depths. His conclusion is that, on the basis of existing friction constitutive relations, only very thin faults (about 1 mm thick) can display the rate dependent characteristics (such as stress drop) necessary for initiation of seismic slip to any signi"cant depth in the crust. Blanpied et al. (1995) also use the multimechanism friction model for frictional slip of granite. They stress that extrapolating the laboratory results to conditions not encompassed by the data set (i.e. to approach the conditions in the crust) is uncontrolled as many mechanisms are competing in a complex way. Even the #uid-assisted deformation mechanisms, such as solution precipitation creep, often invoked to explain the apparent low strength of some major faults actually lead to velocity strengthening stable slip. An honest statement is probably that we do not have enough informations, in view of the complexity of all the competing processes, to determine with some certainty the rheology at seismogenic conditions in the crust. The solution which is often proposed to resolve this problem is to invoke #uid pore pressure. The presence of a #uid decreases the normal stress exerted on a fault and thus decreases the driving shear stress that is necessary to reach the sliding threshold while keeping the most favorable slipping direction unchanged. This last condition ensures the compatibility with the in-situ stress inversion measurements. The problem however remains to "nd a mechanism for pressurizing fault #uids above the hydrostatic value o gh towards the lithostatic value o gh+3 o gh in short time scales      compatible with earthquake cycles. Several scenarios have been proposed in which low fault strength results from high pore pressure. One model invokes the expansion of #uid pores from co-seismic shear heating (Lachenbruch, 1980), but it is hard to understand how this mechanism would work to trigger an earthquake. Another mechanism uses a mantle-derived source of #uids to maintain overpressure within a leaky fault (Rice, 1992). To work, the amount of #uid must be a function of time in a way which is poorly constrained. Another mechanism is suggested by recent laboratory sliding experiments on granite which show that the sliding resistance of shear planes can be signi"cantly decreased by pore sealing and compaction which prevent the communication of #uids between the porous deforming shear zone and the surrounding material (Blanpied et al., 1992). The problem however, not tested in the laboratory experiment, is to understand what causes the change in permeability and the sealing of the pores (which are imposed exogeneously in the experiments). Several works have explored the mechanisms and characteristic time scales of pore and crack sealing, with the conclusion that crack sealing occurs more by pressure solution with mass transfer with time scale of years to centuries rather than by fast self-healing (Gavrilenko and Gueguen, 1993; Renard et al., 1997, 1998; Gratier et al., 1998). While at "rst view, this mechanism seems reasonable, it is at variance with other well-documented e!ects of #uid on rocks under stress, namely hydrolytic weakening and stress-corrosion subcritical crack growth (Scholz, 1990). It is thus hard to justify the postulate (Blanpied et al., 1992) that `a fault remains sealed inde"nitely once pressure seals are in placea. This assumption is also in disagreement with the relatively high values of crust permeability inferred from in situ phenomena which indicate that the rocks are in general close to hydrostatic (Brace, 1980), implying that water at depth is su$ciently connected to the free

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surface. Similar weakening of the shear strength may also be attained by the presence of a relatively soft gouge layer surrounding the harder rock (Lockner and Byerlee, 1993). However, in this last case, the low apparent friction is no more compatible with the orientation of the fault and in-situ stress measurements. Another problem is, as already mentionned, that #uid pressure close to lithostatic value implies #uid trapping and absence of connectivity with larger reservoirs and the upper surface. Thus, the overpressurization can only work in small local fault regions, which become similar to open cracks. It is hard to conceive that the overpressurization will invade the whole length of the fault, in view of the amount of available #uid. The friction would thus be decreased or even suppressed only on localized region and it is not obvious that this would decrease signi"cantly the e!ective macroscopic friction. To summarize, it seems that invoking #uid overpressure to weaken the e!ective friction coe$cient serves a single purpose and creates novel di$culties to interpret other observations. For instance, how to rationalize within the #uid pore pressure scenario the observation by Kanamori et al. (1992) and others that the longer the interval between two earthquakes on the same fault, the larger the stress drop?

6. The heat 6ux paradox 6.1. Statement of the paradox The heat #ow paradox (Hickman, 1991) in a seismically active region was "rst proposed by Bullard (1954): to allow for large earthquakes, a fault should have a large friction coe$cient so that it can store a large amount of elastic energy and overpass large barriers. However, if the dynamical friction coe$cient is large, large earthquakes should generate a large quantity of heat due to the rubbing of the two surfaces, which is not easily dissipated in a relatively insulating earth. Under repetition of earthquakes, the heat should accumulate and either result in localized melting (which should inhibit the occurrence of further earthquakes because slow continuous sliding should then occur) or develop a high heat #ow at the surface. Observations over the entire state of California have shown the absence of anomalous heat #ow across the major faults (Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1980, 1988, 1992, 1995; Sass et al., 1992). Quantitatively, consider a 1906-San Francisco earthquake releasing about E+10J over a rupture length of 400 km and occurring every ¹+100 yr. Assume that a major fraction is dissipated during the rupture in a localized domain around the fault due to friction, rupture, and similar processes. To get an upperbound, we suppose that all the energy is converted into heat. Neglecting the "nite width of the lithosphere, the one-dimensional heat di!usion problem is easily solved (Turcotte and Schubert, 1982) (p. 188). An impulsive heat pulse di!uses with the heat di!usion coe$cient i"k/oc, de"ned in terms of the coe$cient k+2 W m\ K\ of thermal conductivity, the density o and the speci"c heat c of the crust. For the lithosphere, i+1 mm/s  This is probably an upperbound because it is estimated by assuming that the stress drop is equal to the shear stress before the earthquake (Knopo!, 1958), while it is generally thought that the stress drop is a fraction of the pre-rupture stress.

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typically. Over the repeat period of 100 yr, an impulsive source of heat di!uses over the typical scale (it+50 m, which is signi"cantly smaller than the broken zone width (about a few hundred meters to a few kilometers in a mature fault such as the San Andreas). It thus makes sense to use a continuous approximation (low-pass band "lter) and consider that the amount of heat generated by a series of repeating earthquakes with period 100 yr is equivalent to a localized continuous heat source with output power q equal to the average of the impulsive contributions from earthquakes.  This yields E . q +  ¹S

(9)

Using E+10 J, ¹+100 yr+2.5;10 s and S"400 km;15 km (heat is generated in a narrow zone along the rupture surface of depth approximately given by the brittle crust depth), we get q +67 mW/m. Neglecting dissipation, we can calculate the surface heat #ux q(x) perpendicular  to the fault, which is generated by this constant localized source q . It is given by  q(x, t)"q erfc(x/2(it), where erfc is the complementary error function. In the presence of the  upper crust surface, this horizontal heat #ux must also translate into a vertical heat #ux of the same magnitude. We thus see that one should expect an anomalous heat #ux of large amplitude localized in the vicinity of the fault. Our approximations provide an upper bound for this anomalous #ux, equal to q +70 mW/m. We get the same order-of-magnitude estimate by following (Lachen bruch and Sass, 1980, 1988, 1992; Lachenbruch et al., 1995) and calculating the frictional heating as the work v p per unit time and unit volume done by the fault moving at its average tectonic velocity v (a few centimeters per year) under its average stress p. This is also the same as the work < p dt/¹ per unit time and unit volume done by the fault moving during an earthquake at particle velocity <+1 m s\ over a time dt"du/< to produce the slip du. The normalizing factor dt/¹ accounts for the fact that this heat dissipation only occurs during the brief instant of time dt of the duration of the earthquake compared to the large recurrence interval ¹ between earthquakes. The estimation q +70 mW/m is a signi"cant e!ect, comparable to the mean surface heat #ux  in California equal to about 50 mW/m. Over one hundred heat #ow measurements in the San Andreas fault zone give no evidence for any local frictional heating on the main fault trace at any latitude over a 1000 km length. These measurements provide an upper bound for the frictional resistance allocated to heat production of about 10 MPa (Lachenbruch, 1980; Lachenbruch and Sass, 1980, 1988, 1992; Lachenbruch et al., 1995). 6.2. Proposed resolutions Since the previous estimations are upper bounds, we have probably overestimated the fraction of energy allocated to frictional heating and this is the "rst possible explanation. But present measurements are able to detect anomalies of less than 10% of the average #ux and this should be enough to access to the frictional heating. Another standard explanation is that no signi"cant heat is generated because the dynamical friction coe$cient is low on the most rapidly slipping faults in California. Two classes of models are then usually proposed. In the "rst class, the low dynamical friction is produced during the event itself. Various mechanisms are invoked, such as crack-opening modes of slip (Brune et al., 1993;

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Anooshehpoor and Brune, 1994; Schallamach, 1971), dynamical collision e!ects (Lomnitz-Adler, 1991; Mora and Place, 1994; Pisarenko and Mora, 1994; Schmittbuhl et al., 1994), self-organization of gouge particles under large slip (Lockner and Byerlee, 1993; Scott et al., 1994; Scott, 1996), acoustic liquifaction (Melosh, 1996) and so on. In the second class of models, the fault has a low friction before the onset of the event. This may be due to the presence of low-strength clay minerals such as Montmorillonite (the weakest of the clay minerals) (Morrow et al., 1992), to an organized gouge structure similar to space "lling bearings with compatible kinematic rotations (Herrmann et al., 1990), to the presence of phyllosilicates in well-oriented layers (Wintsch et al., 1995) or to the existence of a hierarchical gouge and fault structure leading to renormalized friction (Schmittbuhl et al., 1996). Much attention has also been devoted to the role of overpressurized #uid (Lachenbruch, 1980; Byerlee, 1990; Rice, 1992; Blanpied et al., 1992; 1995; Sleep and Blanpied, 1992; Moore et al., 1996): if water, believed to be ubiquitous within the crust, is connected to the surface, its pressure is given by the hydrostatic value o gh+100 MPa at 10 km depth (o is the density of water); on the other U U hand, if pockets of water are trapped, the #uid pressure in these pockets becomes equal to the lithostatic value ogh, where o is now the density of rocks. This leads to a pressure about three times larger ogh+300 MPa at 10 km depth. In this situation where the #uid pore pressure is elevated to values close to the normal compressive stress (lithostatic pressure), the fault could be weakened enough to exhibit a weak stress threshold and friction. Recall that, in the friction model, a fault is strong if the normal stress is strong to hold the two surfaces of the fault together. If the interstitial #uid pressure decreases signi"cantly the normal stress, the fault becomes weak. Fluid pressure close to lithostatic value implies #uid trapping and absence of connectivity with larger reservoirs and the upper surface. Thus, the overpressurization can only work in small local fault regions, which thus become almost free, similarly to open cracks. It is hard to believe that the overpressurization will invade the whole length of the fault, in view of the amount of #uid that would entail. The friction would thus be decreased or even suppressed only on localized region and not on the whole fault. In addition, laboratory data (David et al., 1994) suggest that a relatively high in#ux of #uid from below the seismogenic layer is necessary to maintain pore pressure at close to the hydrostatic value, as postulated in the Rice model (Rice, 1992). Scholz (1992b) has noticed that the #uid pressure scenario has one major problem that remains to be resolved, namely that a brittle material can never resist a pore pressure in excess of the least compressive stress p without  drainage occurring by hydrofracturing. p could then be the #uid threshold of a rupture instability  which could possibly trigger an earthquake. However, this mechanism does not lead to a small friction coe$cient. Segall and Rice (1995) have developed a model of unstable slip of a #uid in"ltrated fault using a rate and state dependent friction model with dilatancy and pore compaction. For earthquake to nucleate, they "nd that the #uid pressure must not be large and this places contraints to rationalize the absence of heat #ow anomaly on the San Andreas fault. Furthermore, their numerical simulations do not exhibit large interseismic increases of the #uid pressure, in contrast to the often proposed scenario. There is evidence that the main form of dissipation of the released tectonic elastic energy is not due to frictional heating. The total elastic energy released during an earthquake can be estimated using the method of static elasticity. The idea is to calculate the energy di!erence of the crust in two states, before and after the earthquake. The earthquake is modelled by the introduction of a shear fault with a stress on it which is reduced with respect to the preexisting uniform shear stress by an

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amount equal to the earthquake stress drop. The total energy released is a quantity which is not measurable directly but nevertheless is the reservoir of the various possible forms of energy dissipation that take place during an earthquake. The total dissipated energy comprises seismic wave radiation energy, which is the usual measured `earthquake energya by seismic methods, the heat dissipated in friction (not directly accessible), the energy consumed in new cracking and fragmentation in the gouge (King, 1983, 1986) (not directly accessible) and in the growth of the fault (Andrews, 1989) (not directly accessible), as well as other relaxation phenomena, including #uid #ow and ductile deformations and possibly polymorphic transformation, chemical and electrical processes. Friction, ductile deformations and processes with latent heat would generate heat, while radiation carries the energy far away and the rupture and fragmentation consume energy by the creation of new surfaces. The scenario of a low heat generation thus leads to a large dissipation in the other modes, i.e. seismic radiation and fresh rupture. Measurements of the radiated seismic energy give uncertain values, due to the need to correct for attenuation and propagation dispersion (Kanamori et al., 1993; Choy and Boatwright, 1995). For Loma Prieta 1989 earthquake, M "7.1$0.3, the radiation energy is approximately E "10!8;10 J, which must be 1 1 compared with the static elastic calculation using a simple shear model (Knopo!, 1958) giving a total released energy p E + Gd¸ . 2 8

(10)

For a slip d+1.6 m over a rupture length ¸+40 km, we get E +1.3;10 J, which is in the low 2 end of the interval for E . Similar results are found for other large earthquakes. This indicates that 1 the dissipation by seismic radiation is not negligible and could be a dominant mode. Kanamori and Anderson (1975) have proposed that the seismic radiation e$ciency is given by 2(p !p )/(p #p ),     i.e. the ratio of the stress drop over the average stress. In the standard picture, a large seismic radiation would thus need a small absolute shear stress. This seems in contradiction with the stress drop measurements in the range 0.1}10 MPa which are much smaller than the typical lithostatic stresses in the range of hundreds of MPa. Recent stress measurements by boreholes near active faults suggest near-complete stress drop (Barton and Zoback, 1994). We note that a large radiation e$ciency has important consequences on the form of the organization of earthquakes and their repetition (Xu and Knopo!, 1994). Other suggestions to explain the absence of an anomalous heat #ux near active faults include geometric complexity in the San Andreas fault at depth, hydrothermal circulation and missing energy sinks (see Hickman, 1991, and references therein).

7. Water and rocks: metamorphism and faulting 7.1. The ubiquity and importance of water It is more and more recognized that #uids play an essential role in virtually all crustal processes (O'Neil and Hanks, 1980). The solubility of water is typically 0.15% by weight in quartzite. Clay can contain up to 5}10% of water. Hickman et al. (1995) review the historical development of the

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conciousness among researchers of the ubiquitous presence and importance of #uids within the crust. Numerous examples exist that demonstrate water as an active agent of the mechanical, chemical (Wintsch et al., 1995) and thermal processes that control many geologic processes that operate within the crust (National Research Council, 1990). It is now well-established that the presence of water governs essentially all the physical processes in quartz, from the piezoelectric properties to dislocation mobility and to recrystallization. Older experimental results published before this recognition must thus be reassessed with caution. In other words, the mechanical properties of quartz, and probably of other quartzite minerals, are not to be explained in terms of intrinsic mechanisms. In the same vein, we will review below the fact that phase transformations documented in thermodynamic equilibrium may be drastically modi"ed in the presence of water and deformation, thus enriching considerably the possible behaviors of rocks. The bulk of available information on the behavior of #uids comes from observations of exposed rocks that once resided at deeper crustal levels. These outcrops give a series of `snapshotsa (with often multiple exposures) that have the disadvantage of not providing direct samples of #uids and of its role in crustal development. In any case, present day surface exposures indicate that, at all crustal levels, #uids have been present in signi"cant volume. Exposed metamorphic rocks indicate large #uxes of #uids. Fluids have been directly sampled at about 11 km by the Soviets at the Kola Peninsula drillhole. All these data collectively support the existence of #uid circulation and therefore porous rocks and/or fractures to crustal depths of at least 10}15 km. Studies indicate that chemical modi"cation which require signi"cant #uid volume has occurred, contrary to inferences based solely on mechanical properties. The advection of chemical components by #uid #ow and dispersive #uxes from one chemical environment to another causes chemical reactions between minerals and #uids, such as dissolution, precipitation, ion exchange and sorption. The closed nature of many mineralized fractures illustrates the important interaction between the chemistry and physical transport properties of the crust. It is recognized that moving #uids that transport both heat and mass facilitate the precipitation of minerals, dissolution of the rock matrix, ion exchange, and a host of potential chemical reactions. The system is often considered dynamic with reactions occurring in the moving #uid. The #uid can dictate the style of metamorphism, the extend of chemical equilibrium and much about the texture fabric observed in rocks. The evidence that large amounts of #uid #ow occurred in deeply buried rocks consists of documented chemical changes in mineral assemblages. Because the porosity of metamorphic rock is probably less than 1%, the high calculated #uid necessary to produce observed chemical changes suggests that #uid must have been replenished thousands of times. If #uid #ux through the metamorphic rock is not continuous, many generations of microcracking will form. Sibson (1982) and Sibson et al. (1975) document the liquifaction of fault, i.e. the release of #uids during and after faulting. It has been argued that rock permeability and thus microcracking adjusts itself, so that #uid pressure is always close to rock pressure irrespective of the extend of hydration/dehydration (Walther, 1990). However, the mechanism by which this would occur is not clear. To operate in this state, feedback processes must exist. In this vain, Nur and Walder (1990) have proposed a timedependent process to relate #uid pressure, #ow pathways and #uid volumes. This is in order to

 A rock whose original mineralogy, texture or composition has been changed by the e!ects of pressure, temperature or loss of chemical components.

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reconcile the competing ideas of high crustal permeabilities and associated hydrostatic #uid pressures with evidence for hydraulically sealed low-permeability rocks with elevated #uid pressures. They assume a tendency of sealing and healing which produce a progressive decrease of the porosity. If the permeability decays faster than the available volume, #uids will be trapped and will be raised to the lithospheric pressure. This high pressure can then break the surrounding rocks in tension, freeing the #uid which depressurizes down to the hydrostatic pressure. Because the process of porosity reduction continues, the #uid pressure will again build up towards lithostatic, leading to another cycle of hydrofracture, #uid release, and sealing, etc. One must take in addition the release of water from transformed minerals. For instance, Montmorillonite changes to illite with a release of free water from the clay structure at approximately the same depth as the "rst occurrence of the anomalous pore pressure (Burst, 1969). This is the most commonly discussed example of hydration and dehydration of minerals changing the #uid mass and the pore pressure. It has been proposed that gold-quartz vein "elds in metamorphic terranes provide evidence for the involvement of large volumes of #uids during faulting and may be the product of seismic processes (Robert et al., 1995). One can suggest an alternative explanation for the apparently opposed views of a large permeability on the one hand and sealed high pressure #uid on the other hand, which is heterogeneity. It is clear that the crust is very heterogeneous and the previous scenario neglects this heterogeneity. In the presence of a large distribution of pore and fracture lengths and their opening sizes (Walmann et al., 1996, document a power law relationship between crack opening in tension and crack length), the permeability at large scale is controlled by the large size fraction of the population of openings, while trappings and sealing can occur in the dead ends and small cracks. The two processes can thus coexist without invoking additional mechanisms. Direct evidence of the e!ect of #uid on fault stability has come from earthquakes induced in intraplate regions either through direct injection of #uids down boreholes or from the "lling of large reservoirs with subsequent in"ltration of water into the underlying rock mass (distinct from the direct loading e!ect of the reservoir water mass (Grasso and Sornette, 1998, and references therein). Such induced seismicity by reservoir impoundement and #uid injection could also be a signature of the weakening e!ect of #uids on rocks. The magnitude of #uid pressure transients associated to earthquake faulting have been estimated from the examination of fractures which exhibit di!erent phases of induced dilation by escaping #uids (Fleischmann, 1993).

7.2. Ewect of water on minerals: hydrothermal alteration under stress Polymineralic crystalline rocks, comprising most of the lithosphere, exhibit complex strain partitioning that has been little studied but is probably very important in the chemical}mechanical}electrical processes occurring in the crust. Kirby (1984) concluded in his digest of the papers of the special issue on chemical e!ects of water on deformation and strengths of rocks that it would be premature to establish with con"dence the rheological laws for any crustal rock type under hydrothermal conditions on the basis the presented result. This conclusion still holds more than ten years later. He also stressed the fact that the chemical e!ects of aqueous #uids may be heterogeneously distributed in the crust. Our understanding of the possible interplays between earthquakes and rock transformation is still in its infancy.

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7.2.1. Role of water in the presence of cracks: stress corrosion When the chemical environment of a solid is changed, and some active species adsorbs on its surface, then because the electronic structure of the near-surface region of the solid is changed, so are its mechanical properties. This chemomechanical e!ect are been studied for a long time for metals (Westwood and Ahearn, 1984). The e!ect of water on cracks operates similarly. The presence of water enriches the elastic rebound model of Section 2. The simplest scenario for the preparation of an earthquake which takes into account the presence of water is the following: "rst, the slow straining of the crust due to tectonic forces produces many microcracks throughout the rock. Water di!uses into the cracks and "lls them. Water can in fact be also at the origin of the formation of tension microcracks if sealing occurs in the permeative #uid network allowing the #uid pressure to increase towards lithostatic values. During this period, the volume of the region dilates. In fault zones, these modi"cation and damage brought to the rocks "rst weaken them by various processes including stress corrosion. A threshold is eventually reached at which a major crack can develop along the fault. In this way, the elastic rebound of the strained material can occur during an earthquake. As yet, this sequence of events has never been directly observed before an earthquake. The scenario is more based on extrapolations of many laboratory experiments showing the progressive weakening of rocks with preexisting cracks in the presence of water. Other experiments have shown the localization properties of rocks in narrow shear bands under con"ning pressures comparable to those found in the seismogenic crust. These are the elements that are usually advanced to justify the friction model, which includes the action of water under the form of mechanical overpressure and of chemical attack, as in corrosion and hydrolytic weakening. Stress corrosion occurs in the presence of pre-existing cracks in quartz, quartz rocks, calcite rocks, basaltic rocks, granitic rocks and many other geological materials (Atkinson et al., 1981; Atkinson, 1984). Only limited data exist on crack growth in polycrystalline minerals, but it is clear that grain boundary chemistry and structure dominate the process (Freiman, 1984). Several possible mechanisms of subcritical crack growth are relevant to crustal processes. They include stress corrosion, di!usional mass transport, microplasticity, dissolution and ion exchange. Below 250}3003C, stress corrosion is thought to be the main mechanism. Above 3003C, signi"cant crack healing occurs (Atkinson, 1984). There is no data under conditions that simulate realistically those in the bulk of the earth's crust. In particular, a question can be raised on the existence of other competing processes involving genuine chemical reactions or substitutions (Wintsch et al., 1995). Extensive healing occurs at temperatures as low as 4003C in times as short as 1}2 days in presence of #uids (Smith and Evans, 1984). The evidence is that cracks heal by local transport of silica by di!usion along the crack surface and/or through the pore #uid. The subcritical crack growth induced by stress corrosion has been essentially documented for mode I tension stress. Such loading condition can emerge when a crack is sheared and microcracks in tension then tend to grow at the tip in a direction parallel to the main principal direction (Scholz, 1990, p. 27). As discussed above, pore sealing can increase the #uid pressure and lead to tension microcracking. Extrapolation to shear cracks is an open question. More generally, it is important to note that, while almost all quantitative rheological data for rocks have been obtained from axi-symmetric compression experiments, it is widely accepted that most natural rock deformation phenomena are non-coaxial, such as in fault shear zones. Franssen and Spiers (1990) show from experiments on polycrystalline salt that there is no justi"cation that #ow laws derived from coaxial

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experiments can be generalized to non-axi-symmetric deformations. The anisotropy is in addition enhanced by the development of anisotropic mineral textures, such as crystallographic preferred orientations within a deforming material or by microstructural changes such as dynamic recrystallization or grain boundary alignment. 7.2.2. Hydrolytic weakening In triaxial experiments under dry conditions, it is seen that quartz deformation is not observed, in disagreement with the large deformations of quartzites seen at relatively low temperatures. Indeed, quartz appear to have deformed in crustal rock samples while other adjacent minerals, known to be weaker in the laboratory, have not. As discussed by Goguel (1983) and others, what is missing is the in#uence of #uids. Hydrolytic weakening (the weakening of quartz by included water) could well be the resolution of this enigma and there is much continuing research to quantify this phenomenon. Water alters the yield strength of quartz, reducting it from 2 GPa to 200 MPa at 5003C. It is found that water molecules are the dominant hydrogen containing species in synthetic quartz and this water is mainly in tiny atomic inclusions not large enough to form ice when cooled (Aines and Rossman, 1984). It is di$cult to extend our imperfect knowledge of hydrolytic weakening in the laboratory to in situ conditions (Ranalli, 1987). Water may have a chemical e!ect on minerals directly, without the need for pre-existing cracks. The micro-cracks may even result from the chemical action of water according to the three mechanisms of silicon-oxygen bonds replaced with hydrogen bonds, dissolution of chemical impurities and ion exchanges. It is well-documented that the strength of wet rocks is less than the strength of dry rocks under most crustal conditions. For example, marble masons know that marble can be deformed continuously using a humid sponge! Hydrolytic weakening denotes the general softening e!ect of water on rocks and minerals. Its seems to act above a critical temperature which is inversely related to water content. The observed (p,¹)-dependence of weakening in single crystals is consistent with the variations of water solubility and di!usivity in quartz. Griggs and his team (Griggs et al., 1957; Griggs and Handin, 1960) and many other groups have developed a wealth of laboratory experiments of rock deformation under high con"ning pressure and temperature. Modern experiments have shown that, in the absence of non-hydrostatic stress, it is not possible to cause hydrolytic weakening of dry quartz (Kronenberg et al., 1986). This is due to the fact that the penetration of water is extremely small in pure dry quartz and hence the solubility and di!usivity of water in bulk quartz is very low. Molecular water in tiny #uid inclusions gives rise to a broad absorption band while hydroxyls with precise allocation in the crystal lead instead to polarised sharp absorption peaks. The exact position of these peaks depends on the environment of the vibrating OH and this has led to quantitative estimates of the H/Si ratio in wet quartz and other minerals (Paterson, 1982). According to recent measurements, the solubility of water in quartz is probably signi"cantly less than 100 H per 10 Si and this does not appreciably change with heating (up to 1100 K or pressure build-up up to 1.5 GPa). Water di!usivity in quartz appears to be strongly dependent on pressure, stress and temperature. Also, the solubility of water in quartz increases with temperature and pressure. It is also probably  A very hard, nonfoliated, white metamorphic rock formed from a sandstone rich in quartz sand grains and quartz cement.

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dependent on the deviatoric stress, i.e. on the anisotropic distorsion of the crystal mesh. Note that the e!ect is probably weak for quartz, which has a weak anisotropy as measured by the elastic coe$cients or the energies of dislocations as a function of their orientation. For application to the conditions reigning in the earth crust, the presence of numerous interfaces and defects is a favorable circumstance for the exchange of water within minerals. It must also be recognized that water is a very special #uid (Poole et al., 1992, 1993, 1993a, 1993b, 1994; Sciortino et al., 1990) and can be structurally, molecularly or gel bound (Bambauer et al., 1969). A ductile deformation of quartz, which occurs due to the activation of slip systems, is linked to the hydrolytic weakening e!ect and also probably to distorsion and structural transformation in the minerals. The transformation temperature increases with hydrostatic pressure and applied stress but decreases with water content. The change of slip system by weakening of quartz crystal by di!using in water and increasing temperature is coincident with the hydrolytic weakening temperature (Blacic and Christie, 1984). Observations, based on the orientations of optical deformation lamellae, indicate a change of slip systems as well as the deformation mechanism at the onset of hydrolytic weakening. It has been documented that dislocation kink propagate easily on adjacent Si}O}Si bridges which are hydrolyzed, permitting dislocation motion by hydrogen bond exchange. The Peierls'stress is also decreased by hydration. There is not a complete microscopic understanding of hydrolytic weakening in the bulk. There appear to be four possibilities for the action of water (Jones et al., 1992). E Molecular water residing interstitially in c channels. Small amounts of water in minerals can be detected by infrared absorption in the wavenumber range 4000!2500 cm\, which corresponds to the absorption of OH stretching vibrations. E ¹he hydrogarnet defect involving the replacement of a SiO unit with two water molecules (in  other words, this amounts to substitute a Si by four H atoms); Infrared bands seen around 4500 cm\ and 5000 cm\ in wet quartz could be due to this hydrogarnet defect. The stable con"gurations of the hydrogarnet e!ect can be calculated by numerical methods (Purton et al., 1992). E ¹he Frank}Griggs}Blacic defect which involves the hydrolysis of the strong }Si}O}Si} bridges by water molecules: (}Si}O}Si})#H OP(}Si}OH ) HO}Si) . (11)  The detailed impact of this hydrolysis on the glide motion of dislocation by motion of kinks or the climb and recovery rather than glide is not ascertained (Paterson, 1990). E Molecular water found in dislocation cores or at grain boundaries whose concentration is defect dependent. Dislocation mobility in silicates is known to be strongly in#uenced by the presence of structurally bound water. The concentration of structurally bound water in silicates is dependent upon the chemical and physical environment. Discussions of the supposed mechanism of hydrolytic weakening is often by analogy with dislocation processes in metals, which neglect the charged nature of defects in minerals which can have signi"cantly di!erent e!ects on plastic deformation (Hobbs, 1981). Electronic}chemical aspect of deformations in minerals has received less attention (Blacic and Christie, 1984, and references therein). The electronic defect center theory of Hobbs (1981) as well as the Frank}Griggs}Blacic model of a lattice resistance mechanism by dislocation kink nucleation seem both to have

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explanatory powers but none adequately explain all the existing data. The mechanism(s) of hydrolytic weakening is (are) still uncertain and controversial. In addition, one must consider the presence of species which can undergo ion exchange with species in the solid phase. If there is an important mismatch in the atomic size of the species, lattice strain can result from ion exchange. An example is the exchange of H> by Na> in silicate glasses (Atkinson, 1981). The majority of silicates, especially the common rock-forming minerals, are wide-band gap materials, with an optical band gap in excess of about 7 eV (Nitsan and Shankland, 1976; Hobbs, 1984). Since optical bands corresponds to relative motions of atoms within the crystal, this high gap re#ects the rigidity of the atomic structure of silicates. Indeed, their crystal structures are dominated by the (SiO )! tetrahedron. Inclusions of charged atoms (Al, H, Na,  Fe, etc.) disturb silicon sites and sometimes decrease the band gap as well as create impurities levels within the gap. Atomic inclusions act as acceptors or donors of electrons and thus play an important role in the electric properties as well as structural properties of the silicates (Hobbs, 1984). For instance, the incorporation of hydrogen, as a donor of electron, is quite common. The in#uence of an impurity depends on the mode of incorporation, as an interstitial or substitutional atom or group of atoms in the crystal structure of the material. The importance of the inclusions stems from the fact that, in quartz for instance, electrical conductivity is normally ionic, the charge carriers being interstitial sodium, lithium, etc. The main hydrogen defect introduced in silicates seems to be the `hydrogarneta defect, where a (SiO )\ group is replaced by a (H O )\ group    (Hobbs, 1984). Hydrogen in minerals most frequently occurs bonded to oxygen. The resulting OH group is highly polar. We can thus expect a strong coupling between electric and mechanical structures and properties. In a series of works (Jones et al., 1992; Heggie, 1992; Heggie et al., 1992), density functional and quantum mechanical calculations of the structure of water in quartz, its di!usion and the role of dislocations have shown that there are important changes to the quartz crystalline network caused by the nearby water molecules. Calculations of defects in quartz are thus likely to be incorrect unless they take into account the perturbation of the network. The e!ect is less pronounced when the water molecule is located in a dilated dislocation core and this leads to a much reduced activation energy for dissolution and di!usion. Examination of dislocations in the basal plane of a-quartz has shown that the activation energy for di!usion of water molecules can drop below 1 eV due to the interaction with the dislocation core. The structure of the dislocation cores is itself in#uenced by the presence of water molecules. Similarly, reduction in kink-pair formation energy from 5 eV for dry quartz to 1 eV or less for wet quartz is possible, indicating that wet quartz would deform relatively easily, provided that water could di!use to dislocations. It has been shown that water cannot only di!use easily along dislocations in quartz, but also be `pumpeda along them (Heggie, 1992). Numerical simulations indicate that water molecules can dissociate in dislocation cores and the protons and hydroxyl groups become strongly bound to kinks. A shear stress can do work on kinks to move them along a dislocation, dragging with them water-bearing species and sweeping other undissociated water molecules before them. This process underlines the importance of microcracking as stress enhancement at crack tips creates and moves wet dislocations which are fed by a reservoir of water in the crack. The dislocations themselves facilitate the spread of water into the crystal and in so doing catalyse their own motion. These observations underlie the interplay between plastic deformations (involving dislocations) and the presence of water.

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In sum, hydrolytic weakening occurs only when defects (such as dislocations) are present in rocks and in the presence of non-hydrostatic stresses. Hydrolytic weakening has mostly been studied for quartz but occurs also in many other minerals constituting rocks. It belongs to the general domain of `chemomechanicala e!ects discovered by Rebinder in 1928 (see Westwood et al., 1981, and references therein) where several basic mechanisms have been found to operate, none of them su$ciently well-understood. 7.3. Role of water in rock metamorphism and relation to earthquakes There is overwhelming evidence from geological observation of rock that mineral metamorphism is associated to faulting. Faults thus provide an obvious relationship between earthquakes and metamorphism. But is the relationship only structural, with earthquakes and metamorphism occurring on the same geometrical structure at separate times with no real interaction? Rock metamorphism is the domain of study of geology and petrology. It involves long time scales in comparison to the relatively short recurrence times between large earthquakes. We summarize a portion of the literature which presents evidence of the complex chemical and transformation reactions occurring within fault zones in the di!erent (relatively rare) parts of the world where evidence has been brought to the observable earth surface. We note that the available evidence consists in the observation of shape and chemical patterns, which are the signature of maybe a succession of processes. The inverse problem of the inference of the causative factors is obviously quite ill-de"ned and unstable. It is thus necessary to constraint the inversion by a priori information. The interpretations are thus given within a priori given physical models, such as friction or rupture. It is important to keep in mind this procedure for a possible reassessment of the data in the light of di!erent models and new information. The informations extracted from the data are thus heavily model dependent. Chemical e!ects on deformation are generally not understood so well because of their manifold and complex nature. These coupled e!ects are in some instances cooperative and in others competitive. As as been emphasized by many authors (Carter et al., 1990), metamorphic reaction kinetics and thermally activated deformation rate processes are tightly interwoven. There is increasing evidence of strong coupling between deformation and metamorphism (Carter et al., 1990). The metamorphism is known to start at a minimum temperature of 3503C but this depends considerably on the chemical/water conditions. Evans and Chester (1995) have documented the fact that #uid-driven reactions, neomineralization and veining occurred during faulting in a small region near the fault zone less than a meter wide. This extremely narrow fault core contains ultracataclasite and foliated cataclasite (i.e. highly fragmented rocks). Geochemical combined with microstructural and mineralogic data show that the degree and style of #uid-rock interaction in the fault core vary along the strike of the San Gabriel fault. Hydrothermal mineral reactions can occur at temperatures well below those necessary for the plastic #ow of quartz, and can dramatically lower the ductile strength of the granite (through presumably basal plane dislocation glide in the mica) (Janecke and Evans, 1988). Stress-enhanced hydrothermal mineral reactions are also recognized to be important in weakening crustal rocks, even when both the reactant and product phase are strong (Rubie, 1983; Pinkston et al., 1987). Wintsch et al. (1995) have speculated that a hydrated phyllosilicate-rich fault rock slipping

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predominantly by dislocation glide may develop a low and pressure-independent shear strength approaching that of mica single crystals, without #uid overpressuring, if the phyllosilicate grains are preferentially oriented and highly contiguous. This reaction softening is promoted by the presence of Mg-rich wall rocks. Vein textures and rock chemistry indicate that the veins episodically dilated and collapsed during the history of #uid in#ux and out#ow, suggestive of an interconnected fracture network that grew either quasi-statically or suddenly during earthquakes. It must be stressed that stress heterogeneity induced by fault rupture and deformation can lead to considerable uncertainties in inferring past #uid pressures from observations of vein geometry in outcrop. No reliable method currently exists to distinguish the cataclastic products of seismic versus aseismic slip in fault zones. Much less data exist on rock-forming silicates (other than quartz) on their strength and ductile deformation as a function of pressure, stress, deformation rate, temperature and water content. This is why, as we have already pointed out, most of the evidence on the e!ect of water has been documented for quartz. Natural quartz crystals are very strong; by contrast, there is abundant microscopic evidence in quartz-bearing rocks that quartz has #owed extensively in nature and was weaker than other minerals, such as the feldspars, under most conditions in the earth's crust (Blacic and Christie, 1984). As explained above, this paradoxical contrast between high strength of quartz in the laboratory and its apparent weakness in nature has been qualitatively explained in terms of dissolved water content and the e!ect of low strain rates on the #ow behavior (Griggs and Blacic, 1965). There is a strong contrast between very strong, dry natural quartz crystals and their hydrolytic weakening forms. There are many indications that crystal plasticity and super-plasticity occurs at pressure and temperature conditions found in the brittle part of the crust. The plasticity is associated with metamorphism. Fine-grained albite microstructure formed in feldspar prophyroclasts have thus been documented in granite and quartz-felspar mylonite. In these systems, millimetric layer partitioning and grain sliding in a very-"ned-grained mixture have been found (Behrmann and Mainprice, 1987). Microstructures in cataclasites, for instance in basement rocks of the Scandinavian Caledonides, exhibit many forms: microcracks "lled with "ne-grained mineral fragments, quartz crystals with euhedral terminations, undulose extinction, deformation lamellae and grain boundary recrystallization, narrow banding parallel to rhomb and prism planes, vein-like structures. The interpretation is usually that formation of cataclasites is a cyclic operation of brittle failure, crystal growth followed by plastic deformation (Stel, 1981). Deformed calcite relics in deformed marbles from the Mt. Lofty Ranges, South Australia, show abundant evidence of strain-induced migration of twins and grain boundaries, together with static and dynamical recrystallization (Vernon, 1981). Fluid inclusions rapidly change shape and re-equilibrate upon change of temperature and pressure (Pecher, 1981). Deformation tests on aplite at constant strain rate (10\ s\), at 210}250 MPa con"ning pressure and at 2003C and above and at 5603C and

 A very "ne grained metamorphic rock commonly found in major thrust faults and produced by shearing and rolling during fault movement.  High pressure, low temperature structures occurring primarily by the crushing and shearing of rock during tectonic movements and resulting in the formation of powdered rock.

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above with varying water contents show a "rst plasti"cation regime (Paquet et al., 1981). In several tectonic settings, a steady-state dynamical recrystallization of quartz grains has been shown to be essentially independent of post-creep processes. In addition, the stresses were estimated 1.5}2 times higher in these zones (Etheridge and Wilkie, 1981). The discrepancy with the stress and friction values found in experiments has led to the suggestion that laboratory experiments have not been able to reproduce the conditions within seismogenic fault zones. Investigations of the mineral chemistry of shear zones in amphibolite facies metagabbro has shown that the composition of the amphibole and plagioclase varies with deformation. The amphibolite varies progressively from an initial magnesio-horn-blende to a ferroan pargasitic hornblende in the shear zone, with alkalties, titane, ferrous and magnesium ions increasing in concentration. Such changes are more usually associated with increasing temperatures, showing the role of deformation to produce complex metamorphic and chemical changes (Brodie, 1981). As we have described, the `water-weakeninga e!ect of quartz has been suggested to be due to the e!ect of OH ions which act as electron acceptors and so greatly enhances the di!usivity of oxygen and of charged jogs on dislocations. Thus, chemical components such as oxygen, water, aluminium and other ions can control the concentrations and mobilities and point defects in minerals. For applications to earthquakes, it is noteworthy that almost all experiments on silicates, sulphides and carbonates have been carried out under unspeci"ed chemical conditions (Hobbs, 1981). It is remarkable that low-grade metamorphic slates and fold belts show crenulation cleavage which progressively obliterated with further increase of metamorphism and cleavage with syntectonic phyllosilicate recrystallization. The "ne fabric results from progressive solution, crystallization and recrystallization superimposed with bending, kinking, and internal rotation achieved by slip parallel to the basal planes of phyllosilicates (Weber, 1981). Interactions between deformation and metamorphism in slates leading to cleavage involves mechanical rotations, solution and crystallization, recrystallization and metamorphic reactions (Knipe, 1981). Variations of grain chemistry and orientation have been suggested to control the di!erent deformation behaviors of di!erent grains within the strain accomodation zones and dictate whether grains bend, kink, dissolve, recrystallize, react and grow. As much as ten potential chemical reactions are been found to be involved in cleavage, ranging from ionic exchange to crystallization from solution and solid state reordering. These processes are in#uenced by deformation with respect to their location and rate. Large normal fault zones are characterized by intense fracturing and hydrothermal alteration and the displacement is localized in a thin zone of cataclasite, breccia and phyllonite surrounding corrugated and striated fault surface (Bruhn et al., 1994; Wintsch et al., 1995). Hydrothermal alteration of quartzo-feldspathic rock at temperature equal or above 3003C creates mica, chlorite, epidote and alters the quartz content. The alteration processes are of course dependent on the minerals. Intragranular extension microcracks, transgranular cataclasite extension fractures, through-going cataclasite "lled shear microfaults are observed to result from di!use mass transfer and intracrystalline (low temperature plasticity). Young faults consists of blocks of relatively intact rock separated by narrow zones of intense deformation where fracture processes dominate (Knipe and Lloyd, 1994). A clear correlation has been observed between cataclastic deformation and mineral on one hand and elemental distribution on the other hand which has been interpreted to result from a deformation-induced dolomite to calcite transformation in the 1}2 m wide shear zone of intense deformation. The transformation has resulted in removal of magnesium from the shear zone, selective deposition of

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calcite as an intergranular cement in cataclasite/microbreccia units and a relative increase in the concentration of detrital quartz and feldspars (Hadizadeh, 1994). Microstructures and particle size distributions in the damage zone of the San Gabriel fault imply that deformation was almost entirely cataclastic and can be modeled as constrained comminution. Cataclastic and #uid-assisted processes are signi"cant in the core of the faults, as shown by pervasive syntectonic alteration of the host rock minerals to zeolites and clays and by folds, sheared and attenuated cross-cutting veins of laumontite, albite, quartz and calcite (Chester et al., 1993). The structure of the ultracataclasite layer re#ects extreme slip localization resulting from a positive feedback between comminution and transformation weakening. Optical and scanning electron microscopy of textures in rocks show that syntectonic alteration of feldspars, the presence of iron oxides in the faults, and late-stage quartz veins attest to the #ow of water during deformation and to the syntectonic development of quartz (Evans, 1990). Fractures in feldspar associated to cataclastic #ow lead to the development of clay-rich cataclasites at low temperature. In feldspars, the silica SiO tetrahedra form a three-dimensional framework in which cation sites accommodate  calcium, sodium and potassium ions. In contrast, clays are mainly composed of micas in which the SiO tetrahedra are linked with cations to form two-dimensional planar sheets. The transformation  from feldspar to mica is an example of the crystalline transformation occurring in deformation. These transformations occur preferentially along feldspar grain boundaries and intragranular fractures. Evans (1990) documents the fact that hydrogen (which has a much larger mobility than oxygen in silicates) is added to the feldspar by the penetration of water in the rocks along intergranular fractures and the development of mica and quartz result in the release of K> and Na> ions. The potential role of crystal chemistry in determining mechanical behavior in multicomponent mineral systems has been highlighted by the recent proposed resolution of a paradox on con#icting data on the relative mechanical strength of muscovite and biotite (Dahl and Dorais, 1996). The basic suggestion is that interlayer-site energetics of micas exert the dominant kinetic control on various processes, including volume di!usion and basal slip. The crystal-chemical resolution of the paradox is based on the fact that the di!erent biotite samples which were compared had in fact di!ering #uorine F(OH) substitutions. The #uorine substitution \ and concomitant removal of hydroxyl H> increases the mechanical strength of biotite by elimination of K>}H> repulsion. This then strengthens the interlayer bonds, and by a stronger binding of K> in the interlayer cavity, opposes more strongly basal slip. This is a simple example where substitution is closely linked with strength and the style of deformation. Testing of these type of crystal-chemical e!ects on rheology and deformation requires mineral specimens of well-controlled composition.

8. Structure and polymorphism of crustal minerals It is also useful to give some informations on the richness of polymorphic phase transformations in rocks as they may have some e!ect on faulting and earthquakes. X-ray crystallography, spectroscopy, electron-spin resonance and proton-spin resonance methods have been used to study chemical reactions between two solid phases (Lonsdale, 1969). Such a reaction is in general controlled by local di!usion of atoms and is most readily occurring in

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powdered or microcrystalline specimens and in defect structures having vacant sites, or at the interfaces. A chemical reaction within a single crystal is characterized by the nature of the change in the substance itself, with a minimum of atomic or molecular movement, or by decomposition, isomerization, dimerization or polymerization generally. The general form of the crystal often remains unchanged so that the "nal substance is pseudomorph on the original. The basic constitutive SiO\ tetrahedra unit of many mineral rocks is quite favorable  in this respect to these types of chemical reactions. Indeed, silicate minerals form complicated structures in which SiO\ tetrahedra are linked through a common oxygen. In this way, it is  possible to form polymer chains (as for instance in the "brous asbestos minerals); in the micas, two-dimensional crosslinking of chains form anionic silicate sheets which are held together through bonds to cations. In the aluminosilicate minerals, such as the feldspars, the SiO\  tetrahedra are linked in a three-dimensional network, where some of the tetrahedra are replaced by AlO tetrahedra.  Silicates are full of structural solid}solid transitions (Salje, 1990, 1991, 1992). Among them, displacive transitions are di!usionless transformation resulting in a local structural modi"cation of the crystalline structure. The locality of the structural reorganization of the atoms is the reason for the di!usionless nature of the transitions. More generally, martensite transformations denotes any di!usionless structural transformation resulting in a long-lived metastable state with a high degree of short-range order. The existence of these structural phase transitions is a signature of the many di!erent conformations that the SiO structural element can take in response to disturbances  created by deformations and inclusions. Silicate framework structures consist of rather sti! or `rigida units, such as SiO\ tetrahedra  and AlO octahedra, rather #oppily jointed at the corners of shared oxygen atoms (Heine et al.,  1992). One can visualize this sti!ness constrast by looking at a two-dimensional perovskite structure: in this structure, the squares in successive layers can rotate alternatively by angles $h giving a rigid unit phonon mode in which the structural units rotate and translate without any distorsion. In quartz and silicates, the tetrahedra are the rigid units which rotate and translate. In quartz, the tetrahedra rotate by 173 with a relative variation of volume *
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Quartz a also transforms into coesite above 2000 MPa at room temperature with a transition pressure increasing with temperature. The inversion from quartz a to coesite is not displacive as bonds must be broken in contradistinction with the transformation to quartz b. The transformation thus involves atomic di!usion (Nicolas and Poirier, 1976). The arrangement of tetrahedra in coesite is very similar to that found in feldspars. These structural transformations can occur either slowly or quite fast, depending on the degree of strain concentration and alteration. Displacive transformations (such as quartz a to quartz b) can occur very fast due to the small motions of atoms involved in the polymorphic transition. The transformation front can propagate very fast, with a velocity comparable to the velocity of sound, as a kind of `domino wavea from crystalline cell to its adjacent cells (Rao and Sengupta, 1997). In contrast, sliding or rupture involve deformation instabilities that need to activate large domains within the minerals and the nucleation of their dynamics is slower (Dieterich, 1992, 1994; Dieterich and Kilgore, 1994, 1996) as it is controlled by the mobility of dislocation and other defects. The kinetics of the transitions but also the stability of the di!erent phases are also in#uenced by the presence of water.

9. The mechanical rupture: possible role of high frequency acoustic vibrations We now examine the co-seismic rupture. It is expected that intense fragmentation and comminution must occur during the sliding, with the generation of intense shaking and high frequency sound waves. This may lead to a kind of partial acoustic liquifaction of the granular gouge, similarly to what Melosh (1996) proposed. Indeed, the gouge zone at 10 km depth can be thought of as a granular medium with a large polydispersity in grain sizes, highly compacted by the overburden lithostatic pressure. Liquifaction usually refers to the experimental observation that granular material in the presence of an interstitial #uid can liquify when shaken su$ciently strongly (Russo et al., 1995; Chirone et al., 1995). The liquifaction is due to the fact that granular media are "rst compressive for small deformation leading to an increase of the interstitial #uid pressure. This increase in turn decreases the friction between the grains that can eventually become free to shear. For example, liquifaction of sediments by resonant ampli"ed seismic waves have been proposed to be in part responsible for the damage and collapse of certain buildings during the Michoacan earthquake, 1985 (Lomnitz, 1987; Campillo et al., 1989). A similar liquefaction is believed to be responsible for the the damage in the Marina district of San Francisco during the Loma Prieta earthquake of 1989 (Bardet, 1990). Melosh (1996) has suggested that acoustic liquifaction occurs without interstitial -uids when the local acoustic pressure, due to the intense multiply scattering high-frequency acoustic waves, can become of the order of the overburden pressure ogh, thus letting free a loosely cohesive granular material.  The structure of coesite (monoclinic system with density 2.93) is composed of SiO tetrahedrons that are linked into  four membered rings. The rings are then linked together into a chain-like structure. This structure is much more compact than the other members of the quartz group, except stishovite, and is re#ected in the higher density and index of refraction.

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For an earthquake comparable to the 1906 San Francisco earthquake, the total released energy is of order 10} J. Let us compare this value with the energy in the acoustic "eld. The typical acoustic material velocity < (which is di!erent from the shear velocity) is such that p"oc<"ogh. The "rst relation p"oc< relates the acoustic pressure to the acoustic particle velocity (o is the material density and c the compressional acoustic wave velocity). The second equality is the condition that the acoustic pressure be equal to the overburden pressure at depth h. This equation states the condition that the acoustic pressure can liquidify only if it balances the compression. Taking p+200 MPa, a density of 3 times that of water and a velocity of order 4000 m/s, we get the very large value for the acoustic particle velocity <+120 m/s. The total acoustic energy is p/2oc times the volume of the earthquake. If we take a volume of 400 km;10 km (active zone depth);&1 m active fault width, this gives an acoustic energy of 4;10 J, about one order of magnitudes smaller than the total energy released by the earthquake. Thus, this mechanism is compatible with the energy balance since only a (rather signi"cant) fraction of the total released energy needs to be in the form of the high frequency acoustic waves. There is however a problem with the Melosh's mechanism because it predicts a slip velocity during an earthquake more than two orders of magnitudes smaller than the typical meters per second for observed earthquakes. The argument goes as follows. On one hand, the intrinsic quality factor Q of the attenuation of the acoustic waves must be high, in the range of 1000 or more for the creation of the acoustic waves to be self-sustained during the earthquake slip motion (as discussed by Melosh and as seen from Eq. (6) of Melosh, 1996). Recall that the intrinsic Q measures the dissipation to heat and not due to scattering, because scattering is not a loss of energy in the di!usive multiple-scattering transport of the high-frequency acoustic energy (Eq. (2) of Melosh, 1996) that creates the shaking at the origin of the acoustic #uidization mechanism. On the other hand, Melosh estimates that the attenuation length l "(mjQ/4c must be of the order of the H typical wavelength j for the slip velocity u+1.4(q/oc)l /j to be of the order of a meter per second. H q is the stress drop, o is the density, m+cl /3 is the di!usion coe$cient (Ishimaru, 1978) of the C acoustic wave di!usion process in the multiple-scattering regime, l is the so-called elastic mean C free path (Ishimaru, 1978), i.e. scattering length, and c is the wave transport velocity, of the order of the shear wave velocity (Van Albada et al., 1991; Van Tiggelen and Lagendijk, 1993; Turner and Weaver, 1996). By de"nition of the Q factor (Knopo!, 1964), Q,2pE/*E, where *E is the energy loss over a wavelength or equivalently a cycle, we get Q"2pl /k. We thus see that QK1000 and * l +j are incompatible. We stress again that l is distinct from the scattering length l (which can * * C be itself comparable to j for strong scatterers) because l measures the true attenuation of the high * frequency acoustic "eld and because scattering, which is the mechanism by which the acoustic "eld is shaking the gouge, is not a dissipation. For the crust, we expect indeed QK1000 (Melosh, 1996; Knopo!, 1964) and thus l K160 j, leading to a slip velocity uR K7.5 mm/s, using the numerical * example for Eq. (9) of Melosh (1996). We may try to save this mechanism by invoking that the acoustic pressure does not need to reach the overburden pressure but only a small fraction g of it, in order to liquidify the fault. Indeed, it is well-established experimentally (Biarez and Hicher, 1994) that the elastic modulii of granular media under large cyclic deformations are much lower than their static values. This e!ect occurs only for su$ciently large amplitudes of the cyclic deformation, typically for strains e above 10\. At e "10\, the elastic modulii are halved and at e "10\, the elastic modulii are more than "ve times smaller than their static values. As a consequence, the strength of the granular medium is

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decreased in proportion. Extrapolating these properties to the crust, we need to estimate the strain created by the acoustic "eld. The acoustic pressure is related to the acoustic particle velocity v by p"ocv. Assuming p"gogh, this yields v"ggh/c+12 m/s for p+200 MPa, a density o"3;10 kg/m, a velocity c"4000 m/s and g"0.1. At a frequency f, this corresponds to an acoustic wave displacement u "v/2pf+2;10\ m at f&10 Hz. The corresponding strain u /w ? is &2;10\ for a gouge width w of the order of 1 m (Melosh, 1996) over which the intense shaking occurs. These estimations suggest that Melosh's criterion that the acoustic stress #uctuations must approach the overburden stress on the fault for acoustic #uidization to occur is too drastic and smaller shaking can reduce signi"cantly the fault friction. Persuing this reasoning, we see that the fundamental Eq. (6) in Melosh (1996) is changed into





1!erf(g/2(W) dW "W!rR , df 1#erf(g/2(W)

(12)

where g"1 recovers the case treated by Melosh. W is the normalized acoustic energy, f"z/l , z is H the coordinate perpendicular to the fault, the regeneration parameter is r"eQ/2 where e is the acoustic energy conversion e$ciency, R"q/ogh and erf(x) is the error function. This equation expresses the fact that the high frequency acoustic energy density results from the motion of the fault that creates the shaking (last term in the r.h.s.) and is dissipated by the intrinsic absorption ("rst term in the r.h.s.). We see that a factor g(1 implies a more e!ective generation of acoustic waves because the second source term of the r.h.s. is larger. But, since the bracket term saturates to one for large energies and/or small g, this does not lead to a signi"cantly larger slip velocity than found above. This remains a problem of the theory. This problem might be alleviated by treating e selfconsistently as a decreasing function of the friction coe$cient, and thus of the acoustic energy density. The problem then becomes even more non-linear because it re#ects in addition the vibration-weakening strength of the granular gouge. Further improvement should also take into account that the stochastic acoustic "eld is known to be distributed according to a Rayleigh distribution (Ishimaru, 1978) instead of a Gaussian distribution, which modi"es the functional form of the term in bracket in Eq. (12) and thus all numerical estimations.

10. Seismic radiation 10.1. The diwerent regimes There can be several contributions to the measured seismic radiation. If the earthquake is accompanied by a sudden change of shear modulus localized in the fault core, Knopo! and Randall (1970) have shown that this transformation in the presence of a pre-existing shear strain leads to a double-couple source with the correct and usual characteristic radiation pattern found in earthquakes at low frequencies. In fact, it is impossible to discriminate between this source and that of the displacement dislocation from their "rst motions. The size of the corresponding doublecouple moment is (Knopo! and Randall, 1970) M"2 dk de < ,

(13)

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where dk is the change in shear modulus, de is the change in strain inside the volume < in which transformation has occurred. For the sake of illustration, let us take the change in modulus corresponding to a transformation of quartz to another phase such as coesite. This is only o!ered as an illustration of the many possible transformations that can occur. The bulk modulus of coesite is 9.4;10 Pa, compared to that of quartz which is 3.7;10 Pa. We take as an upperbound of the change in strain de one-third the relative change in density (11%). The volume < is the surface S of the fault times the width w+0.1 m, say, of its core over which the transformation occurs. We thus see that this corresponds to the seismic moment of an earthquake with average slip w de less than a centimeter, irrespective of the size of the fault. This contribution is thus negligible for large earthquakes having slips of meters or more. In addition to this double-couple component, the change in bulk modulus and in density radiates isotropically with radial directions for the "rst P motions and no "rst S motions. However, this is again a very small e!ect. The second contribution to seismic radiation comes from the mechanical fault slip. Many models have been studied and the seismic radiation e$ciency is a function of the acceleration and deceleration of the slip. Modelling this phenomenon of fault unlocking is di$cult due to the fact that friction is a complex function of slip, velocity and other parameters. One thus has to solve a nonlinear problem of wave propagation and radiation with moving boundary conditions (Stefan's problem). Another essentially unsolved problem is the stopping of the slip or the arrest of the rupture. The seismic radiation and the resulting post-seismic stress are controlled by the nature of the arrest. Does it occur when the sliding velocity goes to zero as in standard solid friction or abruptly by a sudden collision with a barrier? We do not know. Within the acoustic #uidization mechanism, the sliding motion lasts as long as the high-frequency waves which are trapped within the liqui"ed material with low-acoustic impedence remain of su$cient amplitude to unlock the fault. The detrapping of these waves control the healing of the fault and therefore the static stress drop. This mechanism is expected to produce large variations of static stress drops. 10.2. Earthquake dynamical rupture In this section, we study a simpli"ed case in which a crack suddenly appears within the crust and leads to a given stress drop *p. This example is not the most common view according to which a rupture front propagates from the nucleation center to the edges of the fault. It however allows us to present the salient features of elasto-dynamic e!ects. Physically, it could correspond to the hypothetical case where another mechanism, such as a shock or and electromagnetic impulse would propagate much faster than the shear sound velocity and prepare almost simultaneously the fault for rupture. It is also of interest because the static version has been examined in details (Knopo!, 1958). In the presence of the tectonic loading, this sudden crack formation creates an elastodynamic transient that relaxes the mechanical system to a new stress and strain con"guration,

 The measurement of the shear modulus is more delicate and is not available to the best of our knowledge.

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corresponding to a new equilibrium in the presence of the open crack. The static version of this problem, in which a crack is introduced in a stressed medium, has been considered by Knopo! (1958) to evaluate the total energy released by an earthquake. Here, we are led to examine the dynamical version of this problem for the smaller portion of the fault where the nucleation occurs. Backus and Mulcahy (1976a,b) have shown that any mechanically admissible seismic source should admit a moment tensor representation. A seismic source, i.e. some internal rupture of the crust, is a non-elastic stress and strain distribution. Given the moment tensor density m(r, t), the equivalent body force density derived from this inelastic stress distribution is f 4(r, t)"!R m (r, t) . @ ? ?@

(14)

The divergence expression ensures that the integration of Eq. (14) over an arbitrary volume with no geometrical or stress discontinuity gives zero, which is a "rst requirement for the body to be at equilibrium. A second requirement is that the total torque in the same volume be zero which leads to the property that m(r, t) is a symmetric tensor. It has thus all the properties of a stress tensor. The integration of Eq. (14) over a volume with one part in the solid and one part in the empty interior of the crack (in which the stress is equal to the large scale value minus the stress drop *p) leads to the relation between the surface force and the moment: f 1(r, t)"n m b(r, t) , @ ? ?

(15)

where n is the a component of the unit (out-going) vector normal to the surface. ? What is the moment tensor density m(r, t) which is equivalent to the sudden opening of a crack? The crack is de"ned uniquely by the geometrical locii of points on which the stress is zero. By the linearity of elastic equations and the corresponding principle of superposition, the creation of a stress-free crack is the same as imposing a singular stress source at the position of the crack and which opposes locally the stress "eld p established globally prior to the creation of the crack. The GH sum of the pre-existing stress and the singular stress gives a stress decreased by the amount of the stress drop exactly on the position of the crack. This singular stress source can be called the stress drop due to the creation of the crack. This simple reasoning recovers the general result that the moment tensor density is nothing but the stress drop, when including the surface contributions (Gilbert, 1971; Backus and Mulcahy, 1976). We thus get m (r, t)"!*p d(r!r )H(t) , GH GH 

(16)

where d(r!r ) expresses the fact that the moment tensor source is localized on the crack position. A H(t) is the Heaviside's function ("0 for t(0 and "1 for t50) and 0 is the origin of time at which the crack is created inside the medium. Using the representation theorem (Aki and Richards, 1980), we can write the elastic displacement created by the source (16) as (Madariaga, 1981)

 

R RG (r, t"r , t )   m (r , t ) , d< GH u (r, t)" dt  HI   G Rx  4 I

(17)

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obtained from quiescent initial conditions u(r, t)"Ru(r, t)/Rt"0 and the elasto-dynamic Green function is (Aki and Richards, 1980)





t R R 1 (3c c !d ) H t! !H t! G (r, t"r , t )" GH R GH   v v 4po G H  







1 1 R 1 1 R # c c d t! # (d !c c ) d t! . G H GH G H 4pov R v 4pov R v    

(18)

In Eq. (18), o is the density of the crust, c are the cosine directors of r!r , R""r!r ", G   v "((j#2k)/o and v "(k/o are, respectively, the P (compressive) and S (shear) wave vel  ocities. The "rst term in Eq. (18) is the near "eld term which decays with distance from the source much faster than the two others which correspond to the classical P and S body waves. Inserting Eqs. (16) and (18) in Eq. (17), performing the integrals over the delta in time and in space as well as the derivative with respect to x yields the following expression for the elastoI dynamic displacement due to the P and S waves

 

*p !u (r, t)" HI G 4po

dS

1



p ! HI 4po

d GH!c c (1!d ) G H GH R







1 R 1 R H t! ! H t! v v v v    







ccc 1 R 1 R dS G H I d t! ! d t! R v v v v 1    







p c R 1 R # GI dS I H t! ! d t! 4pov  R v v v  1   

,

(19)

where S denotes the crack surface. This is the seismic signal measured at position r and time t.  We can use these expressions to calculate the stress impulse that the border of the crack will receive and whether it can rupture further due to the stress loading. The on-going rupture is then that of a fault with friction that has been loaded dynamically by the local crack rupture. This regime has been analyzed recently in the literature. Freund (1990) provides the complete treatment of the problem of a suddenly applied loading on a half-plane crack. The solution for the antiplane case is given in Eq. (2.3.16), p. 70 in Freund (1990) and leads to a shear stress close to the crack tip equal to 2*p p(r, t)& p



ct , r

(20)

where r is the distance to the crack tip. The case of the suddenly applied in-plane shear traction, corresponding in our problem to the full tensorial elastic case with stress drop *p over the whole extension of the crack, is treated in Freund (1990), Section 2.6, p. 97. The result (20) still holds, with c replaced by the longitudinal acoustic wave velocity.

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11. Other observations and questions We present a series of questions and/or observations, not all "rmly established (little is found non-controversial in this "eld), but which provide stimulating questions that an understanding of the earthquake process should embrace. 11.1. Laboratory experiments To our knowledge, laboratory experiments of rock friction have not explored fully the parameters that are relevant in seismogenic conditions. In particular, we mentionned already the di$culties in constraining chemical activity. Another relevant question is the role of possible mineralogic transformations of crystalline structures at conditions of temperature, pressure and water content, in the presence of deformation, that would be relevant for shallow seismogenic depths. With respect to the dependence with pressure, recent simulations (Badro et al., 1996) suggest to explore also the relevance of nonhydrostatic compression, for instance in triaxial mechanical set-ups. 11.2. The longer the recurrence time, the larger the stress drop The average slip rate of faults varies very signi"cantly from a small fraction of 1 mm/yr to several cm/yr. Several studies have shown recently a remarkable relationship between the average slip rate on faults and the stress drop associated to earthquakes occurring on these faults (see Scholz, 1990, p. 408; Kanamori, 1994 for reviews). What is obvious (within the standard rebound earthquake model) is that faults with slow (resp. fast) slip rates usually have long (resp. short) repeat times. A number of works (Kanamori and Allen, 1986; Scholz et al., 1986; Houston, 1990) have found that earthquakes on faults with long repeat times (thousands of years) radiate more energy per unit fault length and have a signi"cant larger dynamical stress drop than those with short repeat times (a few decades to a few centuries). This is quite puzzling: how can the loading process, occurring on a very slow time scale of centuries, have an in#uence on the dynamics occurring at the time scale of seconds? The usual explanation is that faults heal progressively (by plasticity and chemical processes) and thus become stronger as more time is given to them between earthquakes. However, we note that this is at variance with the other piece of argument usually brought forth, namely that an earthquake can occur because of a progressive weakening due to corrosion. Also, a stronger fault should produce a larger frictional heating and need a larger stress to break down. 11.3. Shear heating Direct measurements of heat #ux do not exhibit anomalous values and thus no signi"cant frictional heating (Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1980, 1988, 1992; Lachenbruch et al., 1995; Sass et al., 1992). On the other hand, the accumulated weight of evidence on the metamorphism of rocks adjacent to faults in higher grades has been taken as an indication that shear heating can be associated with faulting (Scholz, 1980; Sibson, 1980; Frischbutter and Hanisch, 1991; Peacock, 1992). Interpreted with a pure mechanical point of view, this implies shear stresses of about 100 MPa at least at the initiation of slip, in contradiction with many other

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interpretations. A mechanism has recently been proposed (Harrison et al., 1997) that renders unnecessary the need for exceptional physical conditions (such as high shear stress), to explain the generation of the Himalayan leucogranites. In this example, the apparent inverted metamorphism appears to have resulted from activation of a broad shear zone beneath the Himalayan Main Central thrust which tectonically telescoped the young metamorphic sequence. 11.4. Seismic P-wave nucleation phase Seismic P-wave signals have been reported (Ellworth and Beroza, 1995; Beroza and Ellworth, 1996; Dodge et al., 1996), that seem to precede the arrival of the "rst P-wave (longitudinal compressive acoustic wave). These observations are still controversial (Mori and Kanamori, 1996), not only because the reported signals are weak and the e!ect is hard to establish, but also because their presence is essentially ruled out within the standard picture. Indeed, "rst, one would have to invent the existence of a new mode that propagated faster than the P-wave. Secondly, Ellworth and Beroza (1995), Beroza and Ellworth (1996) and Dodge et al. (1996) make the remarkable claim that the duration of the P-wave precursors is roughly proportional to the total length of the earthquake rupture: it is as if the precursor, supposedly associated to a small nucleation zone, knew in advance about the (large) size of the earthquake; or, that the size of the nucleation zone controlled the size of the large earthquake. Present research attempts to understand these observations using the slip dependence of friction given by the Ruina}Dieterich laws (Dieterich, 1972, 1978, 1979, 1992; Ruina, 1983). 11.5. Earthquakes occur preferentially in strong rocks Fault imaging shows the systematic picture that earthquake rupture takes place in rock with relatively high seismic velocity, while surrounding rocks have velocities that are relatively low. The same pattern emerges in images of the 1994 Northridge earthquake (Zhao and Kanamori, 1994), of the 1966 Park"eld shock of 6.0 and of the 1992 Landers earthquake of magnitude 7.3. Why do earthquakes regularly take place in strong rock? If we consider an heterogeneous fault, which weak and strong rocks distributed along its length, we do expect on mechanical grounds that the large earthquakes will nucleate on the strong rocks. Indeed, only strong asperities can store su$cient stress to produce a rupture that will not be easily stopped. This scenario has been veri"ed in block-spring models with heterogeneous friction along the fault (Knopo!, 1996b). But this does not account for the observation that earthquakes take place in rocks that are stronger than the surrounding rocks in the direction transverse to the fault. 11.6. Earth tides and seismicity remotely triggered at long distances The long-term tectonic loading stress rate of the order of 10\ bar/h is much less than the stress rate up to 0.15 bar/h due to earth tides from gravitational interactions with the Moon and Sun.  Guided waves that have been found to propagate within the core of faults (Li et al., 1992) are slower, by the very constraint that they are trapped. If they were faster, they would leak immediately: this is the usual condition for the existence of evanescent waves.

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Tidal triggering of earthquakes would thus be expected if rupture began soon after the achievement of a critical stress level. Correlations between tides and earthquakes have been investigated repeatedly (Emter, 1997; Cotton, 1992). The most careful statistical studies have found no evidence of triggering (Heaton, 1982; Rydelek et al., 1992; Tsuruoka et al., 1995; Vidale et al., 1998). Two existing theories can explain these observations by invoking high stress rates just before failure. Dieterich's model of state- and rate-dependent friction predicts high stressing rates across earthquake nucleation zones (Dieterich, 1992). Alternatively, changes in #uid plumbing of the fault system could conceivably be more rapid than tidal strains and may trigger failure (Sibson, 1973). There is strong evidence that the Landers earthquake, June 28, 1992, southern California, triggered seismicity at distances equal to many times the source size (Hill et al., 1993), with a rate which was maximum immediately after passage of the exciting seismic waves. The problem is that the dynamical stress created by the seismic waves is very small at these long distances, of the order of the e!ect of lunar tides (0.01 MPa and less) which have not been found to be correlated with earthquakes. Sturtevant et al. (1996) have recently proposed a model in which the earthquakes are triggered by a rapid increase of pore pressure due to recti"ed di!usion of small pre-existing gas bubbles in faults embedded in hydrothermal systems. This model depends on the con#uence of several favorable conditions, in particular supersaturated gas in water, large mode conversion occurring in the geothermal "eld and a very low (50 m/s) shear velocity in the porous media "lled with the bubbly liquid. 11.7. Non-double couple components Some earthquakes with non-double-couple mechanisms have been claimed not to be explained solely by a composite rupture (Frohlich, 1994b). The standard explanation for non-double-couple components relies on the fault zone irregularity (Kuge and Lay, 1994). Could they be witnesses of a compensated linear dipole component associated with other phenomena such as structural transformations? 11.8. Creeping sections and slow earthquakes Some faults appear not to rupture in sudden earthquakes but rather to creep continuously. Slow earthquakes (Linde et al., 1996) are acceleration of creep over a few hours to a few days. This leads to the usual classi"cation in terms of three types of behavior: regions where the seismic coupling is total; those where slip is seismic only when triggered by seismic slip from an adjacent region and which otherwise slip asesimically; and regions where slip is always aseismic. Within the standard stick-slip friction theory, these three regimes can be ascribed to di!erent friction stability conditions, controlled by the slip velocity parameter of the friction law which are material properties (Scholz, 1990). Consider for instance the famous San Andreas fault system. This long fault evidently cuts through many di!erent types of rocks and regional structures along its traverse from the Transverse Ranges to the Mendocino triple junction, and the patterns of seismicity di!er strikingly along various segments of the fault. These di!erences in seismic behavior coincide so closely with certain geologic situations along the San Andreas fault in central California as to suggest a causal relation (Wallace, 1990).

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11.9. Deep earthquakes Solid}solid transformations have been invoked to explain the puzzle of deep-focus earthquakes. The paradox is that brittle fracture and frictional sliding at depths in excess of 100}200 km would require unrealistic rock strength, while the rheology of the rocks becomes more and more ductile as pressure and temperature increases, as found in laboratory experiments. Furthermore, the isostatic control of topographic relief implies that the mantle has no strength on long time scales at depths greater than 70 km or so (Je!reys, 1929). Several mechanisms have been proposed for deep earthquakes (Frohlich, 1994a), such as plastic instabilities (Hobbs and Ord, 1988), shear-induced melting (Post, 1977), instabilities accompanying recrystallization (Ogawa, 1987), polymorphic phase transformation (Kirby, 1987) and olivinePspinel anticracks formation (Green and Burnley, 1989). Each of these proposed mechanisms exhibits certain weakness (Kirby, 1987). While a consensus has not yet emerged, one of the most preferred mechanism involves so-called transformational faulting in which metastable olivine transforms under stress to denser spinel in thin shear zones along planes of maximum shear stress. Then, it is proposed that newly formed, "ne-grained spinel is the `#uid lubricanta and/or the damage (through the operation of `anticracksa) that makes catastrophic faulting possible. A recent analysis of the deep Bolivian earthquake of 9 June 1994 has been proposed to be incompatible with an origin by transformational faulting (Silver et al., 1994), because the rupture extended to more than three times that expected for a metastable wedge of olivine at the core of a subducting slab. These authors suggest that failure during deep-focus earthquakes represent the reactivation of faults that were established when the lithosphere was near the earth surface. 11.10. Precursory phenomena Since the inception of seismology, the search for reliable precursory phenomena of earthquakes has been very active (Mogi, 1969). Current research suggests that the seismic process is preceded by a complex set of physical precursors. However, there is no concensus on the statistical signi"cance of these precursors and their reliability. The lack of agreement among prediction advocates on a single set of `best-candidate precursorsa underscores the weakness and inconsistency of their case (Wyss, 1991, 1997). To validate a prediction method, one must show it to be successful beyond random chance, but such success has not been demonstrated convincingly. The problem is very di$cult because of the complexity of earthquake phenomenology and the low rate occurrence of earthquakes. Seismic activity is one obvious candidate. It is monitored in the di!erent parts of the world by arrays of seismometers which allow for the determination of the epicenter, hypocenter, and rupture parameters (magnitude, seismic moment, source mechanism, orientation of the fault plane and direction of motion) of an earthquake. Foreshocks correspond to an increase of seismic activity at time scales of no more than a few weeks before a `maina shock (Abercrombie and Mori, 1996; Maeda, 1996; Jones, 1994). Only about 30% of earthquakes exhibit foreshocks and the problem is  The deepest earthquakes occur at depth 600 km, which is well within the mantle.

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that they are recognized as such after the fact. Nevertheless, foreshocks as well as aftershocks indicate the existence of a delay mechanism (Shaw, 1993) in the activation of earthquakes which could possibly be used (Sornette et al., 1992). The notion of foreshocks has been extended to that of seismic precursors to describe the surprising and somewhat controversial observations showing that many large earthquakes have been preceded by an increase in the number of intermediate sized events (JaumeH and Sykes, 1998). The relation between these intermediate sized events and the subsequent main event has only recently been recognized because the precursory events occur over such a large area that they do not "t prior de"nitions of foreshocks. In particular, the eleven earthquakes in California with magnitudes greater than 6.8 in the last century are associated with an increase of precursory intermediate magnitude earthquakes measured in a running time window of "ve years (Knopo! et al., 1996). What is strange about the result is that the precursory pattern occurred with distances of the order of 300 to 500 km from the futur epicenter, i.e. at distances up to ten times larger that the size of the futur earthquake rupture. Furthermore, the increased intermediate magnitude activity switched o! rapidly after a big earthquake in about half of the cases. This implies that stress changes due to an earthquake of rupture dimension as small as 35 km can in#uence the stress distribution to distances more than ten times its size. This result de"es usual models. This observation is not isolated (Sornette and Sammis, 1995; Saleur et al., 1996). There is mounting evidence, albeit still controversial, that the rate of occurrence of intermediate earthquakes increases in the tens of years preceding a major event. Sykes and Jaume (1990) present evidence that the occurrence of events in the range 5.0}5.9 accelerated in the tens of years preceding the large San Francisco bay area quakes in 1989, 1906, and 1868 and the Desert Hot Springs earthquake in 1948. Lindh (1990) points out references to similar increases in intermediate seismicity before the large 1857 earthquake in southern California and before the 1707 Kwanto and the 1923 Tokyo earthquakes in Japan. More recently, Jones (1994) has documented a similar increase in intermediate activity over the past 8 years in southern California. This increase in activity is limited to events in excess of M"5.0; no increase in activity is apparent when all events M'4.0 are considered. Ellsworth et al. (1981) also reported that the increase in activity was also limited to events larger than M"5 prior to the 1989 Loma Prieta earthquake in the San Francisco Bay area. Bufe and Varnes (1993) analysed the increase in activity which preceded the 1989 Loma Prieta earthquake in the San Francisco Bay area while Bufe et al. (1994) document a current increase in seismicity in several segments of the aleutian arc. More recently, Bowman et al. (1998) conducted a systematic review of seismicity in California speci"cally aimed to test for the occurrence of accelerating seismic strain release before large earthquakes. They examined all M56.5 earthquakes along the San Andreas fault system since 1950 (total of eight). They found that the cumulative seismic strain prior to all eight M56.5 earthquakes can be better "t by a power acceleration than by a steady seismic rate. Brehm and Braile (1998) has used the same technique as Bowman et al. (1998) to study earthquakes in the New Madrid Seismic Zone in the central United States. Surprisingly, they "nd they can model accelerating seismic strain sequences before mainshocks as small as magnitude 3.5 that yield reasonable predictions of time and magnitude: of 26 post-1979 mainshocks ranging from 3.5 to 5.2, 16 could be modelled by a power law accelerating time-to-failure function. Seismic activity induces other variations of geophysical parameters such as the surrounding electric and magnetic "elds. Under favorable conditions these electromagnetic variations

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may be observed before seismic rupture. A recent workshop on earthquake precursors held at The University of Electro-Communications, Tokyo, and sponsored by The National Space Development Agency of Japan, illustrated a wide range of seismo-electromagnetic precursor radiation (NASDA-CON-960003, 1997). The ability of the Earth's crust in areas of active deformation to transform mechanical energy into electromagnetic waves is a possible contribution to the short-term prediction of seismic hazards. The most important and di$cult problem in seismoelectromagnetic research is the location of source. This problem is related to both the physical processes in the earthquake preparation zone and the choice of monitoring device. In a reexamination of the VAN precursors (Varotsos and Lazaridou, 1991), Mulargia and Gasperini (1992) suggested that electric signals are more readily found to follow rather than precede large events. See also Debate on VAN (1996). Other potential signatures of impending earthquakes involve the behavior of global quantities, such as the conductivity and permeability of the crust at various distances from the epicenter. The change in permeability, controlled by microcracking and a change in porosity, can be monitored through the #uctuations of groundwater. In this spirit, #uctuations in ion concentrations of groundwater issuing from wells located near the epicenter of the recent earthquake of magnitude 6.9 near Kobe, Japan, on January 17, 1995 have been observed (Tsunogai and Wakita, 1995, 1996) and quanti"ed (Johansen et al., 1996). See also Note added in proof.

12. Concluding remarks This review has attempted to bring together di!erent disciplines that have hitherto been developed in large part independently from each other, while they study fundamentally the same `objecta. We thus view the challenge of understanding what is an earthquake as somewhat parallel to the integration of many disparate observations in seismicity, volcanism, mountainbuilding and paleo-magnetism which were "nally made sense of and uni"ed within the theory of plate tectonics. The review has been built on the di$culties to reconcile data with existing models and on the recognition of the fundamental role played by water in fault zone. The possible scenarios for an earthquake that emerge are not simple and invoke a series of mechanisms, which are naturally interwoven. This may suggest directions for a coherent framework to explain a large set of diverse observations in earthquakes. It also raises the exciting promise of providing a guideline for recognizable and reproducible precursors that could be exploited.

13. Note added in proof In concluding Section 11.10, let us mention the Russian school (Keilis-Borok, 1990): Mathematician Gelfand and geophysicist Keilis-Borok have pioneered the development of the mathematics of pattern recognition of sparse data to which standard statistical methods cannot be applied (Gelfand et al., 1976). They and their students have identi"ed a number of new patterns of precursory seismicity, such as long range stress transfer (see also Knopo! et al.,

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1996a,b); the importance of the numbers of foreshocks of intermediate magnitude precursors (Keilis-Borok et al., 1980); precursory swarms; and increased release of seismic moment over large areas prior to large earthquakes (see also Bowman et al., 1998). These are new &&universals'' in seismicity, as they go beyond the obvious statistical statements of the Gutenberg}Richter magnitude}frequency distribution and Omori's law for the rate of aftershocks. These methods for identi"cation of times of strong earthquakes, especially the weaker algorithm of Keilis-Borok and Kossobokov (1990), have not received universal approbation. Several competitive assessments of the methods have been made by Kossobokov and co-workers, with unpersuaded parties on both sides. One can attribute part of the controversy to philosophical di!erences between many American seismologists and the Russian school. The Russian approach is to identify patterns by systematically "ltering seemingly patternless data. The gist of the pattern recognition approach has been to identify the "lters. In the process, there has been little attempt to date to identify the physical processes that underly the patterns so identi"ed. The mathematics of pattern recognition of sparse data is not limited by a requirement that we have a full understanding of the physical processes, and provides novel insights that may guide and suggest novel mechanisms and models.

Acknowledgements I have bene"tted from very helpful discussions with Y. Ben-Zion, Y. Brechet, J.M. Christie, P.M. Davis, J. Dieterich, P. Evesque, J. Gilman, M. Harrison, H. Houston, D.D. Jackson, Y.Y. Kagan, W.D. Ortlepp, G. Ouillon, E. Riggs and J. Vidale. Useful correspondences with J.-C. Doukhan, J. Ferguson, R.J. Geller, I. Main, G. Martin, J.P. Poirier are acknowledged. Special thanks are due to L. Knopo! and A. Sornette for many discussions and interactions over many years.

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B PHYSICS AT THE Z-RESONANCE

S. SCHAEL Max-Planck-Institut fu( r Physik, Werner-Heisenberg-Institut, Fo( hringer Ring 6, D-80805 Mu( nchen, Germany

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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B physics at the Z-resonance Stefan Schael Max-Planck-Institut fu( r Physik, Werner-Heisenberg-Institut, Fo( hringer Ring 6, D-80805 Mu( nchen, Germany Received August 1998; editor: J.V. Allaby

Contents 1. Introduction 2. Experimental facilities 2.1. The LEP collider 2.2. The LEP experiments 2.3. The LEP data sample 3. The Z resonance parameters 4. Z decays into heavy quarks 4.1. The measurement of R @ 4.2. The measurement of R A 4.3. The measurement of R SBQ 4.4. The forward}backward asymmetries 4.5. Standard model "t of the electroweak couplings 5. B hadron lifetimes and masses 5.1. Inclusive measurements 5.2. Exclusive measurements

295 298 298 302 304 304 309 309 316 316 320 323 326 327 330

6. Excited beauty states 6.1. Selection and reconstruction of B hadrons 6.2. B* production 6.3. B* polarisation 6.4. B** production 7. BBM -oscillations 7.1. Time integrated measurements 7.2. BBM  oscillations B B 7.3. BBM  oscillations Q Q 8. The CKM unitarity triangle 8.1. Experimental constraints 8.2. Fit of the unitarity triangle 9. Summary and conclusions Acknowledgements References

335 336 337 338 338 342 344 345 347 351 354 355 359 361 362

Abstract The study of ZPbbM decays allows for precision tests of the electroweak sector of the Standard Model. The vector and axial vector couplings of b-quarks are especially sensitive to physics beyond the Standard Model. Signi"cant deviations from the Standard Model expectations were observed in previous years and have recently been decreased due to new experimental techniques and a better understanding of systematic uncertainties. The experimental development is reviewed in the "rst part of this article and the measurements are confronted with the Standard Model calculations. The second part is devoted to the measurements of the B hadron lifetimes and masses, of BBM  oscillations and the search for B B BBM  oscillations. In combination they lead to interesting constraints on the unitarity of the CKM-matrix. The Q Q determination of the production rates and properties of excited beauty states o!ers new perspectives for CP-violation measurements in future experiments.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.70.Hp; 12.15.!y Keywords: B physics; Electroweak interactions; Z-resonance

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 8 ) 0 0 0 8 9 - 1

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295

1. Introduction For more than two millenia philosophers and scientists have studied the building blocks of matter and the interactions between them trying to "nd an answer to the question `What are things made of ?a. The tremendous advances in experimental techniques and theoretical understanding made during the last two decades allow us today to draw a consistent picture of the evolution of the universe starting approximately 15 billion years ago with the `big banga [1]. The world's best `microscopesa are the Large Electron Positron collider LEP at CERN and the Stanford Linear Collider (SLC) at Stanford, each having a resolution of j"1/E +   1.3;10\ m at a centre of mass energy of E "91 GeV. Each e>e\ collision observed at these   colliders corresponds to a mini "reball at a temperature of 6;10 K, a state which presumably existed in the universe about 10\ s after the big bang. The processes in this very early phase of the universe are described by the so-called `Standard Modela [2]. In the framework of local gauge invariant "eld theories, electroweak and strong interactions between the fundamental fermions can be calculated. The masses and quantum numbers of three generations of quarks and leptons are summarised in Table 1. It was proven in 1971 by 't Hooft [3] that the Standard Model is a renormalizable theory, and hence any experimental observable can be predicted theoretically in terms of a "nite number of parameters. Measurements are called consistent if they can be explained by one set of input parameters. These measurements can then be used to further constrain the input parameters. Inconsistency in this process would be a signal for physics beyond the Standard Model. This review summarises the results of precision measurements which are especially sensitive to physics beyond the Standard Model, i.e. the electroweak couplings of the b-quark to Z- and =! bosons. The b-quark is the heaviest quark with charge !1/3 discovered up to now. A fourth generation of quarks should therefore lead, especially in physical processes involving b-quarks, to signi"cant deviations from the Standard Model calculations. Also most extensions of the Standard Model such as Super-Symmetric Models [4] can either be signi"cantly constrained in their parameter space or even be excluded by measurements in the heavy quark sector due to higher order loop corrections to Standard Model processes. In the framework of the Standard Model the Lagrangian density function which describes the interaction between fermions and the heavy gauge bosons (Z, =!) is given by !e !e Z JH # (=>JH #=\JHR ) , ¸ " H !! H !!  sin h cos h H ,! (2 sin h 5 5 5 where sinh "1!m /m. The neutral current JH 5 5 8 ,! 1 JH ( f )"fM cH (g !g c) f D ,! 2 4D

(1)

(2)

describes the dominant process for the fermion production in e>e\ collisions at the Z resonance: e>e\PZPf fM , f"e, k, q, u, d, c, s, b. The vector and axial-vector couplings, g and g , for all 4D D

 Throughout this article a system of units is used in which "c"1 and e "k "1.  

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Table 1 The masses, charges (Q ), third component of the weak isospin (¹) and the weak hypercharge (>5) of the basic fermions D D D in the Standard Model Generation

Fermion

Mass (GeV)

Q D

¹ D

>5 D

I

l C e u d

(0.8;10\ 5.110;10\ 0.003P0.007 0.007P0.015

0 !1 #2/3 !1/3

#1/2 !1/2 #1/2 !1/2

!1 !1 1/3 1/3

II

l I k c s

(0.17;10\ 0.105 1.3P1.7 0.15P0.3

0 !1 #2/3 !1/3

#1/2 !1/2 #1/2 !1/2

!1 !1 1/3 1/3

III

l O q t b

(0.024 1.777 177$6 4.8P5.2

0 !1 #2/3 !1/3

#1/2 !1/2 #1/2 !1/2

!1 !1 1/3 1/3

Table 2 The standard model predictions for the ZPf fM neutral current parameters using sin h "0.2231 5 Fermion

g D

g 4D

l,l,l C I O e\, k\, q\

1/2 !1/2

1/2 !1/2#2/2 sin h "!0.054 5

u, c, t d, s, b

1/2 !1/2

1/2!4/3 sin h " 0.203 5 !1/2#2/3 sin h "!0.351 5

fermions depend only on one universal parameter sinh : 5 g ,¹ (1!4"Q "sin h ) , 4D D D 5 g ,¹ . D D Here ¹ and Q are, respectively, the third component of the weak isospin and the charge of D D fermion f. In the Standard Model sin h is a free parameter and has to be supplied by measure5 ments. The experimental facilities used for these measurements at LEP and SLC will be described in Section 2. The values of g and g are listed in Table 2 using the current best estimate of 4D D sin h "0.2231$0.0006 [5]. Higher order loop corrections modify this picture as will be 5 discussed in Section 3. The study of the production rates and asymmetries of b-quarks at the

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Z resonance with the LEP and SLC experiments and the determination of g and g for heavy 4D D quarks will be covered in Section 4. While the "rst part of this review covers neutral current couplings of heavy quarks, the second part is related to the charged current interactions


 e




d * * (3) JH "(lN , lN , lN )cH k #(uN , cN , tM )cH< s * * * !)+ * !! C* I* O* * q b * * which have been investigated in the b-quark sector by analysing B hadron decay properties produced mainly in e>e\-collisions. This was done by the ARGUS and CLEO experiment running at the B(4S) resonance (E "10.4 GeV) and by the LEP and SLD experiments running at the   Z resonance (E "91.2 GeV). Recently the CDF experiment running at ppN collisions   (E "1 TeV) has started to contribute signi"cantly to this "eld. < is the Cabibbo}Kobayashi   !)+ Maskawa (CKM) matrix [6]





<



< < SQ S@ < " < < < !)+ AB AQ A@ < < < RB RQ R@ c c s c s e\ B      c c !s s s e B s c " !s c !c s s e B , (4)             s s !c c s e B !s c !s c s e B c c             where c "cos h and s "sin h with 04h 4n/2 and 04d42. The fact that the CKM-matrix GH GH GH GH GH can be complex, depending on the value of d and s , has fundamental implications concerning CP  invariance of the electroweak theory. The symmetries C (particle}antiparticle interchange) and P (space inversion) and the product CP hold for strong and electromagnetic interactions. CP invariance of electroweak interactions can, for example, be investigated by the measurement of the amplitude (A) for the quark scattering process abPcd which is proportional to the CKM-matrix elements < and < : AJ< <* . For this process the theory is CP-invariant if the amplitude for A? B@ A? B@ the CP-transformed process (A ) full"lls the condition: A "AR. This is in general only the case if !. !. < "<* and < "<* , i.e. if the CKM-matrix is real. A? A? B@ B@ Up to now CP-violation was only observed in the neutral Kaon system. Precision tests of the Standard Model description of CP violation does not look possible in the kaon system due to large theoretical uncertainties in non-perturbative QCD corrections. In the B-system the Standard Model can make precise predictions for CP asymmetries in several decay modes [7]. On one hand the measurements of the B hadron lifetimes and masses and the observation of BBM -oscillations are important input parameters to calculate the expected CP-violation in the B-system. On the other hand they can be already used to test the unitarity of the CKM-matrix. The understanding of CP-violation is closely related to the understanding of the origin of the matter}antimatter-asymmetry in our universe [8]. The "rst observation of CP-violation in the beauty sector is expected from CDF or D0 after the collider upgrade at Fermilab in 1999 or from one of the dedicated B-experiments like Belle [9], Babar [10] or Hera-B [11]. The goal of the SB



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LHC-B experiment [12] are precision measurements in this sector. The unitarity of the CKMmatrix, and hence the electroweak couplings of quarks in charged current reactions, will then be investigated with similar precision as the neutral current reactions have been tested by the LEP and SLC data to date.

2. Experimental facilities After the discovery of the Z and =! bosons in 1983 by UA1 [13] and UA2 [14] in ppN collisions, two electron-positron colliders were built to study electroweak interactions in the clean environment o!ered by e>e\ collisions: the SLC in Stanford, USA and the LEP collider in Geneva, Switzerland. The linear collider SLC started data taking in summer 1989 a few weeks before LEP. The LEP collider will be described in more detail in the next section. Most of the precision measurements from the SLD experiment at Stanford are limited by the luminosity delivered by SLC. Nevertheless the high values of the longitudinal polarisation (P +80%) achieved at the SLC and the inherent interaction point properties of linear e>e\ C colliders, namely the tiny and very stable interaction region with a typical size of (p , p , p )"(2.1, V W X 0.6, 500) lm and a low collision rate, have allowed the SLD experiment to instrument more e$cient heavy #avour detection (see Section 4.4) and to present competitive results for key parameters of electroweak physics. 2.1. The LEP collider The LEP collider is the world largest particle accelerator with a circumference of 27 km housed in a tunnel O(100 m) below the surface. In e>e\ storage rings the electrons and positrons are forced into a circular motion by dipole magnets and emit synchrotron radiation in the bending process. The emitted energy e E (5) E" 3m R  is proportional to the fourth power of the beam energy (E) and inverse proportional to the circular radius (R) and the fourth power of the electron rest mass (m ). Inserting the electron charge e and  m leads to the useful relation  E[GeV] *E[keV]"88.5 . (6) R[m] The large radius of the LEP collider is therefore a compromise between construction cost and the cost of supplying the power to replenish the radiated energy. The equipment with 120 roomtemperature copper cavities allows beam momenta up to 55 GeV.

 Throughout this article the right handed coordinate frame is de"ned by the positive z-axis pointing in the direction of the incoming electron beam.

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The main physics goal of the LEP experiments is the investigation of the fermion pair production e>e\ f fM . The large cross section given by the Z resonance, (see Fig. 1) enable precision measurements of the Z boson properties, such as its mass, its total and partial decay widths as well as the search for rare and new phenomena. The test of the description of strong interactions in the framework of Quantum Chromodynamics (QCD), the precision tests of the electroweak sector of the Standard Model, the search for new particles and the study of the production and the decay of q-leptons, D- and B hadrons o!ers a rich "eld for fundamental research. The "rst e>e\ collisions were recorded by the four LEP experiments ALEPH [15], DELPHI [16], L3 [17] and OPAL [18] on the 13th of August 1989, after a construction period of 7 years. Within a few months the four experiments collected 400 000 Z decays and measured the Z boson mass with a precision of 1 part in 3000: M "(91.177$0.031) GeV [19]. This was a tremendous 8 success since M is one of the key parameters needed for precision tests of the Standard Model 8 predictions in the electroweak sector. The improvement compared to the measurement of M "(92.9$1.6) GeV based on only 300 Z's found in ppN collisions by UA1 and UA2 was more 8 than two orders of magnitude. LEP reached its design luminosity of ¸"13;10 cm\ s\ in 1991 after two years of running. The interesting "gure for all research is the integrated luminosity collected by the four experiments. Up to the end of data taking for LEP I each of the four LEP experiments collected nearly 190 pb\ corresponding to more than 4 million hadronic Z decays (see Fig. 2). A signi"cant part of the running time was spent for energy scans $2 GeV around the Z peak in order to measure the Z total decay width and mass to a "nal accuracy of 2 MeV, i.e. 2 part in 100 000. After the "rst two years of data taking it turned out that measurements of the Z boson mass were systematically limited (dM +20 MeV) by the energy calibration of LEP. During running the 8

Fig. 1. e>e\ cross sections as measured by the various experiments. Fig. 2. Number of hadronic Z decays collected by the ALEPH experiment as a function of time (upper plot) and the corresponding integrated luminosity (lower plot).

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beam energy was calculated from the magnetic "eld in a reference magnet which is connected in series with the dipole magnets of LEP. This calibration was complemented by two measurements which are not possible during e>e\ colliding mode: "rst the comparison of the revolution frequencies of protons and positrons in LEP at 20 GeV and second the in situ determination of the magnetic "eld in the magnets. Starting in 1992 a new method based on resonant beam depolarisation was developed [20]. Already in 1961 Lostukov, Korovina and Ternov predicted that electrons and positrons in a storage ring with a homogenous magnetic "eld could be spontaneous polarised [21] due to their synchrotron radiation. The solution of the Dirac equation shows, that on average after emitting of 1.2;10 photons the spin of the electron is #ipped [22]. It was found that the rate for the transition from a spin-vector parallel to the magnetic "eld to the antiparallel direction is much larger than for the inverse process. Therefore a transversal polarisation rate of up to 92.4% can be theoretically reached at LEP. Unfortunately an electron radiates only approximately 10 photons per second, and thus it takes 5 to 6 hours to reach the maximal polarisation. The resonant beam depolarisation using a Compton-Laser polarimeter allows a measurement of the LEP energy with a precision of 0.2 MeV. This procedure could be performed successfully at the end of 40% of the o! peak "lls in 1993 (70% for 1995). For the other "lls and for the time during the "ll an extrapolation model was needed. These models have to include Earth tide e!ects (see Fig. 3), the level of Lake Leman (see Fig. 4) and other e!ects as temperature variations of the dipole magnets. It was also observed with an NMR probe which was installed in 1995 in the LEP tunnel that the beam energy rises with time due to hysteresis e!ects in the LEP magnets which originate from leakage currents through the beam pipe induced by the TGV train in the Geneva region (see Figs. 5 and 6). A model for E was developed [23] to include all the e!ects discussed above to determine *#.

Fig. 3. Variation of the LEP energy due to Earth tide e!ects. The terrestial tides due to the moon and the sun move the earth surface up and down. Since the RF frequency, and thus the orbit length, is "xed, stresses in the local rock structure result in changes to the machine radius and the position of the beams in the quadrupole magnets. This changes the beam energy, since the e!ective dipole "eld changes.

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Fig. 4. Variation of the LEP energy in 1994 together with the expectation from a model based on the level of Lake Leman.

Fig. 5. Schematic view of the current #ow in the train rails and the LEP ring.

the LEP energy as precisely as possible. Taking into account all uncertainties [23] the estimated errors on the LEP energy are 2.0 and 1.5 MeV at the two scan points 1.8 GeV below and above the Z peak respectively. The resulting systematic errors in M and C are dM (LEP)"$1.5 MeV 8 8  and dC (LEP)"$1.7 MeV. This was a tremendous achievement. 8

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Fig. 6. Correlation between trains and LEP energy.

The running period at the Z resonance (E +91.2 GeV) was called LEP I and ended 1995. The   continuous upgrade of LEP I with installation of 192 superconducting Ni-cavities with "eld gradients up to 6 MV/m up to the end of 1998 will allow cms-energies of nearly 200 GeV, well above the =>=\ production threshold. The running of LEP II started in 1996 and is dedicated to the precision measurement of the =! boson mass, its couplings and to the search for new particles, such as the Higgs boson or particles expected in the supersymmetric extension of the Standard Model. 2.2. The LEP experiments All four LEP experiments are general purpose detectors with almost 4n solid angle coverage. The design of the ALEPH and OPAL detector is based on well known technology. The inner tracking chamber of OPAL is a large Jet Chamber with z-coordinate determination by charge division. In ALEPH a large Time-Projection-Chamber (TPC) was chosen as central tracker. It was the "rst experiment to be equipped with a double-sided Silicon Vertex detector (1991). In the DELPHI experiment new technologies like the Ring-Image-Cherenkov counter (RICH) for particle identi"cation or the High-density-Projection-Chamber (HPC) as an electromagnetic calorimeter with very good spacial resolution were tested. The design of the L3 experiment was optimised for lepton identi"cation with an excellent electromagnetic calorimeter and very good muon identi"cation capabilities. During the "ve years data taking period for LEP I all experiments were upgraded. The most important improvement for the physics topics discussed in this review was the implementation of silicon micro-vertex detectors in all four experiments. The "nal performance of the LEP experiments is rather similar. As a typical example the ALEPH detector will be described in the following.

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Fig. 7. The ALEPH detector.

2.2.1. The ALEPH detector The ALEPH detector is shown in Fig. 7. A brief description of the apparatus is given here while a detailed description can be found in Ref. [15]. Charged particles are tracked with three devices inside a super-conducting solenoid which provides an axial "eld of 1.5 Tesla. Closest to the beam-pipe is the vertex detector (VDET), installed in 1991, which consists of silicon wafers with strip readout in two dimensions, arranged in two cylindrical layers at average radii 6.3 and 10.8 cm. This detector covers an angular range down to "cos h"(0.85 for the inner layer and "cos h"(0.69 for the outer layer. The spatial resolution for r } coordinates is 12 lm and varies between 12 and 22 lm for the z-coordinates, depending of the track polar angle. Surrounding the VDET is the Inner-Tracking-Chamber (ITC), a drift chamber giving up to eight measurements in r with a resolution of 150 lm. Outside the ITC, the Time-Projection-Chamber (TPC) lying between radii of 30 and 180 cm provides up to 21 space points for "cos h"(0.79, and a decreasing number of points for smaller angles, with four at "cos h""0.96. The resolutions are 180 lm in the r} view and 1}2 mm in the r}z view. The TPC also gives up to 338 measurements of the speci"c ionisation of each charged track, with a dE/dx resolution of 4.5%, as measured for Bhabha electrons. For charged tracks with two VDET coordinates, a transverse momentum resolution of *p /p "6;10\p 0.005 (p in GeV) is achieved. The impact parameter resolution is 2 2 2 2 2595/p lm (p in GeV) in both the r} and r}z views. An electromagnetic calorimeter (ECAL) inside the coil and a hadron calorimeter (HCAL) outside the coil are used to measure the energies of neutral and charged particles over almost the full 4n solid angle. The ECAL is a 22 radiation-length lead/wire-chamber sandwich operated in proportional mode and is read out via projective towers sub-tending typically 0.93;0.93. The

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HCAL uses the iron return yoke as absorber and has an average depth of 7 interaction lengths. Hadronic showers are sampled by 23 layers of streamer tubes and towers sub-tending approximately 3.73;3.73. Muon chambers consisting of two double layers of streamer tubes surround the HCAL. Electrons and photons can be identi"ed using the ECAL, whilst muons are seen as tracks giving a series of hits on digitally readout strips in the HCAL and in the muon chamber streamer tubes. The mean position of the e>e\ interaction region is determined with accuracies of 20 lm in the horizontal and 10 lm in the vertical directions, for groups of about 75 hadronic events. The r.m.s. size of the luminous region is found vary, year by year, from 110 to 150 lm horizontally, whilst remaining less than 8 lm vertically. 2.3. The LEP data sample The results of the LEP experiments are based on a data sample of more than 4 million hadronic events per experiment, taken at a mean centre of mass energy of 91.2 GeV. Ten percent of these data were taken at energies approximately 2 GeV above and below this Z peak energy. All analysis use Monte Carlo simulations based on the JETSET 7.4 generator [24] to estimate e$ciencies and backgrounds. The production rates, decay modes and lifetimes of heavy hadrons are adjusted to agree with recent measurements [25]. For the fragmentation of heavy quarks the Peterson model [26] is used. Detector e!ects are simulated using the GEANT package [27].

3. The Z resonance parameters At centre of mass energies close to the Z mass the production cross section for the process e>e\Pf fM with fOe is dominated by the Z boson exchange (see Fig. 8). At Born level one obtains therefore sC 8 #c #(c/Z) , p (s)"p    DD DD (s!M)#MC 8 8 8 where the pole cross section p is de"ned as DD 12nC C CC DD . p " DD MC 8 8

Fig. 8. Feynman graph for the reaction e>e\P(c, Z)Pf fM .

(7)

(8)

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Fig. 9. Photonic corrections to the process e>e\P(c, Z)Pf fM , fOe.

Fig. 10. Electroweak corrections for the Z-f fM vertex.

The contributions due to the photon exchange and the cZ interference are of order 1%. The partial decay widths are given by (2G M $ 8 (g #g ) ND C M" DD 4D D ! 12n

(9)

with the vector and axial-vector coupling constants of the fermions as de"ned in Eq. (3). The radiative corrections which have to be applied to this picture can be divided into two parts: E Photonic corrections. They describe all processes which add real or virtual photons to the Born diagram (see Fig. 9). These e!ects are large (+30%) and can be calculated in the framework of QED [28]. Since the corrections depend on the cuts used in the analysis the experimental results on the cross sections presented by the LEP experiments are corrected for the e!ect of initial-state radiation as well as t-channel and s/t-interference in the case of e>e\ "nal states. E Non-photonic corrections. They describe e!ects such as vacuum polarisation and vertex corrections shown in Fig. 10. In order to retain the Born structure of the hard scattering process an s-dependent photon vacuum polarisation correction *a(s) is introduced, and the s-dependence of Z total decay width is approximated by C (s)"s/M ) C (s"M). Additional corrections are 8 8 8 8

 In the following C ,C (s"M ) is used. 8 8 8

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absorbed in so-called e!ective couplings gN and gN using 4D D gN gN "¹D(o , 4D"1!4"Q "sinhD , D  D gN D  D where the leading correction in m /m gives o +1#do with  5 D R 3(2G m I  . do " R (4n)

(10)

(11)

The e!ective mixing angle is here de"ned as






cos h 5 do . sin hD "sin h i +sin h 1# (12)  5 D 5 R sin h 5 Only for the process e>e\PZPbbM vertex corrections involving virtual top quark loops have to be considered since < +1. They lead to an additional correction in gN and gN : R@ @ 4@ (13) gN PgN #do , gN PgN #do . 4@ 4@  R @ @  R This improved Born approximation [29] provides an excellent approximation for many applications. The distribution of the angle h of the outgoing fermion f, with respect to the incident e\ direction is given by 8 dp "1#cos h# A cos h 3 $ d cos h

(14)

for an unpolarised electron beam. The forward-backward asymmetry in the di!erential cross section is de"ned as pD !pD AD "  . (15) $ pD #pD  where pD and pD are the cross sections in the forward and backward directions. At the Z pole A is  $ given by 2gN gN 3 3 2gN gN 4C C 4D D . AD" A A " (16) $ C D 4 4 (gN  #gN  ) (gN  #gN  ) 4C C 4D D The forward}backward asymmetries as measured by OPAL in ZPl>l\ decays is shown in Fig. 11. The large values of the longitudinal polarisation (P +80%) achieved at the SLC have allowed C the SLD experiment to make an extremely precise measurement of the weak mixing angle sin h   from the left}right asymmetry, de"ned as p !p 0. A " * (17) *0 p #p * 0 Here p (p ) is the total cross-section for a left (right) handed polarised incident electron beam. In * 0 terms of vector and axial vector couplings A measures directly the couplings of the Z to the *0

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Fig. 11. Forward}backward asymmetries as function of centre-of-mass energy measured by the OPAL experiment for: (a) e>e\Pe>e\, (b) e>e\Pk>k\ and (c) e>e\Pq>q\. The solid line are the results from the Standard Model "t. The deviation between the experimental data and the expectation from the Standard Model "t assuming lepton universality is shown in (d). The solid circles show the 1993 data, the solid squares the 1992 data, the open squares the 1991 data and the open circles the 1990 data.

electron current 2(1!4 sin h ) 2gN gN  4C C " . A "A " *0 C (gN  #gN  ) 1#(1!4 sin h )  4C C

(18)

The SLD measurement of A yields the most precise determination of sin h  from a single *0  experiment. It turns out to be more convenient for the averaging procedure to extract a set of minimal correlated parameters from the measured cross sections. These parameters are M , C , p  de"ned 8 8  as 12n C C CC   p, F m C 8 8

(19)

and R "C /C . The results, based on 16 million Z decays recorded by the four LEP experiments J  J are summarised in Table 3 [5] together with the Standard Model predictions. The parameter correlations are given in Table 4. The improvement by nearly one order of magnitude in precision compared to the results presented in 1990 [19] after the "rst year of LEP running based on 400 000 Z decays is impressive and beyond anything that was expected before the startup of LEP. The measurements given in Table 3 can be used to determine the number of light neutrino species (N ). The invisible decay width of the Z is J C "N C1+"C !C !(3#d )C ,  J JJ 8  K JJ

(20)

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Table 3 LEP Electroweak data from 1990 and 1997 together with their Standard Model predictions which are determined from a "t [5] in terms of m , m and a (M ) of all available data including the CDF/D0 value of m "(175.6$5.5) GeV  & Q 8  Quantity

Data 1990

Data 1997

Standard Model

Pull

m (GeV) 8 C (GeV) 8 p   R F AD $

91.177$0.031 2.498$0.013 41.77$0.42 21.08$0.19 0.0207$0.0066

91.1867$0.0020 2.4948$0.0025 41.508$0.056 20.788$0.029 0.0174$0.0010

91.1866 2.4966 41.467 20.756 0.0162

0.0 !0.7 0.4 0.7 0.9

Table 4 Correlation matrix for the electroweak parameters given in Table 3 [5]

m 8 C 8 p F Rl Al $

m 8

C 8

p F

Rl

Al $

1.00 0.05 !0.01 !0.02 0.06

0.05 1.00 !0.16 0.00 0.00

!0.01 !0.16 1.00 0.14 0.00

!0.02 0.00 0.14 1.00 0.01

0.06 0.00 0.00 0.01 1.00

where d +!0.0023 accounts for a small correction due to the mass of the tau lepton and C1+ is K JJ the Standard Model prediction for the Z decay width into neutrinos. In the ratio (C /C ) "1.991$0.001 JJ JJ 1+

(21)

all universal electroweak loop corrections cancel. It is therefore more advantageous to compare this quantity with the experimental determined ratio C /C "5.960$0.022  JJ

(22)

to derive the result N "2.993$0.011 [5]. J The determination of the number of light neutrinos is a beautiful example of the interactions between cosmology and elementary particle physics. The upper limit of N (3.5 [30] which is J obtained from the primordial abundance of He in the universe and the big-bang based model of the nucleosynthesis con"rms the collider measurements!

 For the Standard Model prediction of (C /C ) the values for m "(175.6$5.5) GeV, m "115> GeV and JJ JJ 1+  & \ a (m)"0.120$0.003 as obtained from the global "t of the electroweak data (see Section 4.5) is used. Q 8

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4. Z decays into heavy quarks The heaviest quarks which are produced in Z decays are b- and c-quarks. Their electroweak couplings can be derived from the measurements of the partial decay widths of the Z (Eq. (9)) and from the measurements of the forward}backward asymmetries A@A (Eq. (16)). ZPbbM and ZPccN $ events can be experimentally isolated using their characteristic properties (see Table 5) as will be discussed in Sections 4.1 and 4.2 in more detail. In contrast to the light #avours, the Z-bbM vertex (see Fig. 12) is expected to be subject to relative large radiative corrections (+!2%) resulting from "< "+1 and the large top quark mass of R@ m "(175.6$5.5) GeV [31,32,73] (see Fig. 13). In the ratio R ,C(ZPbbM )/C(ZPhadrons)  @ uncertainties which a!ect all quark #avours equally cancel. R isolates therefore the ZPbbM vertex @ corrections and is very sensitive to physics beyond the Standard Model such as supersymmetry [33] or technicolor [34]. Great theoretical interest was triggered by the observation that early measurements [35] for R and R showed signi"cant (see Fig. 14) deviations from the Standard @ A Model expectation. During the last few years e!orts have intensi"ed to measure the partial decay width of the Z to bbM and ccN quarks to high accuracy in order to test the electroweak sector of the Standard Model. 4.1. The measurement of R

@

The experimental techniques which are used to measure R are very similar for the various @ experiments. Di!erences exist only in the geometrical acceptance coverage or in the resolution of the tracking detectors. As the ALEPH collaboration played a leading role in the determination of Table 5 Important properties of B hadrons and D mesons used to isolate heavy #avour decays of the Z boson Property

B

D>

D

Mass (GeV) Lifetime (ps) cq (lm) Decay charged multiplicity Semileptonic branching ratio 1X "E /E 2 #  


5.3 1.6 462 5.5 0.10 0.7

1.9 1.0 317 2.2 0.17 0.5

1.9 0.4 124 2.2 0.07 0.5

Fig. 12. Vertex corrections diagrams in ZPbbM decays involving the top quark.

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Fig. 13. The Standard Model expectation for R "C(ZPf fM )/C(ZPhadrons) for di!erent values of m . D 

Fig. 14. The experimental results for R "0.2209$0.0021 and R "0.158$0.010 together with the Standard Model @ A expectation from the 1995 electroweak data [35].

R this review will summarise the development within the ALEPH experiment to explain the @ analysis techniques. A schematic view of a ZPbbM decay is shown in Fig. 15. In order to measure R one has to @ separate the ZPbbM decays from all other Z decay modes. ZPl>l\ decays with l"e, k, q can

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Fig. 15. Schematic view of a ZPbbM decay. The event is divided into two hemispheres by a plane (dotted line) perpendicular to the thrust axis (dashed line) which contains the interaction point.

easily be rejected with high e$ciency using the invariant mass and the total multiplicity of the events (see Fig. 16). To ensure that events are well contained within the silicon vertex detector, they are required to satisfy "cos h"(0.7, where h is the angle between the thrust and the beam axes. With this requirement the hadronic event selection e$ciency is 61.9% according to the Monte Carlo simulation. Non-hadronic background, which is dominated by q>q\ events, represents (0.30$0.01)% of the sample. To reject the light #avour events one has to `taga desired events by looking at the typical features of events containing B hadrons (see Fig. 64). The hard fragmentation function of the b-quarks lead to an average B-hadron momentum of 32 GeV (see Fig. 47). Together with the long lifetime of q(B)+1.5 ps (see Section 5) this results in a typical #ight length for the B hadrons in Z decays of 2.5 mm. These distances can be resolved with the vertex detectors of the LEP and SLC experiments. The reconstruction of inclusive B-hadron decay vertices turned out to be experimentally very di$cult due to the limited vertex detector resolution (see Section 2.2.1) and due to the existence of tertiary vertices from charm decays in most ZPbbM events. Much more successful was the idea of computing for each event hemisphere the con"dence level that all charged tracks stem from the primary vertex (P ), i.e. the probability that a given event hemisphere contains no long-lived & particles. Based on the three dimensional impact parameter signi"cance (see Fig. 17) the con"dence level for all charged tracks (P ) in the event is computed to stem from the primary vertex. The 2 hemisphere probability, de"ned as:






,\ H , , (23) P , “ P I !ln “ P I /j ! 2 & 2 H I I allows a good separation of ZPbbM events from other hadronic Z decays (see Fig. 18). Here N is the number of charged particles in the hemisphere. This idea was pioneered in 1991 by the MARK II Collaboration [36]. Typical e$ciencies of +25% and purities of +95% were achieved at LEP. The dominant background source were ZPccN events.

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Fig. 16. Invariant mass versus number of particles for all events with at least two charged particle tracks, from the 1992 ALEPH data. Fig. 17. The impact parameter of a charged track is de"ned as the distance of its closest approach to the interaction point. ¸ is the #ight-length of the B hadron.

Fig. 18. The hemisphere probability P in hadronic Z decays as measured by the ALEPH experiment. &

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Early measurements of R used high p leptons as the signature for B-hadron decays. These @ R analysis were systematically limited as they had to obtain the b-tag e$ciency from the Monte Carlo simulation. In order to measure the b-tag e$ciency from the data one had to count in addition to the number of tagged b hemispheres (N ) also the number of events where both hemispheres were R tagged (N ): RR N "2N [e R #e R #e R ] , R  @ @ A A V V N "N [eR (1#o )#eR (1#o )#eR (1#o )] (24) RR  @ @ @ A A A V V V with R ,R "1!R !R and e e e . The hemisphere correlation for the #avour f is here V SBQ @ A @ A V de"ned as o "(eB /e)!1 , (25) D D D where e is the b-tag e$ciency for one hemisphere and eB the e$ciency to tag both hemispheres of D D an event for a given #avour f. Hemisphere correlations are introduced, for example, due to "nal state gluon radiation which reduces the available energy for the fragmentation of both b-quarks. This so called `double-taga method measures the b-tag e$ciency e from the data as can easily be @ seen if one neglects the charm- and light #avour background and uses o +0: @ e +2N /N , R +N/(4N N ) . @ RR R @ R RR   The only quantities which have to be taken from the Monte Carlo simulation are the charm e$ciency of the b-tag (e +1%), the light #avour b-tag e$ciency (e +0.1%) and the correlations in A V the b-tag e$ciency between the two event hemispheres (o +0). Due to the smallness of e and @ A e the hemisphere correlations o and o have a negligible impact on the value obtained for R . The V A V @ value of R must be taken from Standard Model calculations or from direct measurements. A The "rst result based on this method presented by a LEP experiment was from the ALEPH collaboration [37]: R "0.2192$0.0022 $0.0026 . The analysis was based on the 1992 data @    statistics of 600 000 Z decays. The result was limited by the systematic uncertainties. The dominant sources were here the uncertainties in the two key quantities which had to be taken from the Monte Carlo simulation: the charm e$ciency for the b-tag (e ) and the hemisphere correlations (o ). A @ Especially the hemisphere correlations are very di$cult to determine from the Monte Carlo simulation. The value obtained for o is in most cases close to zero and hence indicates that the @ hemisphere correlations are small. But this happens only by chance due to a cancellation of several large e!ects which lead to either positive or negative correlations between the hemispheres. Similar analysis were performed by the other LEP experiments. The world average dominated by the results based on the double tag method from the year 1995 is shown in Fig. 14. Within the next two years the following points led to signi"cant improvements: E The data statistics used in the analysis was increased from 0.6 MZ decays to 4 MZ decays per experiment. E It turned out that the main source for the hemisphere correlations was the use of a common primary vertex for both hemispheres to compute the impact parameter signi"cance for the charged tracks. This procedure led to a correlation in the size of the primary vertex error ellipse between the two hemispheres. The situation was improved in new analysis using a primary vertex position which was computed for each event hemisphere separately. The disadvantage

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was an increase in the size of the error ellipse for the primary vertex position and hence a reduction in the background rejection for the b tagging algorithm based on the impact parameter signi"cance. E A new idea pioneered by the SLD collaboration improved the charm rejection signi"cantly using the fact that the B hadron mass is large compared to the mass of the charmed hadrons. The invariant mass spectrum of all charged particles in an event hemisphere which have impact parameter signi"cance not compatible with the primary vertex is shown in Fig. 19. ALEPH used this information to construct a lifetime-mass tag for b hemispheres. Tracks in a hemisphere are ordered inversely to their probability P to originate from the primary vertex. Tracks are 2 combined, in this order, until the invariant mass of the combination exceeds 1.8 GeV. The quantity k is the P of the last track added. Using Monte Carlo simulations an optimised & 2 b-tagging quantity was constructed as a linear combination of k and P to be: & & B "!(0.7 log k #0.3 log P ) . (26)   &  & The distribution of B is shown in Fig. 20 and the background e$ciencies as a function of the  cut on B in Fig. 21. At the nominal cut B '1.9 used in the analysis the following e$ciencies   were found in the Monte Carlo simulation: e +23%, e "0.4% and e "0.06%. Compared to @ A SBQ the old tag based only on lifetime information the dominant background from charm events is reduced by a factor two while keeping the same e$ciency for b events. E The knowledge about the behaviour of the charm background events had been improved due to a large numbers of measurements performed in the charm sector by the LEP experiments and by the CLEO experiment running at the B(4S) [25].

Fig. 19. Invariant mass of charged tracks which are not compatible with the primary vertex according to the Monte Carlo simulation in a due to a lifetime cut b enriched event sample. The dashed line indicates the D! mass value. Fig. 20. The distribution of dN /dB of the tagging variable B for data hemispheres and for Monte Carlo b, c and    uds hemispheres. The Monte Carlo has been normalised to the same number of events as the data. The last bin includes over#ow entries. The dotted vertical line indicates the cut used in the analysis.

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Fig. 21. Hemisphere tagging e$ciencies for the combined lifetime-mass b-tag. The dotted, vertical line indicates the cut used in the analysis. Fig. 22. Variation of errors on R with the b tag cut from ALEPH [38]. @

The new result published by ALEPH in 1997 was [38]: R"0.2167$0.0011 $0.0013 ! @    0.037. (R!0.172). A breakdown of the systematic error into its various sources as a function of A the b tag cut is shown in Fig. 22. The largest systematic uncertainty comes from the unknown gluon splitting rate into a pair of bbM quarks in light quark events. This situation will improve since "rst attempts to measure this rate from DELPHI and ALEPH show promising results. The total error is reduced by a factor two compared to the previous measurement, and the result has moved closer to the Standard Model prediction of R"0.2158. @ In parallel to the analysis described above a new technique was developed by the ALEPH collaboration to improve the precision of the R measurement. It is based on "ve mutually @ exclusive event hemisphere tags. The basic idea was to use all available experimental information to categorise hadronic event hemispheres either as b-, c- or uds event hemispheres. The "ve tags constructed were the following: E The combined lifetime-mass tag (B ) as described above.  E The output of a neural network (!14N 41) based on event shape variables trained with Monte Carlo data to identify b event hemispheres. E Identi"ed high p leptons as a signature of semileptonic B hadron decays. R E A neural network tag (N ) using event shape and lifetime information trained to identify charm ! event hemispheres. E An anti b-tag to identify light quark events: log P '!0.25 and N (0.  &

 The superscript 0 indicates that a quantity is corrected to the pure Z exchange, e.g. the contribution from the photon exchange is removed.

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The "ve mutually exclusive hemisphere tags result in 20 statistically independent measurements: 5 singly tagged fractions, 5 doubly tagged with the same tag, and 10 doubly tagged with di!erent tags. These were used to determine 14 quantities: R and 13 of the 15 e$ciencies of the 5 tags for b, @ for c and for the combination of light #avours u, d, s. The two backgrounds of the lifetime-mass tag could not be determined experimentally. The 45 hemisphere correlations had to be taken from the Monte Carlo simulation. Only 10 of them had a signi"cant impact on the result obtained [39]: R"0.2159$0.0009 $0.0011 !0.019 (R!0.172) . (27) @    A This measurement is highly correlated to and consistent with the result given above, because both use the same data and have similar systematic uncertainties. The result obtained with the multitag method has the smaller total error and is therefore taken as the "nal result from the ALEPH collaboration. A summary of all R measurements is given in Fig. 23. The world average of R" @ @ 0.2171$0.0009 is now within 1.5 standard deviation of the Standard Model expectation. Without the electroweak corrections for the ZPbbM vertex dominated by top quark e!ects a value of R"0.2186, would have been expected. The precision of the measurement leads to signi"cant @ restrictions for any extension of the Standard Model. Supersymmetric Models were the most promising candidates for an explanation of the previously observed deviation in R . The new mass @ limits for charginos s! set by LEP 2 (m(s!)'65 GeV) imposes signi"cant constraints on such contributions to R : *R(0.0003 [40]. The new results for R are therefore not in contradiction @ @ @ with SUSY Models but they also do not favour such models. 4.2. The measurement of R

A

The methods to measure R are similar to those discussed in the previous section for the A measurement of R . The Z decays into charm quarks are identi"ed by the semileptonic decay @ DPXll or by the reconstruction of exclusive "nal states from D or D* decays. The main background making up to 20% of the tagged c-quark sample originates from ZPbbM decays. The clean identi"cation of c-quark hemispheres is experimentally much more di$cult than the b-quark identi"cation. All LEP measurements of R are therefore either statistically limited if they use A a clean charm tag like cPDH>PDn>, DPK\n> with an e$ciency of +0.5% or they are systematically limited due to uncertainties in the estimate of large background fractions which are based on Monte Carlo simulations. A summary of the R measurements is given in Fig. 24. A The main reasons for the shift of the R value compared to 1995 results (see Fig. 14) were new A measurements of the BR(DPK\n>) and of the production rate of D* mesons in ZPccN decays. The world average of R"0.1734$0.0048 is now in perfect agreement with the Standard Model A expectation of 0.1732. 4.3. The measurement of R

SBQ

ALEPH presented [54] in summer 1996 the "rst attempt to measure directly the Z decay width into light quarks, R ,R "1!R !R . This is an important input to understand possible V SBQ @ A deviations from the Standard Model in the heavy quark sector as they were observed in summer

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Fig. 23. The measurements of R [39,41}49] together with the Standard Model prediction (dotted vertical line) for @ m "(175.5$5.5) GeV. 

1995 (see Fig. 14). A measurement of R to a precision of better than 3% would also provide V interesting constraints on R using R "1!R !R and the measured value of R . A A @ V @ The basic idea, a double tag method, used for the precision measurements of R and R is also @ A applied for the measurement of R . The events are divided into two hemispheres by a plane V perpendicular to the thrust axis. The algorithm for the #avour tag is applied independently to both event hemispheres. This allows a determination of the tagging e$ciency and R directly from V the data. In order to identify Z decays into light #avours in each hemisphere the particle, charged or neutral, with the highest momentum is found. The scaled momentum distribution (X "P /E ) is shown in Fig. 25 for data and Monte Carlo. From this distribution it is clear .


 that highly energetic charged or neutral particles are a signature for light quark events. The reason is the absence of heavy D- or B-hadrons, and therefore all the available energy can be used for the fragmentation into light particles.

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Fig. 24. The measurements of R [46,47,50}53]. A

Fig. 25. Spectrum of the highest energetic particle in a hemisphere in data (dots) and Monte Carlo simulation (line). The di!erent contributions from the various #avours according to the Monte Carlo simulation are indicated.

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Fig. 26. The minimum p distribution measured relative to the thrust axis for data (dots) and Monte Carlo simulation 2 (line). The di!erent contributions from the various #avours according to the Monte Carlo simulation are indicated.

The uds-purity of an event sample tagged with high momentum particles can signi"cantly be improved using: E Lifetime information (see Fig. 18). The signed impact parameter for all charged tracks in the hemisphere is used to compute the probability (P ) that these tracks are compatible with the & primary vertex [37] which is determined for each hemisphere separately [38]. Requiring P to & be large rejects hemispheres with long lived particles (B- and D-hadrons). E A cut on the minimum p relative to the jet axis (Fig. 26). A soft pion with a small p is 2 2 a signature of a D*PDn! decay. This has already been used as a tag for charm events for the direct measurement of R [50]. A From the three quantities given above the combined probability can be calculated that a hemisphere contains a Z decay into light quarks. The performance of this light quark tag (X ) can be  seen in Fig. 27. Since a double tag technique is used to measure R , the tag e$ciency for light V quarks as predicted by the Monte Carlo simulation can be compared to the one obtained from the data. Nice agreement between data and Monte Carlo simulation is observed as can be seen in Fig. 27a. With this method the decay width of the Z boson into light quarks is determined to be R "(61.62$0.45 $1.31 )% . V   

(28)

The systematic error is dominated by uncertainties in the estimate of the charm e$ciency from the Monte Carlo simulation. The value obtained is in good agreement with the Standard Model expectation of R "61.20%. In combination with the ALEPH result of V R "(21.59$0.09 $0.11 )% [39] a value of R "1!R !R "(16.79$0.45 $1.32 )% @    A @ V    is derived, in good agreement with the direct measurement of R "(16.83$0.91)% [50]. A similar A analysis has been presented by the OPAL collaboration [55].

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Fig. 27. (a) The e$ciency of the light quark tag as a function of the tagging variable X for uds-, c- and b-hemispheres.  The e$ciency as measured in the data for uds-events is shown by the black dots. (b) The purity of the light quark tag as a function of X according to the Monte Carlo simulation. 

4.4. The forward}backward asymmetries The forward}backward asymmetries in the production angle of quarks (see Eq. (16)) is another experimentally accessible observable in Z decays which is sensitive to the vector and axial vector couplings of quarks. Measurements have been performed of the inclusive quark charge asymmetry Q [56}59] de"ned as $ Q " d AD R , (29) $ D $ D DSBQA@ where d is a dilution factor which parametrises the sensitivity of the analysis to determine the D initial quark charge in the hemispheres. In an ideal case d would be 2q , q being the quark charge. D D D The measurements are based on the hemisphere charge information pG q (30) Q " 

,G G  pG 

,G using the fact that the primary parton charge tends to manifests itself in the leading hadron. Here q is the charge of particle i and p its momentum projected on the thrust axis. The exponent i G ,G is chosen to optimise the sensitivity of Q to the initial parton charge. The sum runs over  the charged particles in an event hemisphere de"ned by the thrust axis. In the framework of the Standard Model the measurements of Q are translated into measurements of sin h  using $  the ZFITTER package [60]. The average of the LEP measurements is [5] sin h "0.2322$0.0010 . 

(31)

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Fig. 28. The measurements of AA [46,48,61}64] together with the Standard Model prediction as a function of the $ Higgs boson mass m . &

The "rst measurements of A@, (AA) were based on semileptonic b-hadron (c-hadron) decays using $ $ the lepton to identify the ZPbbM (ZPccN ) decay and to tag the charge of the initial quark produced in this hemisphere. A newer but equally successful strategy to measure A@ is an inclusive lifetime $ tag to select the ZPbbM events and to identify the hemisphere containing the quark or anti-quark based on the hemisphere charge information Q (see Eq. (30)) similar to the measurement of Q .  $ AA has also been measured using fully reconstructed D or D* meson decays. The results for the $ measurements of AA and A@ are summarised in Figs. 28 and 29. $ $ The SLD experiment is able to measure A and A [66] directly from the polarised for@ A ward}backward asymmetries. The di!erential cross section for the production of a quark with #avour f with polarised beam is dp J(1!P A )(1!cosh)#2 cos h(A !P )A C C C C D d cos h

(32)

with A and A as de"ned in Eq. (16). The beam polarisation, P , in this expression can be of either C D C sign. Statistically this method has an advantage by a factor (P /A )+25 compared to the C C determination of A and A from the measurements of A@ JA A and AA JA A used by @ A $ C @ $ C A the LEP experiments. For the measurements in the heavy quark sector the SLD experiment

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Fig. 29. The measurements of A@ [46,48,52,62}65] together with the Standard Model prediction as a function of the $ Higgs boson mass m . &

Fig. 30. The b-tagging performance of the various experiments. IP stands for impact parameter based b-tagging, mass for the usage of secondary vertex masses, Vtx for inclusive reconstructed secondary vertices, Pl for secondary vertex momentum and usage of the rapidity distribution of the charged tracks belonging to the secondary vertex and VXD2 is the old SLD vertex detector.

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bene"ts from the small beam pipe and from the excellent CCD-based vertex detector which together give an impact parameter resolution of p(xy)"(929/p (GeV)) lm and p(rz)"(1429/p (GeV)) lm. The advantage of the new SLD vertex detector compared to the LEP experiments can be seen from the comparison of the heavy #avour tagging capabilities as shown in Fig. 30. At the same b purity SLD reaches with its new vertex detector (VXD3) more than twice the e$ciency of the LEP experiments. SLD uses various techniques to measure A and A like lepton tags, @ A lifetime#kaon tag and lifetime#jet-charge tag. The combined values from the SLD experiment are [67] A "0.650$0.058 . A

A "0.900$0.050, @

4.5. Standard Model xt of the electroweak couplings For the averaging procedure of the various results in the heavy quark sector the correlation between the analysis have to be taken into account. This is done by a s minimisation procedure with a statistical and systematic covariance matrix calculated for all measurements. For example the assumed D and D! meson production rates in ZPccN events enter as an important systematic error due to their large lifetime di!erence into all R measurements. Common systematic uncertain@ ties induce correlations between measurements and therefore must be taken into account in the combined value. The averaging procedure is described in detail in Ref. [35]. The combined SLD and LEP results are [5] R"0.2170$0.0009, R"0.1734$0.0048 , @ A A@"0.0984$0.0024, AA"0.0741$0.0048 , $ $ A "0.900$0.050, A "0.650$0.058 . @ A In deriving these results the parameters A and A have been treated as independent of the @ A forward}backward asymmetries A@ and AA [5]. The s/d.o.f for the "t procedure is 53/ $ $ (94-13)"0.65. The correlation matrix is given in Table 6. It is clear from the discussion in the previous sections that the values of R and R are correlated. The correlation plot is shown in @ A Fig. 31. If R is "xed in the averaging procedure to its Standard Model value of R "0.1723, the A A result for R is not changed. @ Table 6 Correlation matrix from the "t of the SLD, ALEPH, DELPHI, OPAL and L3 electroweak measurements

R @ R A A@ $ AA $ A @ A A

R @

R A

A@ $

AA $

A

1.00 !0.23 !0.01 0.02 !0.03 0.01

!0.23 1.00 0.05 !0.06 0.05 !0.05

!0.01 0.05 1.00 0.12 0.04 0.02

0.02 !0.06 0.12 1.00 0.01 0.10

!0.03 0.05 0.04 0.01 1.00 0.12

@

A

A

0.01 !0.05 0.02 0.10 0.12 1.00

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Fig. 31. Contours in the R!R plane derived from the LEP#SLD data, corresponding to 68% and 95% con"dence @ A levels assuming Gaussian systematic errors. The Standard Model prediction using m "(175.6$5.5) GeV is also  shown. The arrow points in the direction of increasing values of m .  Fig. 32. The *s"s!s  curve from the "t of the electroweak data given in Table 7 with m , m and a (m) as free   & Q 8 parameters. The band represents an estimate of the theoretical uncertainties due to missing higher order corrections. The vertical band shows the 95% CL exclusion limit on the Higgs boson mass m from direct search. &

The electroweak data summarised in Table 7 are normally used in a global "t to determine the Standard Model parameters [5] m "(173.1$5.4) GeV, m "115> GeV (see Fig. 32) and  & \ a (m)"0.120$0.003. Q 8 The only measurements which show signi"cant deviations from the Standard Model expectation are sinh (A ) and A@. It is therefore interesting to "t the electroweak data in a di!erent  *0 $ scenario using the e!ective couplings of leptons, c- and b-quarks as free parameters. In addition one has to take m , C and p as free parameters to determine C from the de"nition of p (see 8 8 F  F Eq. (19)) and sinh  from  gN 1 (33) sinh , 1! 4J .  gN 4 J For the computation of the forward}backward asymmetries Eq. (16) is used. The QED and QCD corrections for the partial decay widths R and R are determined with the ZFITTER package @ A [60]. The signs of gN and gN are based on the convention gN (0. This "xes the signs of the J 4J J couplings for all quarks and leptons from the LEP data alone. The results from the "t are m "(91.187$0.020) GeV, C "(2.4948$0.0025) GeV and pF "(41.486$0.053) nb and the 8 8  lepton and quark couplings as given in Table 8. The resulting lepton couplings (see Fig. 33a) and charm quark couplings (see Fig. 33b) are in good agreement with the Standard Model expectations while the Z!bbM couplings show






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Table 7 Summary of electroweak measurements from Ref. [5]. The Standarad Model results in column 3 and the pulls (di!erence between measurement and "t in units of the total measurement error) in column 4 are derived from the Standard Model "t including all data with the Higgs mass treated as a free parameter

a(m)\ [68] 8 (a) ¸EP Line-shape and lepton asymmetries: m (GeV) 8 C (GeV) 8 p (nb) F Rl Al#correlation matrix Table 4 $ q polarisation: A O A C qqN charge asymmetry: sinh  (1Q 2)  $ m (GeV) 5

Measurement with total error

Standard Model

128.896$0.090

128.898

Pull 0.0

91.1867$0.0020 2.4948$0.0025 41.486$0.053 20.775$0.027 0.0171$0.0010

91.1866 2.4966 41.467 20.756 0.0162

0.0 !0.7 0.4 0.7 0.9

0.1411$0.0064 0.1399$0.0073

0.1470 0.1470

!0.9 !1.0

0.2322$0.0010 80.48$0.14

0.23152 80.375

0.7 0.8

(b) S¸D [66] sinh  (A )  *0

0.23055$0.00041

0.23152

!2.4

(c) ¸EP and S¸D heavy -avour R @ R A A@ $ AA $ A @ A #correlation matrix Table 6 A

0.2170$0.0009 0.1734$0.0048 0.0984$0.0024 0.0741$0.0048 0.900$0.050 0.650$0.058

0.2158 0.1723 0.1031 0.0736 0.935 0.668

1.3 0.2 !2.0 0.1 !0.7 !0.3

80.41$0.09 0.2254$0.0037 175.6$5.5

80.375 0.2231 173.1

0.4 0.6 0.4

(d) ppN and lN m (GeV) (ppN [69]) 5 1!m /m (lN [70}72]) 5 8 m (GeV) (ppN [31,32,73]) R

deviations (see Fig. 33c) at the three standard deviation level. This is mainly due to the LEP measurement of A@ which is low compared with the Standard Model expectation, the SLD $ measurement of A which is also slightly lower, and the SLD measurement of A which is high @ *0 compared with the Standard Model. There are still ongoing analysis of the LEP and SLD data which have the potential to lead to signi"cant changes in the world average for A@ and hence also $ in gN and gN . The results obtained for the b-quark couplings might be taken as an indication for @ 4@ physics beyond the Standard Model but it is de"nitely too early to establish a severe crack in the electroweak sector of the Standard Model.

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Table 8 The e!ective couplings of leptons, b- and c-quarks from a "t of the electroweak data listed in Table 7 [5] together with the Standard Model expectations. The uncertainties in the Standard Model predictions are derived from the experimental uncertainties in m , m , a(m) and a (m) only  & 8 Q 8 Coupling

Data

SM

Pull

gN J gN 4J gN A gN 4A gN @ gN 4@

!0.5010$0.0003 !0.0376$0.0005 0.505$0.008 0.188$0.009 !0.520$0.007 !0.314$0.011

!0.5013$0.0014 !0.0370$0.0080 0.5015$0.0003 0.1920$0.0007 !0.4985$0.0003 !0.3438$0.0003

0.2 !0.1 0.4 !0.4 !3.1 2.7

5. B hadron lifetimes and masses Heavy quark decays can be described in the framework of the spectator model (see Fig. 34). The long B lifetime is a consequence of the small Kobayashi}Maskawa coupling between b and c quarks (< ) that neutralises the e!ect of the large mass m in the Fermi decay of the b quark: A@ @






"< "  G m S@ C " $ @ ) "< " ) F(e )#F(e ) A@ A S "< " @ 192n A@

(34)

bPu transitions have a very small impact on B hadron lifetimes due to the small value of < : "< "/"< ""0.08$0.02 [25]. The phase factor F(e) is given by S@ S@ A@ F(e)"1!8e#e!e!24e ln e

(35)

and e,m /m . A similar picture holds in the charm sector and one expects q(D)"q(D>) in the O @ framework of this simple model while the experimental value is q(D>)/q(D)"2.55$0.04 [25] in agreement with detailed calculations [74]. For the beauty sector recent calculations based on Heavy Quark E!ective Theory (HQET) [75] lead to the following predictions (which have a few percent accuracy) [76]:




q(B>)/q(B)"1.0#0.05 q(B)/q(B)"1.0, Q



f  , 200 MeV

q(K )/q(B)"0.9 , @

where f is the B decay constant. Thus the lifetimes should follow the pattern: q(B>)'q(B)+q(B)'q(K ) . B Q @ The B lifetime measurements are important inputs for the calculation of the CKM-matrix element < from the semileptonic branching ratio and for the study of BBM  oscillations and CP violation A@ in the beauty system. From the experimental point of view a precise knowledge of the B lifetime is

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Fig. 33. Contours of 68% (full line), 95% (dashed line) and 99% (dotted line) probability for the vector and axial vector couplings of leptons (a), c-quarks (b) and b-quarks (c) together with the Standard Model Prediction for M "91.1867 GeV, m "173.1 GeV, m "115 GeV, a (M)"0.120 and (M)\"1/128.896. 8  & Q 8 8

needed to calibrate b-tagging algorithms for the measurement of R "C(ZPbbM )/C(ZPqqN ) or for @ the Higgs search in the decay mode HPbbM . 5.1. Inclusive measurements The classical method to determine the inclusive B lifetime (q ) is based on the identi"cation of leptons from semileptonic B decays. From the measurement of the impact parameter (d) of the

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Fig. 34. Heavy quark decay in the spectator model.

leptons (see Fig. 17) the B lifetime can be extracted. The impact parameter is de"ned for a charged track as its distance from the interaction-point at closest approach and can be expressed in terms of experimentally accessible quantities to be d"b c c q "sin h"sinW ,

(36)

where bc is the lorentz boost factor, h is the angle of the track to the incoming electron beam and W the angle of the track relative to the B-hadron #ight direction (see Fig. 17). This technique was developed at PEP and PETRA and is nowadays used for precision measurements at LEP and SLC. It has the advantage that it is rather insensitive to the B hadron boost, since sin WJc\ and b+1 and therefore d&c ) q . Nevertheless, Monte Carlo studies are needed to determine the exact relation between lifetime and impact parameter. The typical #ight length for a B hadron produced in e>e\ annihilation at 91 GeV centre of mass energy is +2.6 mm. The vertex detector resolution for the LEP experiments in the r plane are p "12 lm and p "(12}22) lm [15] along the beam P( X direction. The primary vertex position is reconstructed with a precision of p +50 lm, p +10 lm V W and p +60 lm. The position in the x, y plane is determined from a sample of typically one X hundred events, while the z position has to be calculated on an event-by-event basis. The "nal impact parameter resolution at LEP is p +70 lm. B The lepton impact parameter distribution as measured by the ALEPH experiment is shown in Fig. 35 based on a data sample of 1.5 million hadronic events recorded in the years 1991}1993. Semileptonic B decays are selected by requiring the presence of a lepton candidate with a momentum p greater than 3 GeV and a transverse momentum p , relative to the associated jet axis,  greater than 1 GeV. This rejects most of the leptons from semileptonic charm decays and most of the electrons from photon conversions. The "t of the three-dimensional impact parameter distribution yields an average B hadron lifetime of (1.533$0.013 $0.022 ) ps [77].    Another technique which leads to high precision measurements of the average B lifetime is the inclusive reconstruction of secondary vertices. The DELPHI experiment is able to reconstruct 23 000 vertices from a data sample of 1.5 million hadronic Z events corresponding to an e$ciency of +3.5%. The decay length resolution for these vertices is (301$24) lm. They were selected to have an invariant mass of larger than 1.7 GeV, to contain more than 4 charged tracks and to be separated from the primary vertex by more than 1.5 mm. The b purity of this sample is (93.5$0.3)% according to Monte Carlo simulation. The decay length distribution

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Fig. 35. Impact parameter distribution for high p leptons as measured by the ALEPH experiment.  Fig. 36. The decay length distribution of secondary vertices as reconstructed by the DELPHI experiment in 1.5 million Z decays.

is shown in Fig. 36. The signal was parametrised by a single exponential, exp(!¸/A), with ¸"cbcq and slope A a linear function of the B hadron lifetime. The boost of the B had to be taken from the Monte Carlo simulation. The average B hadron lifetime is measured to be q "(1.582$0.011 $0.027 ) ps [78].    A similar analysis was performed by the SLD experiment. The excellent vertex detector, which is much closer to the interaction point than for the LEP experiments, leads to a much higher e$ciency for the inclusive reconstruction of B vertices. SLD measures the decay length distribution with 5000 vertices reconstructed from only 50 000 hadronic Z decays, i.e. with a reconstruction e$ciency of +23% and determines the inclusive B hadron lifetime to be q "(1.524$0.030 $0.036 ) ps [79].    The results of the inclusive B lifetime measurements are summarised in Fig. 37. For the average value, common systematic errors from the b- and c-fragmentation function, from b- and c-decay models, from the semileptonic branching-ratios (BR(BPl), BR(BPcPl) and BR(cPl)), from the c-hadron lifetime and from the B charged track multiplicity are taken into account [80]. Following the recipe described by Forty [81], the relative errors p /q are used as weights to compute lifetime G G averages throughout this paper. Nearly all analysis are limited by their systematical errors. The dominant sources are the uncertainties in the shape of the lepton spectrum for the bPcPl process and in the modelling of the b-fragmentation function. The average B hadron lifetime measured from 1986 to 1997 is shown in Fig. 38. Although the measurement has become rather stable since 1994, the plot makes it

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Fig. 37. Measurements of the average B hadron lifetime [80]. Fig. 38. The average B hadron lifetime measured from 1986 to 1997.

Fig. 39. Schematic view of a semileptonic B decay.

di$cult to believe that the measurements of the four LEP experiments are really independent. It is obvious that systematical errors were underestimated in the early days. 5.2. Exclusive measurements Semileptonic B decays have also been used for the measurement of exclusive B lifetimes (Fig. 39). It is straightforward to reconstruct the B decay vertex after the lepton has been identi"ed and the charm hadron (X ) reconstructed in one of its known decay modes. The typical decay length A resolution is p +300 lm. To measure the lifetime also the B hadron energy needs to be * determined (q "¸ ) M /P ). Depending on the di!erent experimental setups various techniques

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have been developed for this purpose. Experiments with good hermiticity try "rst to estimate the energy of the missing neutrino from the total visible energy in the hemisphere. Corrections are then determined from Monte Carlo models to estimate the B hadron energy either from the (X , l)- or A from the (X , l, l )-system. The best resolution has been obtained by two dimensional correction A J functions based on the momentum P(X , l) and the mass M(X , l) of the (X , l) system. In general the A A A resolution is in the range p @/P +(10}15)%. . @ The following decay modes have been reconstructed by the various experiments: BPD*\l>l X, B>PDM l>l X, N\PNl\l X , B J S J @ A J BPD\l>l X, K PK>l\l X . Q Q J @ A J Charge conjugation is implicitly assumed throughout this paper. Most of these analysis are limited by statistics due to the small charm branching-ratios to the experimental accessible decay channels. DELPHI recently presented a new measurement of the B lifetime based on a semi-inclusive B reconstruction of the D*>PDn> decay [82]. This approach has a higher statistical power than previous analysis. The D*> signal obtained for the data-taking periods 1991 to 1993 and 1994 is shown in Fig. 40. The data set is divided into two periods due to the vertex detector upgrade in the

Fig. 40. D* signals as found by the DELPHI experiment for the B lifetime measurement using the BPD*\l>l X B B J decay mode for the di!erent data taking periods. The dark shaded area shows the signal background.

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1993/94 shutdown. From a sample of 3520 reconstructed decay vertices a B lifetime of q(B)"(1.532$0.041 $0.040 ) ps [82] is measured.    DELPHI [83] and SLD [84] have measured B and B lifetimes using inclusive reconstructed B Q secondary vertices for which the charge of the vertex indicates the B charge. Care must be taken to exclude events with tracks that are ambiguous between production and decay vertices. DELPHI "nds 1817 B hadron vertices from 1.4 million hadronic Z decays. The B purity was estimated to be (99.1$0.3)%. According to the Monte Carlo simulation 83% (70%) of the events measured as neutral (charged) came from neutral (charged) B's. With assumptions on the B and K production Q @ fractions and lifetimes, the B lifetime can be determined from the neutral B lifetime. B Fully reconstructed B decays have been used by ALEPH [85] and CDF [86] to measure B lifetimes. While the ALEPH result is obtained by adding various decay channels, CDF is able to perform such analysis based only on BPJ/WK decay modes. Although these analysis are currently limited by statistics they have the advantage that they are independent of Monte Carlo models to determine the B hadron boost. The measurements of the B and B> lifetimes are summarised on Fig. 41. For the weighted mean B the background composition (includes D** branching ratio uncertainties), the B hadron momentum estimation, the lifetimes of B and B baryons, and the production fraction of Q B ( f "10.3> %) and of B baryons ( fK@"10.6> %) were considered as correlated systematic \  Q Q \  uncertainties. The B and the B> masses have been measured with very good accuracy in B(4S) decays by B CLEO II [87] and ARGUS [88]. The world averages are m(B)"(5279.2$1.8) MeV and B m(B>)"(5278.9$1.8) MeV [25]. At the B(4s) the centre of mass energy is to low for the production of B mesons and K baryons. The "rst measurements of the B mass have been Q @ Q

Fig. 41. Summary of the B and B> lifetime measurements [80]. B

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Fig. 42. B PJ/t decays as reconstructed by CDF. The left plot shows the invariant mass distribution while the right Q plot shows the decay length distribution in the signal region.

published by ALEPH [89] and CDF [90]. The ALEPH measurement of m(B)"(5368.6$ Q 5.6 $1.5 ) MeV is based on the reconstruction of only two unambiguous events, one with    a BM PD>n\ decay is shown in Fig. 64. Compared to the LEP experiments CDF has the Q Q advantage of the large bbM production cross section and presented recently impressive new measurements [91}93]. They were able to reconstruct 32$6 B PJ/W decays (see Fig. 42) from which Q they measured m(B)"(5369.9$2.3 $1.3 ) MeV [91]. Using their full statistics from the Q    1992 to 1995 data, corresponding to 115 pb\ they also observed K PJ/W decays [92], which @ had been expected for a long time [94]. From a signal of 38 events with an estimated background of 18.3 events the K mass is measured to be: M(K )"(5621$4$3) MeV [92]. This measurement @ @ has a precision which is by a factor two higher than the current world average of M(K )" @ (5624$9) MeV [25]. The K lifetime is determined by CDF to be q(K )"(1.32$0.15) $0.07 ) ps [93] from @ @    a sample of 197$25 right sign K l combinations using the decay chain: K PK ll, K PpK\n> A @ A A (see Fig. 43). The B and K lifetime measurements are summarised on Fig. 44. As correlated Q @ systematic uncertainties for the weighted mean of the B lifetime, the average B lifetime used in the Q background estimation, the B decay multiplicity and the B branching ratios were considered. For Q Q the average of the K lifetime, the systematic error due to the K fragmentation function, the @ @ K polarisation and K decay models were considered to be correlated between the experiments. @ @ The K lifetime is signi"cantly shorter than the average B meson lifetime and even shorter than @ expected from calculations based on HQET [76].

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Fig. 43. K l correlations as observed by CDF. The left plot shows the K signal for the right charge K l combinations A A A (dots) and the wrong charge combinations (shaded histogram). The right plot shows the proper decay length distribution in the K signal region. The shaded area is the signal distribution while the dashed line shows the background A contribution.

Fig. 44. Summary of the B and K lifetime measurements [80]. Q


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6. Excited beauty states The low lying B(1S) states, B and B*, are well established [25]. Due to the small mass splitting between B* and B mesons only electromagnetic decays of the B* to Bc are allowed. At LEP an inclusive reconstruction of B* states is performed by association of a converted photon to a jet selected to have the B meson momentum and direction. The B* mass and production rate is obtained. Replacing the photon by a charged track the technique is extended to a search of B** states. The fraction of b quarks forming a B** state is of great interest since their expected decay modes into B*n! can be used to identify the #avour of the b quark at the production time (see Fig. 45). Recently a method based on this idea has been proposed to measure CP-violation in the decay of the neutral B meson [95]. Due to this proposal much e!ort was made to predict the properties of the B** states based on extrapolations from the K** and D** sectors [96]. They are summarised in Table 9. The "ne structure of the ¸"1 system is dominated by whether the sum ¸#S "j O corresponds to j"1/2 or j"3/2. The relative production rate of B* mesons is expected to be N H/(N #N H)"0.75 from a simple spin counting picture. The maximum energy of the photon from B* decays is 0.8 GeV, while the mean energy is only 1E 2"0.3 GeV. Therefore instead of using the electromagnetic calorimeter, A which for most LEP experiments is designed to measure e$ciently electromagnetic showers with energies above 1 GeV and hence limited in this low energy range in resolution and e$ciency, the tracking system is used to reconstruct and identify converted photons.

Fig. 45. Diagram for the B**PB* p decay. Table 9 Expected properties of ¸"1 bqN states State (J.) H

q"u, d Mass (GeV/c)

=idth (GeV)

Decay mode

1>  0>  1>  2> 

5.76 +5.65 +5.65 5.77

0.020 broad broad 0.024

(B*n) J (Bn) J (B*n) J (B*n) , (Bn) J J

 Throughout the paper N and N

H

refer to the number of primary B* mesons.

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6.1. Selection and reconstruction of B hadrons B hadrons are tagged using the algorithms, described in Section 4.1 developed for the measurement of the Z decay rates into b quarks, in which the impact parameters of charged tracks are used to select long lived particles. For charged tracks two criteria are available to decide whether they belong to the primary vertex or the B decay vertex: the track probability P as obtained from the signed impact parameter and 2 the rapidity computed relative to the jet axis. The di!ering rapidity distributions for particles from B decays compared to particles from the primary fragmentation process (see Fig. 46) can also be used to classify neutral particles. The di!erent experiments have developed several techniques to reconstruct the B hadron momentum vector. ALEPH uses the track probability P as obtained from the signed impact 2 parameter for the charged particles and rapidity for the neutral particles. DELPHI employs only rapidity for charged and neutral particles. OPAL uses inclusive reconstructed secondary vertices for the charged particles and the electromagnetic energy in 233 cone around the jet axis for the neutral particles. To account for neutrino losses, detector ine$ciencies and wrong mass assignments a correction dependent on the measured B hadron mass and relative jet energy is applied, as determined from the Monte Carlo simulation. This improves the relative B momentum resolution signi"cantly. The typical resolution for the reconstructed B momentum is p(P )/P +7}10% plus @ @ a non Gaussian tail for (10}20)% of the B hadron candidates. The direction is reconstructed with a resolution of p( )+p(h)+(14}20) mrad.

Fig. 46. Rapidity distribution according to the JETSET Monte Carlo model measured relative to the thrust axis in ZPbbM events. The shaded area shows the distribution for the B hadrons. Fig. 47. B hadron fragmentation function as measured by DELPHI.

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DELPHI [97] has used the inclusive B hadron momentum reconstruction to measure the fragmentation function of the various B meson states (see Fig. 47). The average B energy, scaled to the beam energy is found to be 1x 2 "0.716$0.0006 $0.007 and the shapes of the # @    fragmentation functions are in good agreement with expectation from the JETSET Monte Carlo simulation. 6.2. B* production L3 was the "rst LEP experiment to report the observation of B*PBc decays [98]. It is the only LEP experiment to reconstruct well the low energetic photons from this decay using their excellent electromagnetic calorimeter. The B hadron boost was "xed to 37 GeV and the B #ight direction was approximated by the jet direction with a resolution of 35 mrad only. The relative B* production rate in e>e\ annihilation was measured to be N(B*) "(76$8 $6 )% .    N(B)#N(B*)

(37)

Fig. 48. (a) The B -B-mass di!erence as measured by ALEPH. The background estimated from the Monte Carlo A simulation, normalised to the same number of qqN -events, is shown by the hatched area. (b) The background subtracted signal for the decay B*PBc "tted with a Gaussian (curve).

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Similar analysis have been performed by ALEPH [99] and DELPHI [100]. Contrary to L3, converted photons were used to reconstruct the low energetic photon from the B*PBc decay as explained above. Using better algorithms to reconstruct the B hadron momentum vector (see Section 6.1) these experiments were also able to measure the B*!B mass di!erence, as seen in Fig. 48. 6.3. B* polarisation The decay angle distribution of the photon in the B* rest frame h* was measured by ALEPH [99] and DELPHI [100]. It can be used to distinguish between transverse (helicity $1) and longitudinal (helicity 0) polarised B* mesons, which have the di!erential cross sections p J(1#cosh*)/2 and p Jsinh*, respectively. If the helicity states are equally populated, i.e. 2 * p : p "2 : 1, a #at helicity angle distribution is expected. 2 * The relative longitudinal contribution is determined from the "t of the decay angle distribution (see Fig. 49) to be p /(p #p )"(33$6 $5 )%, in agreement with the expectation of 1/3 for * * 2    equally populated helicity states. 6.4. B** production For the B** search, the photon in the B*PBc analysis is replaced by a charged pion. This pion is denoted in the following n**. Only a small fraction of B* mesons (+10%) are expected to originate from B** decays. Therefore the search for B** states starting with a reconstructed B*PBc, cPe>e\ decay would reduce the signal by a large factor but not a!ect the background

Fig. 49. The acceptance-corrected number of B*-mesons as a function of the photon decay angle (cos h*) in the B* rest frame. The dashed and the dotted curves are the contributions from the transverse and longitudinal polarised states. The "t of both contributions to the data points is given by the solid curve. Fig. 50. Probability distribution for charged tracks to stem from the primary vertex computed from the signed impact parameter signi"cance. The probability is signed according to the sign of the impact parameter. The shaded area shows the contribution of the tracks from B hadron decays.

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level. Also no improvement for the signal width is expected, due to the fact that the photon is of such low energy compared to the n** in the B** rest frame: the photon lost in the decay chain only shifts the e!ective mass of the (Bn!) system down by 46 MeV from the true B** mass but does not broaden the signal signi"cantly. The B** has a negligible lifetime compared to the average B hadron. This allows the combinatorial background to be reduced using the signed impact parameter to distinguish between tracks from the primary vertex and tracks originating from the long-lived B hadron decay (see Fig. 50). The "rst LEP experiment which reported evidence for B** states was OPAL [102]. Using inclusive reconstructed secondary vertices they were able to distinguish with good purity between B> and B candidates (see Fig. 51). Using the charge correlation in B**PB>n\ decays they could prove the existence of B** states by comparing the JETSET Monte Carlo simulation for the background with the wrong charge combination (see Fig. 52). The ALEPH [99] signal for B** states is shown in Fig. 53. Within the experimental resolution for the inclusive B hadron momentum reconstruction it is not possible to resolve the di!erent

Fig. 51. Secondary vertex charge as measured by OPAL in select B events.

Fig. 52. B** signal from OPAL: (a) right charge Bp combinations, (b) wrong charge Bp combinations.

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Fig. 53. (a) The (Bn)!B-mass di!erence as measured by ALEPH. The background estimated from the Monte Carlo simulation is shown by the hatched area. (b) The background-subtracted signal for the decay B** PB*n! "tted with a Gaussian (curve).

Fig. 54. B** signal from OPAL. (a) right charge BK combinations. (b) wrong charge BK combinations. Q

B** states, for which four are expected (see Table 9). ALEPH has also observed B** states SB combining fully reconstructed B hadron decays with charged pions from the primary vertex [101]. Unfortunately this analysis is limited by the available statistics and is therefore also not able to resolve the masses and productions rates for the four expected B** states. SB With the same inclusive method, but now with charged tracks identi"ed as kaons by dE/dx, B** states have been observed (see Fig. 54) with a relative production rate of Q BR(ZPbM PB**PBH>K\) Q "0.026$0.008 BR(ZPbM PB>)

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and a mass of M(B**)"(5884$15) MeV. Evidence for these states has also been reported by Q DELPHI [104]. One should note that a large B** production rate reduces signi"cantly the number Q of B mesons for which BBM oscillations can be observed (see Fig. 55). Q Q Q In order to "nd RHPK n! decays DELPHI [105] uses a B-baryon-enriched event sample. @ @ Therefore events are selected with an identi"ed fast proton or K from the primary fragmentation process. The observed signal is shown in Fig. 56. The following mass di!erences relative to the K are measured: @ m(R!)!m(K )"(173$3 $8 ) MeV , @ @    m(R*!)!m(Kb)"(229$3 $8 ) MeV @   

Fig. 55. Expected B** decay modes.

Fig. 56. R* signal from DELPHI (Q(K n)"m(K n)!m(K )!m(n)). The expected background from the Monte Carlo @ @ @ @ simulation is shown by the shaded area.

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in agreement with recent calculations. The relative production rate is measured to be p(R )#p(R*) @ @ "0.048$0.006 $0.015    p @ One should note that a large R production rate could reduce signi"cantly the expected @ K polarisation. @ 7. BBM oscillations The decay of the weak eigenstates B and BM  is described by the SchroK dinger equation:


 


R "B2 " i Rt "BM 2







C C "B2  ! i  (38) 2 C* M* M C "BM 2   with M being the mass of the weak eigenstates B and BM  and their decay width. The o! diagonal elements M and C are responsible for B!BM  oscillations. The two CP eigenstates B "1/(2(B$BM ) with masses M "M$*m/2 and width C "C$*C/2 are obtained    from the diagonalisation of the decay matrix. They are equivalent to the mass eigenstates if CP is conserved. The consequence of Eq. (38) is that an initially pure B state can decay either as a B or a BM . The decay probabilities can be computed from Eq. (38) to be M

M

(39) P(B)"[e\CR#e\CR#2e\CRcos ut] ,  (40) P(BM )"[e\CR#e\CR!2e\CRcos ut]  with u,*m/ . For the discussion of particle}anti-particle oscillations it is more advantageous to express the mass di!erence *m in units of ps\. This will be done throughout this article. The mass di!erence *m and the decay rate di!erence *C are given by (41) *m"2 Re((M !C ) (M* !C* ) ,       (42) *C"!4 Im((M !C ) (M* !C* ) ,       In the framework of the Standard Model M and C can be calculated [106]. The particle}antiparticle oscillations proceed by a second order weak interaction, dominated by a top quark exchange, as shown in Fig. 57. M corresponds to virtual BBM  transitions while C  

Fig. 57. Box diagrams for BBM  transitions.

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Fig. 58. BBM bar oscillations for two di!erent values of *m.

describes real transitions due to decay modes which are common to B and BM  mesons, such as B, BM PudM duN , udM dcN or cdM dcN . These decay modes are Cabibbo suppressed in the BBM  system and therefore C can be neglected in most cases. This simpli"es the expression for *m and causes *C to  vanish: *m+2"M " and 

DC+0 .

(43)

One then obtains the following simple expressions for the expected oscillation pattern in the BBM  system: (1!cos *mt) (1#cos *mt) and P(BM )"e\RO . P(B)"e\RO 2 2

(44)

This is shown for two di!erent values of *m in Fig. 58. In agreement with the arguments given above detailed calculations [107] predict the lifetime di!erences in the BBM  and the BBM  system to be small: B B Q Q (*C/C) B41% and (*C/C) Q"(10}20)% .

(45)

One should note however that it could well be that (*C/C) Q"(10}20)% is the largest lifetime di!erence in the B meson sector. Unfortunately some of the QCD corrections needed to extract the CKM-matrix element < from the measurement of the BBM  oscillation rate are poorly known. However, most of these RB B B uncertainties cancel in the ratio

 

<  m *m Q " Qm RQ m Q < *m RB B

(46)

with m"1.18$0.08 [108]. Thus it is very important to measure both BBM  and BBM  oscillations. B B Q Q Together with the existing measurements of the B semileptonic branching-ratios (bPcll and bPull), this would allow the unitarity triangle (see Section 8) of the CKM-matrix to be constrained completely and hence give precise predictions of the CP-violation asymmetries expected in the framework of the Standard Model in the B sector. This will be discussed in detail in Section 8.

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7.1. Time integrated measurements The "rst observation of BBM  oscillation was made by ARGUS [109] at the B(4S) resonance. B B They used the time integrated observable s de"ned as



 1 (*m/C) 1 x " , P  M (t) dt"  2 1#(*m/C) 2 1#x  where x"*m/C. Both at the B(4S) and at the Z resonance B hadrons are produced in pairs. A B meson can decay according to B(bM , d)PDH\(cN , d)l>l while the similar decay mode for a BM meson is BM (b, dM )DH>(c, dM )l\l. The experimental signatures which have been used by all BBM  oscillation measurements up to now are therefore like-sign lepton pairs or unlike-sign lepton DH combinations. The fraction of mixed events (R) is de"ned as s"

N  R" N #N  N(l!l!) " for a lepton}lepton tag N(ll) N(DH!l8) " for a DH}lepton tag . N(DHl)

(47)

In the decay of the B(4S)PBBM  the B meson pair is produced coherently. For this case s is given by






f q  R, (48) s" 1# > > f q   where f and f are the B> and B production rates and q and q their lifetimes. N  and >  >  N  are measured using lepton-lepton correlations and D*-lepton correlations by ARGUS and CLEO [25]. The world average is s "0.156$0.024 [25]. The measurements are dominated by B the systematic uncertainties in f and q . > > At LEP, the two B mesons are produced in an incoherent state and therefore RJ2s(1!s). Also di!erent from the B(4S), not only B> and B mesons are produced in ZPbbM decays but also B B mesons. The quantity which is therefore measured at LEP is [5] Q sN "f s #f s "0.1214$0.0043 , (49) B B Q Q where f and f are the B and B meson production rates. Similar experimental signatures as B Q B Q described above for ARGUS and CLEO are used in the analysis. Using the measurements of the B hadron semileptonic branching ratios bPcll and bPull, the value of s and the strength of the CP violation in the K system (e), the unitarity triangle (see B Section 8) can be used to predict *m to be *m "(15.0> ) ps\. It can be seen in Fig. 59 that s is Q Q \  insensitive to values of x"*m/C larger than six. Therefore new techniques have been developed at LEP to resolve the time structure in BBM  oscillations.

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Fig. 59. s as a function of x"*m/C.

Fig. 60. Schematic view of a ZPbbM event.

7.2. BBM  oscillations B B A schematic view of a ZPbbM event with BM PD*>X decay is shown in Fig. 60. The #avour of B the B meson at decay time is identi"ed by the D*>PDn> decay. The #avour at the production time can either be measured using a lepton from a semileptonic decay in the opposite hemisphere or by jet charge techniques. The jet charge is de"ned by "p ) e "G ) q G, (50) Q " G G (

 "p ) e "G G G ( where p are the momentum vectors of the particles with charge q in the jet and e is the unit vector G G ( along the jet direction. The weights i are chosen to give an optimal charge reconstruction and di!er from analysis to analysis. To deduce the oscillation rate as a function of the proper time (t) one needs a measurement of both the decay length d and momentum p , since t"d /cb"d ) m /p . The proper-time resolution is then the sum of the two terms: p p t p R" B  N , q 1d 2 p q

(51)

where q is the B-hadron lifetime and 1d 2 is the average B #ight length, which is typically 2.5 mm at LEP. The B meson decay length (d ) is taken from the reconstructed charm vertex, and the B B momentum is estimated from the (D*l) momentum with the help of Monte Carlo models. The typical momentum resolution is p /p "(10}20)% and the decay length resolution is p +300 lm B N

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Fig. 61. Like-sign fraction versus proper time, for the D*-tagged analysis from OPAL.

0.2 ps. Such analysis have been performed by ALEPH [111], DELPHI [112] and OPAL [115]. The result from OPAL is shown in Fig. 61. The time dependent fraction of like-sign events R is given by N (t)!N
(t)

 R(t)"  N (t)!N
(t)   "u#(1!2u)

1!cos *m t B , 2

(52)

(53)

where u is the fraction of incorrectly tagged events. A clear oscillation pattern is observed and a value of *m "(0.539$0.060 $0.024 ) ps\ is measured. B    Also events with semileptonic B decays in both event hemispheres have been used to measure the time dependent BBM  oscillation rate. The B decay vertex is reconstructed in this case with an inclusive secondary vertex search starting with the identi"ed lepton. ALEPH [111], DELPHI [112] and OPAL [113] have used inclusive techniques to reconstruct the B boost. L3 uses P "0.85E , which works rather well due to the hard B fragmentation function. L3 [116] "nds
 1107 di-lepton events in a sample of 1.5 million hadronic Z decays. They are able to reconstruct 1429 secondary vertices in this sample. From the fraction of the like-sign lepton pairs they measure: *m "(0.458$0.046 $0.032 ) ps\. B    The measurements of *m from the various experiments are summarised in Fig. 62. The B world average is *m "(0.472$0.018) ps\"(3.11$0.12);10\ eV. This corresponds to a B remarkable precision of 4% and is much better than the time integrated measurements done at the B(4S).

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Fig. 62. Summary [80] of the *m measurements from LEP [111}117] SLD [118] and CDF [119]. B

7.3. BBM  oscillations Q Q The measurement from LEP and SLD of sN "f s #f s "0.1214$0.0043 [5] can be used to B B Q Q constrain s experimentally. The world average for *m , including the LEP, SLD and CDF Q B measurement of *m "(0.472$0.018) ps\ and the CLEO and ARGUS measurement of B sB1"0.156$0.024, gives a value of s "0.172$0.010 using a B lifetime of B B B q(B)"(1.57$0.04) ps. The B fraction, f , can be estimated from the product branching ratio B Q Q Br(bM PB) ) Br(BPD\l>lX"(1.60\ )% [120] to be f "(10.8> )%. The B production rate Q Q Q \  Q \  B in Z decays has also been measured: f "(38.8$1.3$2.1)% [121]. The experimental situation is B summarised in Fig. 63 from which a lower limit of s 50.23 with 95% CL can be derived. If one Q uses on the other hand the constraint from the unitarity triangle of the CKM matrix that *m "(15.0> ) ps\ (Section 8), which corresponds to s "0.5, one can determine the various Q \  Q B hadron production rates in Z decays from the measurements quoted above to be

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Fig. 63. Constraints on the (s , s ) plane. The dark shaded area shows the constraint from the CKM unitarity triangle. B Q

Fig. 64. A fully reconstructed BM PD>n\ decay from ALEPH, where the B oscillates to a BM  before decaying. Q Q Q Q

f "f Q"(10.2> )%, f "f B"f >"(39.5> )% and f "1.0!2f !f "(10.6> )%. Q \  B \ 
\
 B Q \  The precise knowledge of f is one of the key inputs for any measurement of BBM  oscillations in Q Q Q Z decays. Evidence for BBM  oscillations was observed by ALEPH [89] in one of its handful of fully Q Q reconstructed B decays (Fig. 64). The identi"ed K> in the same hemisphere and the e\ in the Q opposite hemisphere tag the production state of a B meson which decays then as BM PD>n\. Q Q Q Unfortunately the statistics of fully reconstructed B decays at LEP is too small to measure the Q oscillation frequency.

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The di-lepton analysis presented in the previous section is also sensitive to BBM  oscillations. The Q Q like-sign fraction as a function of proper time from ALEPH [111] is shown in Fig. 65. In the blown up region in the lower left corner a "t to the data points is shown with a second oscillation frequency *m "8.4 ps\. From a data sample of nearly 10 000 leptons with a proper time Q measurement no signal for BBM  oscillations is observed. Monte Carlo techniques are used to set Q Q a lower limit of *m 56.0 ps\ at 95% CL (see Fig. 65). Q Similar to the technique used for the measurement of *m with (D*l)-events (D l)-events can be B Q used to estimate *m . The B #avour at the decay time is given by the D and the lepton charge. To Q Q Q determine the B #avour at the production time several observables are used (see Fig. 66). The Q

Fig. 65. Left: Like-sign fraction versus proper time from di-lepton events measured by ALEPH. The curve shows a "t to the data points with *m as free parameter and maximal BBM  mixing. The blown up region shows a "t where as example B Q Q *m "8.4 ps\ is used. Right: Log likelihood function for the di-lepton analysis from ALEPH. The dotted curve shows Q the expectation for the Log likelihood function from a Monte Carlo simulation with *m "R. The 95% CL limits as Q a function of *m obtained from Monte Carlo experiments is indicated by the dashed line. Q

Fig. 66. Schematic view of a ZPbbM event. The di!erent possibilities to identify the B #avour at the production time are Q indicated. Q and Q are jet charges.  

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preferred tag is a fast lepton in the opposite hemisphere. If this is not found, a fast kaon in the same hemisphere determines also the B #avour at the production time. In a sizeable fraction of the Q events with D l combination none of the above was found, in which case the jet charge in the sameQ and in the opposite hemisphere is used. ALEPH [122] reconstructs 277 D l combinations from 4 million Z decays using 7 di!erent Q D decay modes. The likelihood function is constructed from the probability distribution Q e\ROH [1#j cos(*m t )] , (54) p (j, t )" H H  2q H where q and *m are the lifetime and oscillation frequency of B hadron source j (with the H H convention that *m "0 for non-oscillating B hadrons). Here t is the true proper time and H  the discrete variable j takes the value !1 for the mixed case or #1 for the unmixed case. The resulting distribution as seen in Fig. 67 indicates that low values of *m are strongly disfavoured Q but that no signal value of *m is signi"cantly preferred. A lower limit of *m '6.8 ps\ at 95% CL Q Q is obtained. The limits for the BBM  oscillation frequency *m are summarised in Fig. 68. None of the LEP Q Q Q experiments observed BBM  oscillations. A mathematical procedure has been proposed [126] to be Q Q able to combine the limits from the various experiments. This is achieved by replacing the probability density function of the B source given in Eq. (54) with Q e\ROQ [1#jAcos(*m t )] (55) Q 2q Q

Fig. 67. (a) *¸(*m ) from the ALEPH D l analysis. The 95% CL limits obtained from Monte Carlo experiments as Q Q a function of *m are shown for statistical errors only (dotted line) and including systematical errors (dashed line). The Q thin line indicates the expected behaviour of the likelihood function for *m "30 ps\. (b) The solid curve shows 1-CL Q determined from the data and fast Monte Carlo simulations to include systematic uncertainties. The dashed curve shows the average behaviour expected for *m "30 ps\. Q

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Fig. 68. Excluded values of *m (shaded bars) and sensitivities (triangles) [80] of the various B oscillation analysis Q Q [113,114,122}125]. Sensitivities are only quoted for analysis based on the amplitude method and represent the frequency at which p "1/1.645. Recent analysis are marked by an arrow. 

which depends now on a new parameter A, the B oscillation amplitude. For each value of *m , the Q Q new negative log-likelihood is then minimised with respect to A, leaving all other parameters (including *m ) "xed. The minimum is well behaved and very close to parabolic. At each value of Q *m one can thus obtain a measurement of the amplitude with Gaussian error, A$pA . If *m is Q Q close to the true value, one expects A"1 within the estimated uncertainty, and if *m is far from Q its true value, a measurement consistent with A"0 is expected. A value of *m can be excluded at Q 95% CL if A#1.645pA41. Measurements of A can then be combined easily mathematically. Nonetheless the combination of the various limits of *m is a non-trivial task since the di!erent Q analysis are statistically and systematically correlated. The combined result of the amplitude "ts from the LEP experiments is shown in Fig. 69 and lead to a limit of *m '10.2 ps\ at 95% CL Q [80]. This lower limit provides already interesting constraints for the unitarity triangle of the CKM Matrix as will be discussed in the next section.

8. The CKM unitarity triangle The Cabibbo}Kobayashi}Maskawa (CKM) matrix [6] is given in its standard parametrisation [25] by




<

SB < " < AB !)+ < RB






< < c c s c s e\ B SQ S@      , < < " !s c !c s s e B c c !s s s e B s c AQ A@             < < s s !c c s e B !s c !s c s e B c c RQ R@             (56)

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Fig. 69. Combined measurements of the B oscillation amplitude as a function of *m [80]. The measurements Q Q are dominated by statistical uncertainties and neighbouring points are highly statistically correlated. Discontinuities in the data, as for example at 15 ps\, are caused by the di!erent frequency ranges provided by the various analysis.

where c "cos h and s "sin h with 04h 4n/2 and 04d42n. The possible violation of the GH GH GH GH GH unitarity of the CKM-matrix is a sensitive test for physics beyond the standard model. Non unitarity of the CKM-matrix would contradict the fundamental requirement of quantum mechanics for probability conservation and therefore directly show the existence of as yet unobserved particles coupling to the six known quarks. In the following recent experimental results and theoretical developments will be used to extract the four free parameters which are needed to determine a unitary CKM-matrix. From these four parameters the strength of the CP-violation in the B-meson system can be predicted. In the near future it will be possible to measure CP-violation in the B-system directly with the new dedicated experiments Belle [9], Babar [10], Hera-B [11] and LHC-B [12]. A comparison between the measured and the predicted CP-violation will be one of the most sensitive tests for physics beyond the Standard Model performed so far. A re-parametrisation of the CKM-matrix is very often used to give a better understanding of the underlying structure s ,j, s ,Aj, s e\ B,Aj(o!ig)   

(57)

with j, A, g and o being real parameters. The unitarity of the CKM-matrix is still satis"ed exactly [128] in this notation, and it follows that s o"  cos d, s s  

s g"  sin d . s s  

(58)

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Fig. 70. The unitarity triangle of the CKM-matrix in the complex (o, g) plane.

Rewriting the CKM-matrix in this notation yields the well known Wolfenstein parametrisation [127]




< " !)+



1!j/2

j

Aj(o!ig)

!j

1!j/2

Aj

Aj(1!o!ig)

!Aj

1

#O(j) .

(59)

From di!erent relations derived from the requirement of unitarity of the CKM-matrix the most often used one is < <* #< <* #< <* "0 SB S@ AB A@ RB R@

(60)

leading in the Wolfenstein parametrisation to the `CP trianglea in the complex (o, g) plane shown in Fig. 70. This parametrisation is correct at the 3% level since only O(j) terms are taken into account [129]. The Wolfenstein parametrisation is only used here to visualise how parameters are constrained by the various experimental results, but in the "tting programs the exact formulae have been used. The three angles of the unitarity triangle are directly accessible by studying CP violation in the B system. The CP asymmetry in neutral B decays is de"ned as C(B(t)Pf )!C(BM (t)Pf ) !. !. , a ( f )" !. C(B(t)Pf )#C(BM (t)Pf ) !. !.

(61)

where f is a CP eigenstate with CP" f 2"$" f 2. The angle sin 2a is derived from !. !. !. the CP asymmetry in decay B Pn>n\, sin 2b from the decay B PJ/WK and sin 2c from the B B Q decay B PD!K8. Experimental results expected for these quantities, for example from the Q Q SLAC B-factory proposal, have a goal for the precision on sin 2a of $0.10 and on sin 2b of $0.08 [10].

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8.1. Experimental constraints The available experimental results constraining the CP-triangle are the following: 1. Nuclear b decays and Semileptonic K-decays < is measured in nuclear b-decays to be [25] SB "< ""0.9736$0.0010 SB and from hyperon decays and semileptonic kaon decays one obtains [25]

(62)

"< ""0.2196$0.0023 . (63) SQ 2. Semileptonic B-decays The most precise determinations of < is based on the analysis of the semileptonic decay A@ BM PDHllN in the framework of HQET [75] yielding a value of "< ""0.0387$0.0021 [130]. A@ Exclusive BPDM l>l decays give a consistent but less precise result. The value of "< " can also be J A@ obtained from the analysis of the inclusive lepton spectrum from B decays. Again the result is consistent with the other measurements. The combined value of [130] "< ""0.0395$0.0017+Aj A@ is now the third most accurately measured CKM matrix element. The ratio "< /< " is obtained from the endpoint region of the inclusive lepton spectrum. The S@ A@ present best estimate is obtained from the combined ARGUS and CLEO data [25] to be

 

< S@ "0.08$0.02+j(o#g. < A@ 3. BBM -oscillations The experimental e!orts to measure BBM -oscillations have been already discussed in detail in Section 7. The box-diagram shown in Fig. 57 describes the second-order weak interaction of BBM -oscillations and can be used to compute the mass di!erence (*m) between the two CP eigenstates B "(1/(2) (B$BM ) to be  G Dm " $ m B( f B(B B)g m f (y )"<* <* "   R RB R@ B 6n G + $ m B( f (B )g m f (y )Aj[(1!o)#g] .   R 6n Here m B is the mass of the B meson, g "0.55 is a QCD correction factor [131], and B y "m /M and f (y ) parametrise the top-quark-mass dependent corrections as given R  5  R in Eq. (68). The B-meson decay constant f has not been measured up to now. The only decay constant measured with good accuracy in the heavy quark sector is f Q" " ) MeV [132]. Recent lattice calculations [133] predict the (280$17 $25 $34    0">QUL> ratio f B/f Q to be f B/f Q"0.76$0.03(stat.)> (sys.)> (quench), where the "rst error is " " \  \  statistical, the second systematic within the quenched approximation and the third accounts for

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the systematical uncertainty due to the quenching. From these two values one obtains f B" (213>) MeV. This results is slightly higher but in agreement with lattice calculations \ f B"(170>) MeV [134]. In the same framework one obtains for the bag factor B B" \ 1.02$0.06 [134]. In the following "ts a value of f B(B B"(210$50) MeV

(64)

will be used. A measurement of *m would lead to a very stringent constrained for the unitarity triangle if Q one uses "< " "< " RB + RB "< " "< " A@ RQ

(65)

and

 

 

*m <  m m fB <  m 1 Q " Q Q Q RQ " Q m RQ " Bm . *m m B f B B B < m B < m B j[(1!o)#g] RB RB B Here m is determined from lattice QCD and QCD sum rules to be m"1.18$0.08 [135]. 4. ¹he CP violation parameter e ) Following [129] "e " is determined by the imaginary part of the box diagram for K!KM  ) mixing G f m m (66) "e "" $ ) ) 5B Ajg(y [g f (y , y )!g ]#g y f (y )Aj(1!o)) , A AR  A R AA RR R  R ) 6(2nDm ) ) where *m has been measured to be [25] ) *m "(0.5304$0.0014);10 ps\. (67) ) Here, g are QCD correction factors, g +0.22, g +0.62, g "0.35 for K "200 MeV, G AA RR AR /!" [136], y ,m/M , and the functions f and f are given by G G 5   1 9 1 1 3 3 x ln x f (x)" # ! ! , (68)  4 4 (1!x) 2 (1!x) 2 (1!x)






y 3y y f (x, y)"ln ! 1# ln y .  x 4(1!y) 1!y

(69)

B is the renormalisation group invariant non-perturbative parameter describing the size of ) 1KM "(sN d) (sN d) "K2 [129]. The values for B , f and m are speci"ed below. The experi4\ 4\ ) ) ) mental value of "e " is [25] ) "e ""(2.258$0.018);10\. (70) ) 8.2. Fit of the unitarity triangle The experimental constraints using the measurements described above are summarised in Table 10 and displayed in Fig. 71. The mass of the top-quark is corrected down by (7$1) GeV

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Table 10 Summary of the input values used for the "t of the unitarity triangle of the CKM-matrix Reference

Source

Status in the "t

Measurements of CKM-matrix elements: < "0.9736$0.0010 SB < "0.2205$0.0018 SQ < "0.0395$0.0017 A@ < /< "0.08$0.02 S@ A@

[25] [25] [25] [130]

Experiment Experiment Experiment Experiment

Varied Varied Varied Varied

Kaon physics input: e "(2.258$0.018);10\ ) *m "(0.5304$0.0014);10\ ps\ ) B "0.90$0.09 ) f "(160.6$1.3) MeV ) m "(497.671$0.031 MeV) )

[25] [25] [129] [25] [25]

Experiment Experiment Theory Experiment Experiment

Varied Varied Varied Fixed Fixed

[80] This paper [25] [25]

Experiment Theory Experiment Experiment

Varied Varied Fixed Fixed

[73] [5] [25]

Experiment Experiment Experiment

Varied Fixed Fixed

B physics input: *m "(0.472$0.018) ps\ B f (B "(210$50) MeV B B m "(5.279$0.002) GeV B m "(5.375$0.006) GeV Q

Other inputs: mN (m )"(167$6) GeV R  m "(80.33$0.15) GeV 5 G "(1.16639$0.00002);10\ GeV\ $

Fig. 71. The unitarity triangle of the CKM-matrix in the complex (o, g) plane. The 1 p contours from BBM  mixing (*m ), B B B the CP-violation in the K-meson system (e ) and from the ratio "< /< " are indicated by the hatched areas. The 95% CL ) S@ A@ lower limit from the search for BBM  on *m (dashed line) constrains the length of the right side of the unitarity triangle. Q Q Q Fig. 72. The "tted unitarity triangle of the CKM-matrix in the complex (o, g) plane. The dark shaded area is the 68% CL contour while the light shaded area represents the 95% CL contour. The constraint from *m '10.2 ps\ at 95% CL is Q indicated by the dashed line.

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from the pole top quark measured by CDF [31] and D0 [32] to (167$6) GeV. In the "t procedure the top quark mass is taken as a free parameter constrained by the value given above. The B parameter and the B decay constant f B(B B are treated in an analogous way in the "t. ) The result of the unitarity triangle "t is shown in Fig. 72. The s/d.o.f for the "t procedure is 1.9/(10!7)"0.6. Here the 95% CL. limit of *m '10.2 ps\ is not included in the "t as this Q cannot be done without the knowledge of the likelihood function used to determine the *m limit. Q The following "t results were obtained j"0.2217$0.0017,

A"0.805$0.036 ,

g"0.35> , \ 

o"0.13>  , \ 

f B(B B"(200>) MeV, B "0.90$0.09 , \ ) mN (m )"(167$6) GeV . R 

(71)

The product f B(B B is constrained signi"cantly by the "t procedure compared to the input value. Concerning our knowledge about the CKM matrix a direct measurement of f B from the decay B>Pq>l would be equivalent to a measurement of *m . Unfortunately the branching ratio O Q BR(B>Pq>l ) is proportional to f  ) "< " and hence expected to be only of the order of 7;10\. O S@ It is therfore very unlikely that this quantity will be measured directly in the near future. In the LEP data sample for example one could "nd only +50 such decays per experiment. However new measurements of the charm decay constant f are expected from the Tau-Charm factory with " a precision of 1.5% and, in combination with improved QCD lattice calculations for f /f , will lead " to a much better knowledge of f in the future. The prediction for the B mixing parameter extracted from the "t is *m "13.8>  ps\. As one Q Q \  can see from Fig. 72 the experimental result of *m '10.2 ps\ at 95% CL. [80] reduces Q signi"cantly the allowed region for the unitarity triangle of the CKM-matrix. The combined amplitude "t (see Fig. 69) used to determine the limit on *m provides more information than just Q the lower limit on *m . This information can be incorporated into the "t procedure [110] as an Q additional contribution to the total s using the compatibility of A(*m ) with the value A"1 (see Q Section 7.3) [A(*m )!1.0] Q . s(*m )" Q p(A(*m )) Q

(72)

As an additional free parameter one has to include m"( f Q(B Q)/( f B(B B)"1.17$0.08 [135]. This approach reduces the negative error on o by more than a factor of two as seen in Fig. 73, while it has a negligible impact on g. The s/d.o.f for the "t procedure is 1.97/(11!8)"0.7 and the "t results are j"0.2217$0.0017,

A"0.805$0.033 ,

g"0.35>  , o"0.13>  , \  \  f B(B B"(200>) MeV , B "0.90$0.09 , \ ) mN (m )"(167$6) GeV , m"1.17$0.08 . R 

(73)

358

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Fig. 73. The "tted unitarity triangle of the CKM-matrix in the complex (o, g) plane with the results of the amplitude "t (see Fig. 69) included in the "t procedure. The dark shaded area is the 68% CL contour while the light shaded area represents the 95% CL contour.

Also the negative error on f B(B B is reduced by a factor of two. The angles of the unitarity triangle can be extracted from these results to be a"(903>33) and b"(21.23> 33) . (74) \ \  Using Gaussian error propagation the corresponding values for the CP-violating phases in B-meson decays are sin(2a)"0.00> , sin(2b)"0.67>  . (75) \  \  In this scenario the preferred value for *m is Q *m "(15.0> ) ps\. (76) Q \  This implies that a direct measurement of *m is just out reach for the LEP experiments even with Q combined analysis as presented in Section 7.3. From the currently running experiments only SLD would be able to measure such large values of *m ; they expect to be sensitive up to *m +18 ps\ Q Q if a total data sample of 500 kZ events is collected. Compared to the LEP experiments, the excellent new SLD vertex detector much closer to the interaction point due to the small beam pipe at SLD wins over the large amount data at LEP. If SLC fails to deliver the expected luminosity one would have to wait for a measurement of *m from the upgraded running of CDF staring in 1999 or even Q later from the new B-factories [9}12]. How signi"cant a measurement of *m would change the Q situation in the o!g plane is demonstrated in Fig. 74. The prediction obtained from the "t for sin(2b) has already a similar precision as the "rst direct measurements expected at the beginning of the next century [9}12]. Starting with the measurement of the CP-asymmetry in the BPJ/tK decay stringent tests of the CP-violation mechanism in B Q the framework of the Standard Model will be possible. Physics beyond the Standard Model will be indicated if the unitarity triangle does not close or if the results obtained in the neutral kaon sector disagree with the measurements in the B system. These results will constrain all possible extensions of the Standard Model, as for example due to supersymmetric particles [110]. The question of whether the world is built of more than 3 generations of quarks or not will then "nd its answer.

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Fig. 74. Improvement in the "tted unitarity triangle of the CKM-matrix in the complex (o, g) plane expected from a hypothetical measurement of *m "(15.0$1.5) ps\ at future accelerators. The dark shaded area is the 68% CL Q contour while the light shaded area represents the 95% CL contour. The current constraints from the measurement of *m is indicated by the two dashed lines and the improvement due to a measurement of *m by the shaded band. B Q

9. Summary and conclusions The electroweak data from LEP, SLC and the Tevatron have reached a remarkable precision. They are sensitive to electroweak radiative corrections which allow indirect measurements of Standard Model parameters. The agreement of the direct and indirect measurements of the top quark mass is a tremendous success of the Standard Model. Only a few results, especially in the heavy quark sector, show deviations at the 2p level. These results have been reviewed in the "rst part of this article. The large e!ort put into this "eld during the last few years lead to impressive improvements in the experimental techniques which reduce systematic uncertainties well below the one percent level. From the electroweak data collected at LEP, SLC and the Tevatron the e!ective axial- and vector couplings of the b-quarks are extracted to be gN "!0.520$0.007, gN "!0.314$0.011 . @ 4@ They show compared to the Standard Model expectations of gN "!0.4985$0.0003 and @ gN "!0.3438$0.0003 deviations at the three sigma level. This is mainly due to the LEP 4@ measurement of A@ which is low compared with the Standard Model expectation, the SLD $ measurement of A which is also slightly lower, and the SLD measurement of A which is high @ *0 compared with the Standard Model. Further data-taking at SLD and the "nal analysis in the qand b-quark sector from the LEP experiments are expected to either decrease these di!erences or to establish a believable crack in the Standard Model. The "nal precision of the b-quark couplings will depend on the data statistics collected by the SLD experiment since the electroweak measurements in the heavy quark sector by SLD could dominate all the LEP measurements. The reason is the small beam pipe size at SLC and the excellent CCD-based vertex detector of the SLD experiment. These in combination allow a reconstruction of the B-hadron and D-hadron decay vertex in ZPbbM events with a high e$ciency and hence a much cleaner separation of ZPccN events from ZPbbM events and from light #avour events.

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The study of electroweak charged current interactions points to one of the fundamental questions of todays particle physics: the origin of the observed CP violation in the neutral kaon system. Together with baryon number violation, CP violation is one of the necessary conditions to generate asymmetry in the amount of matter and anti-matter in the universe [8]. It is believed that the B hadron system is the perfect laboratory to study CP violation. In contrast to the neutral kaon system non-perturbative QCD corrections can be controlled very well in the framework of HQET due to the large b-quark mass. The physics of B decay started in the 1980s and the "rst generation of experiments at CESR, DORIS, PEP, and PETRA made great contributions, including "rst measurements of B hadron masses and lifetimes, the discovery of BBM  mixing and the measurement of the CKM parameters < and < . The second generation of experiments such as CLEO II, the LEP experiments and A@ S@ CDF at the Tevatron started to observe rare decays such as BPK*c, completed the B hadron spectrum with the discovery of B and B mesons, the K and R baryon and the observation of Q A @ @ excited beauty states. The analysis of B hadron decays in the framework of HQET lead to signi"cant improvements in the determination of < and < . A@ S@ The experimental situation for the B lifetime measurements is given in Fig. 75. The K lifetime is @ signi"cantly smaller than the average B meson lifetime and even shorter than expected. The individual lifetime measurements have reached a precision of up to 2%. The lifetime ratio q(B>)/q(B) is within the errors compatible with unity (see Fig. 76) and in agreement with the theoretical expectations. The LEP experiments have studied the production of excited beauty states in hadronic Z decays. The LEP average of the relative production rate of N(B*)/(N(B)#N(B*))"(75$4)% is in good agreement with the expectation from spin counting. The determination of M(B*)!M(B)"(45.7$0.4) MeV has reached a similar precision as the previous world average [25].

Fig. 75. World averages of the B hadron lifetime measurements. Fig. 76. World averages for B-hadron lifetime ratios. The shaded area indicates the theoretical predictions for the lifetime ratios.

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The B** states have been experimentally established by OPAL [102], DELPHI [103] and ALEPH [99,101]. A large production rate for B** states has been observed by all experiments with an average value of N(B**)/N(B )"0.294$0.036. An initial B #avour tag for a CP violation SB SB measurement based on the charge of the pion from a B**>PBn> decay could therefore be interesting for future CP violation measurements. The average mass M(B**)"(5730$9) MeV is found for the B** states, with good agreement between the experiments. None of the experiments is able to resolve the four expected B** states. In principal CDF should be able to do these measurements with their large sample of fully reconstructed B decays. The capability of resolving the time structure of B!BM  oscillations at LEP, SLD and CDF B B improved the precision of the *m measurements during the last years signi"cantly. The measureB ment of *m "(0.472$0.018) ps\, corresponding to an impressive precision of 3.6%, leads, in B combination with the lower limit of *m '10.2 ps\ at 95% CL, to interesting constraints for the Q unitarity triangle of the CKM-matrix. The available experimental data permit today rather precise predictions for the expected CP violation asymmetries in B decays to be made sin(2a)"0.00> , sin(2b)"0.67>  \  \ 

(77)

from a "t of the unitarity triangle of the CKM-matrix. In this framework the BBM  oscillation Q Q frequency is predicted to be *m "(15.0> ) ps\, which is just out of reach for the LEP Q \  experiments. From the currently running experiments only SLD would be able to measure such large values of *m ; they expect to be sensitive up to *m +18 ps\ for a total data sample of Q Q +500 kZ events. The prediction obtained from the "t of the CKM-matrix unitarity triangle for sin(2b) has already a similar precision as the "rst direct measurements expected at the beginning of the next century from the third generation of beauty physics experiments [9}12]. Starting with the measurement of the CP-asymmetry in the BPJ/tK decay stringent tests of the CP-violation mechanism in the B Q framework of the Standard Model will be possible. Precision measurements of all three sides and all three angles of the unitarity triangle are expected from the LHC experiments and will be needed to test the charged-current sector of the Standard Model to a similar level of accuracy as the neutral current sector has been tested today in Z decays. If consistency between all these measurements is found, more of the 18 free parameters in the minimal Standard Model will be determined with good accuracy. Any inconsistency will point to physics beyond the Standard Model.

Acknowledgements I am grateful to my colleagues on ALEPH and to the representatives of the other experiments whose results I have reviewed here, in particular R. Forty, H.G. Moser, M. Feindt and O. Podobrin. Congratulations to the CDF Collaboration for the excellent WWW page which provides all relevant information to review their impressive physics results. It is a pleasure to thank D. Schaile and R. Settles for useful discussions and generous help.

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