Physics Reports vol.302

Physics Reports 302 (1998) 1—65

Dense plasmas, screened interactions, and atomic ionization Michael S. Murillo!,",*, Jon C. Weisheit# ! Physics Department, Rice University, Houston, TX 77005, USA " Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA # Space Physics & Astronomy Department, Rice University, Houston, TX 77005, USA Received December 1997; editor: R. Slansky

Contents 1. Introduction 1.1. Plasma preliminaries 1.2. The dense plasma environment 1.3. Plasma ionization balance 1.4. Atomic transitions in dense plasmas 2. Plasma density fluctuations 2.1. Dynamic structure factor 2.2. Plasma susceptibility and dielectric response function 2.3. Screening models 2.4. Vlasov plasmas with local field corrections 3. Static screened coulomb potentials 3.1. Classical, multicomponent case 3.2. A hybrid potential 3.3. Energy level shifts 3.4. Total elastic scattering cross section 3.5. Number of bound states 4. Generalized oscillator strength densities 4.1. Definitions 4.2. Plane-wave model 4.3. Orthogonalized plane-wave model 4.4. Numerical partial-wave model

4 4 6 7 9 11 13 16 18 20 22 24 28 31 32 33 36 36 37 38 39

5. Ionization rates 5.1. Independent electron impact method 5.2. Stochastic perturbation method 5.3. Plasma impact method 6. Numerical study of projectile screening issues 6.1. Ionization rates for He` (ground state) 6.2. Ionization rates for He` (excited state) 6.3. Ionization rates for Ar`17 (ground and excited states) 7. Numerical study of target screening issues 7.1. Non-orthogonality of initial and final states 7.2. Bound state level shifts 8. Summary and future directions 8.1. Important conclusions for ionization rates 8.2. Dense plasma issues 8.3. Screened interaction issues 8.4. Atomic ionization issues Appendix A. List of frequently used symbols Appendix B. Numerical computation of the dielectric responses function Appendix C. Formulary C.1. Plasma parameters C.2. Plasma potentials for ion of charge z References

42 43 44 46 47 48 51 52 53 53 54 56 56 57 57 58 59 60 61 61 62 63

* Corresponding author. Present address: Plasma Physics Applications Group, Applied Theoretical and Computational Physics Division, Mail Stop B259, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. Tel.: #1 505 667-6767; fax: #1 505 665-7725; e-mail: [email protected]. 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 1 7 - 9

DENSE PLASMAS, SCREENED INTERACTIONS, AND ATOMIC IONIZATION

M.S. MURILLO, J.C. WEISHEIT Physics Department, Rice University, Houston, TX 77005, USA Space Physics & Astronomy Department, Rice University, Houston, TX 77005, USA

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

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

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Abstract There now exist many laboratory programs to study non-equilibrium plasmas in which the electron interparticle spacing n~1@3 is no more than a few Bohr radii. Among these are short-pulse laser heating of solid targets, where e n &1023 cm~3, and inertial confinement fusion experiments, where n '1025 cm~3 can be achieved. Under such e e extreme conditions, the plasma environment is expected to have a strong influence on atomic energy levels and transitions rates. Investigations of atomic ionization in hot, dense plasmas have been motivated by the fact that the instantaneous degree of ionization is a key parameter for the modeling of these rapidly evolving physical systems. Although various theoretical treatments have been presented in the literature, here we focus on the “random field” approach, because it can readily incorporate (quasi-static) level shifts of the target ion as well as dynamic plasma effects. In this approach, the stochastic perturbation of the target by plasma density fluctuations is described in terms of the dielectric response function. Limiting cases of this description yield the familiar binary cross-sectional model, static screening collision models, and the more general dynamical screening models. Screening of the target ion is treated here with several static screening potentials, and bound state level shifts of these potentials are explored. Atomic oscillator strength densities based on these different models are compared in numerical calculations for ionization of He` and Ar`17. Finally, we compile a list of atomic/plasma physics issues that merit future investigation. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 52.20.-j; 52.25.Mq; 52.25.Jm Keywords: Dense plasma; Ionization; Screened interactions; Generalized oscillator strengths

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1. Introduction 1.1. Plasma preliminaries When plasmas are mentioned in non-technical discussions, they often are described with the phrase “the fourth state of matter”, to reinforce the notion that the ionized substances in neon bulbs and lightning bolts differ dramatically from our normal material surroundings. Unfortunately, this phrase is not a particularly good one, because in various circumstances ionized matter can behave very much like solids, or liquids, or ordinary gases. Better, albeit more technical, definitions convey the fact that plasmas are many-body systems, with enough mobile charged particles to cause some collective behavior. Non-neutral (single species) as well as quasi-neutral (electron-ion) plasmas are thereby included. One encounters a great range of conditions in laboratory and natural plasmas whose physical properties and behavior are germane to energy, defense, space, and numerous industrial programs [91]. The upper panel in Fig. 1 marks in temperature—density space the locations of the dense, hot plasmas generated in inertial fusion experiments; the cool, dilute plasma of

Fig. 1. (upper panel) Regimes in temperature—density space characteristic of several interesting and important plasmas: pulsar magnetospheres; tokamaks (MCF); ICF experiments; lightning; cores of white dwarf stars, the Sun, and Jupiter; Earth’s ionosphere; non-neutral (pure electron) plasmas; ultra-short-pulse laser plasmas; and electrons in metals. Classical plasmas are left of the dash line denoting ¶ "1, and weakly coupled plasmas are left of the solid line denoting e C "1. (lower left panel) Familiar phase diagram of a simple element (e.g., argon), in terms of the thermodynamic e variables pressure and temperature; both the critical point (CP) and triple point (TP) are marked. (lower right panel) The same phase diagram recast in terms of density and temperature. When translated to the upper panel this plot occupies only a small rectangular region.

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Earth’s ionosphere; the relativistic plasmas in pulsar (neutron star) magnetospheres; the degenerate plasmas formed by electrons in metals; and a few other, well-studied plasma regimes. The lines in this top panel are associated with two important plasma parameters [43,16]. Coulomb coupling, the ratio of the average potential to kinetic energy, for species a is described by the parameter C "2.3]10~7 z2n1@3/¹ , (1) a a a a where n is the particle density in cm~3, z e is the species charge, and ¹ is the temperature in eV. a a a (In general, various plasma species can have different temperatures.) One can also define an interspecies coupling parameter, C (see, e.g., [49]). The condition C*1 identifies the strong ab coupling regime. Fermi degeneracy is measured by the ratio ¶ of the Fermi temperature ¹ to the F particle temperature. The degenerate regime, where ¶*1, requires the use of quantum statistics. This criterion usually is relevant only for the electrons, where ¶ "2.4]10~15 n2@3/¹ . (2) e e e When C@1 one says that the plasma is ideal, and when ¶@1, that the plasma is classical. Further indications of the expected richness of plasma phenomena can be obtained from consideration of the lower panels of Fig. 1: on the left is a familiar equation-of-state (EOS) diagram for a simple element like argon; shown are the lines in pressure—temperature space that delineate ordinary phase transitions. On the right, this diagram is recast in terms of density and temperature, the state variables of the upper panel. Note first that the parameter ranges in typical EOS plots are much smaller than those in the plasma plot, and second that — at a given density — the plasma state can be achieved by increasing or decreasing the density. (This unusual behavior will be explained below, in Section 1.3.) Moreover in most instances these transitions to the plasma phase occur gradually, as more and more electrons populate positive energy states, in contrast to the abrupt changes that occur when, e.g., a liquid freezes. This Report is concerned with classical plasmas that are hot and dense, i.e., that have high-energy density, and specifically with the influence of such an environment on the elementary process of atomic ionization. Historically, motivation for the study of high-energy-density plasmas first arose in connection with stellar interiors: how were their equations of state, their radiatve opacities, and their nuclear reaction rates affected by densities typically exceeding 102 g/cm3 and temperatures exceeding 103 eV? Similar questions later arose in nuclear weapons research. In recent years, however, there has been a growing interest in such plasmas due to their relevance to short wavelength (EUV & X-ray) lasers [61,99,75], inertial confinement fusion (ICF) research [69,9,14,36,58], and short pulse X-ray sources [77]. In addition, experiments to study the fundamental statistical properties of dense plasmas are being carried out with ultra-short-pulse lasers (USP) [89,28,78], compressive shocks [21], and exploding wires [4]. These laboratory plasmas are characterized by electron temperatures in the range 101~3 eV, and in many instances the electron density exceeds that of a solid. It is expected that even higher energy densities will be created at the proposed National Ignition Facility [67], which should produce plasmas with ¹'10 keV and n '1026 cm~3. e Many important statistical properties of real plasmas can be developed from the OCP, the one component plasma model, in which particles of a single species a are embedded in a homogeneous, neutralizing background whose charge density is !z en . But, when correlations between different a a species are important, this scheme must be generalized to two or more components, as described in

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a series of papers by Ichimaru and colleagues in the mid-1980s [45]. These models form the basis of most of the plasma physics used here. 1.2. The dense plasma environment As we quantify below, for the purpose of understanding atomic processes in plasmas, key criteria for the characterization “high density” are a significant overlap of bound state wavefunctions with those of several plasma particles, or atomic transition energies near that of a plasma collective mode. Strongly coupled plasmas (C'1), with or without degeneracy effects, obviously are in the high-density category, but it will become clear that this true for many weakly coupled plasmas, too. So, how dense is “dense”? We describe here only three simple estimates; more sophisticated treatments are possible [81,80]. First, we consider a hydrogenic ion with nuclear charge Z in an excited state having principal quantum number a. (Here, and throughout this paper, in text we refer to ionic charges as z or Z, with the unit “e” being implied.) The “size” of this ion can be estimated by taking the radius of the electron cloud (for an s state) to be r "5SaDrDaT"(15a2/2Z) Bohr , (3) .!9 where 1 Bohr"+2/me2,a &5.29 nm. The factor of 5 is somewhat arbitrary and is used to 0 identify not the mean radius SaDrDaT but rather an effective “edge” of the ion. Note that the radius of the ion increases as the square of the principal quantum number a. The length r may be .!9 compared with the mean interionic spacing n~1@3 to identify those states DaT which are highly i perturbed by neighboring ions. As an example, for an Al plasma near solid density (typical of short-pulse laser experiments [78], we find from Eq. (3) that all states with a*3 overlap neighboring ions and are therefore strongly perturbed by them. In fact, as it is not clear which ion most affects electrons in these states, we should not consider these electrons to be bound to any particular ion; instead they need to be regarded as part of the continuum. This type of ionization, to be contrasted with thermal ionization, is called pressure ionization. Pressure ionization is well known in solid state physics [38,16,82,62] because it gives rise to energy bands and conduction electrons in metals; its appearance has the important consequence for atoms in a high-density plasma environment of limiting the number of bound states. Next, we consider the influence of free electrons on the atom, and compare an effective interaction volume (i.e., the volume occupied by a bound electron) with the mean volume occupied by a plasma electron. Define a density parameter D by the ratio (4p/3)r3 a6n .!9K3]10~22 e. D" (4) 1/n Z3 e The condition D'1 corresponds to having at least one plasma electron within the atomic volume. When this happens, the plasma electron(s) will screen the nucleus and thereby decrease the binding energy of the atomic electron. Thus, we expect that atomic electrons are more weakly bound in a dense plasma: bound state energy levels are shifted towards the continuum. Together, pressure ionization and energy level shifts are referred to as “continuum lowering” since both phenomena move bound states closer to the positive energy threshold. In Section 3, we describe this screening in terms of the plasma’s dielectric response function.

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Table 1 Principal quantum numbers a of states strongly affected by high-density environments. Shown are various hydrogenic noble gas ions of nuclear charge Z for plasma densities (in cm~3) characteristic of (low-density) MCF and (high-density) ICF experiments

Z"2 10 18 36

n "1015 e

1024

a'17 '39 '53 '75

All '1 '1 '2

Table 1 contrasts values of the principal quantum number that correspond to states satisfying D'1, for plasma densities relevant to magnetic confinement and inertial confinement fusion experiments (MCF and ICF). Although only Rydberg states, and hence processes like dielectronic recombination, are affected in MCF experiments, where densities n (1015 cm~3, nearly all states e are affected in ICF experiments, where greater-than-solid densities occur. In addition to the effects associated with high number density there are effects associated with collective behavior at high density. The primary phenomenon is that of electron plasma oscillations. The energy associated with this oscillation, +u "3.7]10~11Jn eV , (5) e e (n again is in units of cm~3) provides a third measure of “dense”. At an electron density of e 1023 cm~3, for example, this corresponds to an energy +u "11.7 eV, which is on the order of e atomic transition energies. The collective behavior of the ions at high density leads to a similar result, but these energies are characteristic of transitions in the Rydberg states which typically are pressure ionized (by the argument given above). Even at modest plasma densities Rydberg states are broadened into a quasi-continuum [47]. From the scales defined by Eqs. (3)—(5) we conclude that, for the purposes of studying atomic processes in plasmas, the term “high density” corresponds to particular combinations of low Z, high a, high n , and high n . i e 1.3. Plasma ionization balance Knowledge of charge state fractions and populations of excited states is important for determining transport properties of, as well as interpreting spectroscopic emission from, hot plasmas. In thermal equilibrium this knowledge is easily obtained via statistical mechanics, for in such circumstances the population n of state DaT is related to the population n of state DbT by a b n /n "(g /g )e~b(Ea~Eb) , (6) a b a b where the E’s and g’s are energies and statistical weights of the states, and b"1/k ¹. Extension of B this result to a non-degenerate plasma yields n

CA B D

n G 2pm 3@2 z`1 e" z`1 2 e~bIz n G bh2 z z

(7)

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

Table 2 A synopsis of the study of ionization balance in plasmas. Early work involved high-density H plasmas in thermodynamic equilibrium, with later studies being of non-thermal, steady-state plasmas at low density. Recently it has become possible to produce solid density plasmas of moderate nuclear charge Z that evolve on the subpicosecond time scale System

Date

Density

Temp.

Z

Thermal

Star HII region Tokamak ICF X-ray laser SPL

1930 1935 1952 1970 1975 1980

1024 103 1014 1024 1023 1023

1 keV 1 eV 5 keV 1.5 keV 1 keV 1 keV

1 1 1 1 34 13

]

Steady state ] ] ]

Time dependent

] ] ]

where n is the number density of ions with charge z, n is the free electron density, I is the z e z ionization potential of the charge z ion, and the G’s are atomic partition functions [100]. This is the “Saha—Boltzmann Equation” which, as indicated in Table 2, was originally applied to ionization balance in stars. (Note that no reference to underlying ionization/recombination processes need be made in obtaining the Saha—Boltzmann equation.) This equation gives, for example, the mean ion charge zN for an element with nuclear charge Z, +Z zn (8) zN " z/0 z , +Z n z/0 z which is important for obtaining effective Coulomb scattering cross sections used in transport calculations [43]. The Saha—Boltzmann equation can be used to show that, at fixed temperature, the degree of ionization zN increases as n"+ n decreases. And, if one accounts for continuum z z lowering it is clear that (again at fixed ¹) zN increases as n increases much beyond that of normal solids. Approximate ionization balance results for equilibrium plasmas also can be obtained from various density functional schemes, such as the Thomas—Fermi and Average-Atom models (see, e.g., [107,24]. It is too bad that none of these straightforward prescriptions apply to most laboratory plasma experiments: since true thermal equilibrium cannot be realized on the timescales involved, these experiments must be modeled by means of the detailed atomic processes that occur. This procedure, which conceptually is well understood, employs a set of rate equations [87,83,109,3,70] describing the time evolution of atomic populations due to various gains and losses. In the rest frame of a homogeneous plasma, the population of state DaT in an ion of charge z, n (t), obeys the equation z,a dn (t)/dt"(formation rate)!(destruction rate) . (9) z,a Here, “formation”/“destruction” refers to any atomic process that can create/destroy the charge z ion in state DaT. Some of the processes commonly included in Eq. (9) are shown in Table 3. Of course, in principle, there are similar equations coupled to this one for each of the other quantum states of the same ion, plus all the quantum states of the other atomic species; in practice, the total number of states (and equations) must be limited by some combination of physics arguments and computational constraints [111]. In some cases the time evolution is slow enough that the

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Table 3 The key atomic processes which govern ionization balance in plasmas. Excited states of charge z ions are denoted by z* and photons by c. Note that the resonant capture process cannot occur in a ion without doubly excited states Name

Reaction

Type

Inverse

Collisional excitation Collisional ionization Resonant capture Photoexcitation Photoionization

e#zPe#z* e#zP2e#(z#1) e#zP(z!1)* Nc#zPz*#(N!1)c Nc#zPe#(z#1)#(N!1)c

Collisional Collisional Collisional Radiative Radiative

Collisional de-excitation Three-body recombination Autoionization Photoemission Radiative recombination

derivative on the left-hand side can be neglected. As indicated in Table 2, this simplification applies to such steady-state environments as interstellar HII (H`) regions, tokamaks, and some aspects of ICF experiments. Also indicated are more recent experiments which require the full time dependence of Eq. (9). Note that the plasma densities of these recent experiments are quite high. Traditionally, the electron impact rates in these population equations are obtained by determining a cross section p (€) for the single electron process e~#z[state DaT]Pe~#z[state DbT] and ba accounting for the plasma environment by an average involving the flux n € of free electrons. e Calculations of this kind appear as early as 1912, in Thomson’s study of the collisional ionization process, which even predates the development of quantum mechanics. Since that time, much progress has been made in computing accurate cross sections for simple atoms, and full quantum treatments are now widely available to treat complex atomic targets as well as the indistinguishability of incident and bound electrons. Accurate approximate methods also exist and are described in the collisional excitation reviews by Bartschat [2], Fritsch and Lin [23], and Burke et al. [8], and in the collisional ionization reviews by Younger [120] and Bottcher [65]. Once the cross section has been obtained, the rate w for that bound—bound or bound—free process aPb can be ba written as

P

w "n d3v vF(€)p (€),n Sp vT . ba e ba e ba

(10)

Here F(€) is the velocity distribution of plasma electrons, and we emphasize that p refers to a cross ba section for a binary (electron—ion) collision. These collisional rates, together with rates for other important processes (e.g., those in Table 3), allow a model to be constructed, from a set of equations (Eq. (9)), which approximately describes the behavior of a nonequilibrium plasma [57]. 1.4. Atomic transitions in dense plasmas One new complication in many of the current laboratory plasma experiments results from the very short timescales involved. There may be insufficient time for the electron velocity distribution F(€) to relax to its equilibrium, Maxwellian form. Although it can be very difficult to determine what F(€) is [116,29], it is straightforward to incorporate any particular result into the rate integrals of Eq. (10), and this issue, which recently has been explored by Salzmann and Lee [102], is not considered further. A second new complication results from the high particle densities being

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

achieved, because the true atomic collisional rates w can differ from their low-density values. ba Quantitative description of these plasmas requires a set of population equations with rates suitably modified to account for the screening effects at high plasma density. In recent years there actually has been considerable effort devoted to determining collisional rates and spectral line shapes for dense plasmas. (Our focus is on collisional transitions; readers interested in spectral line formation are referred to the monograph by Griem [31] and to proceedings of the conference series on Atomic Processes in Plasmas, and Radiative Properties of Hot Dense Matter, for progress in this important, related field.) However, because the relevant experiments tend to be “integral”, in the sense that no single phenomenon can be isolated for measurement, essentially all the collision work has been theoretical. These efforts have employed a wide variety of techniques within various models to address bound—bound excitation, ionization, and three-body recombination. In each case, dense plasma phenomena were typically incorporated by separating plasma screening effects into two parts: screening of the projectile(s) and screening within the target ion. With this distinction, the models may be divided into two categories, in which the projectile screening is treated statically or dynamically. Calculations of bound—bound excitations within a purely static screening model originated with the work of Hatton et al. [39]. There, the Born approximation was used with an electron—ion interaction potential relevant to an ideal plasma. This was later improved upon by Whitten et al. [117] in calculations done with the distorted-wave approximation and close coupling descriptions using both ideal and nonideal interaction potentials for hydrogenic ions. Davis and Blaha [12] presented a similar model, based on a finite temperature average atom model, for bound—bound excitation within the distorted wave approximation that focuses on atomic energy level shifts. Work directed towards improving the aspherical properties of the potentials used by Whitten et al. (within the Born approximation) has been published by Diaz-Valdes and Gutierrez [13,33], and, using a semiclassical model, by Jung [52,53]. Most recently Jung and Yoon have carried out semiclassical ionization rates for hydrogenic ions in dense plasmas [54]. In each case, the collision cross section (or, equivalently, a collision strength) was computed, from which the rate in Eq. (10) could be obtained. These researchers have found, for example, that the result can in some cases be more sensitive to the collision physics treatment (e.g., Born versus close coupling) than dense plasma effects. Dynamical treatments of the plasma—ion interaction are a generalization of the static case and most often require a more elaborate theoretical approach. Typically, the rate is computed directly rather than via a binary collision cross section. Advantages of these treatments are that collective effects are included and rates can be obtained for transitions that cannot be described in terms of binary collision cross sections. Interestingly, the earliest work incorporating dynamical effects predates that of the static treatments. In the paper by Vinogradov and Shevel’ko [113] a method was proposed in which a bound—bound excitation can be described without recourse to a binary collision cross section, but rather as transition due to an external random field produced by all of the interacting plasma electrons. Weisheit [115] developed a similar model for bound—bound excitation rates in which the transition is driven by plasma density fluctuations (including electrons and ions). Both have shown how their model reduces to the binary collision model in the low-density limit. Dynamic screening has also been included in recombination processes as well. Schlanges et al. [103] have introduced a method for obtaining both ionization and recombination coefficients within a quantum kinetic approach based on the nonequilibrium Green function method [55]. The static screening approximation was made for their numerical computations,

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however. Girardeau and Gutierrez [34,27] treated recombination rates using a second quantization approach in which recombination energy is transferred purely to a collective mode in the form of a plasmon. Later, Rasolt and Perrot [95] computed three-body recombination rates which are enhanced by collective behavior. The excitation model of Weisheit [115] has been extended to ionization by Murillo and Weisheit [84] and Murillo [85]. Schlanges and Bornath [104,6] have extended the quantum kinetic approach and have included some nonideal plasma effects. Bound—bound excitations in a relativistic average-atom model incorporating the random field approach have also been computed [121]. Ebeling, Fo¨rster, and Podlipchuk have implemented a computational technique in which the time evolution of the ionizing electron’s wave packet is computed as it is perturbed by dynamic plasma electrons, these being simulated by a molecular dynamics technique [17]. In this paper, we consider the more general case of dynamic screening in the electron—ion interaction. A model is presented which, in the spirit of previous approaches, also separates the plasma interaction into a piece that modifies the projectile and a piece that modifies the target. Dynamical screening will be described, as in Murillo and Weisheit [84], in terms of plasma density fluctuations. In Section 2 these fluctuations are characterized by the dynamic structure factor and the plasma dielectric response function. Approximations are discussed which recover the low density and static screening cases. Then, in Section 3, screening of the target ion is discussed for various plasma conditions. A new potential is introduced which provides a smooth interpolation between well-known ideal and non-ideal plasma potentials. Effects on ionic energy levels are then discussed. In Section 4 various forms for the oscillator strength of a bound-free transition are considered. Then in Section 5 these pieces are put together in a model for calculating atomic transition rates, with numerical results for ionization being presented in Section 6 and Section 7. Our computations treat only hydrogenic ions of nuclear charge Z in various bound states; application of the basic formulae to many-electron targets should be straightforward. Finally, Section 8 summarizes our principal conclusions and offers our opinions on important issues for future work in this subject. Frequently used symbols are listed in Appendix A.

2. Plasma density fluctuations Before addressing collisional atomic processes, we need to discuss static and dynamic structural properties of dense plasmas and the functions which describe them. Specifically, the static and dynamic structure factors will be defined and related to other important quantities such as the radial distribution function, the dielectric response function, and the susceptibility. The topics covered here are a subset of the general subject of linear response theory; see, e.g., the works by Kubo [63,64]. This formalism applies equally well to a variety of other condensed systems such as liquid metals [56,41,110], Fermi gases [76], Bose gases [119], supercoiled DNA [19], membranes [123], and viruses [74,71]. However, we will have an electron gas in mind for application to the atomic collision problem which is treated in Section 5. Excellent discussions of the functions considered here can also be found in the texts by Goodstein [30], March and Tosi [124], and Hansen and McDonald [37]. A classical ideal gas is characterized by the complete absence of interparticle interactions. Since the particles cannot communicate with each other, they behave independently and may be spatially

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located anywhere, relative to other particles, with equal probability. In a dense system, however, interparticle correlations lead to a non-random spatial structure. A measure of this structure is provided by the radial distribution function g(r), which describes the likelihood that there is a particle at r given that there is another particle at the origin. For a completely random system, a classical ideal gas, g(r) is therefore uniform. We may formally define g(r) for a gas of number density n, containing N particles, by

T

U

1 N N ng(r)" + + d[r!(r !r )] ; i j N i/1 jEi

(11)

due to interactions, g(r) will have maxima if two particles are likely to be separated by particular rvalues and will have minima if particles are unlikely to have particular separations. The averaging denoted by S2T represents an ensemble average and thus g(r) is a quantity which describes the mean static structural properties of the dense system. The definition of Eq. (11) corresponds to an asymptotic (rPR) value of unity for g(r). Fig. 2 shows the OCP g(r) for values of the coupling parameter C"0.1,20,100 [97]. Note that larger C values are reflected in g(r) as larger deviations from the ideal gas result of g(r)"1. The minimum at small r arises from strong Coulomb repulsion whereas maxima occur at preferential “lattice-like” spacings. As C increases from small to large values, the plasma’s structure changes from gas-like to liquid-like to solid-like structure, a trend that is consistent with our discussion related to Fig. 1. The radial distribution function is directly measurable in elastic scattering experiments. For elastic scattering of some probe (e.g. electron, X-ray, neutron) by a many-body target, the differential elastic scattering cross section may be written as dp/dX&D»(k)D2S(k) ,

(12)

Fig. 2. OCP radial distribution functions for coupling parameter values of C"0.1,20,100. The structure that appears at larger C values reflects ordering of the particles.

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

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where +k is the momentum transferred to the probe, »(k) is the Fourier transform of the effective two-particle interaction energy »(r) between the probe and individual target particles, and

P

S(k)"1#n d3r e~*k > r[g(r)!1] .

(13)

Once S(k), the static structure factor, has been measured experimentally several thermodynamic properties, such as the energy

P

3 = º" n¹#2pn2 »(r)g(r)r2 dr 2 0 and the pressure

(14)

P

2n2p =d»(r) P"n¹! g(r)r3 dr , (15) 3 dr 0 can be easily obtained [68]. In both Eq. (14) and (15) the first terms are the ideal gas results and the second terms reflect contributions arising from (spherically symmetric) interactions between the particles, as weighted by the radial distribution function. 2.1. Dynamic structure factor Since the radial distribution function describes static properties of dense systems, a generalization is needed for the description of time-dependent phenomena. Such a generalization is provided by the van Hove correlation function G(r, t) [112,37], defined as

T

U

1 N N G(r, t)" + + d[r!(r (t)!r (0))] . i j N i/1 j/1 In terms of the particle number density n(r, t)"+ d(r!r (t)), G(r, t) can be written i i d3k Sn(k, t)n(!k, 0)Te*k > r . G(r, t)" d3r@Sn(r#r@, t)n(r@, 0)T" (2p)3

P

P

(16)

(17)

Physically G(r, t) can be interpreted as the likelihood that there is a particle at r at time t given that there was a particle (which may be the same particle) at the origin at time t"0. Thus, G(r, t) contains dynamical information regarding the movement of particles in the system. Later, in Section 5, we will see how the microscopic fluctuating potential produced by these movements can excite ions in a dense plasma. Often G(r,t) is broken into two pieces,

T

U T

U

N 1 N N + d[r!(r (t)!r (0))] # + + d[r!(r (t)!r (0))] , i i i j N i/1 i/1 jEi where the first term, the “self” term, is the contribution from the particle at the origin being found at r at time t and the second term, the “distinct” term, is the contribution from a different particle being found at r at time t. Evidently, 1 G(r, t)" N

G(r, 0)"d(r)#ng(r) , which establishes the connection between G(r, t) and g(r).

(18)

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

As with g(r), G(r, t) can be measured experimentally, albeit by inelastic scattering experiments. To do so, one considers the generalization of the static structure factor, the dynamic structure factor (DSF) S(k, u), which is defined as

P P

S(k, u)"N d3r dt G(r, t)e~*(k > r~ut) ,

(19)

where +k is again the momentum transferred and +u is the energy transferred in the collision.1 Another dynamic structure factor can be defined in terms of density fluctuations, dn(r)"n(r)!n, rather than of the density itself. This dynamic structure factor satisfies the sum rule

P

1 = S(k)" du S(k, u) . (20) 2pN ~= We need not distinguish between these two definitions of S(k, u) because, as we show below, for inelastic processes (uO0) they are functionally equivalent. To see how S(k, u), and hence G(r, t), arises in an inelastic scattering experiment consider the pedagogic example of a free electron which is inelastically scattered by a plasma. In the Born approximation we may describe this event as an electron in initial momentum state Dp T scattering a into final momentum state Dp T while the plasma undergoes a transition from state DAT to state DBT. b If the Coulomb interaction energy of this electron at r and the plasma particles of type a in a volume element at r@ is n (r@)U (r!r@)d3r@, the first-order transition rate can be expressed as a ea 2 w "(2p/+) SBDSp D + nL (r@)U (r!r@)d3r@Dp TDAT d(E !E ) , (21) fi b X a ea a f i a where E "E #E and E "E #E . Here, nL (r) is the operator whose diagonal matrix elements f B b i A a a SADnL (r)DAT give the species density n (r) when the plasma is in state SDAT, and the integration a a ranges over the plasma volume X. In this paper we consider only the transitions induced by electron density fluctuations, and therefore we will be concerned only with the term n U . (In e ee Section 8.2 we comment on the neglected terms.) It is possible to write Eq. (21) in a way that is physically more revealing, by decomposing the Coulomb term U into discrete Fourier modes of wavevector k, and then defining the momentum ee and energy transferred from the incident electron to the plasma as

K

P

K

+k"p !p , a b +u"p2/2m!p2/2m , a b respectively. With these manipulations Eq. (21) takes the form w (k, u)"(2p/X2+2)DU (k)D2DSBDnL s(k)DATD2d(u!u ) , fi ee BA where +u "E !E and where nL †(k)"nL (!k) is the kth Fourier mode of n (r). BA B A e

(22)

(23)

1 For the sake of simplicity, we consider here the classical definition of G(r, t). In the quantal definition, r (t) and r (t) are i j non-commuting operators which must be properly ordered. This does not represent a limitation since we never directly employ the definition of G(r, t), but rather its Fourier transform S(k, u). For the precise quantal definition, one is referred to the original literature by van Hove [112].

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

15

For our purposes it is unimportant which particular plasma states DAT and DBT are involved in the transition aPb. Therefore, we perform a sum over final plasma states and a canonical average over initial plasma states to obtain the average rate of transitions aPb: w "Q~1 + e~bEAw (k, u) , (24) ba fi A,B where Q is the plasma canonical partition function. The mean (electron-induced) transition rate also can be expressed in terms of the DSF S(k, u) as w "(1/X2+2)DU (k)D2S(k, u) , ba ee where, from Eq. (23) and (24), it is evident that

(25)

e~bEA DSBDnL s(k)DATD2d(u!u ) . (26) S(k, u),2p + BA Q A,B To develop some useful relationships that S(k, u) satisfies, we begin by writing out the squared matrix element in Eq. (26) and using the integral representation of the Dirac delta function to obtain

P

=

e~bEA dq e*uq + e~*uBAq SADnL (k)DBTSBDnL s(k)DAT. (27) Q ~= A,B The time dependence of the electron density fluctuations can now be highlighted by writing the matrix elements in the Heisenberg picture as S(k,u)"

SADnL (k)DBT"SADe~*HK pq@+nL (k, q)e*HK pq@+DBT "e*uBAqSADnL (k, q)DBT,SBDnL s(k)DAT"BDnL s(k, 0)DAT,

(28)

where HK is the Hamiltonian of the plasma, viz. SHK DAT"SE DAT. Substitution of these quantities p p A into Eq. (27) gives

P

=

e~bEA SADnL (k, q)DBTSBDnL s(k, 0)DAT (29) dq e*uq + Q ~= A,B which, by eliminating the expansion of unity, + DBTSBD"1, yields the result B = = dq e*uqSnL (k, q)nL s(k, 0)T"2pn2d(k)d(u)# dq e*uqSdnL (k, q)dnL s(k, 0)T ; (30) S(k, u)" ~= ~= now, the average S2T,+ Q~1exp(!bE )SAD2DAT. In the second step the density has been A A separated as n(r)"n#dn(r) and indicates that, for inelastic processes (uO0), we can equivalently define S(k, u) with either the density or its fluctuations. This form, together with Eqs. (17) and (19), establishes connection with G(r, t) and shows that G(r, t) can be measured by inelastic scattering experiments. It is easy to see that the DSF is a measure of the amplitude that a density fluctuation of wave vector k created (with the nL s operation) at time zero remains at time q later. Alternatively, one can say that the DSF is the time Fourier transform of the density—density correlation function, S(k,u)"

P

P

16

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

and thus it represents the spectrum of density fluctuations. Eq. (30), with nL s(k, 0)Pn(!k, 0) and nL (k, 0)Pn(k, 0), is the usual classical definition2 [43], wherein the averaging is taken over N-body phase space. The situation above corresponds to the case in which an incident electron transfers energy and momentum to the plasma. At finite temperatures the opposite process can occur as well. Since this can also be viewed as the incident electron transferring momentum !+k and energy !+u to the plasma, we are led to consider the function S(!k,!u), which can be found easily by writing Eq. (26) as S(k, u)"2p + A,B

e~b(EA `EB~EB) DSADnL (k)DBTD2d(u!u ) BA Q

e~bEB "2p + eb+u DSADnL (k)DBTD2d(u!u ) . (31) BA Q A,B The quantity E !E has been re-introduced in the exponential and the matrix element has been B B rewritten in terms of nL (k). The dummy indices B and A can be switched which allows the identification S(k, u)"eb+uS(!k,!u) .

(32)

There is an alternate method of arriving at Eq. (32) which provides some physical insight. Consider a thermal equilibrium system in which the state Dp T has population n and the state SDp T a a b has population n . From Eq. (6) we know how the populations of these levels are related, and we b known that, in equilibrium, specific transition rates back and forth between the levels are equal, viz. w "w . (33) ba ab Eqs. (6) and (25) can be combined with Eq. (33) to yield Eq. (32), which reveals that Eq. (32) embodies the principle of detailed balance for a finite temperature plasma. 2.2. Plasma susceptibility and dielectric response function We next relate the DSF to the plasma’s susceptibility. The linear susceptibility s(k, u) is defined in terms of the Fourier components of an external potential / (k, u) and the ensemble averaged %95 electron density fluctuation Sn (k, u)T it induces, as */$ Sn (k, u)T . (34) s(k, u)" */$ !e/ (k, u) %95 Thus, it measures the ability of an external potential to produce density fluctuations. (In the remainder of Section 2.2 all densities will refer to electron density fluctuations and the subscript

2 Many authors use variants of this definition that differ by factors of n and/or 2p. e

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

17

“ind” will be omitted.) The density fluctuations n(r, t) can be found by formally including the external potential in the equations of motion for the plasma density. Since we are seeking linear response functions we apply first-order perturbation theory to write

P

P

ie = dq h(t!q) d3r@SAD[nL (r@, q), nL (r, t)]DAT/ (r@, q) . n(r, t)"! %95 + ~=

(35)

In this expression, which was obtained by allowing the external perturbation to evolve the plasma from some initial state DAT, the unit step function h(x) has been included to allow for an integration over all times q. This form, the fluctuation of a quantity being written in terms of a commutator, is a general and ubiquitous result of many-body theory [94,63]. If the plasma is in thermal equilibrium, we can average over initial states DAT to find the mean thermal density fluctuation Sn(r, t)T. To simplify this expression a complete set of eigenstates is inserted between density operators and, for the first term in the commutator, yields the result, e~bEA 1 e~bEA + SADnL (r@, q)nL (r, t)DAT" + e*k >(r{~r) + DSADnL (k)DBTD2e*uBA(t~q) , (36) Q Q X2 k A A,B which reveals the translational invariance and stationarity properties of this matrix element. Furthermore, this expression identifies the integrations in Eq. (35) as convolution integrals that may be trivially related to the Fourier-transformed quantities Sn(k, u)T and / (k, u). Note the %95 similarity of the right-hand side of Eq. (36) to Eq. (27). The susceptibility s(k, u) can then be extracted from the defining Eq. (34). Its complete expression is rather lengthy, but we will only need its imaginary part, Im s(k, u)"!(1/2+X)S(k, u)[1!e~b+u] .

(37)

The detailed balance result of Eq. (32) was used to obtain this relation, which is one version of the so-called Fluctuation—Dissipation Theorem [94]. Now that we have obtained S(k, u) in terms of s(k, u) we can proceed to relate S(k, u) to the dielectric response function e(k, u). In this form we will (finally) be prepared to explore the screening properties of the plasma. The dielectric response function is defined via the relation e(k, u)/ (k, u)"/ (k, u) , 505 %95

(38)

where / (k, u)"/ (k, u)#/ (k, u) is the total potential resulting from the external perturba505 */$ %95 tion. These two relations can be combined to give

A B

1 / (k, u) 4pe2 "1# */$ "1# s(k, u) , e(k, u) / (k, u) k2 %95

(39)

where the (Fourier transform of the) Poisson equation, +2/ "4pen , and Eq. (34) have been */$ */$ used. We thus obtain the useful relation k2 1 Im s(k, u)" Im , 4pe2 e(k, u)

(40)

18

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

which can be combined with Eq. (37) to yield the key result, +k2X 1 S(k, u)"! Im . 2pe2(1!e~b+u) e(k, u)

(41)

This form for S(k, u) is exact within the context of linear response theory. 2.3. Screening models In a dense plasma, polarization produces an effective interaction between particles. Various approaches exist for obtaining these effective interactions, and in this section we will briefly discuss four of the more common approximations. Three of these approximations will be explored numerically in Sections 6 and 7, in the context of the atomic transition problem, and will be related to the various approaches discuss in Section 1. Many of the experiments discussed in Section 1 are characterized by long periods during which the plasma can be described by classical statistics. Therefore, the +P0 limit will be assumed for the remainder of this paper, in which case we may use S(k, u)"!(Xk2/2pe2bu)Im[1/e(k, u)] .

(42)

The utility of this formula is that the dielectric function is frequently easier to compute than the DSF, as well as lending itself to intuitive descriptions. 2.3.1. No screening The screening properties of S(k, u) are more easily identified if we express e(k, u) in terms of its real and imaginary parts, so in the classical limit we put Xk2Im e(k, u) S(k, u)" . 2pe2buDe(k, u)D2

(43)

It is instructive to relate this general result to that of an ideal gas. In a nearly ideal gas the induced potential / (k, u) is vanishingly small due to the weak interactions. The ideal gas result can be */$ found be taking the limit3 eP0 in Eq. (43) and noting the limits Re e(k, u)P1#O(e2) and Im e(k, u)PO(e2). It follows that we can write the ideal gas limit of Eq. (43) as S (k, u),(Xk2/2pe2bu)Im e(k, u) . (44) 0 Since this result corresponds to a classical non-interacting system, its use is equivalent to many of the traditional electron—ion collision treatments in which each plasma electron scatters independently. These are the approaches reviewed by Younger [120] and Bottcher [65] and discussed here in Section 5. 2.3.2. Static screening It is common practice in plasma physics to assume that a charge’s effective potential / (r) takes 505 the form / (r)"(q/r)m(r) , 505

(45)

3 In this discussion we are treating e as a dimensionless coupling parameter and not as the fundamental unit of charge.

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

19

where m(r) is a screening function. This type of screening can be thought of in terms of the dielectric theory as follows: a bare charge q produces the “external” potential q/r and gives rise to a “total” potential / (r). The screening function m(r) is obviously related to the dielectric response function 505 via Eq. (39). Since the screening function has no time dependence4 only the static part, e(k, 0), enters this picture. The familiar Debye theory is an example of this type of screening. We may use the ideal gas result, Eq. (44), to write a static screening approximation for S(k, u) as S (k, u) S (k, u)" 0 . 45 De(k, 0)D2

(46)

As will become apparent in Section 5, this prescription for the density fluctuations is related to the problem of electron—ion scattering within a static screening theory. Thus, this is the screening model actually used by Hatton et al. [39], Diaz-Valdez and Gutierrez [13,33], and Jung [52,53]. 2.3.3. Dynamic screening In general, Coulomb interactions among moving charges involve time dependent screening functions. If we write our original result as S(k, u)"S (k, u)/De(k, u)D2 , (47) 0 it is clear that Eq. (43) is the time-dependent generalization of the static screening case, Eq. (46). Only when interactions are negligible or time scales are very long are the approximations of Eq. (44) or Eq. (46) appropriate; the screening of particles that can cause ionization usually requires a dynamical description. This issue will be carefully explored in Section 3. 2.3.4. Dynamic screening: Plasma oscillations only It is a property of (unmagnetized) one component plasmas that a single collective mode can arise. This collective mode is associated with plasma oscillations. Since collective modes correspond to resonances in a many-body system, it is possible that plasma oscillations play an important role in dynamic screening. Collective modes are determined by the condition e(k, u)"0 ,

(48)

which defines a dispersion relationship of the form u"u(k). In principle, one should consider both the real and imaginary parts of e(k, u) separately. However, the imaginary part for an actual system cannot be exactly zero, so only the real part need be used to find the collective mode. To single out the collective mode in S(k, u) we expand Re e(k,u) in a Taylor series,

C

Re e(k, u)+Re e(k, u(k))#

LRe e(k, u) Lu

D

(u!u(k))#2 .

(49)

u/u(k)

4 It is important to clarify two possible interpretations of having no time dependence. It is easy to show via Fourier transform theory that the static dielectric function is associated with a time-averaged quantity. Thus, in the present context, time independent refers to a quantity which has previously been time averaged. In contrast, we could consider a time-independent system in which all particles remain fixed in some configuration. This situation is described by a frequency integral of the dielectric function.

20

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

The first term is zero by the definition of u(k). This expansion can be substituted into Eq. (47) to obtain, to lowest order, S (k, u) 0 S (k, u)" . #0-LRe e(k, u) 2 (u!u(k)) # [Im e(k, u)]2 Lu(k)

C

D

(50)

Note that this has a Lorentzian form with width (FWHM), 2 Im e(k, u) , 2c " k LRe e(k, u)/Lu(k)

(51)

where c is the decay rate associated with the oscillation [43]. In the limit that the imaginary term is k very small, this reduces to

A

B

LRe e(k, u) ~1 k2 d(u!u(k)) . S (k, u)" #0-Lu 2e2bu

(52)

In a hot dense plasma the plasma oscillation will dominate the density fluctuations and this approximate expression is useful. It is the quantized version (plasmon) that has been used by Girardeau and Gutierrez [27] to study electron—ion recombination. 2.4. Vlasov plasmas with local field corrections We now turn to the task of obtaining an explicit formula for the dielectric response function. Here we treat the plasma within classical kinetic theory as a one component system. Generalizations to two-component plasmas [11] and to degenerate systems [44] can be found elsewhere. Strong coupling is treated here via local field corrections. Recall from Eq. (38) that the response function e(k, u) relates an external potential applied to the plasma to the total potential within the plasma, the connection being / (k, u)"/ (k, u)#/ (k, u) , (53) 505 %95 */$ where / (k, u) is the potential induced in the plasma. The induced potential is related to the */$ induced number density (fluctuation) via the Poisson equation, / (k, u)"!(4pe/k2)n (k, u) */$ */$ which yields

(54)

e(k, u)"1#(4pe/k2)n (k, u)// (k, u) . (55) */$ 505 To simplify this expression we must obtain an equation of motion for the number density fluctuations n (k, u). */$ Consider a one component electron plasma described by the phase space density function F(r, €, t). This function represents the probability of finding an electron at point r with velocity € at time t. The normalization of F(r, €, t) is given by

P

n(r, t)"n d3v F(r, €, t) , e

(56)

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

21

where n(r, t) now refers to the total electron density. In the absence of any external potential the total electron density is assumed to have a uniform, stationary value n . This corresponds to an e equilibrium phase-space density of the form F (€). However, in the presence of a small external 0 potential, the phase-space density will be driven from its equilibrium value, and we can write F(r, €, t)"F (€)#F (r, €, t) (57) 0 1 where F (r, €, t) is the fluctuation in the phase-space distribution function. The induced number 1 density can be written in terms of F (r, €, t), which in turn gives 1 4pe e(k, u)"1# n d3v F (k, €, u)// (k, u) (58) 1 505 k2 e

P

for the dielectric response function. We now employ kinetic theory to find a suitable equation of motion for F (r, €, t). In general, the 1 phase-space density function F(r, €, t) satisfies the equation

A

B

P

e n L #€ ) +r# +r/ (r, t) ) +€ F(r, €, t)# e +€ ) F(Dr!r@D)F(r, €; r@, €@; t) d3r@ d3v@"0 . %95 m m Lt

(59)

Here F(r, €; r@, €@; t) is the two-particle joint probability function and F(Dr!r@D)"!+rU (Dr!r@D) is ee the electron—electron interparticle force. In fact, this is just the first in a set of equations that relate N particle density functions to N#1 particle density functions; altogether these equations are known as the BBGKY hierarchy [88]. To proceed we must find an appropriate approximation that truncates the hierarchy and leads to a kinetic equation for F(r, €, t) alone. A reasonable choice for F(r, €; r@, €@; t), motivated by the exact form in thermal equilibrium5 [105], is F(r, €; r@, €@; t)"F(r, €, t)F(r@, €@, t)g(Dr!r@D) ,

(60)

where g(Dr!r@D) is the equilibrium radial distribution function defined in Eq. (11). An integrodifferential equation for the single-particle phase-space density F(r, €, t) is obtained when Eq. (60) is substituted into Eq. (59). It is useful to define an auxiliary two-particle interaction W(Dr!r@D), according to F(Dr!r@D)[g(Dr!r@D)!1]"!+rW(Dr!r@D) .

(61)

From this formula we see that W(r) is a measure of the degree of spatial correlation, which is important in strongly coupled systems. One may obtain g(r) from some separate theory [106,90,37] or, more appropriately, by a self-consistent calculation with the dielectric response function [105]. The self-consistency condition can be seen by noting the relationships between Eqs. (18) and (19), and (41); that is, the g(r) in Eq. (60) should be consistent with the final result for e(k, u) in Eq. (58).

5 This form is often referred to as the STLS (Singwi—Tosi—Land—Sjo¨lander) ansatz.

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

If we then define the induced potential and induced charge density as

P

n / (r, t)"! e U (Dr!r@D)F (r@, €@, t) d3r@d3v@ , */$ ee 1 e

P

n (r, t)"n F (r, €, t) d3v , */$ e 1

(62) (63)

we obtain the linearized kinetic equation

A

B

e L #€ ) +r F (r, €, t)# +r(/ (r, t)#/ (r,t)) ) +€F (€) 1 %95 */$ 0 m Lt

P

1 " +€F (€) ) +r W(r!r@)n (r@, t) d3r@ . 0 */$ m

(64)

This equation may be solved by Fourier transform techniques to yield 1 k ) +€ F (€) 0 [!e/ (k, u)#W(k)n (k, u)] , F (k, €, u)" 1 505 */$ m k ) €!u

(65)

where Eq. (53) has been used. The Fourier transform of the induced density can be gotten by integrating this equation over velocities as in Eq. (56). Notice that this equation already contains an induced density term as a result of the local field correction W(k)/(4pe2/k2) [105]. At this point it is straightforward to combine the induced density terms and form the ratio n (k, u)// (k, u) which can then be substituted into Eq. (55) to obtain the dielectric response */$ 505 function, 4pe2 s(k, u) e(k, u)"1! , k2 1!W(k)s(k, u)

(66)

in terms of the susceptibility,

P

n k ) +€F (€) 0 . s(k, u)" e d3v m k ) €!u

(67)

If we neglect the local field corrections, we obtain the standard Vlasov dielectric response function,

P

4pn e2 +€F (€) e k) 0 d3v . e (k, u)"1! 0 mk2 k ) €!u

(68)

Eq. (68) can be used for obtaining the DSF for ideal (i.e., weakly coupled) plasmas, and in the static limit yields the Debye—Hu¨ckel screening model. Numerical evaluation of Eq. (68) is discussed in Appendix B. 3. Static screened Coulomb potentials In the previous section we considered dynamic screening, which is appropriate for treating the screening of the projectile(s). We now turn to the static screening treatment appropriate for target

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23

ions. Since static screening is often used for both the projectile and the target, we begin this section by analyzing the conditions for which the static approximation can also be made for the projectile. Then, in those cases in which the static approximation is valid, the results of this section will apply equally well to the projectile as well as the target without significant modification. Consider a (classical) test particle of charge q traveling straight through a plasma. For the ionization problem we might imagine that this particle impacts a particular target ion with velocity € and impact parameter b. The charge density of this particle can be written as o (r, t)"qd(r!€t!b) (69) %95 where the time origin is taken to be at the time of closest approach to the target. Alone, this particle produces a potential / (r, t) which will polarize the surrounding plasma and yield the total %95 potential / (r, t) of the charge q and its screening cloud. Of course, these potentials are related by 505 Eq. (38). If the plasma is weakly coupled we may write the complete dielectric response function in terms of the responses of individual species as [e(k, u)!1]"[e (k, u)!1]#[e (k, u)!1] . (70) e i A two-component plasma has been assumed here, but the generalization to many species is obvious. The reason we do not limit this analysis to just the electrons (as in the earlier discussion involving S(k, u)) is that atomic transitions are driven by plasma density fluctuations and ionic motions almost always are too slow to cause ionizations. However, the ionic contribution to test particle screening may not be ignorable. The Poisson equation can be used to obtain / (r, t) from Eq. (69) and then, with Eqs. (38) and %95 (70), the total potential can be written as

A

4pq k b / (k, u)" e~* > 505 k2

BC

D

1 d(u!k ) €) . (71) e (k, k ) €)#e (k, k ) €) ! 1 e i The first factor is the (Fourier transformed) potential arising from the bare charge q with impact parameter b and the second factor incorporates the screening from both the ions and the electrons. From this expression it is clear that the potential / (r, t) which is experienced at the target depends 505 sensitively on the velocity of the incident particle. We now discuss some limiting cases of Eq. (71) to find regimes where static screened Coulomb potentials (SSCP) are valid. By noting that a species dielectric response function e (k, u) is close to unity when the angular a frequency u exceeds that species plasma frequency, u "J4pn (z e)2/m , various simplifications a a a a can be found. This is a consequence of the fact that the plasma particles a cannot respond effectively on time scales shorter than 1/u . We may therefore define three regimes whereby k ) €: (1) exceeds a the plasma frequencies of both the electrons and ions, (2) is between the electron and ion plasma frequencies, and (3) is below both plasma frequencies. This is summarized in Table 4. It is important to recognize that these are strong inequalities. In general, of course, particle velocities in the plasma are distributed according to velocity distributions F(€) and none of these regimes will be applicable to the entire distribution of any species a. We must therefore be cautious in our choice for the screening function. We might expect, however, that in some situations a particular part of F (€) will a dominate and we can safely assume, for that species, just one regime applies to the entire problem. This is the case, for example, when one considers the ionization of a very deeply bound state. Such

24

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

Table 4 The three time scale regimes of an electron-ion plasma, for a fixed k. In each regime some static potential can be defined Regime

Time scale

Description

Screening particles

1 2 3

k ) €Au Au e i u Ak ) €Au e i u Au Ak ) € e i

Fast particle Intermediate velocity particle Slow particle

None Electrons only Electrons and ions

a process requires impact by a very fast particle and we may conclude that little electron or ion screening occurs. If, however, for some reason small k values are important (note the 1/k2 factor arising from the Coulomb potential in Eq. (71) this argument can fail; this will be illustrated in more detail later. In the remainder of this section we will assume that some SSCP is appropriate for the target ion, i.e. that for it e(k, u) is well approximated by e(k, 0). Unfortunately, there is no single SSCP that adequately describes every plasma environment. In a dense plasma the free-electron screening cloud which surrounds the target may be partially degenerate and/or the ions may be strongly coupled. Derivations of some of the more common plasma potentials are given below to indicate their range of validity as the plasma temperature and density vary. Then, we present a hybrid, static potential which has applicability over a wide range of the temperature, density parameter space. This result is especially useful in modeling situations where the plasma evolves rapidly through different regimes. Additional discussion of and references to literature on SSCPs can be found in the review document by Fujima [25] and the recent articles by Gutierrez et al. [35] and Chabrier [125]. 3.1. Classical, multicomponent case In a very hot plasma the degeneracy parameter ¶ is much less than unity and screening can be e treated within the context of classical statistical mechanics. Furthermore, if the interaction is relatively weak, the equations describing the potential can be linearized. This leads to the well-known “Debye—Hu¨ckel” result, which enjoys perhaps the widest use of any of the plasma potentials. We begin with the Poisson equation for the potential near a particular impurity ion of charge z. The total potential /, which is produced by the impurity ion, plasma electrons, and other plasma ions, satisifes a Poisson equation of the form +2/(r)"!4p(!en (r)#zN en (r)#zed(r)) . (72) e i Here, the other ions all have been assumed to have the average charge zN , and it is important to remember that this is to be interpreted as an equation for statistically averaged values of the quantities present, in the sense that n(r),Sn(r, t)T. All quantities are therefore spherically symmetric as well as time-independent. The electron number density at a distance r from the target is given by

T P P P

U

Ne + d(r!(r !r )) e z e/1 Ne " d3r d3Ner d3Nir + d(r!r #r )e~bU z e i e z e/1

n (r)" e

NP P P

d3Ner d3Nir d3r e~bU . e i z

(73)

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25

The denominator of Eq. (73) is the configuration integral, and the potential energy U is a sum over all pairwise interactions, viz. U" + U # + U #+U #+ U #+ U . ee{ ii{ ei ez iz e:e{ i:i{ ei e i Eq. (73) can also be written as [37]

(74)

n (r)"n g (r) , (75) e e ez where g (r) is the electron-target radial distribution function. This radial distribution function is ez similar to the one defined in Eq. (11) except that Eq. (73) measures the likelihood that there are electrons with separation r from a target ion rather than the likelihood that the electrons have some separation r from each other. A similar function g (r) describes the ion positions, iz n (r)"n g (r) . (76) i i iz Because it is not easy to evaluate the radial distribution functions exactly, a mean field theory typically is used to simplify the two-particle terms in the total interaction expression of Eq. (74). This is done by noting that the potential /(r) in Eq. (72) is the total potential from the electrons, the ions, and the target. Therefore, if we make the replacement + U #+ U #+ U P!e+ /(r !r ) , ee{ ei ez e z e:e{ ei e e and invoke a similar relation for the ions, we obtain the mean field results

(77)

n (r)"n ebee((r) , n (r)"n e~bizN e((r) . (78) e e i i (Note that the possibility of a separate ion temperature has been allowed for.) These density expressions may now be substituted into Eq. (72) to obtain +2/(r)"!4pe(!n ebee((r)#zN n e~bizN e((r)#zd(r)) , e i which is known as the Poisson—Boltzmann equation.

(79)

3.1.1. Weak coupling Simple analytic solutions can be obtained by linearizing the exponentials, viz. e~bU(r)+1!bU(r) ,

(80)

which is valid under the condition of weak coupling (cf. Eq. (1)), DbU(r)D@1 .

(81)

At large r-values the interaction is small and this condition can be satisfied even for low temperatures. But, due to the point charge z at r"0, this condition can never be satisfied near the origin. In any event, wherever these conditions have been satisfied, we obtain the equation +2/(r)!k2 /(r)"!4pzed(r) , D

(82)

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Fig. 3. A comparison of various interactions, in which the static screening factor m(r) is shown for several models discussed in the text. The electron density and temperature are n "1024 cm~3 and ¹ "15 eV. Only the electron e e component of the plasma has been included in computing the interactions.

where i2 ,4pe2(b n #b zN 2n ) (83) D e e i i is the square of the Debye wave vector. In this form Eq. (82) is the modified Helmholtz—Green function equation [1], which is easily solved to obtain the interaction potential ze ze / (r)" e~iDr" m (r) . D r r D

(84)

The inverse of the Debye wave vector, j ,i~1, is the (Debye) screening length of the plasma. D D The Debye potential is compared to the bare Coulomb potential of He` in Fig. 3, for an electron plasma with a density of n "1024 cm~3 and a temperature of ¹ "15 eV. The plotted screening e e function m (r) represents the deviation from the bare Coulomb interaction. In this plasma environD ment screening is very strong at distances as small as a Bohr radius and significant modifications to most bound states can be expected. 3.1.2. Partial electron degeneracy For a plasma with degenerate electrons, ¶ "b ¹ '1, we can compute the useful ratio n (r)/n e e F e e by first recalling that the density of non-interacting electrons is given by the standard formula [42]

P

4 j~3 = x2 n" e dx 2 ,j~3f (l) , e e 3@2 ex ~l # 1 Jp 0

(85)

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27

where j "J2p+2b /m is the electron thermal wavelength, l is related to the chemical potential e e k by l"b k, and f is the Fermi—Dirac integral. In the spirit of the finite temperature e 3@2 Thomas—Fermi model [66,18] we take the actual density near an ion to be [101] n (r)/n "f (l#eb /(r))/f (l) . (86) e e 3@2 e 3@2 Thus, the mean field / gives rise to an r-dependent energy shift that entices electrons to move to regions where their total energy is lessened. This expression is a quantum mechanical generalization of Eq. (78) which, for weak interactions, may be expanded about /(r)"0 to give n (r)/n "1#eb /(r) f @ (l)/f (l) . (87) e e e 3@2 3@2 This result, which is valid for any degree of degeneracy in a weakly coupled plasma, can be compared with Eqs. (78) and (80) to arrive at a degeneracy corrected Debye length. (Such an approach has been used by Rose [98] to compute radiative opacities.) For ¶ &(j3n )2@3@1, which e e e is the classical limit, we have f (l)+el (88) 3@2 and Eq. (87) reduces to the weak-coupling result of the previous section. But, in the degenerate limit ¶ A1 we have e f (l)+(4/3p)l3@2 (89) 3@2 and the density ratio becomes n (r)/n "1#e/(r)/¹ . (90) e e F It follows directly that in this situation we get an interaction whose screening is functionally similar to the Debye—Hu¨ckel expression, namely /(r)"ze e~iTFr/r ,

(91)

but where the Debye wave vector is replaced by the Thomas—Fermi wave vector i " TF J4pe2n /¹ [72]. e F It is not always convenient to compute the necessary functions f (l) and f @ (l) for an arbitrary 3@2 3@2 degree of degeneracy. Fortunately, one can define an approximate potential, / (r)"ze e~iDDr/r , DD

(92)

with 4pn e2 e (93) i2 " DD J¹2# ¹2 e F that is trivial to calculate and agrees with results obtained using Eq. (87) to within 5% [7]. (Here, the subscript DD refers to a Debye-like interaction corrected for degeneracy effects.) This approximate potential also is shown in Fig. 3, where its screening function m (r)"exp(!i r) is plotted. DD DD The plasma conditions are the same as previously discussed for the classical Debye interaction. It is obvious that the degeneracy correction is significant for these plasma conditions, with the classical Debye theory overestimating the screening.

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3.1.3. Strongly coupled ions The results obtained so far all depend on the assumption of weak coupling, which permits a linearization of the Poisson equation. If any of the plasma species are strongly coupled this procedure is inapplicable and a different approach is necessary. In a system of strongly coupled ions6 potentials may be computed near the target ion by putting the ions in a lattice. The lattice is divided up into cells, much like a solid state (Wigner—Seitz) approach, and the electrons are divided between the cells to give overall charge neutrality to each cell. In this picture we need to find the electron density in the small cell which surrounds the target ion. That is, in the case of strongly coupled ions we usually consider the situation where the electrons also interact strongly with the ions. If we extrapolate Eq. (78) inward toward the target ion we find that the electron density becomes arbitrarily large. This information can be used to estimate how the Fermi energy changes as a function of r. For slowly varying potentials Eq. (78) suggests that the effective Fermi energy may be estimated as E (r)"3.6]10~15[n ebee((r)]2@3 eV . (94) F e Although this extrapolation is not precise, it does indicate that near the ion the effective Fermi temperature ¹ (r)"2E (r)/3 is much greater than e/(r), whence Eq. (90) predicts a nearly uniform F F electron density. This model, a cell filled with a uniform electron distribution, is described by the Poisson equation +2/ (r)"4pen !4pzed(r) (95) IS e and is known as the ion—sphere model. The radius r of the cell is determined by the constraint of 4 charge neutrality, (4p/3)r3n "z . (96) 4 e For the reasonable boundary condition of zero potential energy at the ion—sphere radius r the 4 solution is

C

A

BD

ze r r2 ze / (r)" 1! 3! " m (r) (97) IS r 2r r2 r IS 4 4 for r(r . Fig. 3 also includes a curve showing the ion—sphere screening function m (r) for the same 4 IS He` plasma conditions. 3.2. A hybrid potential In the previous sections three different potentials have been explored. Two are appropriate for weak coupling and the other, for strong ion coupling. For a particular atomic transition, occurring perhaps under a variety of plasma conditions, it is not clear which potential is “better”. For

6 Only strongly coupled ions are considered here since the higher charge of the ions gives rise to a higher Coulomb coupling parameter.

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29

bound—bound transitions it is likely that the ion—sphere potential is preferable: bound states are highly localized and therefore experience the plasma potential when E (r) is large. Similarly, F a Debye—Hu¨ckel potential, which extends far into the plasma, is probably preferable for free—free transitions because the states are highly delocalized. These simple arguments suggest that there is no obvious choice when treating bound—free transitions (or bound—bound transitions involving a Rydberg state). Therefore, in this section we derive yet another screened interaction, one which simultaneously has appropriate properties for both bound and free states. The approach is motivated by the applicability of the ion-sphere picture at small r and the applicability of the Debye—Hu¨ckel picture at large r. Perhaps the simplest way to construct such an interaction is to match an interior, r(r@, ion—sphere form to an exterior, r'r@, Debye—Hu¨ckel form at some point r@ (to be determined). In general, r@ will depend on the properties of the surrounding plasma. The split proposed here is qualitatively similar to Region A and Region B of the potential used by Stewart and Pyatt to find energy level shifts [108], but differs from it in that degeneracy effects are included. In this section the potential energy U, rather than the potential /, will be considered. The determination of this spherically symmetric hybrid (H) interaction U (r) for an electron near H a test ion z begins, once again, with the Poisson equation,

C

D

1 d2 [rU (r)]"4pe2 + z n (r)!n (r)#zd(r) . (98) H i i e r dr2 i The ion species have various charges z and densities n (r). Near the test ion, r(r@, the electrons will i i have their highest concentration and the ions their lowest. This condition is similar to that expressed in the ion—sphere model in which n (r)+n and n (r)/n +0. Using these approximations e e i e for the densities in Eq. (98) the interaction energy takes the form U (r)"c /r#c !(ze2/2r3)r2 . (99) : 0 1 4 This is similar to U , and as rP0 the interaction is dominated by the point charge z, which gives IS c "!ze2 again. But, the coefficient c must be determined by matching to a boundary condition 0 1 that differs from the ion—sphere model. Distant from the test ion, r'r@, the weak-coupling approximation is valid and, in the spirit of the Debye—Hu¨ckel approximation, we take the ion densities to be n [1#b z U (r)] and the i i i ; electron density to be given by Eq. (87). This yields a Poisson equation of the form

C

D

1 d2 [rU (r)]"4pe2 + z n [1#b z U (r)]!n [1!b U (r) f @(l)/f (l)] ; i i i i ; e e ; 3@2 3@2 r dr2 i "i2U (r) , ; where the inverse screening length i is given by

S C S C

D

i" 4pe2 + z2b n #b n f @(l)/f (l) i i i e e 3@2 3@2 i 1 n . + 4pe2 + z2b n # i i i J¹2 # ¹2 e i e F

D

(100)

(101) (102)

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This wave vector is a generalization of Eqs. (83) and (93) to include partial electron degeneracy and ionic species with differing temperatures. The solution U (r)"(c /r)e~ir (103) ; 3 differs from the usual Debye—Hu¨ckel result in that c is not determined by a charge z at the origin 3 (the usual boundary condition), but rather by matching to the interior solution, Eq. (99), at r"r@. To find the three unknowns c ,c , and r@ we match Eqs. (99) and (103), their derivatives, their 1 3 second derivatives, and require r@'0. This yields 3ze2 c " [[(ir )3#1]2@3!1] , 1 2 i2 r3 4 4 1 r@" [[(ir )3#1]1@3!1] , 4 i 3ze2 r@eir{ . c "! 3 i2 r3 4 The full hybrid interaction, using step functions h(x), can thus be written as

(104) (105) (106)

U (r)"U (r)h(r@!r)#U (r)h(r!r@) ; (107) H : ; this interaction has the correct behavior at small and large r and can safely be used to study both bound and free states. It is interesting to compare Eq. (107) with the usual Debye—Hu¨ckel interaction for r'r@. In this regime the interaction can be written as (108) U (r)"(!z e2/r)e~i(r~r{) , ; %&& where the effective charge z , given by %&& [(ir )3 # 1]1@3!1 4 z "z 3 (109) %&& (ir )3 4 is temperature and density dependent. At moderate to high densities, (ir )'1 and z is less than z, 4 %&& owing to the screening in the region r(r@. This leads to the physical picture of a charge z ion %&& screened by a weakly coupled plasma. The screening plasma is guaranteed to be weakly coupled because the strongly coupled portion of the plasma is automatically incorporated into the region r(r@, and therefore into the definition of z . The shift in the exponential arises from the fact that %&& the z “ion” has effective size r@. %&& Clearly, it is r@ which determines the admixture of U (r) and U (r) in U(r). For very high : ; temperatures, viz. small i,r@+(ir )3/3i is very small and almost everywhere U (r)"U (r). Also, in 4 H ; this limit the effective charge of Eq. (109) reduces to the bare charge z and we recover the Debye—Hu¨ckel expression. Conversely, when the temperature is low, corresponding to large values of i,r@+r and c is essentially independent of i. The first condition indicates that the ion-sphere 4 1 form of U (r) extends to r , and the second condition indicates that U (r) is independent of the : 4 : surrounding ions. Thus, the ion—sphere interaction is recovered in this limit. Another interesting

C

D

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limit is that of high density and high temperature. In this case U (r)"U (r) results most often H ; because of the high temperature. But, we can still accomodate ¹ '¹ as well, which indicates that F e this hybrid interaction seems applicable to the study of atomic transitions under wide variety of plasma conditions. 3.3. Energy level shifts Screened hydrogenic states can be found via the Schro¨dinger equation, !(+2/2m)+2t(r)#U(r)t(r)"Et(r) ,

(110)

once a particular screened interaction U(r) has been chosen. In general this equation must be solved numerically for bound and continuum states. But, if the state under consideration is deeply bound, a simple approximation can be made to obtain eigenvalue information. Deeply bound states predominantly experience their binding interaction at small r, and U can be expanded about this limit. For each of the interactions discussed so far this limit yields ze2 U (r)P! #ze2i , DD DD r

ze2 3ze2 ze2 U (r)P! # , U (r)P! #c , (111) IS H 1 r 2r r 4 where i is defined in Eq. (93), r is defined in Eq. (96), and c is defined in Eq. (104). In each case, DD 4 1 the resulting interaction is the bare Coulomb interaction plus a (positive) constant shift *E. Thus, for tightly bound states Eq. (110) can be approximated as !(+2/2m)+2t(r)!(ze2/r)t(r)"(E!*E)t(r)

(112)

which indicates that the hydrogenic wave functions are unchanged but the states have new energy eigenvalues E"*E!(ze2)2m/2a2+2 .

(113)

Note that these uniform level shifts predict smaller ionization energies but do not predict line shifts. Since no large line shifts have been observed experimentally, we do not consider corrections beyond the uniform level shifts [46]. In the weak-coupling limit c Pze2i , and in the strong coupling limit c P3ze2/2r ; thus, the 1 DD 1 4 hybrid case provides a continuous extrapolation of energy level shifts between the two regimes just as it did for the interaction itself. In fact, because c is the energy level shift of Stewart and Pyatt 1 [108] generalized to include degeneracy corrections in i, the interaction U (r) is consistent with H their shift. For reference, Appendix Cgives numerical formulae for several of the important SSCP results obtained above. The ionization potential of the ground a"1 state of He` is shown versus plasma density in Fig. 4 for various choices of the energy level shift. The classical Debye potential predicts energy level shifts at high plasma density which nearly eliminate the state altogether. The degeneracycorrected Debye potential predicts a smaller shift, indicating a breakdown of the classical Debye picture. The ion—sphere potential, which is the extreme degeneracy limit, predicts an even smaller shift. The hybrid—potential level shift is seen to extrapolate between the Debye and ion—sphere shifts and it predicts the smallest shift at high density. Note the fairly strong disagreement at high density. The same information is shown in Fig. 5 for the a"3 state. (Note the different density

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Fig. 4. The energy of the ground (a"1) state of He` versus plasma density for various screening models. The classical Debye screening model (D) predicts the largest changes with the state almost eliminated at n "1024 cm~3. Note that the e hybrid model (H) agrees with the Debye model at low densities whereas it agrees with the ion—sphere model (IS) at higher densities, as expected.

range.) In all models this state is eliminated near n "1022 cm~3. Because of this, the state does not e exist at densities which require degeneracy corrections, so the degeneracy corrected Debye shift does not differ much from the classical Debye shift. The hybrid shift more closely approximates the Debye shift in this density region for the same reason. 3.4. Total elastic scattering cross section It is instructive to compute the elastic scattering cross section for a screened potential. This serves both to calibrate the potential with a familiar quantity and to aid in interpreting future calculations involving scattering states of this potential. The total elastic cross section can be computed in terms of the scattering phase shifts d (k) as [51] l 4p p (k)" + (2l#1)sin2(d (k))"+ p (k) , (114) 505 l l k2 l l where +k is the momentum of relative motion. The partial-wave cross sections p (k) which have l been defined in the second expression are useful for, e.g., quantum transport calculations [122,60]. We focus here on the hybrid potential. Partial cross sections are shown in Fig. 6 for the hybrid potential corresponding to an Ar`17 impurity in a hydrogen plasma typical of ICF experiments. The most notable feature is the very large partial cross section near k"1 for p (k). This is 3 indicative of a low-energy shape resonance which is common to short-ranged potentials. The total

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Fig. 5. The energy of the a"3 state of He` versus plasma density for various screening models. In all screening models, continuum lowering eliminates this state near n "1022 cm~3. At these lower densities the degeneracy corrected Debye e model (DD) does not deviate appreciably from the classical Debye model (D) and only the classical result is shown.

cross section from Eq. (114) is shown in Fig. 7 where it is compared with a simpler Born result for a similar Debye potential. 3.5. Number of bound states It is often useful to assume that the constant energy level shifts of Eq. (179) apply to all bound states. If this were so, all states within *E of the continuum would be moved into the continuum, leaving a finite number of bound states. In this picture, the uppermost state would have principal quantum number a given by7 .!9 z2/2a2 "*E , (115) .!9 where *E is any of the shifts of Eq. (111) in atomic units. The maximum principal quantum numbers for the three models are given by aDD "Jz/2i , .!9 DD aIS "Jzr /3 , .!9 4 aH "Jzi2r3/3[[(ir )3#1]2@3!1] , .!9 4 4

(116)

7 Of course, fractional principal quantum numbers do not exist and the next highest integer is implied in each of the expressions in Eqs. (115) and (116).

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Fig. 6. The partial cross sections p (k) for the first few l-values. The zeros of the partial-wave cross sections arise from the l presence of the bound states. The large feature in p (k) near k"1 is a low-energy shape resonance. 3

Fig. 7. The total elastic scattering cross section p (k) associated with the Hybrid potential for the same plasma 505 conditions as given in Fig. 6. Also shown as a dashed line is the first Born result for a Debye screened potential under the same conditions.

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Fig. 8. Total number of bound states (neglecting spin degeneracy) of various SSCPs for He` ions in a plasma of temperature ¹"15 eV. At very high densities virtually all bound states are eliminated, and there is good agreement between models; elsewhere, the ion—sphere model predicts far fewer states.

where all quantities in Eq. (116) are in atomic units. The total number of hydrogenic bound states, within this picture, is subsequently given by a.!9 (117) N " + a2"a (a #1)(2a #1)/6 . .!9 .!9 .!9 505 a/1 Once the phase shifts of the scattering states are found, we may obtain information regarding the number of bound states N of a given angular momentum. This is a consequence of Levinson’s l Theorem, which can be simply stated as [51] N "d (0)/p . (118) l l This theorem is exact for the types of potentials we are considering here8 and may be compared with the results predicted by Eq. (117), which represent a relatively crude approximation. The number of bound states for each model discussed is shown in Fig. 8 as a function of plasma density. For a non-degenerate Debye case, Eq. (116) can be compared with the numerical results of Rogers et al. [96]. For a system in which z/i "10, Eq. (116) predicts aDD "2 whereas the D .!9 numerical eigenvalue computations show that the 3s and 3p states are still bound, albeit weakly. Furthermore, the numerical results show that all tightly bound states are approximately shifted equally. For this case (z/i "10), Rogers et al. show that the weakly bound 3s and 3p energies D differ by over a factor of two.

8 The modification N "d (0)/p!1 must be made for a zero energy s-wave bound state. 0 0 2

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4. Generalized oscillator strength densities Having discussed several SSCP for the target ion, we turn to computing oscillator strengths for atomic transitions in such potentials. We focus here on bound—free transitions, in which the energy level shifts play a larger role than they do for bound—bound transitions. The oscillator strengths are first computed in two analytic approximations which incorporate bound state energy level shifts. Then, a partial wave method is described in which the continuum wave function is obtained from a particular SSCP, resulting in an oscillator strength with a more complicated temperature—density dependence. 4.1. Definitions The oscillator strength between two discrete levels a and b is defined as [5] f "(2m/+2)E DSbDxDaTD2 , (119) ba ba where E "E !E . This quantity, which is positive for upward transitions, can be interpreted as ba b a the effective number of classical oscillators participating in the transition [50]. This idea can be extended to give the generalized oscillator strength (GOS), f (k)"(2m/+2)(E /k2)DSbDe*k > rDaTD2 , (120) ba ba which is frequently used in the theory of inelastic collisions [48]. (Here +k is the momentum transferred in the collision.) Since the states b and a are assumed to be orthogonal, it is easy to see that Eq. (120) reduces to Eq. (119) in the kP0 limit. When a transition occurs from a bound to a continuum state the basic definition of the GOS is modified slightly. This is due to the fact that, for continuum states, we must specify the probability of the particle being in range dg of some set of observables g that we are free to choose. That is, we must specify the oscillator strength for the transition aPg as df (k, g)"(2m/+2)(E /k2)G(g)DSgDe*k > rDaTD2 dg . (121) a ga The quantity df /dg is referred to as the generalized oscillator strength density (GOSD) and the a different choices of g are referred to as “g-scale normalization” [48]. The quantity G(g) is the number of continuum states per interval dg. The source of this flexibility can be traced to the orthogonality and completeness relations for continuum states. In general, these conditions can be expressed as [51] SgDg@T"F(g)d(g!g@)

P

dg G(g)DgTSgD"1 .

(122)

In the first expression we are free to choose F(g) and the second expression provides the constraint G(g)"1/F(g). Typically, the set of observables g is taken to be the energy and either the linear or angular momentum. As a specific example, consider the observables to be the energy E of the particle and the direction of its linear momentum, which points into the solid angle dO. If we

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simply wish to integrate our final results over these quantities we would choose the corresponding G to be unity, viz.

P

dE dOSDE, OTSE, OD"1 ,

(123)

which fixes the normalization condition to be SE, ODE@, O@T"d(E!E@)d(O!O@) .

(124)

This particular choice is referred to as “energy-scale normalization”. The information contained in the GOSD is frequently presented as a surface of df (k,u)/du a versus u and ln(k2), where +u"E is the energy transferred. Plotted in this manner, the surface is ga referred to as the “Bethe surface”. The Bethe surface captures all information about an inelastic scattering process, insofar as the collision can be described within the (first) Born approximation. 4.2. Plane-wave model There is one GOSD which is very simple to use and can be expressed analytically. It is constructed by treating the initial state of the target as that of an unperturbed hydrogenic system and the final state as that of a free particle, viz. SrD1sT"(2Z3@2/J4p)e~Zr SrDKT"J(K/(2p)3)e*K > r .

(125)

Here, the bound state has been taken to be a 1s state of an ion with nuclear charge Z, and the normalization of the continuum state has been chosen to be on the energy scale with energy E"K2/2. (All quantities will be in atomic units, e"m"+"1, for the remainder of this section.) Thus, the GOSD can be written as (126) df (k, u)/du"(2u/k2)DSKDe*k > rD1sTD2 dO , 1s where u"E!E . Often we are only interested in the oscillator strength as a function of energy 1s and not the details of the particle’s direction. This is the case when the perturbation causing the transition is isotropic (on average), rendering the GOSD independent of the specific direction of K. We will assume this to be so and perform an integral over the solid angles in Eq. (126). It is also beneficial to write the result in terms of the ionization potential I "u!E of the 1s state. (Recall 1 from the discussion of Section 3.3 that the ionization potential can be shifted due to the presence of surrounding high-density plasma.) Together these manipulations give the plane wave GOSD df (k, u,I ) 16Z5u 1s 1" ([Z2#(k!J2(u!I ))2]~3![Z2#(k#J2(u!I ))2]~3) . 1 1 3pk3 du

(127)

Although this result has been obtained for a transition out of the 1s state, it is readily generalized to excited states. An average over substates for a given level a yields Eq. (127) with the replacement ZPZ/a [20,26]. The final form, valid for all principal quantum numbers a, is 16au df (k, u, I ) a a" 3pZk3 du

AC A 1#

BD

C A

BD B

k!J2(u!I ) 2 ~3 k#J2(u!I ) 2 ~3 a a . (128) ! 1# Z/a Z/a

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Fig. 9. The Bethe surface for the PW GOSD computed for an initial 1s state of He`. The surface has been truncated at a value of one to show smaller details. (Apparent jagged features are an artifact of the shading routine.)

The Bethe surface for a GOSD involving a hydrogenic ground state and plane-wave (PW) continuum states is shown in Fig. 9. Note that there is a narrow ridge which extends out to large energy and momentum transfers. This “Bethe ridge” corresponds to classical-like collisions. A second domain, near the origin, is more sensitive to the electronic structure of the initial state, and classically is associated with scattering at large impact parameters. 4.3. Orthogonalized plane-wave model In our general discussion of quantum mechanical transition rate formulae (Section 2) the initial and final states were assumed to be orthogonal. In the example above this assumption in fact was violated: the initial state was an eigenstate of a hydrogenic Hamiltonian and the final state was an eigenstate of a free-particle Hamiltonian. Physically this corresponds to a transition from some initial state to a final superposition state which contains the initial state. Such a situation occurs frequently in studying rearrangment collisions and techniques for handling this issue have been developed [79]. The method we use here is based on the orthogonalized plane-wave (OPW) approximation [40,10] which has been applied previously to recombination in dense plasmas [34]. The effect of the non-orthogonality of the initial and final states can be exposed by looking at a PW GOSD for small k. In this limit, we can write the GOSD of Eq. (126) as df (k, u) 1s "(2u/k2)DSK D1#ik ) r#2D1sTD2 dO . du

(129)

It is clear that the first term containing the nonzero matrix element SK D1sT will diverge as k~2. This behavior is evident in Fig. 9 where the plotted surface has actually been clipped at a value of 1 to allow the Bethe ridge to be highlighted. This can be remedied simply by subtracting the projection of the initial state with the final state to form an “improved” final state DK@T, DK@T"DKT!D1sT1sDKT.

(130)

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Fig. 10. The Bethe surface for the OPW GOSD computed for an initial 1s state of He`.

In principle, the final state wave function should be DK@T"DKT!+ DblmTSblmDKT, (131) blm where the sum runs over all bound states; this describes the situation in which the bound electron has no amplitude to be in any bound state following the ionization process. We will not pursue this more general form in the present paper since there would be a complicated plasma temperature and density dependence in the sum, and this would require a treatment of states lying near the continuum that is beyond the scope of the OPW [81]. (Our purpose here is to explore the nonorthogonality issue, not to generate a highly accurate GOSD.) Using the improved state DK@T from Eq. (130) we find the angle-dependent GOSD to be

C

16uK df (k, u) 1 1s " Z3p2k2 [1#(º/Z)2]4 du dO

D

32 256 ! # . [1#(º/Z)2]2[4#(k/Z)2]2[1#(K/Z)2]2 [1#(K/Z)2]4[4#(k/Z)2]4

(132)

In this expression K"J2(u!I ) and º"Dk!K D. It is easy to verify that the quantity in square 1 brackets vanishes as k2 in the limit kP0. Subsequently, an integration over final emission directions yields an OPW GOSD with interpretation analogous to that of Eq. (127). The corresponding Bethe surface is shown in Fig. 10 for the same conditions as those pertaining to Fig. 9. The effect of removing the non-orthogonality between the initial and final states is quite dramatic: the Bethe ridge is more prominent and the small k divergence clearly has been eliminated. 4.4. Numerical partial-wave model In the GOSD’s considered above plasma effects could be incorporated only through the (shifted) ionization potential. This allows construction of simple analytic forms for the GOSD that include static screening to lowest order. However, plasma effects on the initial and final state wave functions have not been included. We may expect that a tightly bound state is well approximated

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

by a hydrogenic wave function but not that a continuum state is well approximated by a plane wave. For electrons which have been ejected from an ion a better choice would seem to have the continuum state being an eigenstate of a screened Coulomb potential such as discussed in Section 3. Here we will assume only that the potential has spherical symmetry, which allows a partial wave analysis. In this representation it is convenient to choose a normalization based on the energy and angular momentum observables. In choosing an energy-scale normalization analogous to Eq. (123) we obtain the conditions SElmDE@l@m@T"d(E!E@)d d ll{ mm{

P

dE+ DElmTSElmD"1 . lm

(133)

There is some practical difficulty in ensuring that these conditions have been met for numerically generated wave functions. Let the solution be written as cR (r)½ (rL ) where R (r) is the result El lm El found numerically and c is some constant we must choose to satisfy the normalization criterion. In this case, the normalization condition of Eq. (133) reads

P

= dr r2R (r)R (r)"d(E!E@)d d (134) El E{l{ ll{ mm{ 0 which not easy to solve for c. In practice, this problem is handled by normalizing the asymptotic form of the radial wave function, which is presumed to be known. For the short-ranged potentials considered here the asymptotic form involves a spherical Bessel function and has the energy-scale normalized behavior of J2K/pj (Kr). Correct normalization is thus ensured by requiring the exact l solution to have the asymptotic form DcD2d d ll{ mm{

lim R (r)&J(2/pK) r?= El

sin(Kr!lp/2#d (K)) l , r

(135)

where K"J2E. In what follows, it will be assumed that this procedure has been carried out. In the partial wave representation a sum may be performed over angular momentum substates to obtain a GOSD pertaining to a transition between energy levels, independent of angular momentum. This corresponds to the integration over solid angles dO, discussed earlier. Since we may want to treat excited bound states we also average over initial substates. However, the angular momentum quantum number l is important for selection rules and so we only average over angular momentum projections m to obtain df (k, u) 2u/k2 = l{ l a " (136) + + + DSEl@m@De*k > rDalmTD2 . du 2l#1 l{/0 m{/~l{ m/~l The GOSD of Eq. (136) represents the average strength of a transition from bound states with quantum numbers a and l to all continuum states with energy E. Let the initial state DalmT be hydrogenic, SrDalmT"R (r)½ (rL ), with energy given by the shifted al lm hydrogenic value as in Eq. (111) and let the final state have the form SrDEl@m@T"R (r)½ (rL ), with El{ l{m{ continuous energy eigenvalue E. [If I is the (shifted) ionization potential then u"I #E.] R (r) a a El{ is the radial wave function of the chosen screened Coulomb potential. By writing the exponential

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

41

term in a spherical Bessel function expansion and then evaluating the integration over spherical coordinates in terms of 3!j symbols, we obtain the intermediate result

K

df (k, u) 8pu a " + (2l@#1) + iL½* (kK )J2¸#1 LM k2 du l{m{m LM

P

]

A

=

BA

l@ ¸ l dr r2R (r) j (kr)R (r) El{ L al !m@ M m 0

l@

¸ l

0 0

BK

0

2

.

(137)

Eq. (137) can be considerably simplified by writing out the square and performing the sums over m and m@. This eliminates the other M-sum and gives the compact form df (k, u) 2u a " + + (2l@#1)(2¸#1) k2 du l{ L

CP

DA

=

0

dr r2R (r)j (kr)R (r) El{ L al

B

2 l@ ¸ l 2 . 0 0 0

(138)

Generally this expression represents a fairly extensive computation, due to slow convergence of the two infinite sums. However, when the initial state has l"0 we obtain a greatly simplified GOSD, df (k, u) 2u a " + (2¸#1) k2 du L

CP

D

= 2 dr r2R (r)j (kr)R (r) . EL L a0 0

(139)

Figs. 11 and 12 show Bethe surfaces corresponding to transitions from the Ar`17 ground state in a Hybrid potential for two hydrogen plasma densities and a temperature of 1 keV. The Ar`17 ion was chosen because its spectrum is commonly used in ICF plasma diagnostics [118], and because an unperturbed (hydrogenic) initial state wave-function R (r) simplifies the GOSD calculation. a0 The continuum wave functions R (r) were obtained by numericaly solving the Schro¨dinger EL equation in the Hybrid potential. As few as 20 ¸-values could be used for these GOSDs. It is clear from the figures that there can be a significant density dependence of the GOSD.

Fig. 11. Bethe surface for the Ar`17 Hybrid potential described in the text. Here, the plasma density is 1022 cm~3.

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Fig. 12. Bethe surface for the Ar`17 Hybrid potential described in the text. Here, the plasma density is 3]1024 cm~3.

5. Ionization rates Rates of atomic ionization will now be determined for the simplest case, single-electron (H-like) target ions of nuclear charge Z. Three methods for obtaining these transitions rates will be described. Each derivation begins with a direct calculation of the rate, without recourse to a cross section, but the derivations represent distinctly different physical pictures. In Section 5.1, which is pedagogical, transition rates are computed within the standard framework of independent plasma electrons inelastically scattering from the ion; this is shown to be equivalent to the traditional binary cross-sectional approach. Then, Section 5.2 provides a derivation wherein the ionic transition is driven by the time-dependent stochastic field of the surrounding plasma electrons. The time-independent picture given in Section 5.3, in which the ion ‘‘impacts” the plasma, turns out to be equivalent to the stochastic approach of Section 5.2. The models presented in Section 5.2 and Section 5.3 are contrasted to that of Section 5.1, to emphasize those effects which arise from the consideration of interacting plasma electrons. We begin by recalling that time-independent scattering theory is based on a Hamiltonian of the form HK "HK #HK #»K , X Y

(140)

where HK and HK are the Hamiltonians of the (possibly composite) subsystems undergoing the X Y collision and »K is the interaction between them. Since the full Hamiltonian is used, transitions are induced by allowing the interaction »K to be adiabatically turned on, as »K ect, with the limit c P 0 taken at the end of the calculation. If the interaction is sufficiently weak, a first-order transition rate can be computed from the Golden Rule, w"(2p/+)DSy@DSx@D»K DxTDyTD2 d(E@!E) .

(141)

Eigenstates of HK and HK have been used to describe the asymptotic initial and final (primed) states X Y of the colliding subsystems.

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43

5.1. Independent electron impact method It is useful to start with the simple approach to atomic ionization rates presented in this section, because it connects the traditional (binary) cross-sectional picture to the stochastic model of the next section. Here, all interactions among plasma electrons are neglected, and plasma ions are ignored entirely. Thus, the three parts of the Hamiltonian of Eq. (140) are: p2 Ze2 HK " ! , !50. 2m r p2 HK "+ e , 1-!4.! 2m e »K "+ U (r, r ) . (142) ee e e The perturbation »K is the sum of Coulomb interactions between the plasma electrons at r , e with momenta p , and the bound atomic electron at r, with momentum p. Let the atomic electron e be in state DaT, and the N free electrons be in plane wave states D p T, e " 1,2,N. Since only e »K contains two-particle interactions, the initial and final composite states may be written as products, Dt T"DaTDp T2D p T, i 1 N Dt T"Da@TDp@ T2D p@ T. (143) f 1 N Here the Hartree factorization has been made (electron exchange is neglected). With the initial and final states defined by Eq. (143), the Golden Rule can be written as

K

K

2p 2 w " Sp@ D2Sp@ DSa@D+ U (r,r )DaTDp T2Dp T d(E !E ) . (144) fi N 1 ee e 1 N f i + e It is possible to simplify this transition rate formula by expanding the two-particle interaction in terms of its Fourier components Uk " 4pe2/k2. This leads to the expression

K

K

2p 2 w " + UkSa@De*k > rDaT+ Sp@ De~*k > reDp T < Sp@ Dp T d(E !E ) . (145) fi e e e{ e f i X2+ k e e{Ee It is a consequence of using first-order perturbation theory with this two-particle interaction that only two electrons simultaneously undergo transitions; and, since we are considering an atomic transition, only one plasma electron is involved. Of course, we do not know the initial state of the plasma at the level of detail required to evaluate the above rate, nor do we care in which final state the plasma is left. We do know, presumably, the plasma’s statistical properties and we can therefore compute the more relevant mean quantity for the atomic states DaT and Da@T: w , + 2+ + F( p )2+ F( p ) w , a{a 1 N fi p p p N {1 {N p1

(146)

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which involves a sum over final states and an average over initial plasma states. The F( p )’s are the e probabilities associated with finding electron e in state D p T, as given by, e.g., a Maxwellian e distribution. This averaging reduces Eq. (145) to the simpler form 2pN w " + F( p )+ + DUkSa@De*k > rDaTD2DSp@ De~*k > reDp TD2 d(E !E ) . (147) a{a e p k e e f i +X2 p e {e From this formula it is clear that w is an average over the rate due to a single electron impacting a{a the ion. The matrix element involving plane wave states reduces to a Kronecker delta function times the plasma volume X, so the sum over p@ can be immediately effected. This results in the yet simpler e expression 2pN w " + F( p )+ DUkD2DSa@De*k > rDaTD2 d(E !E #( p !+k)2/2m!p2/2m) . a{a e k a{ a e e +X p

(148)

e

The relationship between this rate and the standard, binary cross-sectional rate can easily be worked out. To get a form suitable for making this connection, we first multiply and divide by the particle flux v/X within the average over p . This average next is written as an average over electron e velocities € " p /m to yield e vX 2pN + F(€) + DUkD2DSa@De*k > rDaTD2 d(E #(+k)2/2m!€ ) +k) . (149) w " a{a a{a Xv k +X € Now, we identify the electron density n " N/X, and denote the velocity averaging by angle e brackets. We may then trivially rewrite the preceeding expression in the familiar form w "n Svp (v)T a{a e a{a where the cross section is defined as 2p p (v), + DUkD2DSa@De*k > rDaTD2 d(E #(+k)2/2m!€ ) +k) . a{a a{a +v k

(150)

(151)

This (Born) cross-section is a sum involving the square of an atomic form factor times a Coulomb term Uk, together with the constraints of energy and momentum conservation. This form for the transition rate includes the physics of the plasma in two ways: the factor n represents the mean density of plasma electrons, and statistical properties of the plasma are e included via the momentum distribution function F( p). As reviewed in Section 1, there are numerous publications in the literature describing transition rate calculations in this framework, but also incorporating additional plasma effects. Most of these use some model of static screening in the interaction »K ; a few use more sophisticated cross-section prescriptions, too. However, as we show in the next section, all such treatments are of limited validity. 5.2. Stochastic perturbation method The physical picture of the previous section was one of plasma electrons interacting independently with the target, and their cumulative effect was simply a density factor multiplying

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45

a transition rate for a single (average) electron. At very high plasma densities this picture is inadequate due to the increasingly strong interactions between the plasma electrons themselves. Any given electron’s trajectory will be modified by the presence of the other electrons and the background ions. One approach to including this physics is to construct basis states not from the products of free particle states D p T (Eq. (143) but, rather, to use eigenstates of i p2 ze2 p2 (152) HK #HK @ " ! #+ e # + U #+ U . ee{ ei !50. 1-!4.! r 2m 2m e e:e{ ei (Note that free electron—target interactions are still being neglected.) In this picture no single plasma electron makes a transition independently of the other plasma electrons, a fact which introduces a complicated density dependence. Moreover, since it is no longer possible to write the plasma state as a product of one-electron states, in the traditional cross-section method this would raise the problem of defining the flux. Fortunately, this problem does not arise when rates are computed directly, as we now show. The stochastic approach extends the above notions, and views the target ion as an atomic system surrounded by a gas of dynamic electrons interacting with each other and the background ions. This approach is explicitly time dependent as the random motions of particles within the gas produce a stochastic field at the position of the ion. On average, the electrons and ions will tend to screen the target nucleus, and dynamical effects, due predominantly to the lighter electrons, are not likely to be highly modified by the presence of the target. The stochastic approach can be described with the atomic Hamiltonian HK @ "!(+2/2m)+2#» (r)#» (r,t) " H #» (r, t) , (153) !50. 0 1 0 1 where all coordinates now refer the bound electron. In this Hamiltonian » (r) represents the mean 0 (spherical) interaction produced by the nucleus and the quasi-static screening from the plasma electrons and ions, and » (r, t) represents the time-dependent portion of the plasma—target interac1 tion associated with the stochastic motion of the electrons. In this section, we will assume that the electrons are weakly coupled. Of course, the fluctuations in the potential arise from fluctuations in the plasma (electron) density n(r, t). With this we can write the time-dependent part of Eq. (153) for some particular realization of the plasma as

P

n(r@, t) » (r, t)"e2 d3r@ . 1 Dr!r@D

(154)

It will be assumed that the states associated with H can be found by methods such as discussed in 0 Section 3. Here, the transition rate due to the perturbation given by Eq. (154) will be sought, and a statistical average involving plasma states will be taken at the end of the calculation. As before we begin with the transition rate determined by first order perturbation theory. Due to the persistent time dependence of the interaction, the first-order rate for DaT P Da@T must be written as

KP

P

K

e4 d t 2 n(r@,q) w"lim dqSa@D d3r@ (155) DaTe*(ua{a~*c)q +2 dt Dr!r@D ~= c?0 which is just a generalization of Eq. (141). This can be rewritten (as in the previous section) in terms of the atomic form factor, by Fourier transforming the interaction » . This allows the rate to be 1

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M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65

written in terms of the Fourier modes of the density fluctuations, n(k, q), of one particular plasma realization as

K

P

K

t 2 1 d w " lim + UkSa@De*k > rDaT dq e*(ua{a~*c)qn(k,q) . (156) X2+2 dt k ~= c?0 In this form we can see distinctly the difference in this approach. The plasma interaction of Eq. (145), essentially a momentum transfer factor, has been replaced here by a complicated plasma time evolution. Simplification can be achieved by expanding the square and then performing the average over plasma states. This yields the mean rate

P

= 1 dq e*(ua{a~*c)qSn(k, q)n(!k, 0)T . + DUkSa@De*k > rDaTD2 (157) w " lim a{a X2+2 k ~= c?0 From Section 2.1 we can identify the time integration of the density—density correlation function as the dynamic structure factor. Employing that result we are able to write the rate as 1 + DUkD2DSa@De*k > rDaTD2S(k, u) , w " a{a X2+2 k

(158)

where u,u !ic and the c P 0 limit is understood. a{a Eq. (158) has a very intuitive interpretation due to the three main pieces into which the problem has factorized. Beginning from the right, we have the power spectrum of density fluctuations, which was covered in detail in Section 2; S(k, u) contains the ‘‘physics” of the plasma electrons. The middle term is the atomic form factor of Section 4, which depends only on the properties of the atomic states involved in the transition; recall, though, that in the stochastic model this factor does include some plasma effects, due to the static screening in » . However, it is independent of the 0 dynamic properties of the plasma. The third term is just the Fourier-transformed Coulomb interaction, which connects the other two factors. Earlier, we noted that the wave vector k in the structure factor corresponds to the spatial Fourier modes associated with inhomogeneities, and the frequency u, to temporal Fourier modes associated with oscillations. In the present context we have additional information. From the atomic form factor we see that +k also has an interpretation as the momentum transferred to the target ion. Thus, short-wavelength density fluctuations correspond to the plasma’s ability to transfer a large amount of momentum to the ion. We also see, from u , u !ic, that plasma fluctuation frequencies near resonance are needed to drive the a{a transition. 5.3. Plasma impact method In this section, the ionization rate will be derived a third time by appealing to a somewhat unusual picture. Recall that the main reason for developing the time-dependent stochastic model was that the scattering event could not be pictured as a simple binary collision with independent plasma electrons impacting the target ion. Having abandoned the need for a flux (and a cross section), we may now turn the problem around and view the event as the bound atomic electron ‘‘impacting” the plasma. That is, we let the atomic electron undergo inelastic scattering by the plasma. It will be seen that this is formally equivalent to the stochastic approach, above.

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In the traditional picture we would take, say, state DxT to be a plasma electron impacting the ion which is initially in state DyT. Now, we take the opposite viewpoint and let DxT"DaT be the atomic electron ‘‘impacting” the entire plasma, which is initially in state DyT"DAT. Since the reference here is not to a flux, but rather to asymptotic states of individual subsystems, this picture is perfectly valid too. For the atomic transition DaT P Da@T, Eq. (141) takes the form

K

P

K

2p e~bEA 2 w " + SA@DSa@D nL (r@)U (r!r@)d3r@DaTDAT d(E@!E) , (159) ee a{a + Q A,A{ where nL (r@) is a density operator, U " e2/Dr!r@D is the Coulomb interaction, and an average over ee initial plasma states DAT and a sum over final plasma states DA@T has been made. For comparison with earlier results, we Fourier transform the interaction to obtain an expression similar to Eq. (21),

K

K

2 2p e~bEA + + UkSA@DnL (k)DATSa@De*k > rDaT d(E@!E) . (160) w " a{a X2+ Q k A,A{ The matrix element can be squared to yield a result which contains a double integral over the Fourier transform variables. If the second Fourier transform variable is q and the plasma is assumed to be translationally invariant, the result will contain the product SA@DnL (k)DATSADnL s(q)DA@T"DSA@DnL (k)DATD2 dkq .

(161)

The delta function dkq arises from the fact that translationally invariant states are momentum eigenstates. Using this information, the transition rate then reduces to 2p e~bEA + DUkD2DSa@De*k > rDaTD2 + DSA@DnL (k)DATD2 d(E@!E) w " a{a X2+ k Q A{,A 1 + DUkD2DSa@De*k > rDaTD2S(k,u) , " X2+2 k

(162)

which is identical with Eq. (158). It is Eqs. (26) and (30) that connect the time-dependent approach of the previous section to this time-independent approach.

6. Numerical study of projectile screening issues In Section 5.2 the plasma’s perturbation of the atomic system was broken into pieces which represent separately static target screening and dynamic projectile screening. Quantitatively, this picture led to the Hamiltonian of Eq. (153) (repeated here in atomic units), HK @ "!1+2#» (r)#» (r, t) . (163) !50. 2 0 1 From the discussions of Section 3 and Section 2, we know that both » (r) and » (r,t) contain effects 0 1 of high plasma density. In this section, the high-density effects of » (r, t) will be illustrated, with 1 those of » (r) being postponed to Section 7. 0

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The interaction between two particles in vacuo is often quite different than the complicated effective interaction arising from the presence of other particles in dense systems. It was remarked in Section 5.1 that this might be taken into account in a static screening model by replacing the bare Coulomb interaction between two particles with a screened interaction, such as »(r , r ) " 1/Dr !r D e~i4@r1~r2@ (164) 1 2 1 2 where i is some screening parameter. Thus, a collision is not just an electron impacting the ion but 4 in fact is an electron with its associated screening cloud impacting an ion. This static screening picture suggests that collisional transition rates will be reduced at high-density as the screening becomes more efficient. Numerical results based on the collisional model of Section 5.2 will be presented here to examine this prediction quantitatively. It is instructive to consider the collisional ionization process for various initial (bound) states. Our model system, chosen purely for illustrative purposes, of He` ions in a 15 eV plasma is considered here for a wide range of plasma densities, and for both the a"1 and a"3 initial states. The ionization rate is determined for these two cases to directly examine projectile screening properties of the plasma at high density. In particular, each of the various screening approximations of Section 2.3 is used to compute the rate coefficient as a function of density, after the dielectric response function of Section 2.4 was employed to obtain e(k, u) in each case. At the highest densities presented here, this scheme breaks down and details of the results become suspect. Nevertheless, these calculations do provide considerable insight into issues regarding projectile screening. 6.1. Ionization rates for He` (ground state) Fig. 13 shows the results of our numerical calculation of the total ionization rate for He` from the a"1 state, in a 15 eV plasma. The PW GOSD of Section 4.2 has been used here both to keep the atomic physics as simple as possible and to use a GOSD form that also is valid for the excited state calculation of Section 6.2. Results are shown for each of the three screening approximations and are plotted relative to the no screening case. In this way, the screening models are easily compared. In the no screening (NS) case the dynamic structure factor has been approximated as that of an ideal gas, (e.g. Eq. (44)) S(k, u)PS (k, u) , (165) 0 which implies that each plasma electron impacting the ion does so independently of the other electrons. Since the electrons are independent, there can be no high density information contained within this picture and the ionization rate coefficient, that is, w/n , is density independent. e Therefore, the flat curve shown in the figure represents the actual NS density dependence and is not merely an artifact of plotting a ratio; our computed NS He`(a " 1) ionization rate coefficient is 1.4 ] 10~9 cm3/s. The static screening (SS) result shows behavior consistent with predictions based on interactions of the form of Eq. (164). In fact, within the context of static screening and the dielectric response function of Section 2.3.2, the functional form of Eq. (164) is exact. This can be seen by writing the interaction as U 4p/k2 k " e(k, 0) 1#i2 /k2 D

(166)

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Fig. 13. The total ionization rate of He` from the a " 1 state versus plasma density, for both dynamic (DS) and static (SS) screening models. The plasma temperature is 15 eV and the atomic transition is treated via the PW GOSD. Specifically, the ratio of the rate for each screening model to the rate for the no screening case is shown (left ordinate) versus the logarithm of plasma density. Also shown (dashed line, right ordinate) is I !+u , the difference in the 1 e ionization potential and the plasmon energy.

which has, with i "i , Eq. (164) as its Fourier inverse. Thus, at high densities, with interactions 4 D weakened by screening, the ionization rate is reduced. This reduction also can be explored graphically by plotting the screening function 1/De(k, u)D2 versus k and u. This is shown in Fig. 14 for the SS screening function, and clearly illustrates both the screening behavior at small k and the absence of any u dependence. Note, however, that this static screening does not become important until the plasma has reached a density of about 1022 cm~3. It is easy to explain this behavior by noting that, within the stochastic model, the wave vector for the plasma density mode k also corresponds to the momentum transferred to the atomic system. From Eq. (166) we see that modes with wave vectors less than i are strongly screened, while those above i are essentially D D unchanged. Thus, screening reduces the contribution from small momentum transfers k(i . We D may use the relation k2 /2"z2/2a2 , .*/

(167)

in which the ionization energy has been equated to the classical energy k2 /2, to estimate the .*/ minimum momentum k required to ionize the electron from level a. The ionization process is not .*/ affected by screening until the density increases enough that the momentum i approaches k . If D .*/

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Fig. 14. The screening function 1/De(k,u)D2 in the static screening approximation. In this approximation the modes below k are screened and there is no u dependence. This was computed at a density of 1022 cm~3 and a temperature of 15 eV. D Atomic units are used.

we estimate the onset of screening effects by the condition k " 5i (see discussion following .*/ D Eq. (3)), a threshold density of + 8 ] 1020(Z2¹/a2) cm~3 (168) .*/ is predicted by Eq. (167), with ¹ in eV. For the case considered in this section, n is about .*/ 5 ] 1022 cm~3, which is in good agreement with the computational result. The dynamic screening (DS) case plotted in Fig. 13 shows unexpected behavior, based on the previous analysis of the SS case. Whereas the SS case showed a decrease in the rate at high density the DS case shows just the opposite, an increase. Recall that the DS model is related to the SS model by the generalization e(k, 0)Pe(k, u) and thus it contains the SS model. However, we do not see any evidence for diminution via screening of the interaction in the DS case. We may understand this new behavior by comparing the energy transferred I to the plasmon energy +u . The quantity 1 e I !+u "2!1.365 ] 10~12Jn , (169) 1 e e which also is plotted in Fig. 13, indicates that the ionization potential exceeds the plasmon energy at all densities considered here. Thus, the ionization process is only sensitive to density fluctuations whose characteristic frequencies exceed u . A high-frequency expansion of e(k,u), e e(k, u)+1!u2/u2!u4/u4k2/i2 !2 , (170) e e D clearly shows that the static screening, which arises from the i dependence, is a higher order and D therefore a less important effect. In this regime, the relevant plasma fluctuations occur at frequencies which cannot be screened on a time scale of order u~1. The expression that replaces Eq. (166) e is n

U /e(k, u)+(4p/k2)/(1!u2/u2!2) , (171) k e which indicates an enhanced interaction, in agreement with the calculated result. Note further that this expansion predicts a divergence in the interaction in the very high-density regime where the plasmon energy becomes comparable to the ionization energy. This expectation is illustrated in Fig. 15 where the dynamic screening factor 1/De(k, u)D2 is shown. It is clear from the

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51

Fig. 15. The full (dynamic) screening function 1/De(k,u)D2 is shown versus k and u. The screening function is about unity over most of the surface indicating that, to a good approximation, no screening occurs. A plasma oscillation peak does appear which serves to modestly enhance the ionization rate. Below the plasma frequency the surface dips below unity reflecting static screening behavior. The plasma conditions are identical to those of previous figure.

plot that this screening factor is unity for most values of k and u. There is a plasma oscillation peak which enhances the interaction above the NS case and at higher densities this peak becomes more pronounced. This indicates, perhaps surprisingly, that the DS case is more similar to the NS case than to the SS case! 6.2. Ionization rates for He` (excited state) The total ionization rate has also been computed for the He`a " 3 state, using the PW GOSD. Results analogous to those in the previous section are presented in Fig. 16. The NS and SS curves are in qualitative agreement with the a " 1 calculations. The static screening now, however, much more severely inhibits the ionization rate at high plasma densities. Furthermore, the screening begins to become important at a density of &1021 cm~3, in agreement with Eq. (168). The a " 3, DS case is not even in qualitative agreement with the previous, a " 1, example. The analysis of the previous section suggests that there would be a divergence of the ionization rate as the plasmon energy approaches the ionization potential. It is easy to see from the graph of I !+u 3 e in the figure that this condition is actually realized for the a " 3 state, but evidently there is no corresponding divergence in the ionization rate. Upon closer inspection of the interaction in this regime we find the screened interaction takes the form

K K

16p2/k4 U 2 16p2/k4 k " + . e(k,u) [Re e(k, u)]2#[Im e(k, u)]2 (1!u2/u2)2#[Im e(k, u)]2 e

(172)

The divergence has been avoided by the damping of the plasma oscillation. Although the damping is often ignored, it plays an important role in circumventing divergences near the plasmon energy. Not only is a divergence not present but the DS ionization rate is in fact dramatically reduced at densities where I !+u (0. In this regime fluctuations at frequencies high enough to ionize the 3 e target now exist below the plasma frequency. These fluctuations are screened and we have

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Fig. 16. As in Fig. 13, for the total ionization rate of He` from the a " 3 state, again in a 15 eV plasma. Also shown (dashed line, right ordinate) is I !+u , the difference in the ionization potential and the plasmon energy. 3 e

a situation entirely analogous to the SS case. To be more exact we must remember that these calculations are of the total ionization rate and we have to consider all possible bound—free transitions. With a temperature of 15 eV and a density of n "1024 cm~3 more than half of the e transitions take place as a result of fluctuations below the plasma frequency. Furthermore, it is in the lower-energy region where the GOSD takes its largest values. Thus, at high plasma densities, the total ionization of this excited state is dominated by fluctuations which are screened. 6.3. Ionization rates for Ar`17 (ground and excited states) Ionization of Ar`17 represents an important case due to its use as an ICF diagnostic [118]. Results are shown in Fig. 17 for the bound states a"1,3,5 at a temperature of 650 eV. Both projectile screening and Hybrid SSCP level shifts are included in these calculations. It is clear that the tightly bound ground state is almost completely unaffected under these conditions (cf. Eq. (3).) However, the ionization rates for the excited states are greatly affected by high plasma density. For a"3 the enhancement in the rate is dominated by (quasi-static) shifting of the level, except at very high density where the dynamic and static projectile screening regimes are entered and the enhancement is diminished. The level shifts greatly enhance the a"5 ionization rate until the state is lost (via continuum lowering) just below n "1024cm~3. For this case, projectile screening effects e at high density cannot be realized before the state is lost. Because each state’s ionization rate changes in a different way, it does not seem likely that one can produce a simple prescription for the overall effect of high density on ionization balance in non-equilibrium plasmas.

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Fig. 17. The total ionization rate for Ar`17 from the ground and excited states a"1,3,5. The results were obtained with Hybrid level shifts and dynamic screening. Plotted is the ratio of the rate to the rate Rate computed with no level shifts 0 or screening. The result for a"5 is shown up to the density for which that bound state is lost via continuum lowering.

7. Numerical study of target screening issues High plasma densities also affect atomic transition rates through quasistatic perturbations of the target. In the Hamiltonian of Eq. (163), » (r) contains the quasistatic effects. In the previous 0 section, we employed a PW GOSD, which assumed that the bound state was unperturbed and the continuum state was a free-particle state. Here we take up issues associated with the modifications of the atomic system due to the departure of » (r) from a pure Coulomb term representing the 0 bound electron’s interaction with the atomic nucleus. As discussed in Section 3, free charges in a plasma tend to screen out charges of the opposite sign, on average. The local potential around an ion, for example, differs from that of the bare ion because free electrons are attracted and other ions repelled. This screening modifies the eigenstates of the atomic system and, therefore, the GOSD for the transition. In Section 7.1, ionization calculations with the OPW GOSD of Section 4.3 are compared with those of the PW GOSD to address the orthogonality issue. Ionization rates are then calculated in Section 7.2 with bound state energy eigenvalues that reflect the quasistatic screening. Level shifts are estimated as in Section 3.3. 7.1. Non-orthogonality of initial and final states The total ionization rate as a function of electron density has been computed for He` (a"1) with the OPW and PW GOSD for a fixed plasma temperature of 15 eV. These results are shown

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Fig. 18. The He` a"1 ionization rate for the different screening models for both the OPW and PW GOSD is plotted in ratio with the PW, NS case. Results with the OPW GOSD are shown as solid lines and those of the corresponding PW GOSD are shown as dotted lines.

together in Fig. 18. The solid lines are the OPW ionization rates normalized to the no screening PW result. With the OPW GOSD, the ionization rates are nearly a factor of two less than the PW GOSD results for each screening model. This change is of the same order as that caused by the various projectile screening models, with nearly an order of magnitude spread in the rates at high density. Thus, a seemingly innocuous change in the GOSD leads to a substantial modification of the predicted ionization rate. A comparison of Fig. 10 with Fig. 9 reveals that, in this case, the reduction in the rate is due to the elimination of the small k divergence in the PW GOSD. In addition to an overall reduction in the ionization rate at all densities, there is also a slight change in the behavior of the different screening approximations. In particular, the rates computed with SS or DS do not differ from the NS case as much in the OPW calculation as they do in the PW calculation. Recall from comments related to Fig. 15 that much of the difference between the screening models arises in the small k regime. This is exactly where the PW approximation overestimates the GOSD, and so the (PW) screening effects we found earlier were actually somewhat exaggerated. 7.2. Bound state level shifts The He` (a"1) ionization rate has also been computed with corrected binding energies for the ground state, using the hybrid level shifts of Eq. (179). These results, together with the PW

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55

Fig. 19. The He` a"1 ionization rate with bound state energy level corrections is shown for the various screening models and the OPW GOSD. Specifically, the rate is shown in ratio with the NS case without level corrections. The dotted lines are the same as those the previous figure.

results of Section 6.1, are plotted in Fig. 19. It is clear from the figure that level shifts play a very important role in determining the ionization rate. At high plasma densities the level approaches the continuum and ionization becomes progressively easier. Now, in each screening model we explored, the ionization rate is enhanced at high plasma density by a substantial factor. In the SS case, there is a competition at high density between the screening of the projectile and the target screening. The shift in the energy level dominates at lower densities, producing first an increased ionization rate as the density rises. At the highest density shown, however, screening of the interaction is quite strong and the ionization rate subsequently drops to become about equal to its low-density value. The DS and NS cases are very similar: as the plasma density increases, the collective mode contained in the DS case serves to enhance the ionization rate slightly over the NS case. This effect was also seen in Fig. 13. But, new behavior now appears at the highest densities, where the two approximations yield rates that are nearly equal. The DS result decreases as a function of plasma density just as with the SS case, because the shift in the energy of the level has put the state into the regime where some of the transitions are statically screened: in essence, the excited state behavior of Fig. 16 is now being seen, albeit weakly, for the shifted ground state.

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8. Summary and future directions 8.1. Important conclusions for ionization rates We have investigated transitions driven by stochastic perturbations for the special case of atomic collisional ionization. The most useful number associated with this process is the total ionization rate from a single bound state to all continuum states. Atomic form factors for bound—free transitions were computed in various approximations that reflect properties of the average plasma interaction. Properties of the plasma electron fluctuations were investigated within three screening models which correspond to no screening, static screening, and dynamic screening. Conclusions based on our numerical computations are: f A naive screening model that replaces the bare Coulomb interaction with a static screened potential is almost always a poor approximation. This is because most atomic ionizations involve energies that exceed +u and are not screened. e f When the transition energy is near +u there is an enhancement in the ionization rate. e Transitions below this energy, if there are any, are essentially statically screened and those above this energy are only weakly screened. Thus, for the total collisional ionization rate we must consider different screening properties for the various transitions. All of this information is contained in the dynamic screening model. Only in cases where all important transitions are either above or below +u can a simplification be made. e f The generalized oscillator strength density (GOSD) must be carefully chosen. It has been shown that merely orthogonalizing the initial and final states produced a factor of two change in the ionization rate, which indicates a strong sensitivity to this effect. Also, deviations from the no screening case are exaggerated with a GOSD having non-orthogonal initial and final states. f A new static screened Coulomb potential was developed for this problem. The usual Debye picture was deemed invalid for treating the initial bound state, and the complementary, ion-sphere potential was deemed invalid for treating the continuum states. A hybrid potential derived from these two limiting models was constructed; it has good small-r and large-r behavior and is applicable over wide ranges of temperature and density. f At high densities, modest changes in the bound state energies can produce large changes in the ionization rate. These changes are generally more significant than those associated with dynamic screening. Furthermore, level shifts can bring a state with ionization potential I A+u , a e which originally was in the ‘‘no screening regime”, into a regime described only by dynamic screening. The results presented here can be extended in many directions. Perhaps the most important extension would be to consider the inverse process of three-body recombination. Then, the problem of ionization kinetics and the importance of density corrections could be ascertained for several cases of experimental interest. There also remains much that can be done to improve the underlying physics for both of these processes. These physics issues can be partitioned into the three areas of plasma physics, atomic structure, and the description of the interaction between plasma and atom. We end this Report with an annotated ‘‘shopping list”.

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8.2. Dense plasma issues 1. Plasma degeneracy. There are experiments in which the plasma may spend some time as a degenerate electron gas. This is the case, for example, in laser produced plasmas at early and late times. In fact, at low laser intensities the electrons may never leave the degenerate regime. The stochastic model can be easily extended to the degenerate regime by simply using the finite temperature Lindhard dielectric response function (DRF), which is the appropriate generalization of the Vlasov DRF used here. References are given in Appendix B. 2. Strong coupling. Strong coupling among the plasma electrons appears under conditions similar to those which result in degeneracy: high density and/or low temperature. Strong coupling is also possible in certain non-degenerate experimental regimes we have considered. Relevant extentions of the model used here are well known; the theory is based on treating correlations more carefully in the underlying kinetic equation for the phase space distribution function. This gives rise to the so-called ‘‘local field corrections” in the DRF and, hence, the dynamic structure factor. An account of this procedure has been outlined in Section 2.4. 3. Ions. The ions in a plasma are more likely to be strongly coupled than the electrons since, in a two-component plasma, one has C "z5@3C . The quasistatic potentials described in this z e Report are invalid under strong ion coupling conditions. Also, ions have been neglected entirely in obtaining the plasma density fluctuations (see Section 8.4 below). 4. Double counting. The common problem of double counting [15] plasma perturbations on the atom has not been addressed in a systematic way. Some double counting is inevitable in the stochastic model as a result of treating the quasistatic and fluctuating perturbations with separate interaction terms » (r) and » (r, t). This issue is closely related to the previous issue of 0 1 treating the ions self-consistently. 5. Non-thermal distributions. It was stated in Section 1 that most plasmas are not in thermal equilibrium. It was useful, however, to use a thermal distribution for the plasma electrons to obtain an analytic form for the DRF. In some experiments where electron—ion temperature differences are important this may still be a good approximation whereas in others, such as some laser-produced plasmas, it is known not to be [102,29,73]. The form of the non-equilibrium distributions is likely to be strongly dependent upon the specific time history of the experiment and simple descriptions may not be possible. 8.3. Screened interaction issues 1. Many-electron ions. Our work here has been restricted to hydrogenic ions which, in some cases, happen to be the most important charge state. Often, however, other charge states are key. For example, some X-ray lasers are based on Ne-like ions, and future ICF X-ray diagnostics are likely to be based on a many-electron charge state of Xe. Not only is the basic atomic structure complicated by many bound electrons, but the collision problem itself is further complicated by the appearance of resonances and the likelihood of several important inelastic scattering channels. These are well known, but still challenging issues even for binary electron—ion collisions. How stochastic perturbations are influenced by correlations among bound electrons is an unexplored topic.

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2. Higher-order bound state corrections. The plasma-induced modifications of the bound states have only been treated as energy level shifts obtained from spherically symmetric quasistatic potentials. This lowest-order correction can be extended, and since a numerical method for treating the GOSD has been detailed, this problem has in principle been solved. A code which calculates the exact bound state energy and wave function is all that is needed to complement the existing codes used for obtaining accurate GOSDs. Such an improvement definitely is needed to obtain accurate results for weakly bound states. It is also known that the (hydrogenic) angular momentum degeneracy is lifted by any screened potential. This fact was ignored in obtaining the PW and OPW GOSDs. 3. Nonspherical quasistatic potentials. It has been assumed that all the quasistatic potentials have spherical symmetry. But, recall that the stochastic model treats the ions as static and the electrons as having both static and dynamic aspects; that is, the ions do not cause transitions. If the ions do not move significantly during the ionization process, then it is impossible that a discrete number of them can produce a completely spherical potential. This issue is important in line broadening theory, and results from that area might be borrowed to explore the importance of a non-spherical potential on the ionization problem [93]. This issue also has been explored by Perrot [92] in a study of external electric field effects on inelastic cross sections and in a microfield stochastic model (MSM) developed by Murillo [86] to treat the perturbations of slowly moving ions on electron impact processes. In such descriptions it is likely that level shifts in agreement with spectroscopy [31] can be obtained. 8.4. Atomic ionization issues 1. Beyond the Born approximation. The range of applicability of the Golden Rule has not been explored in this work. For plasmas which are cool, the slower free electrons may not be in a regime in which the (first) Born approximation is valid. For transitions between states with similar energies, screening relaxes such a constraint because the interaction is weakened by static screening. However, as we have seen, this typically is not the case. In fact, the interaction often becomes stronger at high densities due to collective effects. The collisional ionization process is particularly complicated because one has to consider both slow electrons in the distribution as well as transitions which are modified and possibly enhanced by dynamic screening effects. 2. Coulomb three-body problem. The final state of the ionization process contains (at least) two charged particles in the field of an ion and thus represents a Coulomb three-body problem. This fact has been ignored entirely in the stochastic model whose plasma fluctuations are not affected by the presence of the ion. As the Coulomb three-body problem is insoluble, only modest improvements can be made. Some improvements may, however, be essential for obtaining results in quantitative agreement with experiments. 3. Potential and exchange scattering. The quantum statistical effects of the (identical) electrons have not been accounted for here. These effects appear in both the initial state and the final state. In the initial state this effect arises from the indistinguishability of the plasma electron(s) and the bound state electron(s) [120]. In the final state one often refers to the ‘‘exchange scattering” between the plasma and ionized continuum electrons. A careful study of these effects in

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traditional cross-section calculations, but with plasma and ion parameters of interest, would be useful for assessing the importance of these complications. 4. Distorted waves for plasma electrons. Our description of plasma density fluctuations has been based on the one-component plasma model. In this model the electrons are assumed to interaction with each other and a uniform, positively charged background. Polarization of the plasma due to the presence of a high-z ion requires one to consider dynamic structure factors for two-component, electron—ion plasmas. In the traditional (cross section) approach, such polarization is accounted for by treating the plasma electron within, for example, the distorted-wave approximation.

Acknowledgements Much of this work was performed with National Science Foundation support through grants PHY-9321329 and PHY-9024397 to Rice University. Some of this work was performed under the auspices of the United States Department of Energy through support of the Theoretical Division of Los Alamos National Laboratory. We would like to thank Dr. D.P. Kilcrease for a careful reading of the manuscript with accompanying helpful comments. The Aspen Center for Physics provided us with a stimulating environment during August 1995, when a preliminary draft of this Report was created during the workshop on Elementary Processes in Astrophysical Dense Matter.

Appendix A. List of frequently used symbols Notation a, b A, B E f F F g g G G + I z k K l m,m a nL n a N

principal quantum number, atomic state label plasma state label energy oscillator strength phase-space distribution function electron—electron force statistical weight radial distribution function continuum density of states atomic partition function Planck’s constant ionization potential of a charge z ion plasma spatial Fourier mode continuum electron wave vector angular momentum quantum number mass electron density operator particle species density number of electrons

60

q, q a Q r r 4 ¹,¹ a ¹ F v, € » w ba z, z a Z a b C a D e g i i D i TF j e k m p ¶ / U t W u u a X

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charge (of species a) plasma canonical partition function position ion—sphere radius temperature (of species a) Fermi temperature velocity general interaction (energy) transition rate (aPb) ion charge (of species a) nuclear charge generic species label inverse temperature, energy units coupling parameter (of species a) density parameter dielectric response function continuum state observables inverse screening length Debye wave vector Thomas—Fermi wave vector thermal DeBroglie wavelength chemical potential screening function scattering cross section degeneracy parameter electric potential Coulomb interaction energy wave function effective electron—electron interaction plasma temporal Fourier mode plasma frequency of species a plasma volume

Appendix B. Numerical computation of the dielectric response function Having derived the dielectric response function, Eq. (68), we now obtain a form suitable for numerical computations. As all directions are equivalent in a homogeneous, unmagnetized plasma, k can be oriented as k"kzL . In addition, to be consistent with the use of the Vlasov equation, we must choose F (€) to be the equilibrium ideal gas distribution. This distribution is the familiar 0 Maxwellian, F (€)"(bm/2p)3@2exp(!bm€2/2) . 0

(B.1)

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It is convenient to shift to dimensionless variables K and W defined by K"k/k , W"u/u , (B.2) D e where k is the Debye wave number and u is the electron plasma frequency. In terms of these D e quantities we have e(K,W)"1#(1/K2Jp)

P

=

x exp(!x2)/(x!W/J2K) dx (B.3) ~= for the dielectric response function. The integral can be evaluated with the Dirac identity,

C D

1 1 lim "P Gipd(x!a) , (B.4) (x!a)$ig x!a g?0 and the result can be written in terms of the error function complement with complex argument, ¼(z)"e~z2 erfc(!iz)

(B.5)

which has been well studied [22]. After considerable manipulation, one finally obtains

S

S

pW pW 1 ! Im[¼(W/J2K)]#i Re[¼(W/J2K)]. e(K,W)"1# 2 K3 2 K3 K2

(B.6)

The problem of evaluating plasma density fluctuations through S(k, u) therefore has been reduced to evaluating Eq. (B.5). Efficient routines are available to compute ¼(z) [114]. Calculation of the response function for a finite-temperature degenerate plasma has also been investigated [32,59].

Appendix C. Formulary Here, we collect numerical expressions for several of the important quantities discussed in the text. In this formulary, species number densities n are in cm~3, temperatures ¹ and inverse a temperatures are in eV and eV~1, respectively; masses m and charges z are in units of the electron a a mass and charge; lengths r are in Bohr radii (a +5.29 nm), and energies are in atomic units B (e2/a +27.2 eV) unless otherwise noted. B C.1. Plasma parameters f Strong coupling C "2.3]10~7 z2n1@3b . a a a a f Degeneracy ¶ "2.4]10~15 n2@3b . e e e f Fermi temperature ¹ "2.4]10~15n2@3 eV . F e

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f Plasmon energy +u "3.7]10~11z Jn /m eV . a a a a C.2. Plasma potentials for ion of charge z f Debye—Hu¨ckel model C Debye wave vector (differing species temperatures)

C

D

i "j~1"7.1]10~12 + z2b n #b n D D i i i e e i f Ion—sphere model

1@2 .

C Radius

AB

z 1@3 r "1.2]108 . 4 n e f Hybrid model C Potential energy U (r)"U (r)h(r@!r)#U (r)h(r!r@) , H : ; z z r2 , U (r)"! #c ! 1 2r3 : r 4 c U (r)" 3 e~ir . ; r C Parameters i"7.12]10~12[+ z2b n #n /J¹2#¹2 ]1@2, i i i i e F e 3z c " [[(ir )3#1]2@3!1] , 1 2i2r3 4 4 1 r@" [[(ir )3#1]1@3!1] , 4 i 3z r@eir{ . c "! 3 i2r3 4

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f Bound state level shifts (first-order) *E /z"7.4]10~12Jn /[¹2#5.9]10~30n4@3]1@4 , e e DD e n 1@3 , *E /z"1.3]10~8 e IS z

AB

3 *E /z" [[(ir )3#1]2@3!1] . H 4 2i2r3 4 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

G. Arfken, Mathematical Methods for Physicists, Academic Press, Inc., San Diego, 1985. K. Bartschat, Phys. Rep. 180 (1989) 1. D.R. Bates, A.E. Kingston, R.W.P. McWhirter, Proc. Roy. Soc. A 267 (1962) 297; Proc. Roy. Soc. A 270 (1962) 155. J.F. Benage Jr., W.R. Shanahan, in: H.M. Van Horn, S. Ichimaru (Eds.), Strongly Coupled Plasma Physics, vol. 187, University of Rochester Press, 1993. H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Plenum, New York, 1977. Th. Bornath, M. Schlanges, Physica A 196 (1993) 427. H. Brysk, P.M. Campbell, P. Hammerling, Plasma Physics 17 (1975) 473. P.G. Burke, V.M. Burke, K.M. Dunseath, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 5341. M.D. Cable et al., Phys. Rev. Lett. 73 (17) (1994) 2316. J. Callaway, Phys. Rev. 97 (4) (1955) 933. R. Cauble, D.B. Boercker, Phys. Rev. A 28 (2) (1983) 944. J. Davis, M. Blaha, J. Quant. Spectrosc. Radiat. Transfer 27 (3) (1982) 307. J. Diaz-Valdes, F.A. Gutierrez, Phys. Lett. A 193 (1994) 154. T.R. Dittrich et al., Phys. Rev. Lett. 73 (17) (1994) 2324. J.W. Dufty, Phys. Rev. 187 (1969) 305. W. Ebeling et al., Thermophysical Properities of Hot Dense Plasmas, B.G. Teubner, Stuttgart, 1991. W. Ebeling, A. Fo¨rster, V.Yu. Podlipchuk, Phys. Lett. A 218 (1996) 297. R.P. Feynman, N. Metropolis, E. Teller, Phys. Rev. 75 (1949) 1561. D.M. Fishman, G.D. Patterson, Biopolymers 38 (1996) 535. V. Fock, Z. Physik 98 (1935) 145. V. Fortov, V. Ternovoi, S. Kvitov, I. Lomonosov, in: H.M. Van Horn, S. Ichimaru (Eds.), Strongly Coupled Plasma Physics, vol. 177, University of Rochester Press, 1993. B.D. Fried, S.D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian, Academic Press, New York, 1961. W. Fritsch, C.D. Lin, Phys. Rep. 202 (1991) 1. P. Fromy, C. Deutch, G. Maynard, Phys. Plasmas 3 (1996) 714. K. Fujima, Atomic Models for Hot Dense Plasmas, Inst. Plasma Phys., Nagoya Univ. Report dIPPJ-AM-57, 1988. J.D. Garcia, Phys. Rev. 159 (1) (1967) 39. M.D. Girardeau, F.A. Gutierrez, Phys. Rev. A 38 (3) (1988) 1624. J. Glanz, Science 262 (1993) 1379. T.E. Glover, J.K. Crane, M.D. Perry, R.W. Lee, R.W. Falcone, Phys. Rev. Lett. 75 (1995) 445. D.L. Goodstein, States of Matter, Dover Publications, New York, 1985. H.R. Griem, Principles of Plasma Spectroscopy, Cambridge University Press, Cambridge, UK, 1997. C. Gouedard, C. Deutsch, J. Math. Phys. 19 (1) (1978) 32. F.A. Gutierrez, J. DiazValdes, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 593. F.A. Gutierrez, M.D. Girardeau, Phys. Rev. A 42 (2) (1990) 936.

64 [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76]

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65 F.A. Gutierrez, H. Jonin, E. Cormier, J. Quant. Spectrosc. Radiat. Transfer 51 (1994) 665. B.A. Hammel et al., Phys. Rev. Lett. 70 (9) (1993) 1263. J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, San Diego, 1990. W.A. Harrison, Solid State Theory, McGraw-Hill, New York, 1970. G.J. Hatton, N.F. Lane, J.C. Weisheit, J. Phys. B: At. Mol. Phys. 14 (1981) 4879. C. Herring, Phys. Rev. 57 (1940) 1969. J.P. Hill et al., Phys. Rev. Lett. 77 (1996) 3665. K. Huang, Statistical Mechanics, Wiley, New York, 1987. S. Ichimaru, Statistical Plasma Physics, vol. I: Basic Principles, Frontiers in Physics, vol. 87, Addison-Wesley, New York, 1992. S. Ichimaru, Statistical Plasma Physics, vol. II: Condensed Plasmas, Frontiers in Physics, vol. 88, Addison-Wesley, New York, 1994. S. Ichimaru, S. Mitake, S. Tanaka, X.-Z. Yan, Phys. Rev. A 32 (1985) 1768; plus references in X.-Z. Yan, S. Ichimaru, Phys. Rev. A 34 (1986) 2173. C.A. Iglesias, J. Quant. Spectrosc. Radiat. Transfer 54 (1/2) (1995) 181. D.R. Inglis, E. Teller, Astrophys. J. 90 (1939) 439. M. Inokuti, Rev. Mod. Phys. 43 (3) (1971) 297. N. Itoh, H. Totsuji, S. Ichimaru, H.E. DeWitt, Astrophys. J. 234 (1979) 1079. J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. C.J. Joachain, Quantum Collision Theory, Elsevier, New York, 1987. Y.D. Jung, Phys. Plasmas 2 (1) (1995) 332. Y.D. Jung, Phys. Plasmas 2 (5) (1995) 1775. Y.D. Jung, Y.S. Yoon, J. Phys. B 29 (1996) 3549. L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Green’s Function Methods in Equilibirum and Nonequilibrium Problems, Addison-Wesley, New York, 1989. S. Kambayashi, G. Nowotny, J. Chihara, G. Kahl, J. Non-Crystalline Solids 205—207, Part II, 914, 1996. T. Kawachi, T. Fujimoto, Phys. Rev. E 55 (1997) 1836. C.J. Keane et al., Phys. of Fluids B 5 (9) (1993) 3328. F.C. Khanna, H.R. Glyde, Can. J. Phys. 54 (1976) 648. H. Kitamura, S. Ichimaru, Phys. Rev. E 51 (6) (1995) 6004. A. Klisnick, et al., in: W.L. Rowan (Ed.), Atomic Processes in Plasmas, vol. 201, AIP Press, New York, 1995. D. Saumon, G. Chabrier, Phys. Rev. A 46 (4) (1992) 2084. R. Kubo, J. Phys. Soc. Japan 12 (1957) 570&1203. M. Toda, R. Kubo, N. Saito, N. Hashitsume, Statistical Physics, Springer, New York, 1991. C. Bottcher, in: D. Bates (Ed.), Advances in Atomic and Molecular Physics, Vol. 20, Academic Press, New York, 1985, p. 241. R. Latter, Phys. Rev. 99 (6) (1955) 1854. R.W. Lee, R. Petrasso, R.W. Falcone, Science on High-Energy Lasers: From today to the NIF, Lawrence Livermore National Laboratory Report dUCRL-ID-119170 (1995). R.L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, Prentice-Hall, Englewood Cliffs, NJ, 1990. J. Lindl, Phys. Plasmas 2 (1995) 3933. R.W.P. McWhirter, in: H. Huddlestone, S. Leonard (Eds.), Plasma Diagnostic Techniques, vol. 201, Academic Press, New York, 1965. T. Maeda, Macromolecules 24 (1991) 2740. G. Mahan, Many-Particle Physics, Plenum Press, New York, 1990. R.C. Mancini, et al. in: W.L. Rowan (Ed.), Atomic Processes in Plasmas, vol. 322, AIP Press, New York, 1995, p. 161. C. Martin et al., Journal de Physique II 5 (1995) 697. D.L. Matthews et al., Phys. Rev. Lett. 54 (2) (1985) 106. F. Mazzanti, A. Polls, J. Boronat, Phys. Lett. A 220 (1996) 251.

M.S. Murillo, J.C. Weisheit / Physics Reports 302 (1998) 1— 65 [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125]

65

H.M. Milchberg et al., Phys. Rev. Lett. 67 (19) (1991) 2654. H.M. Milchberg et al., Phys. Rev. Lett. 61 (1988) 2364. M.H. Mittleman, Phys. Rev. 122 (6) (1961) 1930. R.M. More, J. Quant. Spectrosc. Radiat. Transfer 27 (1982) 345. R.M. More, in: D. Bates, B. Bederson (Eds.), Advances in Atomic and Molecular Physics, vol. 305, Academic Press, New York, 1985. N.F. Mott, Metal-Insulator Transitions, Taylor & Francis, London, 1974. J.A.M. van der Mullen, Phys. Rep. 191 (1990) 109. M.S. Murillo, J.C. Weisheit, in: H. Van Horn, S. Ichimaru (Eds.), Strongly Coupled Plasma Physics, vol. 233, University of Rochester Press, 1993. M.S. Murillo, in: W. Goldstein (Ed.), Atomic Processes in Plasmas, vol. 301, AIP Press, AIP CP, New York, 1996. M.S. Murillo, Proc. 1997 Internat. Conf. on Strongly Coupled Coulomb Systems, 1997, to appear. J.K. Nash, J.M. Salter, W.G. Eme, R.W. Lee, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 283. D.R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. A. Ng et al., Phys. Rev. Lett. 72 (21) (1994) 3351. K.C. Ng, J. Chem. Phys. 61 (7) (1974) 2680. National Research Council (U.S.), Plasma Science: From Fundamental Research to Technical Applications, National Academy Press, Washington DC, 1995. F. Perrot, Physica Scripta 37 (1988) 351. E.L. Pollock, J.C. Weisheit, in: F. Rostas (ed.), Spectral Line Shapes, Vol. 3, Walter de Gruyter & Co., New York, 1985. D. Pines, P. Nozie`res, The Theory of Quantum Liquids, Addison-Wesley, New York, 1966. M. Rasolt, F. Perrot, Phys. Rev. Lett. 62 (19) (1989) 2273. F.J. Rogers, H.C. GraboskeJr, D.J. Harwood, Phys. Rev. A 1 (6) (1970) 1577. F.J. Rogers, private communication; see also F.J. Rogers, D.A. Young, H.E. DeWitt, M. Ross, Phys. Rev. A 28 2990 (1983). S.J. Rose, Ap. J., 453 (1995) 45. M.D. Rosen et al., Phys. Rev. Lett. 54 (2) (1985) 100. G.B. Rybicki, A.P. Lightman, Radiative Processes in Astrophysics, Wiley, New York, 1979. E.E. Salpeter, Australian J. Phys. 7 (1954) 377. D. Salzmann, Y.T. Lee, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 339. M. Schlanges, Th. Bornath, D. Kremp, Phys. Rev. A 38 (4) (1988) 2174. M. Schlanges, Th. Bornath, Physica A 192 (1993) 262. K.S. Singwi, M.P. Tosi, R.H. Land, A. Sjo¨lander, Phys. Rev. 176 (2) (1968) 589. J.F. Springer, M.A. Pokrant, F.A. Stevens Jr, J. Chem. Phys. 58 (11) (1973) 4863. L. Spruch, Revs. Mod. Phys. 63 (1991) 151. J.C. Stewart, K.D. Pyatt, Astrophys. J. 144 (3) (1966) 1203. S.R. Stone, J.C. Weisheit, J. Quant. Spectrosc. and Radiat. Transfer 75 (1986) 67. K. Tsuji et al., J. of Non-Crystalline Solids 205—207, Part I, 295 (1996). J.A.M. van der Mullen, Phys. Rep. 191 (1990) 109. L. vanHove, Phys. Rev. 95 (1954) 249. A.V. Vinogradov, V.P. Shevel’ko, Sov. Phys. JETP 44 (3) (1976) 542. W. Gautschi, SIAM J. Numer. Anal. 7 (1) (1970) 187. J.C. Weisheit, Advances in Atomic and Molecular Physics 25, Academic Press, New York, 1988, p. 101. K.G. Whitney, P.E. Pulsifer, Phys. Rev. E 47 (1993) 1968. B.L. Whitten, N.F. Lane, J.C. Weisheit, Phys. Rev. A 29 (2) (1984) 945. N.C. Woolsey et al., Phys. Rev. E 53 (1996) 6396. W.-C. Wu, A. Griffen, Phys. Rev. A 54 (1996) 4204. S.M. Younger, in: T.D. Ma¨rk, G.H. Dunn (Eds.), Electron Impact Ionization, Springer, Wien, 1985. J.K. Yuan, Y.S. Sun, S.T. Zheng, J. Phys. B 29 (1996) 153. J.K. Yuan, Y.S. Sun, S.T. Zheng, Phys. Rev. E 53 (1) (1996) 1059. A.G. Zilman, R. Granek, Phys. Rev. Lett. 77 (1996) 4788. N.H. March, M.P. Tosi, Atomic Dynamics in Liquids, Dover, New York, 1976. J.G. Chabrier, J. Phys. France 51 (1990) 1607.

Physics Reports 302 (1998) 67—142

Multi-wavelength signatures of galactic black holes: observation and theory Edison P. Liang Department of Space Physics and Astronomy, Rice University, Houston, TX 77005-1892, USA Received September 1997; editor: D.N. Schramm

Contents 1. Introduction 1.1. Highlights of some recent developments 1.2. GBH candidates 2. Observational signatures 2.1. X- and gamma-ray properties 2.2. Rapid variability and spectral evolution 2.3. Radio properties 2.4. IR/optical/UV properties 2.5. Long-term variability 3. Basic accretion models 3.1. Accretion flows 3.2. Steady thermal disk models 3.3. Hot optically thin disk versus hot disk corona 3.4. Hybrid models 4. Outstanding issues and recent advances of the steady thermal disk models 4.1. Origin of viscosity and variable-a disks 4.2. Comptonization and the origin of the soft photons

69 69 70 72 72 77 84 87 90 93 94 95 97 104 107 107 110

4.3. Pair processes and relativistic effects 4.4. Thick and slim disks 4.5. Advection-dominated and transonic disks 4.6. Disk winds and outflows 4.7. Role of macroscopic magnetic fields 5. Nonthermal models 5.1. Nonthermal X- and gamma-ray emissions 5.2. Nonthermal radio emissions 5.3. Particle acceleration mechanisms 6. GBH variability and time-dependent disk models 6.1. Dynamical and instability time scales 6.2. Linear radial perturbations 6.3. Global nonlinear radial perturbations 6.4. Thermal limit cycle instability and X-ray Nova outbursts 7. Towards a unified framework 8. The GBH-AGN connection 9. Summary and future directions References

111 113 114 118 120 123 123 124 125 126 126 126 129 130 133 134 134 136

Abstract This review attempts to provide a concise overview of the observational signatures of galactic black hole candidates and highlight some of their theoretical models. We focus on the emerging areas of confrontation between observation and theory. The goal is to lay the foundation of a possible unified framework for the systematic study of the multiwavelength emission and dynamical properties associated with galactic black-holes candidates, ranging from X-ray Nova outbursts to relativistic bipolar jet-like outflows. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 98.35.Jk; 97.60.Lf; 04.70.!s

0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 8 0 - X

MULTI-WAVELENGTH SIGNATURES OF GALACTIC BLACK HOLES: OBSERVATION AND THEORY

Edison P. LIANG Department of Space Physics and Astronomy, Rice University, Houston, TX 77005-1892, USA

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

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1. Introduction 1.1. Highlights of some recent developments Recent observations show that, in addition to their characteristic strong hard X-ray emissions 5100 keV (see Oda, 1977; Liang and Nolan, 1984; Liang, 1993; Tanaka and Lewin, 1995 for historical reviews), galactic black-hole (GBH) candidates also exhibit an exciting variety of multiwavelength and dynamical signatures, from low-frequency quasi-periodic oscillations (LFQPOs, cf. Van der Klis, 1995) to episodic bipolar relativistic outflows (Mirabel et al., 1992; Rodriguez et al., 1992; Mirabel and Rodriguez, 1994; Hjellming and Rupen, 1995). Hence, the study of stellar mass black holes has in the last few years blossomed into a truly multiwavelength endeavor. With the all-sky monitoring capability of BATSE onboard the Compton observatory, and the rapid responses from many other satellites and ground-based observatories, the outburst of X-ray transients and emergence of high states of persistent sources can now be monitored in other wavelengths quickly, from gamma-ray to radio frequencies. This explains why we have accumulated such a wealth of informations on GBHs in the past few years. GBH candidates fall into two main categories: persistent hard X-ray sources such as Cyg X-1, and X-ray Novae (XRNs) or transients such as A0620-00. However, the distinction between the two groups is not always clear cut and there are many examples, such as GRS1915#105, which lie somewhere in between. It is generally believed that the basic underlying accretion and radiation mechanisms of both classes are similar, but for reasons yet unclear, the accretion flow in the transient systems is highly episodic, possibly regulated by some kind of disk-instability-driven thermal limit-cycle similar to the case of dwarf novae (Mineshige and Wheeler, 1989; see Cannizzo, 1993 for review). Cyg X-1, the canonical GBH candidate, is an example of a high-mass system, whereas most of the X-ray novae are believed to be low-mass systems. However, it is still controversial whether that is the primary distinction between persistent and transient sources. One of the challenges of black-hole astrophysics is to understand the similarities and distinctions between the two classes. Within the transient category, the XRNs form a separate homogeneous subgroup: they have characteristic rapid rise (a few days or less) and exponential decay (30—60 days decay constant) during outburst (see Tanaka and Shibazaki, 1996; Chen et al., 1997 for recent reviews). At peak outburst they are as bright as the brightest persistent sources (&1037—1038 erg/s), but they are almost undetectable in X-rays during quiescence which typically lasts several decades. In contrast, irregular transients such as GRS1915 have more symmetrical and slow rise and fall, and the outburst cycle is much shorter, typically on the order of months to years. There are also some evidences that the hard X-ray spectra of GBHs form two distinct classes (power law versus exponential tails, Kroeger et al., 1996; Philips et al., 1996). Among the recent developments, the most dramatic is perhaps the discovery of bipolar radio jets and relativistic ejections of radio-emitting plasmoids, two of which (GRS1915 and GROJ1655) appear to have superluminal motions (Mirabel and Rodriguez, 1994; Hjellming and Rupen, 1995). This greatly strengthens the connection between GBHs and the AGN phenomena. Because the GBHs are much closer and more accessable in terms of high-resolution (in energy, time and space) studies, we have much better chances of getting a deeper understanding of black-hole astrophysics with GBHs.

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Since the field of GBH astrophysics is currently in a rapid state of development, it is impossible to provide an up-to-date comprehensive review. In particular, no serious attempt has been made to perform an exhaustive literature search so many recent references are likely missed. To limit the scope of this review we will concentrate on the multiwavelength manifestations of GBH emissions and dynamics which bear most directly on the theoretical models. For example, we will not delve into the large body of literature on the search for optical counterparts and estimating the mass of the GBHs. (see e.g. Cowley, 1994; van Paradijs and McClintock, 1995; Tanaka and Lewin, 1995 for recent reviews) or the formation and distribution of GBH binaries (see White, 1994 for review). Similarly, we will focus on those theoretical themes which have the best chance of being confronted by multiwavelength observations. We hope this will serve as the first step towards building an eventual unified framework for the understanding of the many diverse and complex phenomena associated with both the persistent and transient GBH sources. Because of this empirical approach we will not dwell heavily on the basic theory of accretion disks such as the origin of disk viscosity, for which there is a large body of excellent recent reviews (e.g. Cannizzo, 1993; Chakrabarti, 1996a; King, 1995; Mineshige and Kusunose, 1993; Vishniac and Diamond, 1993). Rather we will focus on those areas of accretion disk theory that will directly impact observable GBH properties, such as the output spectra, time variability and global dynamical behaviors. Another area we will be glossing over will be general relativistic effects, especially those near a rapidly rotating Kerr black hole (Misner et al., 1973; Cunningham, 1975; Fishbone and Moncrief, 1976). Unlike AGN black holes, most GBHs in binares have not been accreting for long enough to approach the angular momentum limit of a"M. Hence, the Schwarzschild geometry is believed to be a reasonable first approximation for most GBHs. In this case since most of the disk binding energy is released at 56 GM/c2, for many applications the use of Newtonian or pseudoNewtonian approximation (e.g. Pringle and Rees, 1972; Novikov and Thorne, 1973; Shakura and Sunyaev, 1973; Paczynski and Witta, 1980) is adequate except for the disk inner-boundary conditions (e.g. Liang and Thompson, 1980; Abramowicz and Zurek, 1981; Muchotrzeb and Paczynski, 1982). However, recent results from the superluminal source GROJ1655 may suggest that this source has an inner disk edge as small as 2GM/c2, which would require a corotating Kerr hole with a"0.95M (Zhang et al., 1997a). Such results await further confirmation. 1.2. GBH candidates Black-hole astrophysics began with the X-ray studies of Cygnus X-1 by UHURU (see Oda, 1977; Liang and Nolan, 1984 for historical reviews). The simultaneous discovery of a soft X-ray transition (Miyamoto et al., 1971; Tananbaum et al., 1972) and radio outburst (Hjellming and Wade, 1971; Braes and Miley, 1971) allowed astronomers to pin down its precise location. Subsequent identification with the 5.6 day binary system of the blue supergiant companion HDE226868 confirmed the high mass of the unseen object, thereby giving credibility to its BH candidacy (Avni and Bahcall, 1975; Bolton, 1975; Margon et al., 1975; Webster and Murdin, 1972). Because of its brightness especially in the hard X-rays, Cyg X-1 is one of the most studied sources in the sky (see e.g. Liang and Nolan, 1984 for summary of earlier references). Its properties, including the characteristic hard X-ray spectrum (power-law cutting off exponentially above a couple of hundred keV, e.g. Agrawal et al., 1972; Coe et al., 1976; Dolan et al., 1987; Haymes and Harnden, 1968, 1970; Holt et al., 1976; Liang, 1980; Mandrou et al., 1978; Nolan and Matteson, 1983; Oda et al., 1976;

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Overbeck et al., 1967; Steinle et al., 1982; Sunyaev and Trumper, 1979), episodic emergence of an ultrasoft component, anticorrelated soft (e.g. Sanford et al., 1975) and hard X-ray transitions (e.g. Baity et al., 1973; Matteson et al., 1975), chaotic variability down to milliseconds (e.g. Canizares and Oda, 1977; Oda et al., 1976; Ogawara et al., 1982; Meekins et al., 1984; Weisskopf et al., 1975), persistent gamma-ray tail above an MeV (McConnell et al., 1993, 1994), episodic gamma-ray bumps at a few hundred keV—2 MeV (Ling et al., 1987), persistent radio emission and radio flaring correlated with X-ray transitions (see Hjellming and Han, 1995 for review), and low-frequency QPOs (Kouveliotou et al., 1993; see Van der Klis, 1995 for review), have been widely adopted as canonical signatures of GBH candidacy. Even though some neutron stars have exhibited one or more of the above properties (e.g. soft X-ray transitions and rapid chaotic variability, cf. Tanaka and Lewin, 1995), none of them has exhibited a majority of these properties. Hence, it is still a good working hypothesis that if a hard X-ray source manifests a majority of the above properties, it is a likely GBH candidate. Recent mass estimates (see Cowley, 1994; van Paradijs and McClintock, 1995; Tanaka and Lewin, 1995 for review) of several X-ray Novae (XRN) with the above properties have indeed confirmed that their masses are comfortably above the conservative neutron—star mass limits (Hartle, 1978). As of this writing there are over 20 GBH candidates based on their hard X-ray properties. Fig. 1 gives their sky positions in Galactic coordinates (see e.g. Wood et al. (1984) for detailed coordinates) and Table 1 summarizes the observed properties of these sources, including their

Fig. 1. Sky distribution of Galactic black-hole candidates in Galactic coordinates. Note the strong concentration towards the Galactic plane suggestive of a Population I association (see Table 1 for source identities).

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Table 1 Current “Zoo” of Galactic black-hole candidates Mass-estimated Persistent sources

Transients or XRN

1. Cyg X-1 (10M )m`* _ 2. LMC X-3 (9M )m _ 3. LMC X-1 ('2.5M )m _ 8. A0620-00 (6.1M ) _ 9. V404 Cyg/GS 2023#338 (12M ) _ 10. GS 1124-684/Nova Muscae (6M )`* _ 11. GS 2000#25 (7.2M ) _ 12. GROJ1655-40d (4.5M ) _ 13. GRS 1719-25 (7M ) _

Hard X-ray selected 4. 5. 6. 7.

GX 339-4* IE1740.7-2942`j GRS 1758-258d GRS 1915#105d

14. 15. 16. 17. 18. 19. 20. 21.

GRO J0422#32* GRS 1009-45* 4U 1543-47 Briggs Source` EXO 1846-31 GRS 1734-292` GRS 1739-278 Nova Sagattarii

m"massive companion, d"radio jets, *"low frequency QPOs, #"alleged pair-related features (511 or 170 keV lines or MeV bumps) Mass estimates based on review by Barret et al. (1996).

mass estimates whenever they are available (cf. Bailyn et al., 1995; Casares et al., 1994; Charles et al., 1991; Dolan et al., 1992; McClintock and Remillard, 1990; McClintock et al., 1995; Orosz et al., 1995; Remillard et al., 1992; Shahbaz et al., 1994, etc.). Optical and IR searches for the companions of the remaining sources are ongoing. It is likely that the list of “confirmed mass” GBH candidates will grow steadily in the near future. Even though in Table 1 we tentatively divide the GBHs into the persistent versus transient categories, we emphasize that the distinction between them is often blurred. For example, though we list GX339-4 and GRS1915 as persistent sources, they are highly variable in intensity (GX339-4 has a 220/440 day quasi-period, Harmon et al., 1994). Similarly, low-level persistent soft X-ray emissions have been detected from XRN A0620 (McClintock et al., 1995), V404Cygni (Wagner et al., 1994) and perhaps Nova Muscae (Greiner et al., 1994) during quiescence. Hence, we suspect that the difference between persistent and transient sources is a matter of degree rather than something fundamental. There is likely a continuum spectrum of sources spanning the two groups. There is also some indication that the persistent or transient nature may be related to whether the companion is high mass or low mass.

2. Observational signatures 2.1. X- and gamma-ray properties 2.1.1. Hard X-ray (10—200 keV ) spectra The 10—200 keV continuum emission is the main stay of all GBH power output. The spectum in this energy range typically exhibits a featureless power-law continuum (Fig. 2) with photon spectral index in the range &1—2.5. Above 200 keV the spectrum cuts off exponentially for most sources. However, in some XRNs the power-law extends above 500 keV (cf. Kroeger et al., 1996). In

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Fig. 2. Generic spectrum of GBH candidates showing the three most commonly observed components: soft black-bodylike excess (dashed), hard power law with exponential cutoff (solid), soft power law (index&2.5) extending out to 'MeV (dash—dot). The relative intensities of the three components vary from source to source and with time for a given source. Fig. 3. Power per decade of energy (lF ) spectrum of Cygnus X-1 at different intensity states measured by OSSE. This l shows that most of the power comes out at &100—150 keV except during the ultra-low state (from Phlips et al., 1996).

the case of Cyg X-1, there is now strong evidence of a persistent &!2.5 power law tail extending from &MeV to '10 MeV (McConnell et al., 1994, cf. Section 2.1.2). However, if we plot the spectrum in the form of vF where F is specific energy flux, during hard-low states we see that the v v power per decade of energy typically peaks at 50—150 keV (Fig. 3). Hence, GBHs emit their power output predominantly in the hard X-rays. This property distinguishes GBHs from most accreting neutron stars whose power outputs peak at keV ranges (however, a few neutrons stars also exhibit hard X-ray tails out to tens of keV, e.g. 4U1608-52, Zhang et al., 1996a) and gamma-ray bright AGNs (whose power outputs peak at 'MeV, Dermer and Schlickeiser, 1992). For most GBHs the spectral index of the hard X-rays is very robust in the sense that even when the intensity varies by a large factor the index changes by very little. On the other hand there is some evidence that the exponential cutoff energy varies more strongly with intensity, especially for the transient sources (e.g. GROJ0422). The conventional interpretation of the hard X-ray continuum is that it is produced by the inverse Comptonization of a soft photon source by hot thermal electrons (Katz, 1976; Shapiro et al., 1976; Sunyaev and Titarchuk, 1980; Titarchuk, 1994). However, as we will discuss below (cf. Section 5) this interpretation is not unique. Unlike the case of Seyfert nuclei, the hard X-ray spectra of GBHs typically do not show a significant “hump” between 10 and 100 keV which is often interpreted as due to Compton reflection by cold matter (Stern et al., 1995). Hence, we believe the evidence for a reflection component for GBHs is weak. Such a difference between GBHs and AGNs may be due to differences in the surrounding environment (cf. Section 8). 2.1.2. Soft X-ray ((10 keV ) spectra Both persistent and transient GBH candidates exhibit large variations in their soft X-ray ((10 keV) fluxes. During the hard-low states the soft X-ray spectrum below 10 keV is often a simple continuation of the hard X-ray power law, with increasing flattening below &3 keV due to ISM absorption (Fig. 4a). In that case the entire X-ray continuum is likely produced by a single

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Fig. 4. ASCA soft X-ray spectrum of 1E1740.7-2942 (top panel) and Cyg X-1 (bottom panel) in low state. Note the presence of the Fe K-edge at &7 keV for 1E1740.7-2942 and a weak broad Fe line feature at 6—7 keV for Cyg X-1. While the continuum of 1E1740.7 is best fit by a simple power law model with ISM absorption (solid curve), the continuum of Cyg X-1 requires a power law with ISM absorption (middle curve) plus an additional soft black body of k¹&0.5 keV (bottom curve). This suggests that the soft disk-black-body component from Cyg X-1 never totally disappears even during low states (from Sheth et al., 1996).

hot component. However, during some episodes of Cyg X-1 low-state the best fit model still requires a small black-body component of temperature &0.3—0.5 keV on top of the power law (e.g. Fig. 4b). This suggests that there is indeed a soft photon source, as expected in the inverse Compton model, even in the hard-low state (Shapiro et al., 1976). In addition there is often evidence of Fe K-edge absorption at &7 keV (Fig. 4, top panel). However, unlike Seyfert galaxies the evidence for Fe K- or L-fluorescent line emissions from GBHs is weak at best. This plus the lack of a hard X-ray hump remove the need of a cool reflection medium near the X-ray source. During the soft-high-state of the persistent sources or outburst of the XRNs, the soft X-ray flux (10 keV increases by a large amount. It can become much brighter than the hard X-rays. The

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Fig. 5. Comparison of the X-ray spectra of sample GBHs in (a) the soft-high state (GRS1009-45) and (b) the hard-low state (GRS 1716-249). (c) Contrasts the soft X-ray slopes of the two states. The soft X-ray data are from ASCA and the hard X-ray data from OSSE (from Moss, 1997).

spectrum also becomes much softer with a slope '3—4. In many cases it resembles a modified or “disk” black-body predicted by the canonical Shakura—Sunyaev (SS 1973) model with a black-body temperature &keV (Fig. 5a, c (Ebisawa, 1989, 1991)). The rise of the soft X-ray component is sometimes, but not always, accompanied by a slight steepening of the hard X-ray power-law index (e.g. Zhang et al., 1996b, 1997b), leading to a “pivoting” of the spectrum at &10 keV. Comprehensive systematic correlative study of the variation of the two components remains to be performed. This will help to settle the physical and geometrical relations between the soft and hard X-ray emission regions (Ebisawa, 1989). 2.1.3. Gamma-ray (5300 keV ) properties Currently there are only two GBHs with reported emissions 'MeV: Cyg X-1 (McConnell et al., 1994, 1995) and GROJ0422 (Van Dijk et al., 1994, 1995) and the statistics for GROJ0422 is marginal. In both cases the gamma-ray data are consistent with either ultrahot thermal or

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Fig. 6. Comptel spectra of (a) Cyg X-1 and (b) GROJ0422 suggest the presence of a high-energy tail above 1 MeV. At present it is still unclear if this tail is due to thermal or nonthermal emissions (from McConnell et al., 1994; Van Dijk et al., 1994).

power-law emissions (Fig. 6). The spectra between 300 keV and 1 MeV of many sources are better determined, and there is strong evidence that they form two distinct classes: exponential versus power law (Grabelski et al., 1993, 1995; Kroeger et al., 1995, Fig. 7). It is interesting that the photon index of the power-law XRNs (&2.5) is similar to the index of the superlow state (Fig. 3) as well as

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Fig. 7. Comparison of the hard X- and gamma-ray spectra of the thermal (GROJ0422, right) versus the power law (GRS1655, left) type of XRNs (from Kroeger et al., 1996).

the 'MeV tail of Cyg X-1 (Fig. 6), which may hint at a common origin. Despite early rumors, currently there is no evidence of GeV or TeV emissions from any GBH candidate. In addition to the above persistent gamma-ray emissions there have been numerous reports of transient emergence of a “bump” at 500 keV—2 MeV for Cyg X-1 (Fig. 8, Ling et al., 1987) and annihilation-like features at & 400—500 keV for 1E1740.7—2942 (Paul et al., 1991; Bouchet et al., 1991), Nova Muscae (Goldwurm et al., 1992; Sunyaev et al., 1994) and the Briggs source (Briggs et al., 1995) (Fig. 9). However, none of these has been confirmed independently by other groups and the existence of these features remains controversial. This suggests that even if such features are real their duty cycle appears small. 2.2. Rapid variability and spectral evolution The rapid variability (;hours or binary orbital period) of the X-rays of GBH candidates has been studied extensively (e.g. Van der Klis, 1995; Miyamoto et al., 1993; Kouveliotou et al., 1993; Bridgman et al., 1994; Grove et al., 1994 for recent results), using power density spectrum (PDS), autocorrelation function (ACF) and complex cross-correlation function (CCCF) techniques. Sample results are illustrated in Figs. 10—14. Here we summarize the generic properties which

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Fig. 8. HEAO3 spectra of Cyg X-1 in the three gamma-ray states showing the presence of an “MeV bump” during the gamma-1 state suggestive of copious pair production (from Ling et al., 1987).

apply to most GBH candidates: 1. The overall RMS fluctuation level seems to be higher for the hard X-rays than the soft X-rays. 2. Generically the PDS of GBHs consists of three distinct regimes: plateau “red noise” at low frequencies, 1/f-like power-law noise at intermediate frequencies and white noise at higher frequencies. The frequencies separating the three regimes vary from source to source, vary with the intensity of the source, and vary with the energy of the photons. But for most sources the red noise lies below 0.1 Hz and the white noise lies above 10 Hz (Figs. 10 and 13). 3. Superposed on the above generic PDS continuum, some GBHs show “peaked noise” or QPOs in the power-law section of the PDS. Unlike the high-frequency QPOs in neutron star systems, the GBH peaked noises are at lower frequencies, ranging from &25—30 mHz to &10 Hz (Figs. 11 and 12). Second harmonics have also been observed in several cases (Tanaka et al., 1991, Tanaka, 1992, Fig. 12). But for many sources no significant peaked noise or QPO was ever observed, even during high states of the hard X-rays. 4. For Cyg X-1, there was some evidence that the PDS shows a narrow peak near 3 ms. If confirmed it may be an indication of the orbit period of the disk inner edge and this frequency could serve as a measure of the black-hole mass. Hence, it may be worth searching for such high-frequency periods or cutoffs in all GBHs with RXTE. 5. The ACF, which is the Fourier transform of the PDS, provides informations on the shape of individual noise spikes (“shot noise”) and their repetition rate. At near-zero time delay the ACF of most GBHs approaches a quasi-exponential (Fig. 13, Nolan et al., 1981). Hence, many authors argue that the noise of GBHs is made up of exponential shots. However, this interpretation is not unique and more study is needed to understand the implications of the shape of the

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Fig. 9. Examples of annihilation-like features from three GBH candidates. Note that all three features peak at energies slightly below 511 keV (from Chen et al., 1993).

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Fig. 10. Examples of rapid flickering and power density spectra (PDS) of five GBHs showing their similarities (from Tanaka and Lewin, 1995).

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Fig. 11. Examples of low-frequency QPOs in the soft X-rays (upper: ASCA data of 1E1740.7 and middle: GRS1009, from Dobrinskaya et al., 1996), and in the hard X-rays (bottom: OSSE data of GROJ0422, from Grove et al., 1994).

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Fig. 12. PDS of Ginga data of Nova Muscae showing that sometimes only one harmonic is present while other times both harmonics are present (from Tanaka et al., 1991).

ACF. In any case Fig. 13 shows that a single exponential definitely does not fit the entire ACF of Cyg X-1. We need at least two different exponentials with different time constants, (1 s and &10 s, respectively. 6. The CCCF provides informations on the time delay between soft and hard photons and how the time delay varies with frequency (Miyamoto et al., 1988, 1993). This of course depends on the energy channels we choose to cross-correlate. For Cyg X-1 it has been established that the hard photons generally lag the soft ones by several to tens of milliseconds at the highest frequencies, and the time delay increases almost linearly with Fourier period (Fig. 14). While these results remain to be confirmed, they promise to pose challenges to any detailed accretion disk models of the X-ray emissions. However, the claim by some authors (e.g. Miyamoto et al., 1993) that these results are in direct conflict with the inverse Compton model (Payne, 1980) is premature since they assume that the soft photon source must be imbedded inside or uniformly in the Comptonizing region. It is equally probable that some or most of the soft photons come from outside the Comptonizing zone, say the outer parts of the disk, in which case the above correlation is not unexpected (see Nowak, 1994 for alternative models). The above three classical techniques, while useful in studying the time variability of specific energy channels, provide insufficient information on the details of rapid spectral evolution. Other new techniques more geared towards the energy domain are badly needed to explore rapid spectral evolution. For example, the rapid evolution of the peak power energy (peak of vF ) and how it v correlates with other physical parameters may provide fresh insights into the physical processes of GBH variability. Other modern methods, such as wavelet and spectrogram techniques, should be further explored for applications to the analysis of GBH variability.

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Fig. 13. Rapid time variability of Cyg X-1 as manifested in its auto-correlation function (ACF), PDS, cross-correlation function (CCF) and light curves (from Nolan et al., 1981). The ACF shows multi-exponential components and the CCF shows hard lagging soft.

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Fig. 14. The soft-to-hard phase lag and time lag of Ginga data on Cyg X-1 as functions of the Fourier period. Notice the almost linear relation between time lag and period (from Miyamoto et al., 1988, 1993).

2.3. Radio properties There are basically three types of radio sources associated with GBH candidates: (a) low-level persistent compact core sources (e.g. Cyg X-1, GX339-4) that do not seem to vary much with X-ray intensity, (b) compact core sources that flare up during X-ray flares, though often with a time delay (e.g. Nova Muscae, A0620, GRS1915), (c) radio-emitting plasmoid ejections and extended radio lobes (Fig. 15, e.g. 1E1740.7, GRS1758, GRS1915, GROJ1655). The last two sources exhibit superluminal ejections (Fig. 16). A comprehensive review of GBH radio properties was given in Hjellming and Han (1995). Here we only highlight a few major results. The radio continuum emission from all sources are consistent with synchrotron emission from relativistic electrons (Hjellming et al., 1975) while the low-frequency turnover in some cases is likely caused by free—free absorption (Fig. 17). To radiate synchrotron in the radio band the magnetic field in the emission region must be weak and the Lorentz factor c must be high, 'hundreds. In many cases one can estimate B, c, electron (pair) density n, etc. using energy equipartition and pressure balance arguments (e.g. Mirabel et al., 1992). Even the core radio emission region must be at a substantial distance away from the accretion disk itself. The nonthermal relativistic particles may be produced by stochastic, wave or electrostatic accelerations in an extended corona above the accretion disk, or they may be accelerated in the shocks of outflows and winds, either from the accretion disk or the companion. It is tempting to associate the radio-emitting nonthermal particles with those that may be responsible for the power-law gamma-ray tails of Cyg X-1 and some XRNs, but such associations remain to be established. For the persistent core sources the energy of the relativistic electrons must be continuously replenished (Hjellming and Han, 1995) whereas in the plasmoid ejecta

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Fig. 15. Radio images of 1E1740.7 (lower left) and GRS1758 (right) showing the core jets structure similar to those of AGNs. Upper left: variation of the core radio intensity seems correlated with the hard X-ray intensity (from Mirabel et al., 1992; Rodriguez et al., 1992).

the particle energy is likely converted from the bulk motion via shocks and boundary layer turbulence. These are the so-called synchrotron bubble events (Hjellming and Johnston, 1988). In many transient sources as well as during state transitions of the persistent sources, the radio intensity often varies correlated in time with the hard X-ray fluxes with time delays of hours to days (e.g. Figs. 15 and 18). This suggests a possible causal coupling between the hard X-ray and radio source, both physically and geometrically. For Cyg X-1 there is also evidence of anticorrelation between the radio flux and the soft X-ray flux (Zhang et al., 1997c). Possible scenarios to explain this are discussed in Section 5. As for the recently discovered relativistic ejections of radio-emitting plasmoids the key issues are: (a) acceleration mechanism of the bulk flow; (b) mechanism of large scale collimation; (c) positron/proton ratio of the ejecta material; (d) origin of the pairs if the plasmoids are indeed pair-dominated; (e) in situ energization mechanism of the radiating particles in the plasmoids. Some of these issues will be addressed in Section 5.

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Fig. 16. Time-lapse radio images of GRS1915 (left, from Mirabel and Rodriguez, 1994) and GRS1655 (right, from Hjellming and Rupen, 1995) showing the superluminal motions of the plasmoid ejecta.

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Fig. 17. Sample radio spectra of Cyg X-1 (top: from Hjellming and Wade, 1971, details see original Ref.) and Nova Muscae (bottom, from Kesteven and Turtle, 1991). If these are optically thin synchrotron spectra the high-frequency spectral break may be associated with the synchrotron critical frequency.

2.4. IR/optical/º» signatures GBHs have companions that are either high mass (M'10M ) or low mass (M(M , Tanaka _ _ and Lewin, 1995). For high mass systems the optical output is always dominated by the companion and the contribution of the accretion disk is negligible. For low mass systems the situation is less clear. During outburst the XRN optical output is clearly dominated by the accretion disk. But during quiescence there is increasing evidence that the optical output is dominated by the low-mass companion. Recent comparison of the quiescent optical spectrum of several systems with those of standard K-M stars tend to substantiate this (Fig. 19, Orosz and Bailyn, 1995). Optical studies of GBH binary companion is of course the primary tool to constrain the BH mass (e.g. Bailyn et al., 1995; Callanan et al., 1992; Charles et al., 1991; Cowley, 1992; Hutchings et al., 1987; Kemp, 1981,

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Fig. 18. Comparisons of BATSE hard X-ray light curves (top panels) with contemporary radio light curves (bottom panels) of GRS1915 (left) and GRS1655 (right) show that radio flares typically follow hard X-ray flares by a few days. Arrows mark the time zeros of the superluminal ejection events (from Harmon et al., 1995).

1987; McClintock et al., 1992, 1995; Remillard et al., 1992; Reynolds and Jauncey, 1994; Shahbaz et al., 1994). This subject is extensively reviewed elsewhere (e.g. Cowley, 1992, 1994; van Paradijs and McClintock, 1995; Tanaka and Lewin, 1995). Here, we concentrate on the optical emissions from the accretion disk itself. For most black-hole XRNs the optical Nova light curve often tracks the X-ray light curve (*»'5—8 mag, Fig. 20) showing that almost all of the optical output is related to X-rays (see Brandt, 1991 for Nova Muscae). The normal stellar features are absent in their spectra. Instead, they show emission lines indicative of X-ray ionization and heating (e.g. the j4640 line, cf. Fig. 21) and the line splitting observed in many cases is consistent of emission from a rotating disk. The optical/X-ray ratio is &10~4. The few second time delay between optical and X-ray flares are also consistent with reprocessing at the outer disk, probably close to the Roche lobe. Since the outer disk subtends a much larger solid angle than the companion, reprocessing by the companion is believed unimportant. This is confirmed by the lack of large amplitude periodic optical brightness variations. At the outer disk the internal energy flux goes as r~3 whereas the incident X-ray heating goes as r~2. Hence, X-ray heating dominates the energetics and structure of the outer disk. Physical models will be discussed in Section 4. Black-body fitting to the optical continuum typically give temperatures of few ]104 K, consistent with temperatures from X-ray heating. A sample spectrum is shown in Fig. 21 showing the H Balmer series, HeI and HeII lines, Na D line and NIII Bowen lines. In contrast to HII regions, planetary nebulae, SNRs and other low-density photoionized sources, forbidden lines are conspicuously absent in the X-ray irradiated disks of XRNs, hinting

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Fig. 19. Similarity of the quiescent optical spectrum of GROJ0422 to that of a M0V star suggests that the quiescent optical emission of GBH transients is dominated by the low-mass companion and not by the accretion disk (from Orosz and Bailyn, 1995).

Fig. 20. The fast rise exponential decay light curves of XRNs is observed in both the optical (left, from van Paradijs and McClintock, 1995) and soft X-rays (right from Tanaka and Lewin, 1995).

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Fig. 21. The optical spectrum of GRS1655 during outburst shows strong H emission lines suggestive of X-ray heating of the outer accretion disk (from Bailyn et al., 1995).

at a high-density emission region. Hence, the optical spectra of XRNs are closest to the BLRs of SyI galaxies and quasars. In addition, the V404 Cyg spectra also showed a P Cygni profile for several HeI and Balmer lines, suggestive of outflow from the system. The P Cygni profile first appeared in the higher excitation lines and later moved to the low excitation lines suggesting that the outflow cooled as it leaves the system (van Paradijs and McClintock, 1995). The optical light curves of most XRNs show strong variability in addition to the overall exponential decay. V404 Cyg showed large RMS variations down to &minute time scales and exhibited quasi-periodic oscillations in the mHz ranges. While this may be extreme, most transients do show variability on all time scales from ms to days. The optical flux of XRNs usually tracks the X-rays even during the re-flares and secondary outbursts (Fig. 20), but sometimes its overall decay can be much slower (e.g. Harrison et al., 1994; Shrader et al., 1994). However, relation between optical and X-rays in other types of GBH transitions is less clear. For example Motch (1985) reported anti-correlated optical/X-ray transitions for GX339-4 (Fig. 23). One unsettled issue is whether the optical rises before the X-rays in the main outburst. This is predicted by models in which the accretion instability starts at the outer edge of the disk and moves inward. Fig. 22 shows the multiwavelength spectrum of Nova Muscae during outburst (Grebenev et al., 1991). 2.5. Long-term variability Fig. 24 shows examples of the long-term hard X-ray light curve of Cyg X-1 (Paciesas et al., 1997, see also Ling et al., 1996). It is relatively stable with the luminosity (10—200 keV) varying by less than a factor of 4. In contrast, the X-ray luminosity of XRNs (e.g. Fig. 20) varies by several orders of

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Fig. 22. Multiwavelength spectrum of Nova Muscae during outburst (from Grebenev et al., 1991).

Fig. 23. Simultaneous X-ray (Hakucho) and optical (V CCD from ESO) observations of GX339-4 during the June 1981 transition show the anti-correlation between the optical and X-rays (from Motch et al., 1985).

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Fig. 24. Long term hard X-ray light curves and spectral history of Cyg X-1 from BATSE observations. The upper panel shows the integrated 20—100 keV photon flux derived by fitting a power-law model to the average count spectra. The lower panel shows the corresponding photon number spectral index (from Paciesas et al., 1997).

magnitude between quiescence (which can last for decades) and outburst. These differences in the long-term behaviors are likely caused by differences in the companion mass, binary separation and eccentricity which regulate the accretion flow from the companion to the BH. In addition, we speculate that the strength of the accreted magnetic field and upstream angular momentum may also play a role. However, even for the persistent sources such as Cyg X-1 the soft component below 10 keV also exhibits large episodic outbursts (the soft “high state”), during which the soft X-rays increase in intensity by almost a factor of 10 (cf. Liang and Nolan, 1984). This may be similar to the XRN phenomenon. In contrast, LMC X-1 and LMCX-3 seem to be always in the soft-high state and show little variability in either soft or hard X-rays. The interplay between the soft and hard components for the different sources are clearly very complex and we are far from having a clear picture (see Section 3). Fig. 20 shows typical soft X-ray light curves of XRNs during outburst (Tanaka and Lewin, 1995). They are consistent with fast rise ((few days) and exponential decay (FRED, decay constant &months (Kitamoto et al., 1992)). There are also secondary peaks or “reflares” in the decay phase that may be related to instabilities of the mass flow at the inner Lagrange point or from the companion. However, such FRED outburst profiles are not universal and some XRNs exhibit rather deviant behaviors (Chen et al., 1994, 1997). The XRN outburst duty cycle also seems to vary a lot between different systems. Moreover, at late times the light curve of the hard X-rays usually decays more slowly than that of the soft X-rays (Fig. 25). Hence, a key challenge for the theorist is to derive from first principles the multiwavelength profiles of the XRN light curves.

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Fig. 25. Comparison of the hard X-ray light curve (data points) with the soft X-ray light curve (solid) of Nova Muscae 1991 (from Gilfanov et al., 1991).

3. Basic accretion models There have been numerous recent reviews of the theory of accretion flows onto black holes (e.g. Cannizzo, 1993; Mineshige and Kusunose, 1993; Vishniac and Diamond, 1993; Chakrabarti, 1996a to name just a few). Hence, this paper does not intend to go into the details of what is already discussed in these other articles. Rather we try to highlight the key results and issues that are most relevant to current multiwavelength observations, with the ultimate goal of building a single unified framework for the systematic confrontation between theory and data, both for the persistent and transient sources. In this section we first review the baseline steady thermal disk (STD) models. In subsequent sections we will discuss generalizations of these STD models, nonthermal models and time-dependent models.

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3.1. Accretion flows Depending on the amount of organized momentum in the accreted matter, accretion flow onto compact objects are divided into quasi-spherical flows and accretion disks (see e.g. review by King, 1995). Historically, quasi-spherical flows are assumed to have negligible organized angular momentum. At least on an average sense the accreted matter is assumed to follow the Bondi (1952) solution (e.g. Michel, 1972; Maraschi et al., 1980). In the absence of macroscopic turbulence or internal shock dissipation, most of heat generated by adiabatic compression is advected into the black hole and only a tiny fraction, estimated to be &10~4 M Q c2, is radiated away. Hence, it was not favored as a viable mechanism for the energy supply of observed BH candidates. However, the works of Meszaros (1976), Meszaros and Ostriker (1983), Babul et al. (1989), Park and Ostriker (1989) and others suggest that if the spherical flow can be efficiently dissipated by turbulent magnetic fields or collisionless shocks it could give rise to much higher radiation efficiency and hotter temperature. While such models may work for low accretion rates such that the overall radial optical depth is low, it remains to be demonstrated that it is viable for the moderate to high accretion rates (&few % of Eddington luminosity) needed for most GBHs, since intuitively one expects that most photons would be trapped and advected into the hole at high accretion rates. Fig. 26 gives examples of the velocity profiles of spherical accretion. In contrast, the early disk models assume that the accreted matter has well-defined organized and persistent angular momentum. In this case tidal force from the BH makes the disk physically thin. Hence most workers (Lynden-Bell, 1969; Pringle and Rees, 1972; Novikov and Thorne, 1973; Shakura and Sunyaev, 1973) constructed first-principles disk models by assuming: (a) vertical

Fig. 26. Mach number profiles of spherical flows around black holes show that they are divided into four regimes separated by the two transonic solutions labeled with arrows: the Parker wind solution and Bondi accretion solution (from Chakrabarti, 1996a).

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averaging of all physical variables, (b) Keplerian angular momentum distribution, (c) angular momentum transfer via azimuthal stress which is proportional to the total local pressure (the a-viscosity assumption) and solely responsible for energy dissipation. Effectively the accretion proceeds because angular momentum is transferred to the outer edge of the disk and eventually deposited on the companion star or the outflowing circumstellar medium. Implicit in such disk models are (a) negligible radial drift velocity or radial pressure gradient; (b) negligible vertical outflow or mass loss; (c) angular momentum transfer and dissipation of energy occur locally; (d) all dissipated energy are thermalized and the disk cools by radiation. These assumptions simplify the mathematics drastically to the point that the vertically averaged structure of the disk can be solved algebraically as a function of radius. Those models in which the disk is optically thick and radiate locally as a black body are referred to as Shakura—Sunyaev or SS disks (Shakura and Sunyaev, 1973), whereas those which are optically thin and radiate via inverse Comptonization of soft photons are called Shapiro—Lightman—Eardley or SLE disks (Shapiro et al., 1976). Many other variant models exist with varying assumptions of radiative cooling mechanisms, equation of state, electron—ion coupling mechanisms, etc. The bifurcation of the local thin Keplerian STD solutions into the optically thick and optically thin branches was a historical enigma. It was later clarified by the unified treatment of Liang and Wandel (1991) and Chen et al. (1995). The distinction between quasi-spherical flows and thin Keplerian accretion disks was blurred by the introduction of “slim-disk” models (Abramowicz et al., 1988) and “advection-dominated” disk models (Narayan and Yi, 1994, 1995; Chen et al., 1995; Abramowicz et al., 1995a), in which the accretion flow is radially quasi-spherical but with finite sub-Keplerian angular momentum. Hence, such flows are intermediate between purely spherical flow and thin Keplerian disks, which represent the limiting cases. These new developments are discussed in Section 4. Quasi-spherical flows are believed to result from accreting from the interstellar medium with little organized rotation, whereas thin Keplerian disks are formed in Roche lobe overflow in LMXBs (Novikov and Thorne, 1973). For BH binaries with a massive companion, however, the situation is less clear since the companion usually has a strong wind. Hence, accretion from the wind will compete with the Roche lobe overflow. In Cyg X-1 for example, it is believed that a thin accretion disk may not extend beyond a few thousand Schwarzschild radii (Eardley et al., 1978), far from the Roche lobe of the black hole. Whether or not such differences between HMXBs and LMXBs is the sole cause of the rather different behaviors of persistent versus transient sources remains to be seen, but it must play some role. 3.2. Steady thermal disk models The elegant STD solutions first obtained by Pringle and Rees (1972), Shakura and Sunyaev (1973) and Novikov and Thorne (1973) solve the vertically averaged disk structure using purely algebraic relations. Since they form the basis of all subsequent developments we list their key assumptions and equations in this section. The assumptions are: (a) disk scale height h;r, (b) gas elements follow Keplerian orbits with u , u ,;u , (c) azimuthal stress n "aP where P is the total z r ( r( pressure and a is assumed constant, (d) no global ordered magnetic field, (e) radiative cooling dominates over other forms of cooling (e.g. conduction, convection, waves etc.). With these assumptions the generic steady disk structure as a function of radius r is determined by six equations for six vertically averaged quantities which are usually taken as density o, pressure scale

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height h, radiative flux F, electron temperature ¹ , ion temperature ¹ and radial drift velocity v . % * 3 Other combinations (e.g. vertical column density R"2oh) are also used. These equations are then expressed in terms of the global input parameters: black-hole mass M, accretion rate M Q and viscosity a: Conservation of energy: F"3u2M Q J/8p ,

(3.1)

where u"(GM/r3)1@2 is Keplerian angular velocity and J"1!b(r /r)1@2 (r "6GM/c2 for 0 0 a Schwarzschild hole) is an ad hoc general relativistic correction factor. Here 04b41 is the Novikov and Thorne (1973) fudge parameter for the residual angular momentum of the gas at the disk inner edge r . b"1 corresponds to Keplerian angular momentum and b"0 corresponds to 0 zero angular momentum (radial infall). Hydrostatic equilibrium: P"oh2u2 ,

(3.2)

where the total pressure P"ok(¹ #Z¹ )/Am #q F/c. (Z, A) are mean atomic charge and * % 1 %4 weight respectively, k is Boltzmann constant and q is Thomson scattering depth. Note that this %4 form of writing the radiation pressure is valid for both optically thick and thin cases. Here we have ignored other possible contributions to the pressure such as magnetic fields (&B2/8p), coulomb interactions or quantum degeneracy at high densities. Conservation of angular momentum: M Q uJ"4paP .

(3.3)

In some models the stress is assumed to be proportional to the gas pressure only (the “beta” models). Electron—ion coupling: F"3l koh(¹ !¹ ) (1#(k¹ /m c2)1@2)/2m , %* * % % % 1 where the electron—ion coulomb coupling frequency is given by

(3.4)

l "2.4]1021o¹~3@2 ln K/s . (3.5) %* % Here the coulomb logarithm ln K&20 for most regimes relevant to a hot disk. Some authors (e.g. Begelman and Chiueh, 1988) have argued that collective plasma processes would couple the electrons and ions much more efficiently than pure coulomb interactions. Hence, Eq. (3.4) should be treated as the minimum energy transfer rate between ions and electrons. At the same time if ¹ truly equals ¹ then the predicted electron temperatures would be too hot to account for the % * observed exponential spectral cutoff energy (&100—200 keV). We speculate that in reality some level of collective coupling may be operative, but not enough to bring ¹ &¹ . More likely % * ¹ &10¹ . Hence, it may be prudent to parametrize the rather uncertain ¹ /¹ ratio instead of * % * % using Eq. (3.4) in future models. Radiative cooling: F"4p¹4/(3q ) for q*<1 %4

(3.6)

and F"F #F for q*;1 , (3.7) #" #4 where q*"(q q )1@2 is the true absorption depth (q is the mean free—free absorption depth). Here %4 && && F "4p¹4q A/3 is the optically thin Comptonized bremsstrahlung flux where A is the Compton #" && enhancement factor (Rybicki and Lightman, 1979; Dermer et al., 1991). F is the optically thin #4

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Comptonized soft photon flux (Katz, 1976; Shapiro et al., 1976; Sunyaev and Titarchuk, 1980; Titarchuk, 1994) where the soft photon flux can consist of both external (e.g. disk black body) photons and internal synchrotron photons. Various interpolative formulas that reduce to Eqs. (3.6) and (3.7) in the opposite limits have been proposed (e.g. Liang and Wandel, 1991; Wandel and Liang, 1991; Artemova et al., 1996). They seem to give similar results over the relevant ranges of q*. Mass conservation: 4prv ho"M Q . (3.8) 3 The above equations can be solved explicitly for the six unknowns in terms of M, M Q , a and r. Table 2 summarizes the solutions in various limits. For intermediate q* the solutions can only be obtained numerically. It turns out that global solutions (i.e. valid for all r) exist only for accretion rates below a critical value M Q that depends on a. When M Q exceeds M Q , the solution first .!9 .!9 disappears near r*"rc2/GM&32/3 (radius of maximum disk flux) and then spread in both directions as M Q is further increased (Fig. 27). This situation can be better visualized in the M Q —R plane for a given radius (Fig. 28). At low M Q the solution at a given r bifurcates cleanly into the optically thick SS branch and the optically thin SLE branch, both with positive slope. But at high M Q the SS branch is radiation pressure dominated and secularly unstable, with a negative slope. Hence, it is unavoidable that the SLE branch will merge with the SS branch at a finite MQ above .!9 which there is no steady state solution. M Q has a minimum at r*"32/3 and increases for both .!9 smaller and larger r*. This explains why the “zone of no solution” spreads from r*"32/3 as M Q is increased. The dependence of M Q on a was obtained by Luo and Liang (1994b) and Artemova .!9 et al. (1996) for different r* and reproduced in Figs. 21—29. We see that M Q can drop below M Q , .!9 %$$ the Eddington accretion rate, for a near 1. What happens to the disk solution when M Q exceeds M Q ? This is a subject of intense current .!9 research and is tied to the existence of alternative disk solutions which violate the Keplerian or negligible radial drift assumption. For example Narayan and Yi (1994), Abramowicz et al. (1995a) and Chen et al. (1995) propose “advection-dominated” solutions which radiate little of their energy output. Disk winds may also be an alternative solution at high M Q . These will be further discussed in Section 4. For most GBH candidates however, empirical estimates typically give MQ below M Q at .!9 least for a not too small (Chen et al., 1995). 3.3. Hot optically thin disk versus hot disk corona From Table 2 we see that the effective surface temperature of the optically thick SS disk is typically &keV for stellar mass BHs. Hence it is conventional to interpret the soft black-body-like component below &10 keV for both the soft-high state of the persistent sources and the XRNs during outburst as coming from a SS disk. On the other hand it is more controversial as to where the hard X-ray continuum (and its extension to the soft X-rays in the hard-low state of persistent sources) originates. Within the context of hot thermal models there were two popular early scenarios: optically thin hot inner disk versus hot corona. (Fig. 30, nonthermal models will be discussed in Section 5). First proposed by Thorne and Price (1975) and Shapiro et al. (SLE, 1976), the hot inner disk model assumes that the secularly unstable radiation-pressure-dominated inner SS disk (Lightman and Eardley, 1974) will expand vertically into an optically thin 2-temperature region with ¹ &109 K (cf. Table 2). Provided that it is fed by a copious supply of soft photons it will %

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Table 2 Summary of local solutions (from Wandel and Liang, 1991; Narayan and Yi, 1995) 1. Hot comptonized Bremsstrahlung solutions The hot two-temperature solution: ¹ "(5.8]1010 K)a~4@9¸2@9A~1@3H5@9r~1@3, e * * q "600a~1@9¸5@9A~1@3H~1@9r~5@6, %4 * *

(A1a) (A1b)

h/R"0.17a~4@9¸2@9A1@6H1@18r1@6, * *

(A1c)

o"(5.8]10~10 g cm~3)a~1@3¸1@3M~1A1@2H~1@6r~2, * 8 * ¹ "(3.2]1011 K)a~8@9¸4@9A1@3H1@9(1!P )r~2@3. i * * *

(A1d) (A1e)

The hot one-temperature solution: ¹ "¹ "(4.9]1011K)a~1¸1@2A1@2(1!P )5@4r~3@4, e i * *

(A2a)

q "390¸1@2A~1@2(1!P )~1@4r~3@4, %4 * * h/R"0.19a~1@2¸1@4A1@4(1!P )1@8r~1@4M~1@2, * * 8

(A2b)

o"(4.1]10~10 g cm~3)a1@2¸1@4M~1A~3@4r~3@2(1!P )~3@8, * 8 *

(A2d)

(A2c)

2. Comptonized soft photons solutions The hot two-temperature solution: ¹ "(3.8]108 K)g1@2(1#4H )~1@2H1@3a~1@6¸~1@6r1@4, e e * * q "3.9g1@2(1#4H )~1@2H~1@3a1@6¸1@6r~1@4, %4 e * *

and

h/R"0.24g~1@4H1@6a~7@12¸5@12(1#4H )1@4r~1@8, * * e o"3.1]10~13 g cm~3 g3@4(1#4H )~3@4H~1@2a3@4M~1¸~1@4r~9@8, e * 8 *

¹ "4.1]1013 K g~1@2H1@3(1!P )a3@2¸5@6(1#4H )1@2r~5@4, i * * * e where g~1 is the ratio of the soft photon flux to the luminosity of the hot disk.

(A3a) (A3b) (A3c) (A3d) (A3e)

The hot one-temperature solution:

and

¹"(2.5]1019 K)g~1(1#4H )a~2¸2 r~3, e * q "7.7]10~6g(1#4H )~1a¸~1r3@2, %4 e * h/R"40g~1@2a~1¸ r~1, *

(A4b)

o"(8.5]10~22 g cm~3)g3@2(1#4H )~3@2a2M~1¸~2r3@2. e 8 *

(A4d)

(A4a)

(A4c)

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Table 2 (Continued) 3. Cool solutions The cool radiation pressure-dominated a disk: ¹"(4.5]105 K)a~1@4M~1@4r~3@8, 8 q "0.02a~1¸~1r3@2, %4 * h/R"20¸ r~1, * o"(1.7]10~16 g cm~3)a~1¸~2M~1r3@2. * 8

(A5a) (A5b) (A5c) (A5d)

The cool b disk: ¹"(5.0]107 K)b~1@5¸2@5M~1@5r~9@10, * 8 q "3.9]106b~4@5¸3@5M1@5r~9@10, %4 * 8 h/R"1.6]10~5¸ r~1, * o"(2.5]10~8 g cm~3)b~4@5¸~2@5M~4@5r~3@5. * 8

(A6a) (A6b) (A6c) (A6d)

The cool black-body gas pressure-dominated disk: ¹"(8.3]107 K)a~1@5¸3@10M~1@5r~3@4, * 8 q "1.9]106a~4@5¸7@10M1@5r~3@4, %4 * 8 h/R"3.0]10~3a~1@10¸3@20M~1@10r1@8, * 8 o"(1.2]10~4 g cm~3)a~7@10¸11@20M~7@10r~15@8. * 8

(A7a) (A7b) (A7c) (A7d)

4. Advection-dominated accretion (see Narayan and Yi (1995) for definitions of constants) v"!2.12]1010ac r~1@2 cm s~1, 1 X"7.19]104c m~1r~3@2 s~1, 2 c2"4.50]1020c r~1 cm s~2, s 3 o"3.79]10~5a~1c~1c~1@2m~1mR r~3@2 g cm~3, 1 3 p"1.71]1016a~1c~1c1@2m~1mR r~5@2 g cm~1 s~2, 1 3 B"6.55]108a~1@2(1!b)1@2c~1@2c1@4m~1@2mR 1@2r~5@4G, 1 3 q`"1.84]1021e@c1@2m~2mR r4 erg cm~3 s~1, 3 n "o/k m "2.00]1019a~1c~1c~1@2m~1mR r~3@2 cm~3, e e u 1 3 q "2n p h"12.4a~1c~1mR r~1@2, %4 e T 1 ¹ #1.08¹ "6.66]1012bc r~1 K i e 3 M Q "mR M Q , M"mM E$$ _ Notations: P*"P /P ; ¸*"¸J/(0.057¸ ); r"r* of text; M "M/108M ; H*"(1!P*)(1#H1@2)] 3!$*!5*0/ 505!%$$ 8 _ % (¹ !¹ )/(¹ #¹ ); A "Compton enhancement factor of bremsstrahlung (cf. Rybicki and Lightman, 1979); g" i % i % Compton enhancement factor of soft photons; H "k¹ /m c2 (original typos corrected). % % %

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Fig. 27. (a) Vertical Thomson depth radial profiles of thin Keplerian accretion disks for samples values of accretion rate and viscosity a, showing that no global steady disk solution exists above a critical accretion rate (curve 4) for a given a; (b) critical accretion rate mR as a function of a (physical regimes are restricted to a(1, from Artemova et al., 1996). .!9

naturally produce an inverse Compton spectrum similar to the hard-low state spectrum of most GBHs (cf. Section 2). Origin of the postulated copious soft photon source has never been settled but external black-body photons from the outer disk and internal synchrotron photons from an equipartition magnetic field (Lightman and Eardley, 1975) are viable candidates. In this model the global disk spectral output is determined by the thinning radius r which separates the optically 53 thick and thin zones, magnetic field B, M Q and a. An alternative scenario, first proposed by Liang and Price (1977) and Bisnovatyi-Kogan and Blinnikov (1977), assumes that the optically thick disk dissipates part of its energy in a rarefied

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Fig. 28. Accretion rate as functions of surface column density of thin Keplerian disk solutions at r"12GM/c2 and a"1 without soft photons (solid triangles) and with soft photons (open triangles at ¹ "5]106 K, stars at ¹ "106 K). "" "" Dashed line indicate potential effects of including electron—positron pairs (from Luo and Liang, 1994b). The branch with a negative slope is secularly unstable.

optically thin corona. Besides helping to stabilize the disk itself the corona will naturally produce an unsaturated inverse Compton spectrum since the underlying SS disk will be a copious source of soft photons. Because of the feedback between the disk and the corona it is also easy to achieve a Komponeets (1957) parameter y&1 (Sunyaev and Titarchuk, 1980; Titarchuk, 1994). However, it is more difficult to make the soft and hard X-ray components to vary independently in this case since most of the soft photons must pass through the Comptonizing region. Detailed thermal corona models have been investigated by many authors over the years (Abramowicz et al., 1995b; Galeev et al., 1979; Haardt and Maraschi, 1991, 1993; Melia, 1991; Melia and Misra, 1993; Sakimoto and Coroniti, 1989; Stern et al., 1995, 1996; Svensson and Zdziarski, 1994). One consequence of the disk corona model is the presence of a “Compton reflection hump” at tens of keV in the continuum spectrum due to the reprocessing of downward-scattered hard X-rays by the cold disk (Fig. 31). While this hump is often seen in Seyfert galaxies, there is little evidence of it in GBH spectra (cf. Section 2). Some authors have come up with partial disk corona models (e.g. Stern et al., 1996) that avoid a significant reflection hump in the continuum and provide more flexibility to the independent variation of soft and hard X-rays. In addition to M Q and a, the disk corona model is usually specified by the fraction f of energy output dissipated in the corona versus (1!f ) in the SS disk. Table 3 compares the pros and cons of the hot inner disk versus the hot disk corona model in the context of current observational data.

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Fig. 29. Effects of varying a on the MQ —R plots of the local disk solutions (from Luo and Liang, 1994b).

Fig. 30. Artist conception of the (a) hot inner disk versus (b) hot corona model of the hard x-ray emissions of Cyg X-1 (from Liang and Nolan, 1984).

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Fig. 31. Left: spectrum of Seyfert galaxy IC4329A showing the “hump” at tens of keV interpreted as a Compton reflection component due to reprocessing by cold gas (from Magdziarz and Zdziarski, 1995); right: disk corona model output spectra showing the Compton reflection component (from Stern et al., 1996).

Table 3 Compatibility of the hot corona model versus the hot inner disk model with observations

Soft X-rays uniquely fit by disk blackbody Independent variation of soft and hard X-rays Komponeets parameter y&1 natural Hard X-ray time lag proportional to Fourier period Weak reflection hump and Fe fluorescence line Thermal stability Predicts LFQPOs

Hot corona

Hot inner disk

Yes No! Yes No No Yes No

No Yes No Yes Yes No Yes

! unless corona has only partial coverage.

If the soft photon source in the innermost region of either of the above models is ever quenched (e.g. due to the decrease of the local magnetic field to (10~2 of equipartition field, Dermer and Liang, 1989; the moving out of r in the hot inner disk model, or disruption of the underlying SS 53 disk in the corona model), the hot inverse Compton region will superheat to an electron temperature of ¹ &m c2 (¹ &0.1m c2) when pairs are produced copiously and swell up into a thick % % * 1 torus. Such an ultrahot torus will emit a Wien-like “MeV bump” via Comptonized bremsstrahlung and pair annihilation (Zdziarski, 1984; Liang and Dermer, 1988). Episodic “MeV bumps” may have been detected in Cyg X-1 (see Ling et al., 1987 and references to earlier claims) but remain to be confirmed by other observations (e.g. Harris et al., 1994).

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3.4. Hybrid disk models To account for the multi-component generic GBH spectra and their complex temporal variation patterns, many authors have combined the above models into a variety of hybrid disk models with each model occupying a different zone of the disk. For example, Wandel and Liang (1991), Melia and Misra (1996), Luo and Liang (1994a), Skibo and Dermer (1995), and Ling et al. (1997) have constructed multi-zone disk models (Fig. 32) to account for the 1 keV to 'MeV continuum spectra of Cyg X-1, GX 339-4 and GROJ0422-42. Such global models require the iterative computation of the spectral output, electron temperature and Thomson depth of each zone. The radially integrated spectrum, when fitted to the observed data, can be used to extract the mean viscosity parameter a and other disk parameters. Luo and Liang (1994a) found in most cases that the best fit model requires a&1 (Fig. 33) instead of the a;1 found in dwarf Novae systems. The SS disk plus homogeneous corona model have also been used extensively to fit the 1—200 keV spectral data of various GBHs (e.g. Haardt and Maraschi, 1991, 1993; Melia and Misra, 1993). Recently, Stern et al. (1996) have modeled self-consistently the spectral output of an inhomogeneous disk corona with “pill boxes” or hemispheres of hot plasma sitting on top of an SS disk and found that one can suppress the reflection component successfully. Earlier, to explain the “MeV bump” of Cyg X-1 observed by HEAO3 in Fall 1979 (Ling et al., 1987), Liang and Dermer (1988) invoked a multi-zone model with an ultrahot pair-dominated cloud surrounding the black hole and protected from the external soft disk photons by a hot transitional region. Sample output from such a hybrid model is reproduced in Fig. 34 and compared to the HEAO3 1979 data.

Fig. 32. Hybrid accretion disk models of AGNs with 2 radial zones (left) and 3 radial zones (right). The Monte Carlo composite spectra are computed self-consistently allowing for the cooling of the inner hot zones by soft photons from the outer cool black-body disk. CS denotes Comptonized soft photons and CB denotes Comptonized bremsstrahlung (from Wandel and Liang, 1991).

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Fig. 33. Modeling of OSSE data of Cyg X-1 and GX339-4 with 3-zone hybrid disk models leads to an a value &1. The hot Comptonized bremsstrahlung innermost zone is needed to fit the hard tail above 200 keV (from Luo and Liang, 1994a).

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Fig. 34. Modeling of the HEAO3 “MeV” bump of Cyg X-1 in 1979 with a pair-dominated ultrahot inner disk leads to a first-principles pair temperature of &400 keV, Thomson depth of 2 and pair density of &1017/cm3 (for details see Liang and Dermer, 1988).

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However, self-consistent hybrid models with both a cold SS disk plus hot corona and an ultrahot inner torus remain to be constructed. Most such multi-zone models, while having enough flexibility to fit the 1—1000 keV spectra of many GBH sources, still have great difficulty when it comes to explaining spectral variability. It is unclear if any hybrid model can completely accommodate all of the observed spectral state transitions of any GBH source.

4. Outstanding issues and recent advances of the steady thermal disk models 4.1. Origin of viscosity and variable-a disks The original introduction of the a-viscosity model by SS (1973) was motivated by dimensional considerations and our ignorance of the detailed physics in such complex environments. The arguments leading to a constant ratio between azimuthal stress and total pressure were based mainly on dimensional analysis: l(viscosity coefficient)&ahc since the sound speed c and pressure 4 4 scale height h are the only natural dimensional parameters in a local formulation of the viscosity coefficient. Amazingly such a simplistic phenomenological model remains the most popular and useful one more than a quarter of century later. This is mainly due to the difficulty of formulating a first-principles theory of viscous stress of accretion disks. Currently there are two main approaches towards a fundamental theory of the origin of turbulent disk viscosity: convectively driven turbulence (Lin and Papaloizou, 1980) and magnetic shearing instability (Balbus and Hawley, 1991). In addition global spiral shocks (Shu, 1976) have been proposed as a dissipation and angular momentum transfer mechanism. The pros and cons of the three mechanisms have been reviewed by Vishniac and Diamond (1993). Both convectively driven turbulence and magnetic shearing instability predict that the effective local a-viscosity parameter scales as some power law of h/r; where h is the disk half thickness and r is the distance to the black hole (e.g. Vishniac and Diamond, 1992, 1993). For disks that are sufficiently optically thick, the vertical diffusion of radiation could be so slow that the disk becomes convectively unstable, in which case vertical convection will arise (Liang, 1977). When such convective flow is coupled with the azimuthal differential rotation it has been hypothesized that fully developed turbulence may persist, leading to macroscopic dissipation and transport of angular momentum (e.g. Lin and Papaloizou, 1980; Papaloizou and Pringle, 1985). However, it has been questioned by many authors whether convectively driven turbulence, even if it can persist, transport angular momentum in the right direction (e.g. Ruden et al., 1988). Other authors however are more optimistic, especially when they consider the anisotropy of the turbulence (i.e. the power spectrum has different indices for the vertical and horizontal wave vector distributions). Some convective models predict an effective a&(h/r)1@2 (e.g. Merielles et al., 1996). In any case convection is unimportant in optically thin disks since the vertical temperature gradient will be small. Turbulence driven by the growth of the magnetic shearing instability (Balbus and Hawley, 1991, 1992; Hawley and Balbus, 1991a,b) in which the turbulent magnetic field grows to saturation, leading to efficient angular momentum transport, is now generally believed to be more promising (Fig. 35). This results in an effective a&(h/r)4@3 (Vishniac and Diamond, 1993). This scaling law has also recently become popular with workers studying the thermal limit cycle instability

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Fig. 35. Numerical simulation of the magnetic shearing instability of a disk showing its time development: (a) initial poloidal magnetic field lines: (b) poloidal field lines after 3.3 orbits; (c) toroidal field intensity after 3.3 orbits; (d) angular momentum after 3.3 orbits (from Balbus and Hawley, 1991).

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(Cannizzo, 1993). However, the magnetic shearing instability requires the existence of a dynamo to sustain the seed field, and the origin of this dynamo is unsettled. For example, Vishniac and Diamond (1993) proposed internal waves as a possible driving mechanism of this dynamo. Much earlier, Eardley and Lightman (1975) and subsequently Coroniti (1981), Pudritz (1981) and many others studied self-consistent models of magnetic shear viscosity, in which the continuous amplification of the field by differential rotation is balance by semi-continuous magnetic reconnection. This leads to transport of angular momentum and nonlocal dissipation of energy which cannot be represented by a local a-model. Alternatively, Heyvearts (1991) advocated that the small-scale turbulent fields in a disk, instead of continuously reconnecting, may in fact inversely cascade into large-scale ordered magnetic arcades connecting macroscopic regions of the disk, as it is often observed on the sun. Such macroscopic ordered fields with size scale &r will provide an effective stress that is highly nonlocal. As a result the disk becomes highly irregular and nonKeplerian. For example in one model belts of rigid rotation alternate with rings of sharp jumps in angular velocity (Fig. 36). However, prospects of large-scale emergent buoyant magnetic flux loops high above the disk have been dampened by Vishniac and Diamond (1993) since the Parker instability is at least partially suppressed by the magnetic shearing instability. Fundamental physics issues aside, the nature and origin of the disk viscosity have important observational consequences. Within the context of the local a-viscosity models, how a scales with h/r and M Q has obvious consequences for the observable spectral output and time variability. For example, the TLC model for XRN outbursts works best when a scales as (h/r)4@3 (Cannizzo, 1993, Cannizzo et al., 1995; Mineshige and Kusunose, 1993). Luo et al. (1994) suggested that a new type of TLC model for hard X-ray disks may be constructed if a varies with M Q in a special way. More importantly, the structure and spectral output of both optically thin and optically thick disk models depend on the scaling of a with h/r, especially when advection and transonic drift effects are included (cf. Section 4.5). Hence, it is important for future modelers to systematically compare GBH data with predictions of variable-a models. Magnetic disks models with nonlocal stresses are more difficult to construct and firm predictions of their spectral outputs

Fig. 36. Disk structure of the Heyvearts model with large scale magnetic arcades transferring angular momentum nonlocally. Note the alternating bands of rigid rotation and sharp jumps in the angular velocity (from Heyvearts, 1991).

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and variability patterns are even more remote. However, recent detection of GBH radio activities correlated with X-ray flares (Harmon et al., 1995) clearly suggests that global magnetic fields must play important roles (cf. Section 4.7) in some stages of the disk evolution and their contributions to the disk viscosity must be included in the models before the multi-wavelength output can be reliably computed. 4.2. Comptonization and the origin of the soft photons The hard-low state spectrum of GBHs can best be interpreted as the unsaturated Comptonization of soft photons by hot thermal electrons (Katz, 1976; SLE 1976). This inverse Compton disk model of GBHs hard X-rays requires a copious supply of soft photons to cool the hot plasma in the quiescent hard-low state. The origin of this soft photon source has never been completely settled. It may be a combination of at least three different sources: internal synchrotron photons from an equipartition magnetic field (Eardley and Lightman, 1976); external black-body photons from the outer part of the SS disk (SLE, 1976) or black-body photons from the underlying optically thick disk in a corona model (e.g. Liang and Price, 1977). During the soft-high state of persistent sources and the outburst of XRNs there is general concensus that the black-body-like soft X-ray spectral component comes from a SS disk. In this case the hot Comptonizing region must be intercepting some fraction of that blackbody flux, so it is likely that at least some of the Comptonized soft photons originate from outside the hot region. The precise fraction of the soft flux intercepted by the hot region depends on the geometry and relative location and can only be probed by detailed rapid spectral evolution studies. But the relative independence of the inverse Compton spectral index (Sunyaev and Titarchuk, 1980; Pozdnyakov et al., 1983) and temperature from the soft X-ray intensity in most cases (cf. Liang and Nolan, 1984) would seem to suggest that the external fraction of Comptonized soft photons intercepted in the hard-low state is small, and a disk corona with almost 100% coverage of the optically thick disk seems unlikely. Therefore, during the hard-low state of the persistent source and epochs of the XRN when the soft flux is low, it is plausible that a significant fraction of the cooling soft flux is internal. Due to the turbulent nature of the accretion disk and the steep vertical density gradient of the accretion disk it seems natural for the disk to possess a hot rarefied corona. However, the key question is whether this corona radiates only a tiny fraction of the total energy, as the solar corona does, or whether it radiates the bulk of the total energy output, as assumed in many disk corona models (e.g. Liang and Price, 1977; Galeev et al., 1978; Haardt and Maraschi, 1991, 1993; Stern et al., 1996; Melia and Misra, 1993). If the hot corona indeed covers most of the optically thick disk then all soft photons would be subject to reprocessing by the corona and one would expect a well-defined relation between the soft and hard flux and spectral index and temperature. This is not observed in most cases, especially during the evolution of the XRN outbursts. Moreover, downward-scattered hard X-rays from the hot corona would produce a strong Fe-fluorescent line broadened by Compton scattering and a strong “Compton-reflection hump” (Zdziarski et al., 1993). These have not been observed in GBH candidates (Ebisawa, 1991; Ebisawa et al., 1994), though there are evidences for them in Seyferts galaxies (Magdziarz and Zdziarski, 1995). Hence, we conclude that if there is a disk corona in GBHs, it can at most partially cover the optically thick disk (Stern et al., 1996). Such a partial corona could also give us enough flexibility to have the independent variations of the soft and hard components in both persistent sources as well as XRNs.

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If the disk is globally radially-partitioned into an optically thin hot inner disk and optically thick cool outer disk, and the soft photons for the hot Comptonizing region comes purely from the external disk (SLE, 1976), then one expects a deterministic relation between the two spectral components which is mainly a function of the thinning radius and the overall accretion rate (Thorne and Price, 1975). In this case the intensities of the two spectral components will uniquely constrain these two parameters. On the other hand, internal synchrotron soft photons from the hot region may contribute significantly during hard-low states (Eardley and Lightman, 1976) while external soft photons dominate only during soft-high states. This requires that the magnetic field be close to equipartition during hard-low states and is consistent with the result that the viscosity a estimated from spectral fitting is often close to unity during hard-low states (Luo and Liang, 1994a). Another potential probe of the relation between the soft and hard photon sources is rapid time variability (Nowak, 1994). Current results suggest that the RMS fluctuation level is higher for the hard X-rays than the soft X-rays, especially during soft high states. This implies that the soft X-rays are from the more stable optically thick disk and the hard X-rays are emitted further in, possibly in a thermally unstable region. The correlation of the hard time lag with Fourier period (Miyamoto et al., 1993) is also more consistent with the soft photons coming from further out than the hard photons. Both results slightly favor the radially zoned disk picture over the corona picture in which both the hard and soft photons originate from the same radial range. In summary, the often independent variations of the soft black-body-like and hard power-lawlike components, and the lack of a strong Compton reflection hump or Fe K fluorescence line in a the spectra of most GBHs, favor either a corona model with only partial coverage of the underlying SS disk, or a multi-radial-zone thin disk model in which at least part of the Comptonized soft photons originate internally within the hot region during hard-low states. Reality could well be a combination of both scenarios. Detailed 0.1—1000 keV spectral evolution and rapid variability studies during XRN outbursts and state transitions of the persistent sources will help to tightly constrain the model parameters. But there is now general concensus that the distinct spectral states of most GBHs are likely related to the movement of the optical thinning radius r , independent of 53 the origin of such movements. 4.3. Pair processes and relativistic effects At electron temperatures 5100 keV, relativistic processes leading to the production and annihilation of electron—positron pairs become important. In steady state, the pair creation and annihilation rate must balance otherwise it would lead to run away pair production. Such pair-balanced thermal plasmas have unique equilibrium properties first explored by BisnovatyiKogan et al. (1971), who pointed out that if charged-particle pair production processes dominate there exists a fundamental upper limit to the lepton temperature (&20mc2) above which there can be no pair-balance. In the opposite regime where gamma—gamma pair production dominates, Liang (1979) first explored hot thermal disks in Wien equilibrium and found multi-valued solutions to the disk structure. Since then pair-balanced plasmas have been extensively studied by many authors (e.g. Lightman, 1982; Lightman and Zdzianski, 1987; Svensson, 1982, 1984; Zdziarski, 1984; Takahara and Kusunose, 1985; White and Lightman, 1989). More recently pair-balanced hot disks have been studied systematically and their solution topology is now basically understood

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(e.g. Kusunose, 1996; Kusunose and Takahara, 1990; Kusunose and Mineshige, 1992, 1995; Mineshige, 1993; Bjornsson and Svensson, 1991, 1992; Liang, 1991). In brief, the general pairbalanced solutions without soft photons may contain one or three branches, at least one of them is hot and pair-dominated, another is hot but pair-deficient, and the third one, which may be either stable or unstable, is usually cool and pair-deficient. The pair-dominated branch may exist only locally so its relevance to real disks is questionable. Topologically, the solution at fixed r in the M Q —R plane forms an S-shape kink corresponding to its multiple valued nature (Fig. 37). But since the domain of this multiple-valueness is quite restricted it affects little the global property of the solution. However, if the hot pair-dominated branch does exist and is stable it may be responsible for the MeV bump of Cyg X-1 (Fig. 34, Ling et al., 1987), the radio plasmoid bipolar outflow of the jet sources (Mirabel et al., 1991, 1994; Hjellming and Rupen, 1995) and the alleged annihilation line-like features (e.g. Paul et al., 1991). In general the most promising signature to search for associated with thermal pair production events should be a broad “bump” around 0.5—2 MeV (Liang and Dermer, 1988). Details of this feature can in principle provide valuable diagnostic of the black-hole mass and angular momentum (Liang, 1990) as well as the accretion plasma conditions near the horizon. A narrow 511 keV line (FWHM &keVs) associated with pairs produced by GBHs will be more difficult to detect, since such a line can only come from pairs already cooled in the ISM, after diffusing far away from their production site and long after the initial production event (time delay up to &years, Ramaty et al., 1992). For a Schwarzschild hole, since most of the observable radiation originates outside 6GM/c2, general relativistic effects such as gravitational redshift and bending of light rays on the observable continuum radiation tend to be washed out and are usually difficult to detect in real data. One exception is the effect on line profiles (e.g. Fe K ) which would be broadened and asymmetrized a due to the intensity enhancement of the blue wing relative to the red wing if such lines are emitted

Fig. 37. Local disk solutions in the MQ —R (left) and ¹—R (right) planes including the effects of pairs. Note the regime of multi-valued pair-balanced solutions giving rise to three distinct accretion rates at a given column density (from Mineshige and Kusunose, 1993).

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Fig. 38. Left: observed ASCA line profile of the Fe 6.4 keV line (from Tanaka et al., 1995); right: theoretical line profiles at different view angles (k "cosine of angle between disk normal and line of sight, from Matt et al., 1996). 1

close to the inner disk edge. The resultant blue-shifted asymmetric line profile can be used as diagnostics of the disk orientation and inner edge (Fig. 38, Payne and Eardley, 1977; Matt et al., 1996). Such Fe K line profile may have been detected for several Seyfert galaxies (Fig. 38, Tanaka et al., a 1995) but not yet for GBH candidates. Of course for the K line to form the Fe must not be fully a ionized and the disk must remain relatively cool all the way to the inner disk edge. This is in conflict with the hot inner disk model and, if confirmed, favors the corona model for Seyfert galaxies. More significantly, the Seyfert Fe-line profiles (Iwasawa et al., 1996; Laor, 1991; Tanaka et al., 1995) suggest that in some cases the blueshifting required may be larger than allowed by a Schwarzschild hole, in which case a rotating Kerr hole must be invoked (Fig. 38, Matt et al., 1996). For Kerr holes with large angular momentum (a&M) the Keplerian disk penetrates much more deeply into the BH gravitational potential well and general relativistic effects become important for the observable radiation (Cunningham, 1975). The structure of disks around Kerr holes had been explored by Fishbone and Moncrief (1976), Moncrief (1980). Dragging of inertial frames near a Kerr hole can in principle be diagnosed by studying the polarization of scattered radiation in the inner disk (Stark and Connors, 1977; Connors et al., 1980; Long et al., 1980). 4.4. Thick and slim disks Two key assumptions of the standard thermal disk model are its physical thinness and small radial drift velocity. When the disk becomes very hot the thinness assumption breaks down. Also

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when the accretion rate exceeds the Eddington luminosity a radiation-pressure-dominated optically thick disk becomes physically very thick. Abramowicz et al. (1980) and Paczynski and Abramowicz (1982) first explored physically thick disks and the related issue of non-Keplerian angular momentum distribution. Self-similar thick disk solutions have also been obtained by Begelman and Meier (1982) and Liang (1988). While in principle radiation dominated superEddington thick disks with non-Keplerian angular momenta can be self-consistently constructed, their global stability and relevance to GBHs are doubtful. Later, Abramowicz et al. (1988) invented “slim disks” which, though not physically very thick, have sub-Keplerian angular momenta so that radial advection terms become important (Fig. 39). Such models are stable and optically thick. Together with the standard SS branches they form an S-shape curve in the MQ —R plane for a lower than a certain critical value (Chen et al., 1995). Hence in principle an SS disk with a rising accretion rate can make a discontinuous jump from the secularly unstable radiation pressure-dominated branch to the stable slim-disk branch with superEddington accretion rates (Fig. 39). However, the true relevance of such solution to GBHs remains doubtful since we have not seen any evidence of superEddington accretion rates in GBH candidates, even during soft-high states (cf. Section 1). Part of the original motivation for considering thick accretion disks was the existence of axial funnels around the hole provided by such disks (Rees et al., 1982), which can be used to accelerate and collimate jets via the superEddington radiation flux (Paczynski and Abramowicz, 1982). Subsequent works (e.g. Phinney, 1987) show that radiation acceleration cannot achieve high bulk Lorentz factors (C'10) such as those found in many AGN jets so interests in such models faded away. However, recent discoveries of GBH jets with only mild Lorentz factors (C&2) have revived interests in the thick superhot ion tori (Rees, 1984). Gamma-rays emitted by a superhot torus can be readily reconverted into outflowing pairs if the compactness is large (say l"¸ /3.7]1028R'100, c Li and Liang, 1996), and optically thin radiation pressure from the disk can further accelerate them to the observed terminal Lorentz factors (Liang and Li, 1995). Multi-wavelength (radio—gamma rays) coverage of relativistic ejection events from GBHs will be critical in establishing the connection between the accretion inflows and relativistic outflows. 4.5. Advection-dominated solutions and transonic disks The standard SS disk model, together with its optically thin cousins (e.g. SLE models) assume that the disk was quasi-Keplerian and ignore all radial advection or pressure gradient terms. This assumption is valid so long as the disk is physically thin and v ;v . However, in the high-M Q limit 3 ( of both the SLE and SS models this assumption often breaks down because the high-M Q requires a high R ) v , while the surface density R is limited (cf. Figs. 28 and 29, near where the two branches 3 merge) so that v must be large. The slim disk solution of Abramowicz et al. (1988) takes into 3 account the radial advection of energy and entropy and leads to a new topology of the optically thick solution: the S-shaped M Q —R curve which no longer crosses over to the optically thin branch. But then what happens to the SLE solution at high MQ ? Recently, Abramowicz et al. (1995a), Honma et al. (1991) and Narayan and Yi (1994, 1995) generalized the SLE optically thin solutions to include the energy advection terms and found that, indeed there is another branch to the optically thin solution which is advection-dominated (AD). Using a parameterized self-similar form for the radial drift velocity they were able to obtain solutions which demonstrate the bifurcation of the optically thin solutions into two branches: the SLE branch and the AD branch, which is even

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hotter and optically thinner (Chen et al., 1995). These advection-dominated solutions radiate away a very small fraction of their energy release (most energy goes down the hole) and are therefore thermally metastable. The topology in the M Q —R plane of all the local solutions as functions of viscosity a is illustrated in Fig. 40. At high a(&1) the inverted-U topology of the Keplerian branch discovered by Liang and Wandel (1991) was recovered, and a separate branch of optically thin AD solution runs parallel to the SLE branch but never intersecting it. As a decreases the AD branch approaches the Keplerian branch and reconnects with it after a drops below a critical value a , #3

Fig. 39. Structure of slim disk solutions. Top: height versus radius for sample accretion rates; bottom: local disk solutions in the M Q —R plane exhibiting the S-shape topology. The upper stable branch of the S requires superEddington accretion rates which have not been observed in GBHs (from Abramowicz et al., 1988).

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Fig. 40. Topology of local disk solutions including advection in the MQ —R plane at different radii (R/R "5 and 30 where ' R "2GM/c2) and using different cooling models. (a, b versus c, details see Chen et al., 1995). Note that at high a the ' advection dominated branch (upper left) is detached from Keplerian branch (inverted U). But at low a the advection dominated branch “reconnects” with the Keplerian branch to form the S-shaped optically thick solution to the right and inverted-V shaped optically thin solution to the left. In this regime the optically thin solution exists only for MQ below certain accretion rates. The critical value of a at which the “reconnection” occurs is a function of the radius but is typically a fraction of unity (from Chen et al., 1995).

whose exact value depends on the radius r. For a below a the lower half of the AD branch merges #3 with the SLE branch forming an inverted-V, while the upper half of the AD branch merges with the SS branch to form the S-shaped branch (cf. Fig. 40). Hence at low a, the optically thin solution ceases to exist above a certain critical accretion rate for a given r and a. For stellar mass black holes this upper M Q cutoff occurs low enough that the optically thin solution exists only within a finite radius r , unless a is close to 1. Narayan and coworkers proposed the AD solution for low 53 luminosity AGNs and our Galactic center where the radiation output is low (Narayan et al., 1995a) and XRNs in quiescence (Narayan et al., 1995b, 1996, 1997), and that it can be used to test for the presence of event horizons, since neutron stars must convert the advected accretional energy into thermal radiation at their hard surface while black holes do not. Narayan (1996) and Esin et al. (1997) have further proposed that the AD solution can be used to explain the spectral state transitions of high luminosity sources such as Nova Muscae and Cyg X-1 via the movement of r with accretion rate. At low M Q , r is large (up to thousands of Schwarzschild radii) so that most of 53 53 the disk is AD, hot and optically thin, giving rise to the low-hard state. As M Q increases, r moves 53 inward, so that the optically thick SS disk output increases relative to the AD region, eventually leading to the soft-high state. However, this picture cannot explain the very high state. The above AD solutions, based on parametrized self-similar solutions, are globally incomplete because they have ignored the radial drift and pressure gradient terms in the Euler equation and the transonic boundary condition for v near the horizon (e.g. Abramowicz and Zurek, 1981; Chen 3 and Taam, 1993; Liang and Thompson, 1980; Muchotrzeb and Paczynski, 1982; Thompson, 1981; Chakrabarti, 1990; Yang et al., 1994; cf. Fig. 41). Recently, several groups have begun to obtain global transonic solutions with restrictive cooling mechanisms (Chen et al., 1996; Narayan et al., 1996) or idealized angular momentum distributions (Luo and Liang, 1998, Fig. 42). This involves replacing the Keplerian angular momentum assumption in Eqs. (3.1), (3.2), (3.3), and (3.8) with the

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Fig. 41. Examples of critical point structures in relativistic transonic thin disk solutions. The number of critical points is related to the number of roots of the numerator function N(r) (see Liang and Thompson (1980) for definition of variables).

radial Euler equation: v dv /dr"!dP/odr#j2/r3!j2 /r3, (4.1) 3 3 K where j is specific angular momentum and j is its Keplerian value, and adding advective terms to K the energy equation (3.1). Realistic, complete transonic models remain to be constructed and their spectral output computed to see if such global solutions are indeed relevant to the observed spectral and temporal properties of GBHs. One interesting result emerges independent of the detailed radiative mechanisms: inclusion of the transonic radial drift and radial pressure gradient effects makes the global solution sensitive to the angular momentum and viscosity of the flow. This new parameter may play a key role in shaping the global optical thickness profile of the disk and be

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Fig. 42. Examples of transonic solution velocity profiles: left: advection-dominated solutions of Narayan et al. (1997) for different M Q . Dashed curves are the sound velocity profiles; right: cooling dominated solutions of Luo and Liang, 1997 for different angular momenta in units of Keplerian values. Both classes of solutions approach the self-similar form at large radii.

relevant to the spectral state transitions of GBHs. Also the issue of existence or nonexistence of shocks in the transition from subsonic to supersonic drift remains to be settled (e.g. Chakrabarti, 1996b). 4.6. Disk winds and outflows There are two regimes of a disk when we expect winds and outflows to become important. One regime is when the (optically thick) disk radiative flux is close to or exceeds the Eddington limit. Then the radiation pressure can drive out a wind (Eggum et al., 1987, 1988; Liang, 1988; Fig. 43, see also Ryu et al., 1995 for polytropic flows). For proton-dominated plasma this corresponds to ¸ "1.25]1038M erg/s. But for a pair-dominated plasma this could be as much as a thousand %$$ _ times smaller, depending on the proton loading. Hence the ordinary GBH luminosity, which is only a few percent of the Eddington limit, can still drive out a pair wind. The other regime is when the disk gets superhot (e.g. during thermal instability or when the inverse Compton disk is starved of soft photons). In this case a wind could arise either due to thermal evaporation (Maxwellian tail

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Fig. 43. Comparison of the 2-dimensional numerical simulations of superEddington axisymmetric accretion flows of (Eggum et al., 1988), (left) with the self-similar solution of Liang (1988), (right). Radiation pressure turns low-latitude accretion inflow into supersonic outflow near the polar axis. Note the presence of convective eddies in mid-latitudes in the numerical simulations which roughly correspond to the density inversion layer of the self-similar solution. Such optically thick radiation driven bipolar outflows may be relevant to the jets of SS433.

particles with vertical velocities exceeding escape velocity, Takahara et al., 1989; Kusunose, 1991; Moscoso and Wheeler, 1994) or due to copious gamma—gamma pair production. In the former case a hot proton-dominated wind could arise. In the latter case a pair wind will arise. In the above discussion the wind is assumed to be purely hydrodynamical without magnetic fields. However, in many cases, especially those with collimated bipolar outflows (jets, cf. Section 1), it is highly likely that global, ordered magnetic fields play an important or even dominant role. MHD winds driven by the disk rotation along open field lines could be a source of the observed jets (Fig. 44, Begelman et al., 1984, 1987; Rees, 1982, 1984; Blandford and Payne, 1982). Disk winds and jets carry away mass, energy and angular momentum. If the wind loss becomes significant, the structure and radiative output of the underlying disk may become altered. In that case we need to modify the disk structure equations in Section 3 to accomodate the effects of wind loss. Whether the wind is proton or pair-dominated, it will initially leave the disk with roughly thermal velocity which is at most mildly relativistic (C&1). However, the radio jets we observe have significantly higher bulk Lorentz factors (C&2.5 for GBHs detected so far and C up to &10 for AGNs). Hence, the wind must be further accelerated by an external agent after it leaves the disk.

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Fig. 44. Examples of field line geometry of a magnetically driven MHD wind from a disk: (a): self-similar solution; (b): numerical simulations (from Li et al., 1991).

Currently there are two favorite ideas: electrodynamic acceleration such as rotationally driven magnetic fields (Fig. 44, Blandford and Payne, 1982; Begelman et al., 1984; Begelman and Li, 1994; Chiueh et al., 1991; Konigl, 1989; Li et al., 1992; Levinson and Blandford, 1996; Lovelace and Romanova, 1996) or optically thin radiation pressure (Figs. 45 and 46; Phinney, 1987; Liang and Li, 1995). GBH jets with mild Lorentz factors can be accelerated by either mechanism, whereas large-C AGN jets can only be accelerated electrodynamically (Phinney, 1987). 4.7. Role of macroscopic magnetic fields In the conventional thin disk models (e.g. SS or SLE) the disk is assumed to have no macroscopic ordered magnetic fields. Whatever field there is, is assumed to be turbulent with maximum cell sizes comparable to the disk-scale height. Such small-scale turbulent fields play double roles: (a) they contribute to the local effective viscosity: a &(B/B )2 via semi-continuous reconnection (Eardley B %2 and Lightman, 1975) or saturation of the magnetic shearing instability (Hawley and Balbus, 1991b)

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Fig. 45. Schematic picture of an accretion disk with a superhot inner torus emitting gamma rays (a) and a hot disk emitting hard X-rays (b). Pairs produced by the collision of X- and gamma-rays are preferentially driven out along the axes by the optically thin radiation pressure of the disk photons, reaching asymptotic Lorentz factors of a few (from Li and Liang, 1996).

Fig. 46. Terminal Lorentz factor of the jet bulk motion in the model of Liang and Li (1995) as a function of the disk luminosity in units of Eddington luminosity. The curves a...e correspond to increasing proton loading (a"pure pairs, e"no pair); inset: bulk Lorentz factors as functions of proton loading for sample luminosities (from Li and Liang, 1996).

where the equipartition field B &(8pP)1@2 is typically &106—107 G for the inner part of a GBH %2 accretion disk; (b) they contribute to the production of synchrotron soft photons which help to cool the hot inner disk. In fact, if the local B-field is close to equipartition then Eardley and Lightman (1976) showed that the (self-absorbed) synchrotron soft photon source may be adequate to cool the inverse Compton disk even in the absence of external soft photons. Dermer and Liang (1988) further showed that synchrotron soft photons will dominate over bremsstrahlung photons whenever the local B'10~2B . Hence, in order for a superhot gamma-ray emitting thick torus to arise %2 (say to explain the MeV bump of Cyg X-1 observed by HEAO3, Ling et al., 1987) we need to have a field so weak that it contributes little to the local viscosity.

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In the picture of Galeev et al. (1979), Sakimoto and Coroniti (1989), Ichimaru (1976, 1977) and others, when the disk field is continuously stressed by differential rotation it will often exceed B . %2 Due to buoyancy the field will likely bubble out of the disk and form coronal loops above the disk (Fig. 47). Such loops presumably will heat the corona via waves, magnetic reconnection or dissipation of currents generated by the disk differential rotation and turbulence. In analogy with the solar corona, the closed field regions will be hot and quasi-static-like solar active regions while the open field regions will have strong outflows, similar to solar corona holes. In the opposite picture of Heyvearts (1991), Fig. 36, when the magnetic loops bubble out of the disk, instead of maintaining a small-scale height via continuous reconnection, they may inversely cascade into larger and larger loops, finally forming an arcade of macroscopic ordered field lines with size scale l&r
Fig. 47. Artist conception of a magnetically dominated accretion disk corona with bouyant flux tubes penetrating the corona. Energy dissipated in the rarefied magnetic loops heats the gas to temperatures much higher than the disk. These loops then emit hard X-rays via inverse Comptonization of the disk soft X-rays (from Galeev et al., 1979). However such models predict strong Fe fluorescence lines and Compton reflection humps which have not been detected from most GBHs.

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overall disk structure and spectral output would depend on the strength of the dynamo and how it is sustained. Finally, macroscopic open field lines connecting the disk to the ISM can serve both as the driving force and the collimating agent for bipolar jets (Begelman et al., 1987; Blandford, 1993; Levinson and Blandford, 1996; Li et al., 1992).

5. Nonthermal models 5.1. Nonthermal X- and gamma-ray emissions As we see in Fig. 7 there are two distinct classes of XRN hard X-ray spectra: those with &!1.5 power laws and exponential cutoffs above &200 keV (usually interpreted as the inverse Compton component) and those with &!2.5 power laws extending all the way to the highest energies measurable. Even Cyg X-1, the prototype of persistent sources, also exhibits both spectral components (Figs. 3 and 6). When its hard X-ray intensity is unusually low (the so-called gamma-0 state, Ling et al., 1997; Phlips et al., 1996), only the steeper power law is detected, whereas the inverse Compton component appears superposed on top of the steeper power law during other times. At least for Cyg X-1 and may be also for GROJ0422, there is evidence that the steeper power law extends beyond 2 MeV and maybe as high as '10 MeV (McConnell et al., 1994, 1995; cf. Fig. 6). If this is confirmed then the steeper power law is likely produced by nonthermal emissions. One can imagine various scenarios in which the source is made up of a nonthermal emission region coexisting with the hot thermal emission region responsible for the quiescent inverse Compton spectrum. In analogy with the sun, the hot thermal region may be the innermost optically thin accretion disk or the inner disk corona, and the nonthermal emission region may be a nonthermal extended corona (NEC) energized by magnetic reconnections or wave turbulence propagating from the disk. Alternatively, the nonthermal emission may come from shocks in the accretion flow (e.g. Abramowicz and Chakrabarti, 1990; Chakrabarti and Titarchuk, 1995; Meszaros and Ostriker, 1983; Shu, 1976; Ryu et al., 1996) or shocks due to the interaction of an outflowing wind or jet with the circumstellar or interstellar matter. In such nonthermal emission models one may want to explain both the radio and gamma ray emissions together. Very few nonthermal models have been systematically pursued, mainly because the data is sketchy and nonthermal acceleration processes are notoriously inefficient. Most nonthermal models constructed for AGNs invoke an ad hoc particle distribution and only consider the emission and cooling processes without fully explaining the origin of the nonthermal particles from first principles. Recently however, some self-contained nonthermal models have begun to appear. One example is the converging inflow model of Chakrabarti and Titarchuk (1995). They hypothesize that the disk somehow turns into semi-spherical infall interior to a certain radius via a standing shock (Fig. 48). Multiple Compton upscattering of soft photons by the freely falling converging test particles leads to a power-law spectrum of upscattered particles (Blandford and Payne, 1981). Chakrabarti and Titarchuk (1995) claim that under reasonable conditions the power-law photon index is close to 2.5. However, such a model works only for low accretion rates since the scattering medium must have low density to be optically thin, and the infall is close to free fall velocity rather than following collisional hydrodynamics.

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Fig. 48. The nonthermal accretion model of Chakrabarti and Titarchuk (1995) postulates a standing shock which converts the accretion disk into quasi-radial infall. The converging radial infall then Compton upscatters external soft photons into hard X-rays with a power-law index of &2.5.

Another model is that of Li et al. (1996) in which they assume that plasma wave (Alfven and Whistler) turbulence generated by the disk fills the nonthermal extended corona or NEC. Electrons and pairs in the tail of a thermal distribution are stochastically accelerated by such wave turbulence (Dermer et al., 1996) and cooled by inverse Compton, bremsstrahlung and synchrotron. By solving the coupled Fokker—Planck and radiative cooling equations they obtain a self-consistent distribution of nonthermal particles plus thermal particles. However, due to efficient radiative cooling the nonthermal population does not achieve a power law. The resultant photon output spectrum is a combination of the thermal inverse Compton spectrum plus a high energy “knee”. The authors show that such spectra can fit the current gamma ray data of Cyg X-1 and GROJ0422 up to a few MeV (Fig. 49). Future observation at higher energies should more critically test such models. However, the !2.5 power-law spectra of many XRNs and the superlow (gamma-0, Ling et al., 1997) state of Cyg X-1 cannot yet be fit by such nonthermal models. 5.2. Nonthermal radio emissions To radiate nonthermal radio photons by the synchrotron process the magnetic field must be very weak and the electron Lorentz factor high. This means that the nonthermal radio sources associated with GBHs must be sufficiently far away from the central BH or the accretion disk. Hence, such nonthermal particles are unlikely energized within the disk or its inner corona since they would be residing in much stronger field regions (up to 10 MG for equipartition fields) and higher densities. Shocks and boundary layer turbulence caused by the interaction of winds and jets with the circumstellar medium are more favorable places to produce such radio emitting relativistic particles. However, since the core radio sources associated with many persistent GBHs are relatively steady, this would imply the existence of a quiescent disk wind even in the absence of any

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Fig. 49. Output spectra (left panels) and electron distributions (right panels) of the nonthermal model of Li et al. (1996) compared with the OSSE and Comptel data of Cyg X-1 and GROJ0422. The nonthermal tails are computed by balancing wave turbulence acceleration with radiative cooling (from Li et al., 1996).

eruptive plasmoid ejections. Little work has been done in modeling particle acceleration in such wind interactions with the circumstellar matter. One way in which the radio and hard X-rays fluxes can become tightly correlated (except for a time delay corresponding to light travel time from the disk to the radio emitting region) is the following scenario. Suppose that relativistic nonthermal particles are initially accelerated primarily along magnetic field lines so they do not radiate synchrotron radiation to first order. When a large flux of hard X-rays comes along they will Compton scatter with the energetic particles. The Compton recoil of the particles then produces finite pitch angles which result in synchrotron radiation. In this case the amount of synchrotron radio emission will be directly proportional to the incident hard X-ray flux. This model also predicts a nonthermal gamma ray tail due to the Compton upscattering of the hard X-rays. The gamma-ray flux will also be correlated with the radio flux. This model can be cleanly tested with future multiwavelength observations of radio flares of GBHs. The anticorrelated behavior of the radio with the soft X-rays of Cyg X-1 may be interpreted as due to free—free absorption in the companion wind (Hjellming and Wade, 1971). 5.3. Particle acceleration mechanisms Many nonthermal particle acceleration processes around black holes have been considered. Among these the most popular are (a) stochastic acceleration by wave turbulence (e.g. Dermer et al., 1996) generated by the disk turbulence and at boundary layers between outflows and the circumstellar medium (e.g. due to Kelvin—Helmholtz instabilities); (b) shock acceleration at the termination and internal shocks of relativistic outflows and jets, or shocks in quasi-spherical accretion (e.g. Meszaros and Ostriker, 1983; Park and Ostriker, 1989) or in the accretion disk itself, (Chakrabarti and Titarchuk, 1995; Shu, 1976; Ryu et al., 1995); (c) parallel electric field acceleration

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due to magnetic reconnection (e.g. Tavani and Liang, 1996) or current instabilities (e.g. double layers, Hamilton et al., 1993) in the NEC. It seems likely that some or all of these processes may be operating in or near GBH accretion disks and outflows. At this point it seems remote that observations will be able to distinguish between different microscopic acceleration mechanisms. However, the key question is how efficient can these processes be: what fraction of the overall accretional energy can be converted into nonthermal relativistic particles by such processes? Conventional wisdom (e.g. from solar analogies) suggests only a small fraction. But in the superlow state of Cyg X-1 and many XRNs if the hard X-rays and gamma-rays are of nonthermal origin, we must pump most of the total energy output into nonthermal particles. In addition the nonthermal particles, once accelerated, must avoid rapid thermalization via coulomb or collective processes before they radiate away most of their energy. This rules out, for example, bremsstrahlung as the primary cooling process. These issues remain major challenges for future nonthermal models of black-hole accretions. Of course one alternative is to hypothesize that the nonthermal component derives its energy not from accretion at all, but from the rotation of the blackhole (e.g. Blandford and Znajek, 1977; Znajek, 1978). AGN models in which particles are accelerated by the poynting flux associated with the rotation of the hole may be scaled to GBHs. Any accretional energy release will then produce a thermal component on top of the nonthermal emission produced by the rotation of the hole.

6. GBH variability and time-dependent disk models 6.1. Dynamical and instability time scales Excluding periodic variations due to binary orbit and precession, etc., we tend to distinguish between two regimes of GBH variability time scales. One is the time scales associated with the long-term ('days) behavior of the accretion rate, such as the XRN outburst cycle and decay profile or the intensity state transitions of the presistent sources. The other one is associated with the short term ((hours) behavior of disk inhomogeneities, such as the millisecond to &minutes chaotic fluctuations seen in all GBHs, and the peaked noises and LFQPOs (cf. Section 2.2). The former is now generally associated with the thermal limit cycle (TLC) instability of the quiescent cool disk (due to hydrogen dissociation and ionization) or intrinsic variation of the accretion feed rate from the companion. The latter is generally associated with the fluctuations and instabilities of the inner disk or an inner region of quasi-spherical flow (e.g. Fortner et al., 1989). Table 4 summarizes sample time scales of disk dynamics of the inner disk. We see that they cover the range from milliseconds to minutes. Hence, a comprehensive study of disk dynamics will involve solving the disk time evolution on many different time scales. This poses a particularly difficult problem for numerical studies. In the following sections we briefly describe some recent works geared separately towards the rapid variabilities and the TLC. 6.2. Linear radial perturbations Beginning with the pioneering works by Lightman and Eardley (1974), Pringle (1976, 1981), Shakura and Sunyaev (1976), Piran (1978) and many others, the study of the time evolution of

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Table 4 Sample time scales for a 10M black hole _ Disk Radius

12GM/c2

120GM/c2

Light crossing time Orbital period Thermal time

0.6 ms 10 ms 4 ms

6 ms 0.4 s 0.15 s

Radial drift time Compton cooling time

20 a~1 ms 2 ms

14 a~1 s 2s

Electron-ion Coulomb Coupling time

10 ms

;ls

accretion disks around black holes has concentrated on the linear perturbations of standard thermal Keplerian disk models (SS or SLE disks). Moreover, most of the works incorporating details of the disk radiative cooling treat only the axisymmetric radial perturbations (e.g. above References), whereas most of the nonaxisymmetric perturbations studies assume simple polytropic or adiabatic equation of state (e.g. Nowak and Wagoner, 1993). Due to the short orbital periods near a stellar mass black hole (&ms) and the inherently noisy nature of the disk emissions, nonaxisymmetric perturbations are believed to be much more difficult to detect than axisymmetric modes. The recent launch of RXTE may change that. In this review we will first concentrate on the results for axisymmetric perturbations. Of course nonradial perturbations play important roles in generating the disk turbulence in the first place, and some unique signatures of nonradial perturbations may provide diagnostics of the nature of the BH (e.g. Nowak and Wagoner, 1993 and references therein). At long wavelengths (j
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Fig. 50. Sample dispersion relations of linear radial perturbations of thin Keplerian thermal disk models for various accretion rates. Left panels: perturbations of the optically thick SS disk; right panels: perturbations of the optically thin SLE disk. Solid curves: instability growth rates, stars: oscillation frequencies. Note that only in the secularly unstable upper SS branch (cf. Fig. 29) is there unstable oscillation at short wavelengths (lower left panel) (from Luo and Liang, 1994b).

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Fig. 51. Plot of the maximal unstable oscillation frequency of a SS disk at R"6R (solid) and R"12R (dotted) as ' ' functions of accretion rate for a"1. The constancy of this oscillation frequency over several decades of accretion rate and its low value ((14 Hz for a 10M GBH) make such oscillations strong candidates for the low frequency QPOs detected _ in many GBHs (from Luo and Liang, 1994b).

rates on temperature and density (see Table 5 for summary of instability criteria). On the other hand, Lightman and Eardley (1974) proved that the secular instability results from a negative mass diffusion coefficient when radiation pressure dominates. The merging of the two modes into a single unstable oscillating mode at short wavelengths involves a delicate balance between the driving forces of the two instabilities. 6.3. Global nonlinear radial perturbations The above linear perturbation results, though intuitive, do not address the issue of what happens globally and in the nonlinear regime when the perturbation amplitude is finite. To study this one must consider nonlinear global evolution of the disk. Lightman (1974) made an initial attempt on the nonlinear evolution of the secular instability and found that the disk tends to clump into rings and choke off the accretion. More recently, Chen and Taam (1995) and Abramowicz et al. (1995b) studied the nonlinear evolution of inertial acoustic modes and found QPO-like global oscillations. Following the approach of Lightman (1974), Luo (1996) and Luo and Liang (1995) studied the fully nonlinear global evolution of perturbations of unstable SS disks, including both the thermal and secular modes. To allow for the transition from the optically thick to optically thin regimes he

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Table 5 Stability criteria of STD models (adapted from Luo and Liang, 1994b) (Below Q` is the local heating flux and Q~ is the local cooling flux of the disk. ­/­h is evaluated at constant R and ­/­R is evaluated at constant h) Thermally stable

Thermally unstable

Secularly stable

(­Q`/­h)((­Q~/­h) (­Q`/­h)(­Q~/­R)((­Q~/­h)(­Q`/­R)

(­Q`/­h)'(­Q~/­h) (­Q`/­h)(­Q~/­R)'(­Q~/­h)(­Q`/­R)

Secularly unstable

(­Q`/­h)((­Q~/­h) (­Q`/­h)(­Q~/­R)'(­Q~/­h)(­Q`/­R)

(­Q`/­h)'(­Q~/­h) (­Q`/­h)(­Q~/­R)((­Q~/­h)(­Q`/­R)

used the bridging prescription of Liang and Wandel (1991) for the radiative flux. Starting with numerical noise he was able to evolve the perturbations into the nonlinear regime. He found that at early times when the perturbation amplitude was less than 10—15%, the numerical noise coalesce into coherent oscillations at a frequency approximately equal to that given by linear theory, with a slowly growing amplitude. However, once the oscillation amplitude reaches &10—15% the oscillations stop and the amplitude grows monotonically. The fastest growth occurs at r&12 GM/c2 as predicted by linear theory. The temperature and scale height exponentiates on the thermal time scale while the column density profile has a single node at around 12 GM/c2. Interior to this the column density continues to decrease as material is drained into the hole while just outside this radius the column density increases as material piles up to form a dense ring. The column density contrast grows on the viscous time scale. As time goes on the disk around 12 GM/c2 explodes into a hot, thick, rarefied bubble while just outside it a dense ring is formed. This hot thick bubble expands in the radial direction both inward and outward. Eventually the bubble scale height becomes so large that the thin disk approximation breaks down, and we see a transformation of the inner disk from the SS disk into a hot optically thin solution. Eventually radial advection, winds, magnetic fields and pair production will become important in the dynamics. The thermal instability may persist, but saturate at a finite amplitude, so that the disk, while steady on long time scales, continues to fluctuate dynamically and in its radiative output on short thermal time scales. The increased level of RMS fluctuations in the hard X-rays compared to the soft X-rays for most GBHs may well be a manifestation of such saturated thermal instabilities. Evolution and saturation of nonlinear thermal modes in advection-dominated and transonic accretion flows remain to be studied in detail. 6.4. Thermal limit cycle instability and X-ray Nova outbursts As discussed in Section 2 X-Ray Nova (XRN) outbursts are triggered by instabilities in the accretion flow of low-mass X-ray binaries onto black holes, with typical recurrence times measured in decades, short rise times ((days) and decay times of the order of months (Fig. 20), even though the detailed profiles differ from source to source and between different wavelengths. Motivated by the similarity to dwarf Nova outbursts (Hoshi, 1979; Bath and Pringle, 1982; Meyer and MeyerHofmeister, 1984; Cannizzo et al., 1982; Faulkner et al., 1983) a number of authors proposed an accretion disk thermal limit cycle (TLC) instability as the cause of this transient behavior (Cannizzo et al., 1985; Huang and Wheeler, 1989; Tuchman et al., 1990; Mineshige and Wheeler, 1989;

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Kim et al., 1992). Even though the competing model of X-ray heating-induced mass overflow instability (Hameury et al., 1986) may still be relevant in some special cases, most of the observational evidence now favors the thermal limit cycle instability. Hence, it seems worthwhile to review the basic physics here. For details see reviews by Cannizzo (1993) and Mineshige and Kusunose (1993). The TLC instability is a global instability of cool optically thick accretion disks at low accretion rates (M Q (1016 g/s for solar mass holes) caused by the partial ionization of hydrogen. Due to the well known behavior of the hydrogen opacity as a function of temperature in this regime, the local disk solution in the ¹ (or M Q )—R plane has the famous S-shape structure (Fig. 52), just as in the SS-slim disk case. The upper (lower) stable branches correspond to the fully ionized (neutral) limits of H while the middle branch corresponds to partially ionized H: it is both thermally and secularly unstable. Hence, as the accretion flow feeds material onto a quiescent cold disk (stable lower branch) it eventually drives the local M Q above the lower turning point at which point the disk jumps to the upper stable hot branch with much higher local M Q . The higher local MQ drains the material from the disk, allowing more material to flow in until M Q reaches the upper turning point, leading to a mass diffusion wave propagating inward. At the same time the radial radiation diffusion causes a thermal (heating) wave to propagate inward as well, gradually converting the entire disk into the high-M Q warm state. This corresponds to the soft X-ray outburst. Eventually, the higher MQ drains the material from the companion and the local disk M Q starts to fall until M Q reaches the upper turning point (Fig. 52) at which point M Q drops to the lower cool branch. This sets in a cooling front (Fig. 53). However, the residual radiation heating by X-rays from the inner disk continues to inhibit the outer disk from cooling and prolongs the optical light curve of the XRN. Detailed calculations of the light curve (Fig. 54) and soft X-ray spectra seem to be capabable of reproducing many features of the data.

Fig. 52. Effective temperature versus R curves of cool disks for various a values. The S shapes are caused by partial ionizations of hydrogen (from Cannizzo, 1993) and are believed to be responsible for the TLC instability of XRNs.

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Fig. 53. Numerical simulation of the propagation of the TLC instability cooling front (from Cannizzo et al., 1997). Fig. 54. Long term evolution of global disk parameters (m is visual magnitude, M is total disk mass) in one version of V $*4, the TLC instability model (see Cannizzo et al., 1997 for details).

However, the current TLC instability model of XRN fails to address the origin of the hard X-rays since the disk gets only as hot as the optically thick SS disk but not the optically thin disk. Moreover, in many XRNs the hard X-rays appear before the soft X-rays so it is important to have a model that can make a direct transition from the cold neutral state to the optically thin state, at least for part of the disk, without having to first pass through the optically thick SS state. This problem remains a major challenge for the TLC model of XRN. Another issue for the TLC model is the effect of X-ray irradiation from the inner disk, which tends to suppress the TLC instability since it helps to ionize the hydrogen. Estimates (Tuchman et al., 1990) show that irradiation stabilization is unimportant as long as ¸ 43]1035 erg/s for a radiation interception fraction of 10~2. Recent x compilation of quiescent X-ray data for stable versus unstable systems seem to support this picture (van Paradijs, 1996). In order to match the observational data, most TLC models of XRN outbursts use disk models with a viscosity prescription of the form a&(h/r)n, where n is assumed to be 51. This is consistent with the viscosity model of Vishniac and Diamond (1993). With increasing multi-wavelength observations of XRN systems, both in outburst and in quiescence, it seems promising that models of XRNs will be confronted in detail.

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7. Towards a unified framework The multi-wavelength signatures of GBHs, especially the XRNs, are clearly very rich and dynamical. The traditional simple picture of quasi-static emission from thermal, thin Keplerian accretion disks is no longer adequate to provide even the first-order approximation of these phenomena. It is now clear that we have to include, at some level, dynamical and nonthermal processes into any eventual unified framework for GBHs. On the other hand, the steady thermal thin accretion disk (STD) models have their merits and remain useful in explaining the quiescent low state emission of the soft and hard X-rays, which is the dominant output for the persistent sources most of the time. Instead of abandoning the STD model completely and restart from scretch, a more fruitful approach would be to keep the STD as a cornerstone of any eventual unified framework, but adding more superstructures. Fig. 55 shows an artist conception of one such scenario. In addition to the STD model that emits the soft and hard X-rays, it contains a transient central ultrahot bulge which may be either a hot ion torus deficient in soft photons (Liang and Dermer, 1988) or an advection-dominated inflow (Abramowicz et al., 1995a; Chen et al., 1995; Chen and Taam, 1996; Narayan and Yi, 1994, 1995). Outside the STD this picture incorporates a nonthermal extend corona which may be responsible for the steeper power law emissions in the gamma-0 state of Cyg X-1 and many XRNs. Such a NEC would still derive it primary energy from the accretion disk. But in this case the disk energy is not dissipated locally but instead pumped into the NEC via waves and winds, and converted via shocks and turbulence into nonthermal particles. In the hard-low state of the persistent sources the NEC output amounts only to (10% of the disk radiative output (cf. Fig. 2), but in the power-law type of XRNs and ultralow

Fig. 55. Artist conception of the unified GBH disk, corona and outflow picture (b) showing the possible origins of the different spectral components (a) (NEC stands for nonthermal extended corona, HOT stands for hot optically thin).

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state of Cyg X-1, the NEC output could dominate over the disk radiative output. In addition to the static NEC Fig. 55 pictures additional outflows, both episodic and quiescent. Such outflows may be dominated by hadrons or pairs, and magnetically collimated along ordered open field lines. The X-ray spectral state transitions of persistent GBH sources do not follow a single universal pattern: sometimes the hard and soft fluxes are correlated, sometimes they are anticorrelated, and often they are uncorrelated, for a single source. It is likely that they correspond to changes of different accretion parameters, e.g. overall accretion rate, the magnetic field strength, viscosity a, angular momentum etc. A unified framework must be able to explain the triggering mechanism and criteria for the different types of transitions from first principles. Similarly, it must be able to explain the different types of XRN multiwavelength evolution patterns.

8. The GBH-AGN connection The recent discoveries of radio jets (Mirabel et al., 1992) and superluminal relativistic plasmoid ejections (Mirabel et al., 1994; Hjellming and Rupen, 1995) associated with GHB candidates have greatly strengthened the connections between GBHs and the AGN phenomena, and indirectly, the common nature of the central compact object. An important property of accreting black holes is the simple mass scaling of most observable quantities, from spectral temperature to dynamical time scales. Hence understanding the multiwavelength signatures and dynamical properties of GBHs, which are much closer and more accessible to high resolution studies, will shed important light on the workings of AGN central engines. However, there are also some fundamental differences between the two classes of objects. For example, most GBHs are dominated by their X-ray output, and radio-optical and gamma-ray ('MeV) outputs are very weak, whereas among AGNs, only Seyfert galaxies have similar property (e.g. Dermer and Schlickeiser, 1992). In fact, the hard X-ray spectra of Seyfert galaxies look very much like those of the persistent GBHs (Fabian, 1979). Other classes of AGNs are dominated by their radio-UV and/or gamma-ray output. If we associate the hard X-rays with a hot optically thin STD and the radio and gamma-rays with the NEC or jets, then GBHs and Seyferts seem to have stronger STD output, whereas radio galaxies and blasars are dominated by the nonthermal components. Also the relativistic ejections of GBHs discovered so far seem to be much milder (bulk Lorentz factor C&2.5 which can be achieved with radiation acceleration, Phinney, 1987; Li and Liang, 1996) than the extreme AGN jets (C510 which cannot be achieved by radiation acceleration, Phinney, 1987). Such distinctions may be due to environmental differences (e.g. circumstellar matter or companion wind may suppress the nonthermal components in binaries) or intrinsic properties (e.g. some AGN black holes may have much higher angular momentum per unit mass so that the rotational power output of the hole may dominate over the accretional power output). All of these differences need to be taken into account if we are to eventually build a unified framework for both GBHs and AGNs.

9. Summary and future directions The multi-wavelength signatures of GBH candidates are clearly very rich, diverse and dynamical, manifesting both thermal and nonthermal properties. Yet there are common underlying

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properties that link all persistent and transient GBH sources. Examples are the predominance of the X-ray power output over other wavebands, episodic emergence of a black-body-like soft X-ray component, chaotic variability on all time scales, multi-spectral state transitions and characteristic shapes of the power density spectrum. While the historical thermal Keplerian accretion disk models is no longer able to explain all of these signatures, a new unified framework based on the generalizations of the thermal disk model, that include a nonthermal extended corona, radial advection, winds and outflows, is definitely worth pursuing. In particular, global disk solutions with the correct transonic inner boundary condition and appropriate outer boundary conditions need investigation. At the same time, nonlinear time evolution of global disk perturbations should be systematically studied for comparison with the observed variabilities over all time scales. With the ongoing observations of CGRO, RXTE, ASCA and many other satellites it is clear that real-time coordinated multi-wavelength studies of GBHs will be increasingly common and sophisticated, revealing for the first time the details of short term spectral variations of these sources. A good example is the recent multiwavelength observation of the soft-high state transition of Cygnus X-1 (Harmon et al., 1996) which provides clear evidence of the relation between the soft X-rays, hard X-rays and gamma rays above 1 MeV. Another example is the relations between the radio, X-ray and gamma ray of the superluminal sources GRS 1915 and GROJ1655 (Harmon et al., 1995). Such multi-wavelength studies will definitely help to define and refine the unified framework theorists hope to construct for all GBHs. Such a theoretical framework will have broad applications to the understanding of AGNs as well. In addition to studying the accretion flow outside the black hole, multiwavelength observations may eventually provide diagnostics of the black hole itself and critical tests of general relativity. Examples include measurements of the black-hole mass and angular momentum using the gamma-ray continuum spectrum, blueshifted profile of the Fe K fluorescence and high-frequency a cutoff of the power density spectrum. Dragging of inertial frames in a Kerr hole may also be measured using the polarization of scattered radiation. Even though BATSE onboard CGRO has provided invaluable all-sky hard X-ray monitoring of GBHs over the last few years, to make further advances we will need new experiments which greatly improve on BATSE both in sensitivity and spectral range. Recent searches of faint BATSE sources may have uncovered additional new XRN candidates at the 0.1 Crab level (Grindlay et al., 1994). As we discussed above, the most critical measurements will come in the soft gamma-ray energies '1 MeV. Since only the brightest source, Cyg X-1, has been marginally detected in this range by Comptel, it is reasonable to surmise that many other fainter GBHs are just below the detection threshold of OSSE and Comptel. Hence a sensitivity improvement of, say, a factor of 50 over OSSE and Comptel, should allow us not only to confirm the existence of such emissions for many sources at '10p level, but also measure the overall spectral shape with some confidence. This would allow us to distinguish between the thermal versus nonthermal nature of the gammaray tail. If the spectrum is indeed thermal with a very high-energy cutoff, the detailed spectral shape will help to constrain the temperature, Thompson depth and pair loading of the plasma. If the spectrum is nonthermal, coordinated gamma-ray and radio-optical studies of these sources will also shed important light on the nature of the source region. In addition to spectroscopy, gamma-ray polarimetry may also help to distinguish the thermal versus nonthermal nature of the source.

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Acknowledgements The author acknowledges partial support from NASA grant NAG5-1547.

References Abramowicz, M. et al., 1980. Astrophys. J. 242, 772. Abramowicz, M.A. et al., 1988. Astrophys. J. 332, 646. Abramowicz, M.A., Chakrabarti, S.K., 1990. Astrophys. J. 350, 281. Abramowicz, M.A. et al., 1995a. Astrophys. J. 438, L37. Abramowicz, M.A. et al., 1995b. Astrophys. J. 452, 379. Abramowicz, M.A., Zurek, W.H., 1981. Astrophys. J. 246, 314. Agrawal, P.C. et al., 1972. Astrophys. Sp. Sci. 18, 408. Artemova et al., 1996. Astrophys. J. 456, 119. Avni, Y., Bahcall, J., 1975. Astrophys. J. 197, 675. Babul, A. et al., 1989. Astrophys. J. 347, 59. Bailyn, C.D., 1995. Nature 374, 701. Baity, W.A. et al., 1973. Nature Phys. Sci. 245, 90. Balbus, S.A., Hawley, J.F., 1991. Astrophys. J. 376, 214. Balbus, S.A., Hawley, J.F., 1992. Astrophys. J. 392, 662; Balbus, S.A., Hawley, J.F., 1992. Astrophys. J. 400, 610. Barret, D. et al., 1996. Astrophys. J. 473, 963. Bath, G.T., Pringle, J.E., 1982. Mon. Not. R. Astron. Soc. 199, 267. Begelman, M., Meier, D.L., 1982. Astrophys. J. 253, 873. Begelman, M. et al., 1984. Rev. Mod. Phys. 56, 265. Begelman, M. et al., 1987. Astrophys. J. 329, 698. Begelman, M., Chiueh, T., 1988. Astrophys. J. 332, 872. Begelman, M., Li, Z.Y., 1994. Astrophys. J. 426, 269. Bisnovatyi-Kogan, G.S. et al., 1971. Sov. Astron. 15, 17. Bisnovatyi-Kogan, G.S., Blinnikov, S.I., 1977. Astron. Astrophys. 59, 111. Bjornsson, G., Svensson, R., 1991. Mon. Not. Roy. Astron. Soc. 249, 177. Bjornsson, G., Svensson, R., 1992. Astrophys. J. 394, 500. Blandford, R.D., 1993. In: Friedlander, M. et al. (Eds.) AIP Conf. Proc. No. 280, AIP, New York, p. 533. Blandford, R.D., Payne, D.G., 1981. Mon. Not. Roy. Astron. Soc. 194, 1033; Blandford, R.D., Payne, D.G., 1981. Mon. Not. Roy. Astron. Soc. 194, 1041. Blandford, R.D., Payne, D.G., 1982. Mon. Not. Roy. Astron. Soc. 199, 883. Blandford, R.D., Znajek, R.L., 1977. Mon. Not. Roy. Astron. Soc. 179, 433. Bolton, C.T., 1975. Astrophys. J. 200, 269. Bondi, H., 1952. Mon. Not. Roy. Astron. Soc. 112, 195. Bouchet, L. et al., 1991. Astrophys. J. 383, L45. Braes, L.L.E., Miley, G.K., 1971. Nature 232, 246. Brandt, S. (Ed.) 1991. Nova Muscae Workshop Proc., DSRI, Lyngby. Bridgman, T. et al., 1994. In: Fichtel, C. et al. (Eds.), AIP Conf. Proc. No. 304, AIP, New York, p. 225. Briggs, M. et al., 1995. Astrophys. J. 442, 638. Callanan, P.J. et al., 1992. Mon. Not. Roy. Astron. Soc. 249, 573. Canizares, C.R., Oda, M., 1977. Astrophys. J. 214, L119. Cannizzo, J.K. et al., 1985. In: Lamb, D.Q., Patterson, J. (Eds.), Proc. Camb. Workshop on CV’s and LMXB’s, Reidel, Dordrecht, p. 307. Cannizzo, J.K., 1993. In: Wheeler, J.C. (Ed.), Accretion Disks in Compact Stellar Systems, World Scientific, Singapore, p. 6.

E.P. Liang / Physics Reports 302 (1998) 67—142 Cannizzo, J.K. et al., 1995. Astrophys. J. 454, 880. Cannizzo, J.K., et al., 1997. Astrophys. J., submitted. Casares, J., Charles, P., 1994. In: Holt, S., Day, C. (Eds.), AIP Conf. Proc. No. 308, AIP, New York, p. 107. Chakrabarti, S.K., 1990. Theory of Transonic Astrophysics Flows, World Scientific, Singapore. Chakrabarti, S.K., 1996a. Phys. Rep. 266, 299. Chakrabarti, S.K., 1996b. Astrophys. J. 464, 664. Chakrabarti, S.K., Titarchuk, L.G., 1995. Astrophys. J. 455, 623. Charles, P.A. et al., 1991. Mon. Not. Roy. Soc. 245, 567. Chen, X., Taam, R.E., 1993. Astrophys. J. 412, 254. Chen, X., Taam, R.E., 1995. Astrophys. J. 441, 354. Chen, X., Taam, R.E., 1996. Astrophys. J. 466, 404. Chen, X. et al., 1995. Astrophys. J. 443, L61. Chen, X., et al., 1996. Astrophys. J., submitted. Chen, W. et al., 1993. Astrophys. J. 426, 586. Chen, W. et al., 1994. In: Holt, S., Day, C. (Eds.), AIP Conf. Proc. No. 308, AIP, New York, p. 67. Chen, W. et al., 1997. Astrophys. J., in press. Chiueh, T. et al., 1991. Astrophys. J. 377, 462. Coe, M.J. et al., 1976. Nature 259, 544. Connors, P.A. et al., 1980. Astrophys. J. 235, 224. Coroniti, F.V., 1981. Astrophys. J. 244, 587. Cowley, A., 1992. Ann. Rev. Astron. Astrophys. 30, 287. Cowley, A., 1994. In: Holt, S., Day, C. (Eds.), AIP Conf. Proc. No. 308, AIP, New York, p. 45. Cunningham, C., 1975. Astrophys. J. 202, 788. Dermer, C.D., Liang, E., 1988. In: Gehrels, N. et al., (Eds.), AIP Conf. Proc. No. 170, AIP, New York, p. 326. Dermer, C.D., Liang, E., 1989. Astrophys. J. 339, 512. Dermer, C.D., Schlickeiser, 1992. Science, 257, 1642. Dermer, C.D. et al., 1991. Astrophys. J. 369, 410. Dermer, C.D. et al., 1996. Astrophys. J. 456, 106. Dobrinskaya, J. et al., 1996. In: Makino, F. (Ed.), ASCA Symp. Proc., ISAS, Japan. Dolan, J.F. et al., 1987. Astrophys. J. 322, 324. Dolan, J.F. et al., 1992. Astrophys. J. 384, 249. Eardley, D.M., Lightman, A.P., 1975. Astrophys. J. 200, 187. Eardley, D.M., Lightman, A.P., 1976. Nature 262, 196. Eardley, D.M. et al., 1978. Comments Astrophys. 7, 151. Ebisawa, K., 1989, Pub. Astron. Soc. Japan 41, 519. Ebisawa, K., 1991. Ph.D. Thesis, ISAS, University of Tokyo. Ebisawa, K. et al., 1994. Pub. Astron. Soc. Japan 46, 375. Eggum, G.E. et al., 1987. Astrophys. J. 323, 634. Eggum, G.E. et al., 1988. Astrophys. J. 330, 142. Esin, A.A. et al., 1997. Astrophys. J., submitted. Fabian, A.C., 1979. Proc. Roy. Soc. London A 366, 449. Faulkner, J. et al., 1983. Mon. Not. Roy. Astron. Soc. 205, 359. Fishbone, L., Moncrief, V., 1976. Astrophys. J. 207, 962. Fortner, C. et al., 1989. Nature 342, 775. Galeev, A. et al., 1979. Astrophys. J. 229, 318. Gilfanov, M. et al., 1991. In: Brandt, S. (Ed.), Nova Muacae Workshop Proc., DSRI, Lyngby, p. 51. Goldwurm, A. et al., 1992. Astrophys. J. 389, L79. Grabelsky, D.A. et al., 1993. In: Friedlander, M. et al., (Eds.), AIP Conf. Proc. No. 280, AIP, New York, p. 345. Grabelsky, D.A. et al., 1995. Astrophys. J. 441, 800. Grebenev, S.A. et al., 1991. Sov. Astron. Lett. 17 (6), 413. Grebenev, S.A. et al., 1991. In: Brandt, S. (Ed.), Nova Muacae Workshop Proc., DSRI, Lyngby. p. 19.

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138

E.P. Liang / Physics Reports 302 (1998) 67—142

Greiner, et al., 1994, Astron. Astrophys. 285, 509. Grindlay, J. et al., 1994. BAAS 1851, 1607. Grove, E. et al., 1994. In: Fichtel, C. (Ed.), AIP Conf. Proc. No. 304, AIP, New York, p. 192. Haardt, F., Maraschi, L., 1991. Astrophys. J. 380, L51. Haardt, F., Maraschi, L., 1993. Astrophys. J. 413, 507. Hameury, J.-M. et al., 1986. Astron. Astrophys. 162, 71. Hamilton, R. et al., 1993. In: Friedlander, M. et al. (Eds.), AIP Conf. Proc. No. 280, AIP, New York, p. 438. Harmon, A. et al., 1994. In: Fichtel, C. et al. (Eds.), AIP Conf. Proc. No. 304, AIP, New York, p. 210. Harmon, A. et al., 1995. Nature 374, 703. Harmon, A. et al., 1996. IAUC 6436. Harris, M., 1994. Astrophys. J. 433, 87. Hartle, J., 1978. Phys. Rep. 46, 201. Harrison, T. et al., 1994. In: Fichtel, C. (Ed.), AIP Conf. Proc. No. 304, AIP, New York, p. 319. Hawley, J., Balbus, S., 1991a. Astrophys. J. 376, 223. Hawley, J., Balbus, S., 1991b. Astrophys. J. 400, 595. Haymes, R.C., Harnden, F.R. Jr., 1968. Astrophys. J. 151, L125. Haymes, R.C., Harnden, F.R. Jr., 1970. Astrophys. J. 159, 1111. Hjellming, R., Han, X., 1995. In: Lewin, W.H.G. et al. (Eds.), X-Ray Binaries, Cambridge, Cambridge, p. 308. Hjellming, R., Rupen, M., 1995. Nature 375, 464. Hjellming, R., Wade, C.M., 1971. Astrophys. J. 168, L21. Hjellming, R. et al., 1975. Nature 256, 111. Hjellming, R., Johnston, 1988. Astrophys. J. 328, 600. Holt, S. et al., 1976. Astrophys. J. 203, L63. Honma, F. et al., 1991. Pub. Astron. Soc. Japan 43, 261. Huang, M., Wheeler, J.C., 1989. Astrophys. J. 343, 229. Hutchings, J.B. et al., 1987. Astron. J. 94, 340. Heyvearts, J., 1991. In: Priest, E., Wood, (Eds.), Advance in Solar System Magnetohydrodynamics, Cambridge, UK. Hoshi, R., 1979. Prog. Theor. Phys. 61, 1307. Ichimaru, S., 1976. Astrophys. J. 208, 701. Ichimaru, S., 1977. Astrophys. J. 214, 840. Iwasawa, K. et al., 1996. MNRAS 279, 837. Katz, J.I., 1976. Astrophys. J. 206, 910. Kemp, J.C. et al., 1981. Astrophys. J. 244, L73. Kemp, J.C. et al., 1987. Sov. Astron. 31, 170. Kesteven, M., Turtle, A., 1991. In: Brandt, S. (Ed.), Nova Muscae Workshop Proc., DSRI, Lyngby, p. 115. Kim, S.W. et al., 1992. Astrophys. J. 384, 269. King, A., 1995. In: Lewin, W.H.G. et al. (Eds.), X-Ray Binaries, Cambridge, Cambridge, p. 419. Kitamoto, S. et al., 1992. Astrophys. J. 394, 609. Kouveliotou, C. et al., 1993. In: Friedlander, M. et al. (Eds.), AIP Conf. Proc. No. 280, AIP, New York, p. 319. Konigl, A., 1989. Astrophys. J. 342, 208. Kompaneets, A.S., 1957. Sov. Phys. JETP 4, 730. Kroeger, et al., 1996. Astron. Astrophys., to appear. Kusunose, M., 1991. Astrophys. J. 370, 505. Kusunose, M., 1996. Astrophys. J. 457, 813. Kusunose, M., Takahara, 1990. Proc. Astron. Soc. J. 42, 347. Kusunose, M., Mineshige, S., 1992. Astrophys. J. 392, 653. Kusunose, M., Mineshige, S., 1995. Astrophys. J. 440, 100. Laor, A., 1991. Astrophys. J. 376, 90. Levinson, A., Blandford, R., 1996. Astrophys. J. 456, L29. Li, H., Liang, E., 1996. Astrophys. J. 458, 514. Li, H. et al., 1996. Astrophys. J. 460, L29.

E.P. Liang / Physics Reports 302 (1998) 67—142

139

Li, Z.Y. et al., 1992. Astrophys. J. 394, 459. Liang, E., 1977. Astrophys. J. 218, 243. Liang, E., 1979. Astrophys. J. 234, 1105. Liang, E., 1980. Nature 283, 642. Liang, E., 1988. Astrophys. J. 334, 33. Liang, E., 1990. Astron. Astrophys. 227, 447. Liang, E., 1991. Astrophys. J. 367, 470. Liang, E., 1993. In: Friedlander, M. et al. (Eds.), AIP Conf. Proc. No. 280, AIP, New York, p. 396. Liang, E., Dermer, C., 1988. Astrophys. J. Lett. 325, L39. Liang, E., Li, H., 1995. Astron. Astrophys. 298, L45. Liang, E., Nolan, P., 1984. Space Sci. Rev. 38, 353. Liang, E., Price, R.H., 1977. Astrophys. J. 218, 247. Liang, E., Thompson, K.A., 1980. Astrophys. J. 240, 271. Liang, E., Wandel, A., 1991. Astrophys. J. 376, 746. Lightman, A.P., 1974. Astrophys. J. 194, 419. Lightman, A.P., 1982. Astrophys. J. 253, 842. Lightman, A.P., Eardley, D.M., 1974. Astrophys. J. 187, L1. Lightman, A.P., Zdziarski, A., 1987. Astrophys. J. 319, 643. Lin, D.N.C., Papaloizou, J., 1980. MNRAS 191, 37. Ling, J. et al., 1987. Astrophys. J. Lett. 321, L117. Ling, J. et al., 1996. JPL BATSE Earth Occultation Source Catalog (JPL, Caltech). Ling, J. et al., 1997. Astrophys. J., submitted. Long, K.S. et al., 1980. Astrophys. J. 238, 710. Lovelace, R., Romanova, N., 1996. In: Carilli, Harris (Eds.), Cyg. A Workshop Proc., Cambridge, UK. Luo, C., 1996. Ph.D. thesis, Rice University. Luo, C., Liang, E., 1994a. In: Fichtel, C. et al. (Eds.), Second Comp. Symp. Proc., AIP, New York. Luo, C., Liang, E., 1994b. Mon. Not. Roy. Astron. Soc. 266, 386. Luo, C., Liang, E., 1995. In: Stepinski, T., Liang, E. (Eds.), Proc. LPI Workshop on Accretion Disks around Young and Compact Stars, LPI, Houston. Luo, C., Liang, E., 1998. Astrophys. J., to appear. Luo, C., Meirelles, C., Liang, E., 1994. In: Fichtel, C. et al. (Eds.), Second Comp. Symp. Proc., AIP, New York. Lynden-Bell, D., 1969. Nature 233, 690. Magdziarz, P., Zdziarski, A., 1995. MNRAS 273, 837. Mandrou, P. et al., 1978. Astrophys. J. 219, 288. Maraschi, L. et al., 1980. Astrophys. J. 241, 910. Margon, B. et al., 1975. Nature 254, 461. Matt, G. et al., 1996. MNRAS 278, 1111; Matt, G. et al., 1996. MNRAS 280, 823. Matteson, J., 1975. In: X-ray Binaries, GSFC, NASA SP-389, NASA, DC. McClintock, J.E., Remillard, R.A., 1990. Astrophys. J. 350, 386. McClintock, J.E. et al., 1995. Astrophys. J. 442, 358. McConnell, M. et al., 1993. In: Friedlander, M. et al. (Eds.), AIP Conf. Proc. No. 280, AIP, New York, p. 336. McConnell, M. et al., 1994. Astrophys. J. 424, 933. Meekins, J.F. et al., 1984. Astrophys. J. 278, 288. Melia, F., 1991. Astrophys. J. 377, L101. Melia, F., Misra, R., 1993. Astrophys. J. 411, 797. Melia, F., Misra, R., 1996, Astrophys. J. 465, 869; Melia, F., Misra, R., 1996, Astrophys. J. 467, 405. Merielles, C. et al., 1996. Astron. Astrophys., to appear. Meszaros, P., 1976. Nature 258, 583. Meszaros, P., Ostriker, J.P., 1983. Astrophys. J. 273, L59. Meyer, F., Meyer-Hofmeister, E., 1984. Astron. Astrophys. 132, 143. Michel, F.C., 1972. Astrophys. Space Sci. 15, 153.

140

E.P. Liang / Physics Reports 302 (1998) 67—142

Mineshige, S., Wheeler, J., 1989. Astrophys. J. 343, 241. Mineshige, S., Kusunose, M., 1993. In: Wheeler, J.C. (Ed.), Accretion Disks in Compact Stellar Systems, World Scientific, Singapore, p. 370. Mineshige, S., 1993. Astrophys. Space Sci. 210, 83. Mirabel, F. et al., 1992. Nature 358, 215. Mirabel, F., Rodriguez, L.F., 1994. Nature 371, 46. Misner, C. et al., 1973. Gravitation. Freeman, San Francisco. Miyamoto, S. et al., 1971. Astrophys. J. 168, L11. Miyamoto, S. et al., 1988. Nature 336, 450. Miyamoto, S. et al., 1993. Astrophys. J. Lett. 403, L39. Moncrief, V., 1980. Astrophys. J. 235, 1038. Moscoso, M., Wheeler, J.C., 1994. In: Holt, S. and Day, C. (Eds.), Evolution of X-ray Binaries, AIP, New York, p. 83. Moss, M., 1997. Ph.D. Thesis, Rice University. Motch, C. et al., 1985. Space Sci. Rev. 40, 219. Muchotrzeb, B., Paczynski, B., 1982. Acta Astron. 32, 1. Narayan, R., 1996. Astrophys. J. 462, 136. Narayan, R., Yi, I., 1994. Astrophys. J. 428, L13. Narayan, R., Yi, I., 1995. Astrophys. J. 452, 710. Narayan, R. et al., 1995a. Astrophys. J. 457, 821. Narayan, R. et al., 1995b. Nature 374, 623. Narayan, R. et al., 1996. Astrophys. J. 457, 821. Narayan, R., Barret, D., McClintock, J., 1997. Astrophys. J. 482, 448. Nolan, P. et al., 1981. Astrophys. J. 246, 494. Nolan, P., Matteson, J.L., 1983. Astrophys. J. 265, 389. Novikov, I.D., Thorne, K.S., 1973. In: DeWitt, C., B. (Eds.), Black Holes, Gordon & Breach, New York, p. 343. Nowak, M., Wagoner, R.V., 1993. Astrophys. J. 418, 183. Nowak, M., 1994. Astrophys. J. 422, 688. Oda, M., 1977. Space Sci. Rev. 20, 757. Oda, M. et al., 1976. Astrophys. Space Sci. Rev. 42, 223. Ogawara, Y. et al., 1982. Nature 295, 675. Orosz, J., Bailyn, C., 1995. Astrophys. J. 446, L59. Overbeck, J.W. et al., 1967. Astrophys. J. 150, 47. Paciesas, W. et al., 1997. In: Dermer, C. et al. (Eds.), Proc. 4th Compton Symp., AIP, New York. Paczynski, B., Witta, P.J., 1980. Astron. Astrophys. 88, 23. Paczynski, B., Abramowicz, M., 1982. Astron. Astrophys. 253, 897. Papaloizou, J.C.B., Pringle, J.E., 1985. Mon. Not. Roy. Astron. Soc. 213, 799. Park, M.G., Ostriker, J.P., 1989. Astrophys. J. 347, 679. Payne, D.G., 1980. Astrophys. J. 237, 951. Payne, D.G., Eardley, D.M., 1977. Astrophys. Lett. 19, 39. Paul, J. et al., 1991. In: Durouchoux, P., Pranzos, N. (Eds.), AIP Conf. Proc. No. 232, AIP, New York, p. 17. Phlips, B. et al., 1996. Astrophys. J. 465, 907. Phinney, S., 1987. In: Zensus, J., Pearson, T. (Eds.), Superluminal Radio Sources, Cambridge, Cambridge, p. 301. Piran, T., 1978. Astrophys. J. 221, 652. Pozdnyakov, L.A. et al., 1983. Sov. Sci. Rev. 2, 189. Pringle, J.E., 1976. Mon. Not. Roy. Astron. Soc. 177, 65. Pringle, J.E., 1981. Ann. Rev. Astron. Astrophys. 19, 137. Pringle, J.E., Rees, M.J., 1972. Astron. Astrophys. 21, 1. Pudritz, R.E., 1981. Mon. Not. Roy. Astron. Soc. 195, 881; Pudritz, R.E., 1981. Mon. Not. Roy. Astron. Soc. 195, 897. Ramaty, R. et al., 1992. Astrophys. J. 392, L63. Rees, M.J. et al., 1982. Nature 295, 17. Rees, M.J., 1984. Ann. Rev. Astron. Astrophys. 22, 471.

E.P. Liang / Physics Reports 302 (1998) 67—142

141

Remillard, R.A. et al., 1992. Astrophys. J. 399, L145. Reynolds, J., Jauncey, D., 1994. IAUC 6063. Rodriguez, L. et al., 1992. Astrophys. J. Lett. 401, L15. Rybicki, G., Lightman, A.P., 1979. Rad. Processes in Astrophysics, Wiley, New York. Ryu, D. et al., 1995. Astrophys. J. 452, 364. Ruden, S.P. et al., 1988. Astrophys. J. 329, 739. Sakimoto, P., Coroniti, F., 1989. Astrophys. J. 342, 49. Sanford, P.W. et al., 1975. Nature 256, 109. Shahbaz, T. et al., 1994. In: Fichtel, C. (Ed.), AIP Conf. Proc. No. 304, AIP, New York, p. 123. Shakura, N.I., Sunyaev, R.A., 1973. Astron. Astrophys. 24, 337. Shakura, N.I., Sunyaev, R.A., 1976. MNRAS 175, 613. Shapiro, S. et al., 1976. Astrophys. J. 204, 187. Shrader, C. et al., 1994. In: Fichtel, C. (Ed.), AIP Conf. Proc. No. 304, AIP, New York, p. 365. Sheth, S. et al., 1996. Astrophys. J. 468, 755. Shu, F., 1976. In: Eggleton, P. (Ed.), Structure and Evolution of Close Binary Systems, Reidel, Dordrecht, p. 253. Skibo, J., Dermer, C.D., 1995. Astrophys. J. 455, L25. Stark, R.F., Connors, P.A., 1977. Nature 266, 429. Steinle, H. et al., 1982. Astrophys. J. 107, 350. Stern, B.E. et al., 1995. Astrophys. J. 449, L13. Stern, B.E. et al., 1996, Astrophys. J., to appear. Sunyaev, R.A., Titarchuk, L.G., 1980. Astron. Astrophys. 86, 121. Sunyaev, R.A., Trumper, J., 1979. Nature 179, 506. Sunyaev, R.A. et al., 1994. Sov. Astron. Lett. 20, 777. Svensson, R., 1982. Astrophys. J. 258, 321; Svensson, R., 1982. Astrophys. J. 258, 335. Svensson, R., 1984. Mon. Not. Roy. Astron. Soc. 209, 175. Svensson, R., Zdziarski, A.A., 1994. Astrophys. J. 436, 599. Takahara, F. et al., 1989. Astrophys. J. 346, 122. Takahara, F., Kusunose, M., 1985. Prog. Theo. Phys. 73, 1390. Tanaka, Y. et al., 1991. In: Brandt, S. (Ed.), Nova Muacae Workshop Proc., DSRI, Lyngby, p. 125. Tanaka, Y., 1992. In: Makino, F., Nagase, F. (Eds.), Ginga Memorial Symp. Proc., ISAS, Tokyo, p. 19. Tanaka, Y., Lewin, W.H.G., 1995. In: Lewin, W.H.G. et al. (Eds.), X-Ray Binaries, Cambridge, UK, p. 126. Tanaka, Y. et al., 1995. Nature 375, 659. Tanaka, Y., Shibazaki, N., 1996. Ann. Rev. Astron. Astrophys. 34, 607. Tananbaum, H. et al., 1972. Astrophys. J. 177, L5. Tavani, M., Liang, E., 1996. Astron. Astrophys., in press. Titarchuk, L.G., 1994. Astrophys. J. 434, 570. Thompson, K.A., 1981. Ph.D. Thesis, Stanford University. Thorne, K.S., Price, R., 1975. Astrophys. J. Lett. 195, L101. Tuchman, Y. et al., 1990. Astrophys. J. 359, 164. Van Dijk, R. et al., 1994. In: Fichtel, C. et al. (Eds.), AIP Conf. Proc. No. 304, AIP, New York, p. 197. Van Dijk, R. et al., 1995. Astron. Astrophys. 296, L33. Van der Klis, M., 1995. In: Lewin, W.H.G. et al. (Eds.), X-Ray Binaries, Cambridge, Cambridge, p. 252. van Paradijs, J. McClintock, J.E., 1995. In: Lewin, W.H.G. et al., (Eds.), X-ray Binaries, Cambridge, Cambridge, p. 58. van Paradijs, J., 1996. Talk given in Aspen Winter Astrophysics Workshop. Vishniac, E.T., Diamond, P.H., 1992. Astrophys. J. 398, 561. Vishniac, E.T., Diamond, P.H., 1993. In: Wheeler, J.C. (Ed.), Accretion Disks in Compact Stellar Systems, World Scientific, Singapore, p. 41. Wagner, et al., 1994. In: Holt, S. and Day, C. (Eds.), AIP Conf. Proc. No. 308, AIP, New York, p. 111. Wandel, A., Liang, E., 1991. Astrophys. J. 380, 84. Webster, B.L., Murdin, P., 1972. Nature 235, 37. Weisskopf, M.C. et al., 1975. Astrophys. J. 199, L147.

142

E.P. Liang / Physics Reports 302 (1998) 67—142

White, T., Lightman, A., 1989. Astrophys. J. 340, 1024. White, N.E., 1994. In: Holt, S. and Day, C. (Eds.), AIP Conf. Proc. No. 308, AIP, New York, p. 53. Wood, K.S. et al., 1984. Astrophys. J. (Suppl.) 56, 507. Yang, R. et al., 1994. In: Fichtel, C. (Ed.), AIP Conf. Proc. No. 304, AIP, New York, p. 395. Zdziarski, A., 1984. Astrophys. J. 283, 842. Zdziarski, A. et al., 1993. Astrophys. J. 414, L93. Zhang, S.N. et al., 1996a. Astron. Astrophys. (Suppl.) 120, C279. Zhang, S.N. et al., 1996b. IAUC 6447 & 6462. Zhang, S.N. et al., 1997a. Astrophys. J. 479, in press. Zhang, S.N. et al., 1997b. Astrophys. J. Lett. 477, in press. Zhang, S.N. et al., 1997c. Astrophys. J. Lett., submitted. Znajek, R.L., 1978. Mon. Not. Roy. Astron. Soc. 185, 833.

Physics Reports 302 (1998) 143—209

The large-N expansion of unitary-matrix models Paolo Rossi, Massimo Campostrini*, Ettore Vicari Dipartimento di Fisica dell+Universita% and I.N.F.N., I-56126 Pisa, Italy Received October 1996; editor: R. Petronzio

Contents 1. Introduction 1.1. General motivation for the 1/N expansion 1.2. Large N as a thermodynamical limit: factorization 1.3. 1/N expansion of vector models in statistical mechanics and quantum field theory 1.4. 1/N expansion of matrix models: planar diagrams 1.5. The physical interpretation: QCD phenomenology 1.6. The physical interpretation: twodimensional quantum gravity 2. Unitary matrices 2.1. General features of unitary-matrix models 2.2. Chiral models and lattice gauge theories 2.3. Schwinger—Dyson equations in the large-N limit 2.4. Survey of different approaches 3. The single-link integral 3.1. The single-link integral in external field: finite-N solution 3.2. The external field problem: large-N limit 3.3. The properties of the determinant 3.4. Applications to mean field and strong coupling 3.5. The single-link integral in the adjoint representation 4. Two-dimensional lattice Yang—Mills theory 4.1. Two-dimensional Yang—Mills theory as a single-link integral

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4.2. The Schwinger—Dyson equations of the two-dimensional Yang—Mills theory 4.3. Large-N properties of the determinant 4.4. Local symmetry breaking in the large-N limit 4.5. Evaluation of higher-order corrections 4.6. Mixed-action models for lattice YM 2 4.7. Double-scaling limit of the single-link integral 4.8. The character expansion and its large-N limit: SU(N) vs. U(N) 5. Chiral chain models and gauge theories on polyhedra 5.1. Introduction 5.2. Saddle-point equation for chiral ¸-chains 5.3. The large-N limit of the three-link chiral chain 5.4. The large-N limit of the four-link chiral chain 5.5. Critical properties of chiral chain models with ¸44 5.6. Strong-coupling expansion of chiral chain models 6. Simplicial chiral models 6.1. Definition of the models 6.2. Saddle-point equation for simplicial chiral models 6.3. The large-N d"4 simplicial chiral model 6.4. The large-d limit 6.5. The large-N criticality of simplicial models

* Corresponding author. E-mail: [email protected].

0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 0 3 - 9

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THE LARGE-N EXPANSION OF UNITARY-MATRIX MODELS

Paolo ROSSI, Massimo CAMPOSTRINI, Ettore VICARI Dipartimento di Fisica dell+Universita% and I.N.F.N., I-56126 Pisa, Italy

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

P. Rossi et al. / Physics Reports 302 (1998) 143–209 6.6. The strong-coupling expansion of simplicial models 7. Asymptotically free matrix models 7.1. Two-dimensional principal chiral models

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7.2. Principal chiral models on the lattice 7.3. The large-N limit of SU(N) lattice gauge theories References

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Abstract The general features of the 1/N expansion in statistical mechanics and quantum field theory are briefly reviewed both from the theoretical and from the phenomenological point of view as an introduction to a more detailed analysis of the large-N properties of spin and gauge models possessing the symmetry group SU(N)]SU(N). An extensive discussion of the known properties of the single-link integral (equivalent to YM and one-dimensional 2 chiral models) includes finite-N results, the external field solution, properties of the determinant, and the double scaling limit. Two major classes of solvable generalizations are introduced: one-dimensional closed chiral chains and models defined on a d!1 dimensional simplex. In both cases, large-N solutions are presented with emphasis on their double scaling properties. The available techniques and results concerning unitary-matrix models that correspond to asymptotically free quantum field theories (two-dimensional chiral models and four-dimensional QCD) are discussed, including strongcoupling methods, reduced formulations, and the Monte Carlo approach. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 11.10.Kk; 11.15.Pg; 64.60.Fr; 75.10.Hk

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It will be good enough that these (results) be judged useful by those willing to find the truth of the facts. ¹hey were put together as a lasting possession, rather than in competition for temporary audience. [Thucydides, ¹he Peloponnesian ¼ar, I, 22]

1. Introduction 1.1. General motivation for the 1/N expansion The approach to quantum field theory and statistical mechanics based on the identification of the large-N limit and the perturbative expansion in powers of 1/N, where N is a quantity related to the number of field components, is by now almost thirty years old. It goes back to the original work by Stanley [1] on the large-N limit of spin systems with O(N) symmetry, soon followed by Wilson’s suggestion that the 1/N expansion may be a valuable alternative in the context of renormalizationgroup evaluation of critical exponents, and by ’t Hooft’s extension [2] to gauge theories and, more generally, to fields belonging to the adjoint representation of SU(N) groups. More recently, the large-N limit of random-matrix models was put into a deep correspondence with the theory of random surfaces, and therefore it became relevant to the domain of quantum gravity. In order to understand why the 1/N expansion should be viewed as a fundamental tool in the study of quantum and statistical field theory, it is worth emphasizing a number of relevant features: 1. N is an intrinsically dimensionless parameter, representing a dependence whose origin is basically group-theoretical, and leading to well-defined field representations for all integer values, hence it is not subject to any kind of renormalization; 2. N does not depend on any physical scale of the theory, hence we may expect that physical quantities should not show any critical dependence on N (with the possible exception of finite-N scaling effects in the double-scaling limit); 3. the large-N limit is a thermodynamical limit, in which we observe the suppression of fluctuations in the space of internal degrees of freedom; hence we may expect notable simplifications in the algebraic and analytical properties of the model, and even explicit integrability in many instances. Since integrability does not necessarily imply triviality, the large-N solution to a model may be a starting point for finite-N computations, because it shares with interesting finite values of N many physical properties. (This is typically not the case for the standard free-field solution which forms the starting point for the usual perturbative expansions.) Moreover, for reasons which are clearly, if not obviously, related to the three points above, the physical variables which are naturally employed to parameterize large-N results and 1/N expansions are usually more directly related to the observables of the models than the fields appearing in the original local Lagrangian formulation. More reasons for a deep interest in the study of the large-N expansion will emerge from the detailed discussion we shall present in the rest of this introductory section. We must however anticipate that many interesting review papers have been devoted to specific issues in the context of

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the large-N limit, starting from Coleman’s lectures [3], going through Yaffe’s review on the reinterpretation of the large-N limit as classical mechanics [4], Migdal’s review on loop equations [5], and Das’ review on reduced models [6], down to Polyakov’s notes [7] and to the recent large commented collection of original papers by Brezin and Wadia [8], not to mention Sakita’s booklet [9] and Ma’s contributions [10,11]. Moreover, the 1/N expansion of two-dimensional spin models has been reviewed by two of the present authors a few years ago [12]. As a consequence, we decided to devote only a bird’s eye overview to the general issues, without pretension of offering a selfcontained presentation of all the many conceptual and technical developments that have appeared in an enormous and ever-growing literature; we even dismissed the purpose of offering a complete reference list grouped by arguments, because the task appeared to be beyond our forces. We preferred to focus on a subset of all large-N topics, which has never been completely and systematically reviewed: the issue of unitary-matrix models. Our self-imposed limitation should not appear too restrictive, when considering that it still involves such topics as U(N)]U(N) principal chiral models, virtually all that concerns large-N lattice gauge theories, and an important subset of random-matrix models with their double-scaling limit properties, related to two-dimensional conformal field theory. The present paper is organized on a logical basis, which will neither necessarily respect the sequence of chronological developments, nor it will keep the same emphasis that was devoted by the authors of the original papers to the discussion of the different issues. Section 2 is devoted to a presentation of the general and common properties of unitary-matrix models, and to an analysis of the different approaches to their large-N solution that have been discussed in the literature. Section 3 is a long and quite detailed discussion of the most elementary of all unitary-matrix systems. Since all essential features of unitary-matrix models seem to emerge already in the simplest example, we thought it worthwhile to make this discussion as complete and as illuminating as possible. Section 4 is an application of results obtained by studying the single-link problem, which exploits the equivalence of this model with lattice YM and principal chiral models in one 2 dimension. Section 5 is devoted to a class of reasonably simple systems, whose physical interpretation is that of closed chiral chains as well as of gauge theories on polyhedra. Section 6 presents another class of integrable systems, corresponding to chiral models defined on a d-dimensional simplex, whose properties are relevant both in the discussion of the strongcoupling phase of more general unitary-matrix models and in the context of random-matrix models. Section 7 deals with the physically more interesting applications of unitary-matrix models: two-dimensional principal chiral models and four-dimensional lattice gauge theories, sharing the properties of asymptotic freedom and “confinement” of the Lagrangian degrees of freedom. Special issues, like numerical results and reduced models, are considered. 1.2. Large N as a thermodynamical limit: factorization As we already mentioned briefly in the introduction, one of the peculiar features of the large-N limit is the occurrence of notable simplifications, that become apparent at the level of the quantum

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equations of motion, and tend to increase the degree of integrability of the systems. These simplifications are usually related to a significant reduction of the number of algebraically independent correlation functions, which in turn is originated by the property of factorization. This property is usually stated as follows: connected Green’s functions of quantities that are invariant under the full symmetry group of the system are suppressed with respect to the corresponding disconnected parts by powers of 1/N. Hence when NPR one may replace expectation values of products of invariant quantities with products of expectation values. One must however be careful, since factorization is not a property shared by all invariant operators without further qualifications. In particular, experience shows that operators associated with very high rank representations of the symmetry group, when the rank is O(N), do not possess the factorization property. A very precise characterization has been given by Yaffe [4], who showed that factorization is a property of “classical” operators, i.e., those operators whose coherent state matrix elements have a finite NPR limit. It is quite interesting to investigate the physical origin of factorization. The property lim SABT"SATSBT N?= implies in particular that

(1.1)

lim SA2T"SAT2 , (1.2) N?= i.e., the vacuum state of the model, seen as a statistical ensemble, seems to possess no fluctuations. To be more precise, all the field configurations that correspond to a nonvanishing vacuum wavefunction can be related to each other by a symmetry transformation. This residual infinite degeneracy of the vacuum configurations makes the difference between the large-N limit and a strictly classical limit +P0, and allows the possibility of violations of factorization when infinite products of operators are considered; this is in a sense the case with representations whose rank is O(N). More properly, we may view large N as a thermodynamical limit [13], since the number of degrees of freedom goes to infinity faster than any other physical parameter, and as a consequence the “macroscopic” properties of the system, i.e., the invariant expectation values, are fixed in spite of the great number of different “microscopic” realizations. This realization does not rule out the possibility of searching for the so-called “master field”, that is a representative of the equivalence class of the field configurations corresponding to the large-N vacuum, such that all invariant expectation values of the factorized operators can be obtained by direct substitution of the master field value into the definition of the operators themselves [3]. There has been an upsurge of interest on master fields in recent years [14,15], triggered by new results in non-commutative probability theory applied to the stochastic master field introduced in Ref. [16]. 1.3. 1/N expansion of vector models in statistical mechanics and quantum field theory The first and most successful application of the approach based on the large-N limit and the 1/N expansion to field theories is the analysis of vector models enjoying O(N) or SU(N) symmetry.

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Actually, “vector models” is a nickname for a wide class of different field theories, characterized by bosonic or fermionic Lagrangian degrees of freedom lying in the fundamental representation of the symmetry group (cf. Ref. [12] and references therein). A quite general feature of these models is the possibility of expressing all self-interactions of the fundamental degrees of freedom by the introduction of a Lagrange multiplier field, a boson and a singlet of the symmetry group, properly coupled to the Lagrangian fields, such that the resulting effective Lagrangian is quadratic in the N-component fields. One may therefore formally perform the Gaussian integration over these fields, obtaining a form of the effective action which is nonlocal, but depends only on the singlet multiplier, acting as a collective field; in this action N appears only as a parameter. The considerations developed in Section 1.2 make it apparent that all fluctuations of the singlet field must be suppressed in the large-N limit (no residual degeneracy is left in the trivial representation). As a consequence, solving the models in this limit simply amounts to finding the singlet field configuration minimizing the effective action. The problem of nonlocality is easily bypassed by the consideration that translation invariance of the physical expectation values requires the actionminimizing field configuration to be invariant in space-time; hence the saddle-point equations of motion become coordinate-independent and all nonlocality disappears. As one may easily argue from the above considerations, the large-N solution of vector models describes some kind of Gaussian field theory. Nevertheless, this result is not as trivial as one might imagine, since the free theory realization one is faced with usually enjoys quite interesting properties, in comparison with the naı¨ ve Lagrangian free fields. Typical phenomena appearing in the large-N limit are an extension of the symmetry and spontaneous mass generation. Moreover, when the fundamental fields possess some kind of gauge symmetry, one may also observe dynamical generation of propagating gauge degrees of freedom; this is the case with twodimensional CPN~1 models and their generalizations [17,18]. The existence of an explicit form of the effective action offers the possibility of a systematic expansion in powers of 1/N. The effective vertices of the theory turn out to be Feynman integrals over a single loop of the free massive propagator of the fundamental field. In two dimensions, where the physical properties of many vector models are especially interesting (e.g., asymptotic freedom), these one-loop integrals can all be computed analytically in the continuum version, and even on the lattice many analytical results have been obtained. The 1/N expansion is the starting point for a systematic computation of critical exponents, which are nontrivial in the range 2(d(4, for the study of renormalizability of superficially nonrenormalizable theories in the same dimensionality range, and for the computation of physical amplitudes. Notable is the case of the computation of amplitude ratios, which are independent of the coupling in the scaling region, and therefore are functions of 1/N alone; hopefully, their 1/N expansion possesses a nonvanishing convergence radius. The 1/N expansion was also useful to explore the double-scaling limit properties of vector models [19—21]. The properties of the large-N limit and of the 1/N expansion of continuum and lattice vector models were already reviewed by many authors. We therefore shall not discuss this topic further. We only want to stress that this kind of studies can be very instructive, given the physical interest of vector models as realistic prototypes of critical phenomena in two and three dimensions and as models for dynamical Higgs mechanism in four dimensions. Moreover, some of the dynamical properties emerging mainly from the large-N studies of asymptotically free models (in two

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dimensions) may be used to mimic some of the features of gauge theories in four dimensions; however, at least one of the essential aspects of gauge theories, the presence of matrix degrees of freedom (fields in the adjoint representation), cannot be captured by any vector model. 1.4. 1/N expansion of matrix models: planar diagrams The first major result concerning the large-N limit of matrix-valued field theories was due to G. ’t Hooft, who made the crucial observation that, in the 1/N expansion of continuum gauge theories, the set of Feynman diagrams contributing to any given order admits a simple topological interpretation. More precisely, by drawing the U(N) fundamental fields (“quarks”) as single lines and the U(N) adjoint fields (“gluons”) as double lines, each line carrying one color index, a graph corresponding to a nth-order contribution can be drawn on a genus n surface (i.e., a surface possessing n “holes”). In particular, the zeroth-order contribution, i.e., the large-N limit, corresponds to the sum of all planar diagrams. The extension of this topological expansion to gauge models enjoying O(N) and Sp(2N) symmetry has been described by Cicuta [22]. Large-N universality among O(N), U(N), and Sp(2N) lattice gauge theories has been discussed by Lovelace [23]. This property has far-reaching consequences: it allows for reinterpretations of gauge theories as effective string theories, and it offers the possibility of establishing a connection between matrix models and the theory of random surfaces, which will be exploited in the study of the doublescaling limit. As a byproduct of this analysis, ’t Hooft performed a summation of all planar diagrams in two-dimensional continuum Yang—Mills theories, and solved QCD to leading nontrivial order in 2 1/N, finding the meson spectrum [24,25]. Momentum-space planarity has a coordinate-space counterpart in lattice gauge theories. It is actually possible to show that, within the strong-coupling expansion approach, the planar diagrams surviving in the large-N limit can be identified with planar surfaces built up of plaquettes by gluing them along half-bonds [26—28]. This construction however leads quite far away from the simplest model of planar random surfaces on the lattice originally proposed by Weingarten [29,30], and hints at some underlying structure that makes a trivial free-string interpretation impossible. 1.5. The physical interpretation: QCD phenomenology The sum of the planar diagrams has not till now been performed in the physically most interesting case of four-dimensional SU(N) gauge theories. It is therefore strictly speaking impossible to make statements about the relevance of the large-N limit for the description of the physically relevant case N"3. However, it is possible to extract from the large-N analysis a number of qualitative and semi-quantitative considerations leading to a very appealing picture of the phenomenology predicted by the 1/N expansion of gauge theories. These predictions can be improved further by adopting Veneziano’s form of the large-N limit [31], in which not only the number of colors N but also the number of flavors N is set to infinity, while their ratio N/N is kept finite. We f f shall not enter a detailed discussion of large-N QCD phenomenology, but it is certainly useful to quote the relevant results.

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1.5.1. The large-N property of mesons Mesons are stable and noninteracting; their decay amplitudes are O(N~1@2), and their scattering amplitudes are O(N~1). Meson masses are finite. The number of mesons is infinite. Exotics are absent and Zweig’s rule holds. 1.5.2. The large-N property of glueballs Glueballs are stable and noninteracting, and they do not mix with mesons; a vertex involving k glueballs and n mesons is O(N1~k~n@2). The number of glueballs is infinite. 1.5.3. The large-N property of baryons A large-N baryon is made out of N quarks, and therefore it possesses peculiar properties, similar to those of solitons [32]. Baryon masses are O(N). The splitting of excited states is O(1). Baryons interact strongly with each other; typical vertices are O(N). Baryons interact with mesons with O(1) couplings. 1.5.4. The g@ mass formula The spontaneous breaking of the SU(N ) axial symmetry in QCD gives rise to the appearance of f a multiplet of light pseudoscalar mesons. This symmetry-breaking pattern was explicitly demonstrated in the context of large-N QCD by Coleman and Witten [33]. However, the singlet pseudoscalar is not light, due to the anomaly of the U(1) axial current. Since the anomaly equation g2N f Tr FI Fkl ­ J5" k k 16p2 kl

(1.3)

has a vanishing right-hand side in the limit N PR with N and g2N fixed (the standard large-N # f # limit of non-Abelian gauge theories), the leading-order contribution to the mass of the g@ should be O(1/N ). The proportionality constant should be related to the symmetry-breaking term, which in # turn is related to the so-called topological susceptibility, i.e., the vacuum expectation value of the square of the topological charge. The resulting relationship shows a rather satisfactory quantitative agreement with experimental and numerical results [34—38]. 1.6. The physical interpretation: two-dimensional quantum gravity In the last ten years, a new interpretation of the 1/N expansion of matrix models has been put forward. Starting from the relationship between the order of the expansion and the topology of two-dimensional surfaces on which the corresponding diagrams can be drawn, several authors [39—43] proposed that large-N matrix models could provide a representation of random lattice two-dimensional surfaces, and in turn this should correspond to a realization of two-dimensional

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quantum gravity. These results were found consistent with independent approaches, and proper modifications of the matrix self-couplings could account for the incorporation of matter. The functional integrals over two-dimensional closed Riemann manifolds can be replaced by the discrete sum over all (piecewise flat) manifolds associated with triangulations. It is then possible to identify the resulting partition function with the vacuum energy E "!log Z , (1.4) 0 N obtained from a properly defined N]N matrix model, and the topological expansion of twodimensional quantum gravity is nothing but the 1/N expansion of the matrix model. The partition function of two-dimensional quantum gravity is expected to possess well-defined scaling properties [44]. These may be recovered in the matrix model by performing the so-called “double-scaling limit” [45—47]. This limit is characterized by the simultaneous conditions NPR,

gPg , (1.5) # where g is a typical self-coupling and g is the location of some large-N phase transition. The limits # are however not independent. In order to get nontrivial results, one is bound to tune the two conditions (1.5) in such a way that the combination x"(g!g )N2@c1 (1.6) # is kept finite and fixed. c is a computable critical exponent, usually called “string susceptibility”. 1 According to Ref. [44], it is related to the central charge c of the model by 1 c " [25!c#J(1!c)(2!c)] . 1 12

(1.7)

An interesting reinterpretation of the double-scaling limit relates it to some kind of finite-size scaling in a space where N plays the role of the physical dimension ¸ [21,48,49]. Research in this field has exploded in many directions. A wide review reflecting the state of the art as of the year 1993 appeared in the already-mentioned volume by Brezin and Wadia [8]. Here we shall only consider those results that are relevant to our more restricted subject.

2. Unitary matrices 2.1. General features of unitary-matrix models Under the header of unitary-matrix models we class all the systems characterized by dynamical degrees of freedom that may be expressed in terms of the matrix representations of the unitary groups U(N) or special unitary groups SU(N) and by interactions enjoying a global or local U(N) ]U(N) symmetry. Typically we shall consider lattice models, with no restriction on the L R lattice structure and on the number of lattice points, ranging from 1 (single-matrix problems) to infinity (infinite-volume limit) in an arbitrary number of dimensions. In the field-theoretical interpretation, i.e., when considering models in infinite volume and in proximity of a fixed point of some (properly defined) renormalization group transformation, such models will have a continuum counterpart, which in turn shall involve unitary-matrix valued fields

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in the case of spin models, while for gauge models the natural continuum representation will be in terms of hermitian matrix (gauge) fields. A common feature of all unitary-matrix models will be the group-theoretical properties of the functional integration measure: for each dynamical variable the natural integration procedure is based on the left- and right-invariant Haar measure dk(º)"dk(º»)"dk(»º),

P

dk(º)"1 .

(2.1)

An explicit use of the invariance properties of the measure and of the interactions (gauge fixing) can sometimes lead to formulations of the models where some of the symmetries are not apparent. Global U(N) invariance is however always assumed, and the interactions, as well as all physically interesting observables, may be expressed in terms of invariant functions. It is convenient to introduce some definitions and notations. An arbitrary matrix representation of the unitary group U(N) is denoted by D(r)(º). The characters and dimensions of irreducible ab representations are s (º)"D(r)(º) and d respectively. (r) is characterized by two sets of (r) aa (r) decreasing positive integers MlN"l ,2, l and MmN"m ,2, m . We may define the ordered set of 1 4 1 t integers MjN"j ,2, j by the relationships 1 N j "l , (k"1,2, s), j "0, (k"s#1,2, N!t) , k k k j "!m , (k"N!t#1,2, N) . (2.2) k N~k`1 It is then possible to write down explicit expressions for all characters and dimensions, once the eigenvalues exp i/ of the matrix º are known: i detE exp Mi/ (j #N!j)NE i j s (º)" , (2.3) (j) detE exp Mi/ (N!j)NE i < (j !j #j!i) j d " i:j i "s (1) . (2.4) (j) (j) < ( j!i) i:j The general form of the orthogonality relations is

P

1 dk(º) D(r)(º) D(s) *(º)" d d d . (2.5) ab cd d r,s a,c b,d (r) Further relations can be found in Ref. [50]. The matrix º itself coincides with the fundamental representation (1) of the group, and enjoys ab the properties s (º)"Tr º, (1)

(2.6) + º º* "d . bc ab ac a The measure dk(º) (which we shall also denote simply by dº), when the integrand depends only on invariant combinations, may be expressed in terms of the eigenvalues [51]. d "N, (1)

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2.2. Chiral models and lattice gauge theories Unitary matrix models defined on a lattice can be divided into two major groups, according to the geometric and algebraic properties of the dynamical variables: when the fields are defined in association with lattice sites, and the symmetry group is global, i.e., a single U(N) ]U(N) L R transformation is applied to all fields, we are considering a spin model (principal chiral model); in turn, when the dynamical variables are defined on the links of the lattice and the symmetry is local, i.e., a different transformation for each site of the lattice may be performed, we are dealing with a gauge model (lattice gauge theory). As we shall see, these two classes are not unrelated to each other: an analogy between d-dimensional chiral models and 2d-dimensional gauge theories can be found according to the following correspondence table [52]: spin

gauge

site, link

link, plaquette

loop

surface

length

area

mass

string tension

two-point correlation

Wilson loop

While this correspondence in arbitrary dimensions is by no means rigorous, there is some evidence supporting the analogy. In the case d"1, which we shall carefully discuss later, one can prove an identity between the partition function (and appropriate correlation functions) of the two-dimensional lattice gauge theory and the corresponding quantities of the one-dimensional principal chiral model. Both theories are exactly solvable, both on the lattice and in the continuum limit, and the correspondence can be explicitly shown. Approximate real-space renormalization recursion relations obtained by Migdal [53] are identical for d-dimensional chiral models and 2d-dimensional gauge models. The two-dimensional chiral model and the (phenomenologically interesting) four-dimensional non-Abelian gauge theory share the property of asymptotic freedom and dynamical generation of a mass scale. In both models these properties are absent in the Abelian case (XY model and U(1) gauge theory respectively), which shows no coupling-constant renormalization in perturbation theory. The structure of the high-temperature expansion and of the Schwinger—Dyson equations is quite similar in the two models. It will be especially interesting for our purposes to investigate the Schwinger—Dyson equations of unitary-matrix models and discuss the peculiar properties of their large-N limit. 2.3. Schwinger—Dyson equations in the large-N limit In order to make our analysis more concrete, we must at this stage consider specific forms of interactions among unitary matrices, both in the spin and in the gauge models. The most dramatic restriction that we are going to impose on the lattice action is the condition of considering only

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nearest-neighbor interactions. The origin of this restriction is mainly practical, because non nearest-neighbor interactions lead to less tractable problems. We assume that, for the systems we are interested in, it will always be possible to find a lattice representation in terms of nearestneighbor interactions within the universality class. Let us denote by x an arbitrary lattice site, and by x, k an arbitrary lattice link originating in the site x and ending in the site x#k: k is one of the d positive directions in a d-dimensional hypercubic lattice. A plaquette is identified by the label x, k, l, where the directions k and l (kOl) specify the plane where the plaquette lies. The dynamical variables (which we label by º in the general case) are site variables º in spin models and link variables º in gauge models. x x, k The general expression for the partition function is

P

Z"
(2.7)

where b is the inverse temperature (inverse coupling) and the integration is extended to all dynamical variables. The action S(º) must be a function enjoying the property of extensivity and of (global and local) group invariance, and respect the symmetry of the lattice. Adding the requisite that the interactions involve only nearest neighbors, we find that a generic contribution to the action of spin models must be proportional to + s (º ºs )#h.c. , (r) x x`k x, k and for gauge models to

(2.8)

+ s (º º ºs ºs )#h.c. , (2.9) (r) x, k x`k,l x`l,k x,l x, k,l where (r) is in principle arbitrary, and the summation is extended to all oriented links of the lattice in the spin case, to all the oriented plaquettes in the gauge case. In practice we shall mostly focus on the simplest possible choice, corresponding to the fundamental representation. In order to reflect the extensivity of the action, i.e., the proportionality to the number of space and internal degrees of freedom, it will be convenient to adopt the normalizations S(º)"! + N(Tr º ºs #h.c.) (spin) , x x`k x, k

(2.10)

S(º)"! + N(Tr º º ºs ºs #h.c.) (gauge) . (2.11) x, k x`k,l x`l,k x,l x, k,l Once the lattice action is fixed, it is easy to obtain sets of Schwinger—Dyson equations relating the correlation functions of the models. These are the quantum field equations and solving them corresponds to finding a complete solution of a model. It is extremely important to notice the simplifications occurring in the Schwinger—Dyson equations when the large-N limit is considered. These simplifications are such to allow, in selected cases, explicit solutions to the equations. Before proceeding to a derivation of the equations, we must preliminarily identify the sets of correlation functions we are interested in. For obvious reasons, these correlations must involve the dynamical fields at arbitrary space distances, and must be invariant under the symmetry group of

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the model. Without pretending to achieve full generality, we may restrict our attention to such typical objects as the invariant correlation functions of a spin model

T

1 n G(n)(x , y ,2, x , y )" Tr < º iºsi x y 1 1 n n N i/1

U

(2.12)

and to the so-called Wilson loops of a gauge model

T

U

1 ¼(C)" Tr < º , l N l|C

(2.13)

where C is a closed arbitrary walk on the lattice, and < C is the ordered product over all the links l| along the walk. It is worth stressing that the action itself is a sum of elementary Green’s functions (elementary Wilson loops). More general invariant correlation functions may involve expectation values of products of invariant operators similar to those appearing in the r.h.s. of Eqs. (2.12) and (2.13). The already mentioned property of factorization allows us to express the large-N limit expectation value of such products as a product of expectation values of the individual operators. As a consequence, the large-N form of the Schwinger—Dyson equations is a (generally infinite) set of equations involving only the above-defined quantities. For sake of clarity and completeness, we present the explicit large-N form of the Schwinger—Dyson equations for the models described by the standard actions Eqs. (2.10) and (2.11). For principal chiral models [54], 0"G(n)(x , y ,2, x , y ) 1 1 n n # b+ [G(n`1)(x , x #k, x , y ,2, x , y )!G(n)(x #k, y ,2, x , y )] 1 1 1 1 n n 1 1 n n k n # + [d 1 4 G(s~1)(x , y ,2, x , y ) G(n~s`1)(x , y ,2, x , y ) x ,x 1 1 4~1 4~1 4 4 n n 4/2 !d 1 4 G(s)(x , y ,2, x , y ) G(n~s)(x , y ,2, x , y )] . 1 1 4 4 4`1 4`1 n n x ,y

(2.14)

For lattice gauge theories [55,56],

C

D

) " + d ¼(C ) ¼(C ) , b + ¼(C )!¼(C x, kl x~k,kl x,y x,y y,x k y|C

(2.15)

where ¼(C ) is obtained by replacing º with º º ºs in the loop C, and C , C x, kl x,l x, k x`k,l x`l,k x,y y,x are the sub-loops obtained by splitting C at the intersection point, including the “trivial” splitting. Eqs. (2.15) are commonly known as the lattice Migdal—Makeenko equations. The derivation of the Schwinger—Dyson equations is obtained by performing infinitesimal variations of the integrand in the functional integral representation of expectation values and exploiting invariance of the measure.

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2.4. Survey of different approaches Schwinger—Dyson equations are the starting point for most techniques aiming at the explicit evaluation of large-N vacuum expectation values for nontrivial unitary-matrix models. The form exhibited in Eqs. (2.14) and (2.15) involves in principle an infinite set of variables, and it is therefore not immediately useful to the purpose of finding explicit solutions. Successful attempts to solve large-N matrix systems have in general been based on finding reformulations of Schwinger—Dyson equations involving more restricted sets of variables and more compact representations (collective fields). As a matter of fact, in most cases it turned out to be convenient to define generating functions, whose moments are the correlations we are interested in, and whose properties are usually related to those of the eigenvalue distributions for properly chosen covariant combinations of matrix fields. By “covariant combination” we mean a matrix-valued variable whose eigenvalues are left invariant under a general SU(N)]SU(N) transformation of the Lagrangian fields. Such objects are typically those appearing in the r.h.s. of Eqs. (2.12) and (2.13) before the trace operation is performed. Under the SU(N)]SU(N) transformation ºP»º¼s, these operators transform accordingly to OP»O»s, and therefore their eigenvalue spectrum is left unchanged. Without belaboring on the details (some of which will however be exhibited in the discussion of the single-link integral presented in Section 3), we only want to mention that the approach based on extracting appropriate Schwinger—Dyson equations for the generating functions is essentially algebraic in nature, involving weighted sums of infinite sets of equations in the form (2.14) or (2.15), identification of the relevant functions, and resolution of the resulting algebraic equations, where usually a number of free parameters appear, whose values are fixed by boundary and/or asymptotic conditions and analyticity constraints. The approach based on direct replacement of the eigenvalue distributions in the functional integral and the minimization of the resulting effective action leads in turn to integral equations which may be solved by more or less straightforward techniques. These two approaches are however intimately related, since the eigenvalue density is usually connected with the discontinuity along some cut in the complex-plane extension of the generating function, and one may easily establish a step-by-step correspondence between the algebraic and functional approach. Let us finally mention that the procedure based on introducing invariant degrees of freedom and eigenvalue density operators has been formalized by Jevicki and Sakita [57,58] in terms of a “quantum collective field theory”, whose equations of motion are the Schwinger—Dyson equations relevant to the problem at hand. A quite different application of the Schwinger—Dyson equations is based on the strong-coupling properties of the correlation functions. In the strong-coupling domain, expectation values are usually analytic in the coupling b within some positive convergence radius, and their boundary value at b"0 can easily be evaluated. As a consequence, it is formally possible to solve Eqs. (2.14) and (2.15) in terms of strong-coupling series by sheer iteration of the equations. This procedure may in practice turn out to be too cumbersome for practical purposes; however, in some circumstances, it may lead to rather good approximations [59,60] and even to a complete strong-coupling solution. Continuation to the weak-coupling domain is however a rather nontrivial task. As a special application of the strong-coupling approach, we must mention the attempt (pioneered by Kazakov et al. [61]) to construct an effective action for the invariant degrees of

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freedom by means of a modified strong-coupling expansion, and explore the weak-coupling regime by solving the saddle-point equations of the resulting action. This technique might be successful at least in predicting the location and features of the large-N phase transition which is relevant to many physical problems, as mentioned in Section 1. A numerical approach to large-N lattice Schwinger—Dyson equations based on the minimization of an effective large-N Fokker—Plank potential and suited for the weak-coupling regime was proposed by Rodrigues [62]. Another relevant application of the Schwinger—Dyson equations is found in the realm of the so-called “reduced” models. These models, whose prototype is the Eguchi—Kawai formulation of strong-coupling large-N lattice gauge theories [63], are based on the physical intuition that, in the absence of fluctuations, due to translation invariance, the space extension of the lattice must be essentially irrelevant in the large-N limit, since all invariant physics must be already contained in the expectation values of (properly chosen) purely local variables. More precisely, one might say that, when NPR, the SU(N) group becomes so large that it accommodates the full Poincare` group as a subgroup, and in particular it should be possible to find representations of the translation and rotation operators among the elements of SU(N). As a consequence, one must be able to reformulate the full theory in terms of a finite number of matrix field variables defined at a single space—time site (or on the d links emerging from the site in the case of a lattice gauge theory) and of the above-mentioned representations of the translation group. This reformulation is called “twisted Eguchi—Kawai” reduced version of the theory [64,65]. We shall spend a few more words on the reduced models in Section 7. Moreover, a very good review of their properties has already appeared many years ago [6]. In this context, we must only mention that the actual check of validity of the reduction procedure is based on deriving the Schwinger—Dyson equations of the reduced model and comparing them with the Schwinger—Dyson equations of the original model. Usually the equivalence is apparent already at a superficial level when naı¨ vely applying to correlation functions of the reduced model the symmetry properties of the action itself. This procedure however requires some attention, since the limit of infinitely many degrees of freedom within the group itself allows the possibility of spontaneous breakdown of some of the symmetries which would be preserved for any finite value of N. In this context, we recall once more that large N is a thermodynamical limit: N must go to infinity before any other limit is considered, and sometimes the limiting procedures do not commute. It is trivial to recognize that, when the strong-coupling phase is considered, symmetries are unbroken, and the equivalence between original and reduced model may be established without further ado. Problems may occur in the weak-coupling side of a large-N phase transition. An unrelated and essentially numeric approach to solving the large-N limit of lattice matrix models is the coherent state variational algorithm introduced by Yaffe and coworkers [66,67]. We refer to the original papers for a presentation of the results that may be obtained by this approach. 3. The single-link integral 3.1. The single-link integral in external field: finite-N solution All exact and approximate methods of evaluation of the functional integrals related to unitarymatrix models must in principle face the problem of performing the simplest of all relevant

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integrations: the single-link integral. The utmost importance of such an evaluation makes it proper to devote to it an extended discussion, which will also give us the opportunity of discussing in a prototype example the different techniques that may be applied to the models we are interested in. A quite general class of single-link integrals may be introduced by defining

P

Z(AsA)" dº exp [N Tr(Asº#ºsA)] ,

(3.1)

where as usual º is an element of the group U(N) and A is now an arbitrary N]N matrix. The U(N) invariance of the Haar measure implies that the one link integral (3.1) must depend only on the eigenvalues of the Hermitian matrix AsA, which we shall denote by x ,2, x . The function 1 N Z(x ,2, x ) must satisfy a Schwinger—Dyson equation: restricting the variables to the U(N) singlet 1 N subspace, the Schwinger—Dyson equation was shown to be equivalent to the partial differential equation [68,69]

A

B

1 ­2Z 1 ­Z 1 ­Z ­Z x 4 x # # + ! "Z , (3.2) k N2 ­x2 N ­x N2 ­x x !x ­x k 4 k k 4 4Ek k with the boundary condition Z(0,2, 0)"1 and the request that Z be completely symmetric under exchange of the x . i It is convenient to reformulate the equation in terms of the new variables z "2NJx , and to k k parameterize the solution in terms of the completely antisymmetric function ZK (z ,2, z ) by 1 N defining ZK (z) Z(z)" . < (z2!z2) i:j i j The equation satisfied by ZK can be shown to reduce to

C

(3.3)

D

­2 ­ 2 + z2 #(3!2N)+ z !+ z2# N(N!1)(N!2) ZK "0 . (3.4) k ­z2 k ­z k 3 k k k k k Eq. (3.4) has the structure of a fermionic many-body Schro¨dinger equation. With some ingenuity it may be solved in the form of a Slater determinant of fermion wavefunctions. In conclusion, we obtain, after proper renormalization [70] (see also [71]),

A

B

N~1 detEzi~1I (z )E j i~1 j , (3.5) < k! detEz2(i~1)E j k/0 where I (z) is the modified Bessel function. Eq. (3.5) is therefore a representation of the single-link i integral in external field for arbitrary U(N) groups. By taking proper derivatives with respect to its arguments one may in principle reconstruct all the cumulants for the group integration of an arbitrary string of (uncontracted) matrices [72,73]. Some special limits of the general expression (3.5) may prove useful. Let us first of all consider the case when A is proportional to the identity matrix: A"a1 and therefore z "2Na and i Z(2Na,2, 2Na)"detEI (2Na)E . (3.6) i~j Z(z ,2, z )"2N(N~1)@2 1 N

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As we shall see, this is exactly Bars’ and Green’s solution for U(N) lattice gauge theory in two dimensions [74]. When only one eigenvalue of A is different from zero the result is Z(2Na, 0,2, 0)"(N!1)! (Na)1~N I (2Na) . N~1 The large-N limit will be discussed in the next subsection.

(3.7)

3.2. The external field problem: large-N limit For our purposes it is extremely important to extract the limiting form of Eq. (3.5) when NPR. In principle, it is a very involved problem, since the dependence on N comes not only through the z but also from the dimension of the matrices whose determinant we must evaluate. It i is however possible to obtain the limit, either by solving separately the large-N version of Eq. (3.2), or by directly manipulating Eq. (3.5). In the first approach, we introduce the large-N parameterization Z" exp N¼ ,

(3.8)

where ¼ is now proportional to N; we then obtain from Eq. (3.2), dropping second-derivative terms that are manifestly depressed in the large-N limit [69],

A B

A

B

­¼ 2 ­¼ 1 ­¼ ­¼ x 4 x # # + ! "1 . (3.9) k ­x ­x N x !x ­x ­x k k 4 k 4 k 4Ek It is possible to show that in the large-N limit Eq. (3.9) admits solutions, which can be parameterized by the expression

C

D

­¼ 1 1 1 " 1! + , c50 . (3.10) ­x 2N k Jxk#c 4 Jxk#c#Jx4#c Substitution of Eq. (3.10) into Eq. (3.9) and some algebraic manipulation lead to the consistency condition

C

D

1 1 + !1 "0 , 2N Jx #c 4 4 which in turn admits two possible solutions: (a) c is determined by the condition c

(3.11)

1 1 + "1 , (3.12) 2N Jx #c 4 4 implying c41; this is a “strong coupling” phase, requiring that the eigenvalues satisfy the bound 4 1 1 + 51 , (3.13) 2N Jx 4 4

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i.e., at least some of the x are sufficiently small; 4 (b) when 1 1 + 41 , (3.14) 2N Jx 4 4 then the solution corresponds to the choice c"0; this is a “weak coupling” phase, and all eigenvalues are large enough. Direct integration of Eq. (3.10) with proper boundary conditions leads to the large-N result [69] 1 3 ¼(x)"2+ Jx #c! + log(Jx #c#Jx #c)!Nc! N , (3.15) k k 4 2N 4 k k,s which must be supplemented with Eq. (3.12) in the strong-coupling regime (3.13), while c"0 reproduces the weak-coupling result by Brower and Nauenberg. Amazingly enough, setting c"0 in Eq. (3.15) one obtains the naı¨ ve one-loop estimate of the functional integral, which turns out to be exact in this specific instance. It is possible to check that Eq. (3.15) is reproduced by carefully taking the large-N limit of Eq. (3.5), which requires use of the following asymptotic limits of Bessel functions [70]

AB

k!

CA S

BD A

B

z2 1~k z2 ~1@4 2 k 1 1# 1# 1# I (z) P k k2 k2 z 2 z?= ]exp(Jk2#z2!k) (strong coupling) ,

1 exp z I (z)+ k J2pz

(weak coupling) .

(3.16) (3.17)

An essential feature of Eq. (3.15) is the appearance of two different phases in the large-N limit of the single-link integral. Such a transition would be mathematically impossible for any finite value of N; however it affects the large-N behavior of all unitary-matrix models and gives rise to a number of interesting phenomena. A straightforward analysis of Eq. (3.15) shows that the transition point corresponds to the condition 1 1 t, + "1 . (3.18) 2N Jx 4 4 It is also possible to evaluate the difference between the strong- and weak-coupling phases of ¼ in the neighborhood of t"1, finding the relationship [69] ¼ !¼ &(t!1)3 . 4530/' 8%!, As a consequence, we may classify this phenomenon as a “third order phase transition”.

(3.19)

3.3. The properties of the determinant The large-N factorization of invariant amplitudes is a well-established property of products of operators defined starting from the fundamental representation of the symmetry group. Operators

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corresponding to highly nontrivial representations may show a more involved pattern of behavior in the large-N limit. Especially relevant from this point of view are the properties of determinants of covariant combinations of fields [52,75]; we will consider the quantities D(x)"det[º ºs] 0 x for lattice chiral models and

(3.20)

D(C)"det < º (3.21) l C l| for lattice gauge theories. The expectation values of these operators may act as an order parameter for the large-N phase transition characterizing the class of models we are taking into consideration. Indeed the determinant picks up the phase characterizing the U(1) subgroup that constitutes the center of U(N). Moreover, since SU(N) U(N)+U(1)] , Z N SU(N)PU(N) as NPR because Z PU(1); therefore the determinant of the U(N) theory in the N large-N limit reflects properties of the center of SU(N). In lattice models this Abelian U(1) subgroup is not decoupled, as it happens in the continuum theory, and therefore SDT does not in general have on the lattice the free-theory behavior it has in the continuum. The basic properties of the determinant may be explored by focusing once more on the external field problem we discussed above. Let us introduce a class of determinant operators, and define their expectation values as [76] :dº det ºl exp[N Tr(ºsA#Asº)] D(l)"Sdet ºlT" . :dº exp[N Tr(ºsA#Asº)]

(3.22)

In order to parameterize the SU(N) external-source integral, besides the eigenvalues x of AAs, i a new external parameter must be introduced, that couples to the determinant: i h" (log det As!log det A) . 2N

(3.23)

Because of the symmetry properties, D(l) may only depend on the eigenvalues z and on h. It was found that, when º enjoys U(N) symmetry (with finite N), ZK D(l)"exp(iNlh) l , (3.24) ZK 0 where ZK is the solution of the following Schwinger—Dyson equation, generalizing Eq. (3.4): l ­2 ­ 2 1 + z2 #(3!2N)+ z !+ z2# N(N!1)(N!2) ZK "l2ZK ; (3.25) k ­z2 k­z k 3 l l N k k k k k

C

D

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ZK satisfy the property l @l@ 1 ­ @l@ < ZK " < z ZK "detEzj~1I (z )E . (3.26) k l 0 i j~1~l i z ­z k k k k When the weak-coupling condition t,+ 1/z 41 is satisfied, the leading contribution to the k k large-N limit of all ZK is the same: l 1 (3.27) ZK PZK (=)"exp + z ! + log 2pz # + log(z !z ) . k 2 k i k l k k i:k In order to determine the large-N limit of D(l), one therefore needs to compute the O(1) factor in front of the exponentially growing term (3.27). It is convenient to define

A BA

B

C

D

ZK X" l , l ZK (=)

(3.28)

whose Schwinger—Dyson equation may be extracted from Eq. (3.25) and takes the form

C

A

BD A B

­2X ­X ­X 1 z z ­X l#2+ z2 l# + l! l k i + z2 k k ­z2 ­z ­z N z !z ­z k k kEi k i k i k k

1 " l2! X . 4 l

(3.29)

Let us introduce the large-N Ansatz X "X (t) , l l reducing Eq. (3.29) to

(3.30)

A B

dX 1 1 1 d2X l" l2! X . l#2(t!1) + (3.31) dt 4 l N z2 dt2 k k Removing terms that are depressed by two powers of 1/N, we are left with a consistent equation whose solution is 1

1

X "(1!t)2(l2~4) . l Finally we can compute the weak-coupling large-N limit of D(l):

(3.32)

D(l) P exp(iNlh)(1!t)l2@2, t41 . N?= From the standard strong-coupling expansion we may show that

(3.33)

D(l) P 0 when t51 . (3.34) N?= An explicit evaluation, starting from the exact expression (3.26), expanded in powers of 1/z for k arbitrary N, allows us to show that the quantities ZK may be obtained from Eqs. (3.27) and (3.28) by l expanding Eq. (3.32) up to 2nd order in t with no O(1/N2) corrections. D(l) according to this result violate factorization; in turn, they take the value which would be predicted by an effective Gaussian theory governing the U(1) phase of the field º.

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3.4. Applications to mean field and strong coupling The single-link external-field integral has a natural domain of application in two important methods of investigation of lattice field theories: mean-field and strong-coupling expansion. Extended papers and review articles have been devoted in the past to these topics (cf. Ref. [77] and references therein), and we shall therefore focus only on those results that are specific to the large-N limit and to the 1/N expansion. Let us first address the issue of the mean-field analysis, considering for sake of definiteness the case of d-dimensional chiral models, but keeping in mind that most results can be generalized in an essentially straightforward manner to lattice gauge theories. The starting point of the mean-field technique is the application of the random field transform to the functional integral:

P P

G

Z " dº exp Nb+ Tr(º ºs #º ºs) N n n n`k n`k n n,k

H

G

H

" d» dA exp Nb+ Tr(» »s #» »s)!N+ Tr(A »s#» As) n n n n`k n`k n n n n n n,k n

P

G

H

] dº exp N+ Tr(A ºs#º As) , (3.35) n n n n n n,k where » and A are arbitrary complex N]N matrices. Therefore the integration over º is just n n n the single-link integral we discussed above. As a consequence, the original chiral model is formally equivalent to a theory of complex matrices with effective action 1 ! S (A, »)"b+ Tr(» »s #» »s)!+ Tr(A »s#» As)#+ ¼(A As) . (3.36) n n`k n`k n n n n n n n N %&& n,k n n The leading order in the mean-field approximation is obtained by applying saddle-point techniques to the effective action, assuming saddle-point values of the fields A and » that are n n translation-invariant and proportional to the identity. We mention that, in the case at hand, the large-N saddle-point equations in the weak-coupling phase are A "a"2b dv, n

1 » "v"1! , n 4a

(3.37)

and they are solved by the saddle-point values

A S

B

1 aN "bd 1# 1! , 2bd

S

1 1 1 vN " # 1! , 2 2 2bd

(3.38)

leading to a value of the free and internal energy 1 1 Fd "aN ! log 2aN ! , 2 2 N2¸

1 ­ Fd "vN 2 . 2d ­b N2¸

(3.39)

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The strong-coupling solution is trivial: v"a"0, and there is a first-order transition point at vN "1, aN "1 . (3.40) b d"1, # 2 # 2 # 2 One may also compute the quadratic fluctuations around the mean-field saddle point by performing a Gaussian integral, whose quadratic form is related to the matrix of the second derivatives of ¼ with respect to the fields, and generate a systematic loop expansion in the effective action (3.36), which in turn appears to be ordered in powers of 1/d. Therefore mean-field methods are especially appropriate for the discussion of models in large space dimensions, and not very powerful in the analysis of d"2 models. The very nature of the transition cannot be taken for granted, especially at large N. However, when d53 there is independent evidence of a first-order phase transition for N53. We mention that a detailed mean-field study of SU(N) chiral models in d dimensions appeared in Refs. [78,79]. When willing to extend the mean-field approach, it is in general necessary to find a systematic expansion of the functional ¼(AAs) in the powers of the fluctuations around the saddle-point configurations. Moreover, one may choose to consider not only the large-N value of the functional, but also its expansion in powers if 1/N2, in order to make predictions for large but finite values of N. The expansion of ¼ up to fourth order in the fluctuations was performed in Ref. [80], where 0 explicit analytic results can be found. A technique for the weak-coupling 1/N2 expansion of ¼ can be found in Ref. [81]. We quote the complete O(1/N4) result:

C

D

¼ 1 1 z #z 3 3 1 b! N2#log (1!t)~1@8# (1!t)~3+ " + z ! + log a a 2 N N2 2N 4 27 z3 a a,b a a 1 #O , N6

A B

(3.41)

where t"+ 1/z . Eq. (3.41) can also be expanded in the fluctuations around a saddle-point a a configuration. Extension to SU(N) with large N was also considered. A discussion of large-N mean field for lattice gauge theories can be found in Refs. [79,82—85]. Let us now turn to a discussion of the main features of the large-N strong-coupling expansion. A preliminary consideration concerns the fact that it is most convenient to reformulate the strong-coupling expansion (i.e., the expansion in powers of b) into a character expansion, which is ordered in the number of lattice steps involved in the effective path that can be associated with each nontrivial contribution to the functional integral. The large-N character expansion will be discussed in greater detail in Section 4.8. Here we only want to discuss those features that are common to any attempt aimed at evaluating strong-coupling series for expectation values of invariant operators in the context of U(N) and SU(N) matrix models, with special focus on the large-N behavior of such series. The basic ingredient of strong-coupling computations is the knowledge of the cumulants, i.e., the connected contributions obtained performing the invariant group integration of a string of uncontracted º and ºs matrices. U(N) group invariance insures us that these group integrals can be non-zero only if the same number of º and ºs matrices appear in the integrand. SU(N) is slightly different in this respect, and its peculiarities will be discussed later and are not relevant to the present analysis.

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It was observed a long time ago that the cumulants, whose group structure is that of invariant tensors with the proper number of indices, involve N-dependent numerical coefficients. The asymptotic behavior of these coefficients in the large-N limit was studied first by Weingarten [86]. However, for finite N, the coefficients written as function of N are formally plagued by the so-called DeWit—’t Hooft poles [87], that are singularities occurring for integer values of N. The highest singular value of N grows with the number n of º matrices involved in the integration, and therefore for sufficiently high orders of the series it will reach any given finite value. A complete description of the pole structure was presented in Ref. [72]; not only single poles, but also arbitrary high-order poles appear for large enough n, and analyticity is restricted to N5n. Obviously, since group integrals are well defined for all n and N, this is only a pathology of the 1/N expansion. Finite-N results are finite, but they cannot be obtained as a continuation of a large-N strongcoupling expansion. However, it is possible to show that the strict NPR limit of the series exists, and moreover, for sufficiently small b and sufficiently large N, the limiting series is a reasonable approximation to the true result, all nonanalytic effects being O(b2N) in U(N) models and O(bN) in SU(N) models. As a consequence, computing the large-N limit of the strong-coupling series is meaningful and useful in order to achieve a picture of the large-N strong-coupling behavior of matrix models, but the evaluation of O(1/N2) or higher-order corrections in the strong-coupling phase is essentially pointless. The large-N limit of the external-field single-link integral has been considered in detail from the point of view of the strong-coupling expansion. In particular, one may obtain expressions for the coefficients of the expansion of ¼ in powers of the moments of AAs: setting 1 = o " Tr(AAs)n, ¼" + + ¼ 1 2 noa12oan , n N a , ,a 1 n 2 n/1 +a1, , an k kak/n one gets

C

(3.42)

D

(2n#+ a !3)! (2k)! ak 1 k k < ! . (3.43) (2n)! (k!)2 a ! k k Further properties of this expansion can be found in the original reference [88]. A character-expansion representation of the single-link integral was also produced for arbitrary U(N) integrals in Ref. [73]. Strong-coupling expansions for large-N lattice gauge theories have been analyzed in detail by Kazakov [26,89], O’Brien and Zuber [27], and Kostov [28], who proposed reinterpretations in terms of special string theories. ¼ 1 2 n"(!1)n a , ,a

3.5. The single-link integral in the adjoint representation The integral introduced at the beginning of Section 3 is by no means the most general single-link integral one can meet in unitary-matrix models. As mentioned in Section 2, any invariant function of the º’s is in principle a candidate for a lattice action. In practice, the only case that has been considered till now that cannot be reduced to Eq. (3.1) is the integral introduced by Itzykson and Zuber [50]

P

I(M , M )" dº exp Tr (M ºM ºs) , 1 2 1 2

(3.44)

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where M and M are arbitrary Hermitian matrices. This is a special instance of the single-link 1 2 integral for the coupling of the adjoint representation of º to an external field. The result, because of U(N) invariance, can only depend on the eigenvalues m and m of the 1i 2i Hermitian matrices. Several authors [50,90,91] have independently shown that

A

B

N~1 detEexp(m m )E 1i 2j I(M , M )" < p! , (3.45) 1 2 D(m ,2, m )D(m ,2, m ) 11 1N 21 2N p/1 where D(m ,2, m )"< (m !m ) is the Vandemonde determinant. A series expansion for 1 N i;j i j I(M , M ) in terms of the characters of the unitary group takes the form 1 2 1 p (r)s (M )s (M ) , I(M , M )"+ (3.46) 1 2 DnD! d (r) 1 (r) 2 (r) (r) where p is the dimension of the representation (r) of the permutation group; we will present an (r) explicit evaluation of p in Eq. (4.105). Eq. (3.45) plays a fundamental roˆle in the decoupling of the (r) “angular” degrees of freedom when models involving complex Hermitian matrices are considered. An interesting development based on the use of Eq. (3.45) is the so-called “induced QCD” program, aimed at recovering continuum large-N QCD by taking proper limits in the parameter space of the lattice Kazakov—Migdal model [92] S"N+ Tr»(U )!N + Tr(U º U ºs ) , (3.47) x x x, k x`k x, k x x, k where º is the non-Abelian gauge field and U is a Hermitian N]N (matrix-valued) x, k x Lorentz—scalar field. The Itzykson-Zuber integration (3.44) allows the elimination of the gauge degrees of freedom and reduces the problem to studying the interactions of Hermitian matrix fields (with self-interactions governed by the potential »). Discussion of the various related developments is beyond the scope of the present report. It will be enough to say that, while one may come to the conclusion that this model does not induce QCD, it is certainly related to some very interesting (and sometimes solvable) matrix models (cf. Ref. [93] for a review).

4. Two-dimensional lattice Yang—Mills theory 4.1. Two-dimensional ½ang—Mills theory as a single-link integral The results presented in the previous section allow us to analyze the simplest physical system described by a unitary-matrix model. As we shall see, one of the avatars of this system is a Yang—Mills theory in two dimensions (YM ), in the lattice Wilson formulation. Notwithstanding 2 the enormous simplifications occurring in this model with respect to full QCD, still some nontrivial features are retained, and even in the large-N limit some interesting physical properties emerge. It is therefore worth presenting a detailed discussion of this system, which also offers the possibility of comparing the different technical approaches to the large-N solution in a completely controlled situation.

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The lattice formulation of the two-dimensional U(N) gauge theory is based on dynamical variables º which are defined on links; however, because of gauge invariance, in two dimensions x, k there are no transverse gauge degrees of freedom, and a one-to-one correspondence can be established between link variables and plaquettes. A convenient way of exploiting this fact consists in fixing the gauge [94] º "1 (4.1) x,0 (the lattice version of the temporal gauge A "0). An extremely important consequence of the 0 gauge choice (4.1) emerges from considering the gauge-fixed form of the single-plaquette contribution to the lattice action: Tr(º º ºs ºs )PTr º ºs . (4.2) x,0 x`0,1 x`1,0 x,1 x`0,1 x,1 This is nothing but the single-link contribution to the one-dimensional lattice action of a principal chiral model whose links lie along the 0 direction. When considering invariant expectation values (Wilson loops), we then recognize that they can be reduced to contracted products of tensor correlations of variables defined on decoupled one-dimensional models. As a consequence, YM 2 factorizes completely into a product of independent chiral models labeled by their 1 coordinate. Not only the partition function, but also all invariant correlations can be systematically mapped into those of the corresponding chiral models. The area law for non self-interacting Wilson loops in YM and the exponential decay of the two-point correlations in one-dimensional chiral models are 2 trivial corollaries of these results [94]. The above considerations allow us to focus on the prototype model defined by the action S"!N+ Tr(º ºs #ºsº ) , (4.3) i i`1 i i`1 i where i is the site label of the one-dimensional lattice. By straightforward manipulations we may show that the most general nontrivial correlation one really needs to compute involves product of invariant operators of the form Tr(º ºs )k , (4.4) 0 l where l plays the roˆle of the space distance, and k is a sort of “winding number”. An almost trivial corollary of the above analysis is the observation that YM and principal chiral 2 models in one dimension enjoy a property of “geometrization”, i.e., the only variables that can turn out to be relevant for the complete determination of expectation values are the single-plaquette (single-link) averages of products of powers of moments [95] < [Tr (º ºs )k]mk (4.5) 0 1 k and the geometrical features of the correlations (in YM , areas of Wilson loops and subloops; in 2 chiral models, distances of correlated points), such that all coupling dependence is incorporated in the expectation values of the quantities (Eq. (4.5)). This result is sufficiently general to apply not only to the Wilson action formulation, but also to all “local” actions such that the interaction depends only on invariant functions of the single-plaquette (single-link) variable, i.e., any linear combination of the expressions appearing in Eqs. (2.8) and (2.9) [95—97].

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In order to proceed to the actual computation, it is convenient to perform a change of variables, allowed by the invariance of the Haar measure, parameterizing the fields by » "º ºs ; l l~1 l the action (4.3) explicitly factorizes into

(4.6)

S"!N+ Tr(» #»s) . (4.7) l l l It is now easy to get convinced that in the most general case a Wilson loop expectation value (correlation function) can be represented as a finite product of invariant tensors, each of which is originated by a single-link integration of the form :d» f (» ) exp [Nb Tr(» #»s)] l l l l ,S f (» )T , (4.8) l :d» exp [Nb Tr(» #»s)] l l l where f (» ) is any (tensor) product of » ’s and »s’s, and the only nontrivial contributions to the full l l l expectation value come from integrations extended to plaquettes belonging to the area enclosed by the loop itself (in chiral models, links comprised between the extremal points of the space correlation). For the sake of definiteness, we may focus on the correlators [98] 1 ¼ , STr(º ºs)kT , 0 l l,k N

(4.9)

and find that :d» 2d» (1/N) Tr(» 2» )k exp [Nb+l Tr(» #»s)] 1 l 1 l i/1 i i . (4.10) ¼ " l,k < :d» exp [Nb Tr(» #»s)] i i i i This problem can be formally solved for arbitrary N by a character expansion, which we shall discuss in Section 4.8. It is however immediate to recognize that we are ultimately led to computing the general class of group integrals whose form is

P

d»< (Tr»k)mk exp [Nb Tr(»#»s)] (4.11) k (where the product runs over positive and negative values of k), and in turn it is in principle an exercise based on the exploitation of the result for the external field single-link integral introduced in Eq. (3.1). By the way, integrals of the form (4.11) can easily be expressed as linear combinations of integrals belonging to the class

P

d» s (») exp [Nb Tr(»#»s)] , (j)

(4.12)

where j labels properly chosen representations of U(N). Eq. (4.12) is in turn related to the definition of the character coefficients in the character expansion of exp [Nb Tr(»#»s)]. For arbitrary N, as a matter of principle, s (») has a representation in terms of the eigenvalues exp i/ of the matrix », (j) i

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while Tr(»#»s)"2+ cos / and the measure itself can in this case be expressed in terms of the i i eigenvalues as dk(»)&< d/ D2(/ ,2, / ) , i 1 N i where

(4.13)

D(/ ,2, / ),det expEi(i/ )E , 1 N j / !/ j. (4.14) D2(/ ,2, / )"< 4 sin2 i 1 N 2 i:j As a consequence, it is always possible to express all U(N) integrals in the class (Eq. (4.12)) in terms of linear combinations of products of modified Bessel functions I (2Nb), with k(N. k Let us now come to the specific issue of evaluating the relevant physical quantities in the large-N limit of U(N) models, and comparing the procedures corresponding to different possible approaches. Basic to most subsequent developments is the observation that the large-N factorization property allows us to focus on a very restricted class of interesting correlations, which we label by

T

U

1 w, Tr»k ,¼ . k 1,k N

(4.15)

The first explicit solution to the problem of evaluating w in the large-N limit was offered by Gross k and Witten [94]. To this purpose, they introduced the eigenvalue density 1 o(/)" + d(/!/ ) , (4.16) i N i and considered the group integral defining the partition function of the single-link model

P

A

B

(4.17) Z(b)& < d/ D2(/ ,2, / ) exp 2Nb+ cos / . i 1 N i i i The integral (Eq. (4.17)) can be evaluated in the NPR limit by a saddle-point technique [99] applied to the effective action

P

P

2b o(/)cos / d/# o(/) o(/@)log sin

/!/@ d/ d/@ , 2

(4.18)

with the constraint :o(/) d/"1. The support of the function o(/) is dynamically determined. The saddle-point integral equation is

P

2b sin /"

(#

d/@ o(/@) cot

/!/@ , 2

(4.19) ~(# and it is possible to identify two distinct solutions, corresponding to weak and strong coupling. When b is small, it is easy to find out that 1 o(/)" (1#2b cos /), 2p

!p4/4p ;

(4.20)

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o(/) is positive definite whenever b41. When b is large, / (n and 2 # 2b / 1 / / 1 o(/)" cos !sin2 , sin2 #" , p 2 2b 2 2 2b

S

171

(4.21)

submitted to the condition b51. Therefore it is possible to identify the location of the third-order 2 phase transition [94]: b "1 . (4.22) # 2 By direct substitution, one finds the values of the free and internal energy (per unit link or unit plaquette):

G

b2, b41 , F 2 " N2 2b!1 log 2b!3, b51 , 2 4 2 b, b41 , 1 ­ F 2 " w " 1 1 2 ­b N2 1! , b51 . 2 4b More generally, one may evaluate w from o(/), thanks to the relationship k 0, b41, k52 , 2 (# d/ cos k/ o(/)" 1! 1 2 1 P(1,2) 1!1 , b51 , w" k 2 2b k!1 k~2 b ~(#

G

P

GA

B

A B

(4.23) (4.24)

(4.25)

where P(a,b) are the Jacobi polynomials. All w are differentiable once in b"b , but their second k k # derivatives are discontinuous. Let us notice that Eqs. (4.22), (4.23) and (4.24) are an immediate consequence of Eq. (3.12) and Eq. (3.15) for the special choice x "b2 . 4

(4.26)

4.2. The Schwinger—Dyson equations of the two-dimensional ½ang—Mills theory It is interesting to obtain the above results from the algebraic approach to the Schwinger—Dyson equations of the model. We can restrict Eq. (2.15) to the set of Wilson loops C consisting of k turns k around a single plaquette, in which case by definition ¼(C )"w . Formally, the Schwinger—Dyson k k equations do not close on this set of expectation values; however, one may check by inspection, using the factorization property of two-dimensional functional integral for the Yang—Mills theory, that contributions from other Wilson loops cancel in the equations for w (this is strictly k a two-dimensional property). As a consequence, we obtain the large-N relationships [100] n b(w !w )" + w w , (4.27) n~1 n`1 k n~k k/1 with a boundary condition w "1. The solution is found by defining a generating function 0 = U(t), + w tk (4.28) k k/0

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and noticing that Eq. (4.27) corresponds to t Ut2!(U!1!w t)" (U2!U) , 1 b

(4.29)

which is solved by b U(t)" 2t

SA

B

A

B A

B

t w b t 2 1# #t2 !4t2 1! 1 ! 1! !t2 . b b 2t b

(4.30)

The condition Dw D41 implies that U(t) is holomorphic within the unitary circle. On the boundary k of the analyticity domain, t"e*( and

P

1 p [Re U(e*()!1]cos k/ d/ , w" 2 k p ~p and as a consequence we may identify

(4.31)

1 [Re U(e*()!1 ]"o(/) . 2 p

(4.32)

The positivity condition on o(/) leads to a complete determination of the solution, implying either w "b, w "0 (k52), 1 k or o(p)"0, which in turn leads to 1 w "1! , 1 4b

b41 2

!/ 4/4/ , # #

b41 , 2

(4.33)

(4.34)

and / is given by Eq. (4.21). It is immediate to check that the resulting eigenvalue densities are the # same as Eqs. (4.20) and (4.21). Let us mention that these methods may in principle be applied to more general formulations of the theory based on “local” actions, and in particular Wilson loop expectation values can be computed for the fixed-point version of the model, corresponding to the continuum action [95]. In the general “local” formulation, the expression of the effective action (Eq. (4.18)) is replaced by

P

P

/!/@ 2+ b o(/)cos k/ d/# o(/) o(/@)log sin d/ d/@ , (4.35) k 2 k where b is an (infinite) set of couplings, to be eventually determined by the fixed-point conditions. k The solution of the corresponding saddle-point equation + kb sin k/"Im U(e*() (4.36) k k leads to an identification of o(/) by the use of Eq. (4.32). As a consequence, we may express all w in k terms of the couplings b . As long as a strong-coupling phase exists, the solution is simply k (4.37) w "kb . k k

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At the fixed point, a quite explicit transcendental equation can be found for the function U(t): U(t)!1 U(t)!1 2"t , exp U(t) 2b

(4.38)

which leads to the continuum solution

C D A B

1 k k w " exp ! ¸(1) , k k 4b k~1 2b

(4.39)

where ¸(a) are generalized Laguerre polynomials. k The fixed-point action in YM in turn is nothing but the “heat kernel” action [101], discussed in 2 the large-N context in Ref. [102]. Large-N continuum YM is slightly beyond the purpose of the 2 present review. We must however mention that in recent years a number of interesting results have appeared in a string theory context. It is worth quoting Refs. [103—105] and references therein. While the problem of evaluating the more general expectation values ¼ is solved in principle, l,k in practice it is not always simple to obtain compact closed-form expressions whose general features can be easily understood. In the strong-coupling regime b(1, it is not too difficult to 2 determine from finite-N results the large-N limit in the form [98]

A

B

(!1)k~1 lk!2 lim ¼ " bkl , (4.40) l,k k k!1 N?= and one may show that the corresponding Schwinger—Dyson equations close on the set ¼ for l,k any fixed l and are solved by Eq. (4.40). As a matter of fact, by defining = U (t), + ¼ tk, l l,k k/0 one may show that the strong-coupling Schwinger—Dyson equations reduce to

(4.41)

[U (t)!1][U (t)]l~1"blt . l l For the interesting values l"1 and l"2, Eq. (4.40) reduces to

(4.42)

U (t)"1#bt , 1 consistent with the strong-coupling solution (Eq. (4.30)), and

(4.43)

U (t)"1(J1#4b2t#1) , 2 2 related to the generating function for the moments of the energy density

(4.44)

T

U

1 1 = Tr "1#2tb2#2 + (bt)2k¼ "2b2t#J1#4b4t2 . (4.45) 2,k N 1!bt(» #»s ) n n`1 k/1 Eq. (4.45) is related to a different approach for solving large-N unitary-matrix models, based on an integration of the matrix angular degrees of freedom to be performed in strong coupling [61,106]. The corresponding weak-coupling problem is definitely more difficult. As far as we can see, the Schwinger—Dyson equations close only on a larger set of correlation functions, defined by the

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generating function [107]

C

D

1 1 , 0(k4l, n50 , D(l) (t)" Tr (» )n`1» 2» k k`1 l 1!t» 2» k,n N 1 l

(4.46)

such that U (t)"1#tD(l) (t) . l 1,0

(4.47)

The explicit form of the equations is n~1 (4.48) + w D(l) (t)#D(l) (t) D(l) (t)#b[D(l) (t)!D(l) (t)]"0, 14k4l . j k,n~j l,n~1 k,0 k,n`1 k,n~1 j/0 When l"1, 2 it is possible to find explicit weak-coupling solutions, but the general case l'2 has not been solved so far. More about the calculability of Wilson loops with arbitrary contour in two-dimensional U(R) lattice gauge theory can be found in Ref. [108]. The corresponding continuum calculations are presented for arbitrary U(N) groups in Ref. [109]. 4.3. Large-N properties of the determinant It is quite interesting to apply the results of Section 3.3, concerning the properties of the determinant, to YM and principal chiral models in one dimension. Exploiting the factorization of 2 the functional integration and the possibility of performing the variable change (4.6) in the operators as well as in the action, we can easily obtain the relationship D ,det[º ºs]"det[» 2» ]"det » 2det » , l 0 l 1 l 1 l

(4.49)

and, as a consequence, SD T"Sdet » Tl . l 1

(4.50)

The problem is therefore reduced to that of evaluating Sdet»T in the single-plaquette model. It is immediate to recognize from Eqs. (3.31) and (3.32) that 1 Sdet »TP 1! , 2b

S

b51 , 2

(4.51)

Sdet »TP0,

b41 . 2

(4.52)

Apparently, this expectation value acts as an order parameter for the phase transition between the weak- and strong-coupling phases. More precisely, according to Green and Samuel [110,111], one must identify the order parameter with the quantity SD T1@N l

(4.53)

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and notice that SD T1@NP1 in weak coupling , (4.54) l SD T1@NP exp (!pl) in strong coupling , (4.55) l where p acts as a U(1) “string tension”. Eqs. (4.54) and (4.55) generalize to higher dimensions, when replacing l with the (large) area of the corresponding Wilson loop. Notice that the weak-coupling result is consistent with the decoupling of the U(1) degrees of freedom from the SU(N) degrees of freedom, and with the interpretation of U(1) as a free massless field. It is therefore interesting to compute 1 p"! logSdet »T N

(4.56)

in the case of the single-matrix model; this requires taking the large-N limit only after the strong-coupling calculation of Sdet »T has been performed. Since the technique of evaluation of p has some relevance for subsequent developments, we shall briefly sketch its essential steps. Standard manipulations of the single-link integrals for finite N allow to evaluate

P

(2Nb)E . A (b)" d» exp [Nb Tr(»#»s)](det »)m"detEI k~l~m m,N

(4.57)

These quantities can be shown to satisfy the recurrence relations [112] A2 !A A "A A . m,N m`1,N m~1,N m,N~1 m,N`1 Willing to compute expectation values, we define

(4.58)

A D (b)"S(det »)mT" m,N . m,N A 0,N Eq. (4.58) implies that

(4.59)

D2 !D D "D D (1!D2 ) . (4.60) m,N m`1,N m~1,N m,N~1 m,N`1 1,N Since all D are known, it is possible to reconstruct all D from Eq. (4.60) once D is m,1 m,N 1,N determined. Now D is exactly Sdet »T, and it is possible to show that it obeys the following 1,N second-order differential equation [113]

CA

B

D

d 2 N2 1d d 1 ! s D # D D # (1!D2 )D "0 , (4.61) 1,N 1,N 1,N 1,N ds s2 1,N s ds ds 1!D2 1,N where s"2Nb. Eq. (4.61) can be analyzed in weak and strong coupling and in the large-N limit. In particular the weak-coupling 1/N expansion leads to

S

A

B

A B

1 1 1 5@2 1 1 1! , #O D P 1! ! 1,N 2b N2 128b3 2b N4

(4.62)

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thus confirming Eq. (4.51), while in strong coupling one may show that D "J (2Nb)#O(b3N`2) P J (2Nb) , (4.63) 1,N N N N?= where J is the standard Bessel function, whose asymptotic behavior is well known. As an N immediate consequence, we find 1#J1!4b2 , b(1 . !p"J1!4b2!log 2 2b

(4.64)

This result was first guessed by Green and Samuel [111], and then explicitly demonstrated in Ref. [113]. 4.4. Local symmetry breaking in the large-N limit Another interesting application of the external-field single-link integral to the large-N limit of two-dimensional Yang—Mills theories is the study of the possibility of breaking a local symmetry, as a consequence of the thermodynamical nature of the limit. If we introduce an infinitesimal explicit U(N) symmetry breaking term in the action [114] S"!bN[Tr »#JN»ij#h.c.] ,

(4.65)

corresponding to replacing A Pb[d #NJd d ] lm lm lj mi in Eq. (3.1), we find that the eigenvalues of AAs are x "b2[1#1N2J2$1JJ4N4#4J2N2] , 1,2 2 2 x "b2, l'2 . l When taking the large-N limit of the free energy, we find log Z "F (b)#2bDJD , lim 0 N2 N?= and in the limit JP0B we then find SRe »ijT"$1 .

(4.66)

(4.67)

(4.68)

(4.69)

We therefore expect that, for finite k, the U(k) global symmetries of large-N chiral models and U(k) gauge symmetries are broken in any number of dimensions [114]. This phenomenon cannot occur for any finite value of N in two dimensions. 4.5. Evaluation of higher-order corrections In the context of large-N two-dimensional Yang—Mills theory, it is worth mentioning that it is possible to compute systematically higher-order corrections to physical quantities in the powers of

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1/N2. It is interesting to notice that the weak-coupling corrections to the free energy [115] (see also [82])

C

A

1 1 1 1 1 !A! log N! log 1! F"F # 0 N2 12 12 8 2b #

C

A

B

D

1 1 ~3 3 1 1! #2 , ! N4 1024b3 2b 240

A

B

BD (4.70)

A

B

A B

1 1 1 1 ~1 1 ~4 1 1 1 º"1! ! 1! 1! , ! #O 4b N2 32b2 2b 2b N2 1024b4 N4

(4.71)

where A"0.248752, are well defined, but become singular when bP1. In turn, when evaluating 2 higher-order corrections in the strong-coupling phase, one finds out that there are no corrections proportional to powers of 1/N, while there are contributions that fall off exponentially with large N, as expected from the general arguments discussed in Section 3.4 in connection with the appearance of the DeWit—’t Hooft poles. Let us however mention that Eqs. (4.60) and (4.61) are also the starting point for a systematic 1/N expansion of the free energy in the weak-coupling regime, alternative to Goldschmidt’s procedure. The basic ingredient is the observation that, defining the free energy at finite N by F (b)"log A (b) , N 0,N one may show that

(4.72)

A

B

d D N d 1,N (log F !log F )" D # D , (4.73) N N~1 ds 1!D2 ds 1,N s 1,N 1,N and this allows for a systematic reconstruction of F , whose strong-coupling form is [112] N = F (b)"N2b2! + kJ2 (2Nb)#O(b4N`4) . (4.74) N N`k k/1 4.6. Mixed-action models for lattice ½M 2 Another instance of the problem of the single-link integration for matrix fields in the adjoint representation of the full symmetry group occurs in the discussion of the so-called “mixed action” models. Consider the following single-link integral [116], resulting from a different formulation of lattice YM , 2

P

Z(b , b )" dº exp MNb Tr(º#ºs)#b DTr ºD2N . & ! & !

(4.75)

It is possible to show that, in the large-N limit, the corresponding free energy can be obtained by the same saddle-point technique presented in Section 4.1, i.e., by introducing a spectral density o(h) for the eigenvalues of º. This spectral density turns out to be precisely the same as the one obtained when b "0, if one simply replaces b by an effective coupling ! & (4.76) b "b #b w (b ) , %&& & ! 1 %&&

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where w can be evaluated in terms of o(h) as 1

P

w (b )" dh cosh o(h) . 1 %&&

(4.77)

Eq. (4.77) is a self-consistency condition for w , which allows a determination of b (b , b ). Finally, 1 %&& & ! by substitution into the effective action, one finds the relationship F(b , b )"F(b (b , b ),0)!b w2(b (b , b )) , (4.78) & ! %&& & ! ! 1 %&& & ! where F(b, 0) is nothing but the free energy obtained in Section 4.1. The strong- and weak-coupling solutions are separated by the line 2b #b "1. In strong & ! coupling one obtains b b "w " & , %&& 1 1!b ! while in weak coupling

b2 F" & , 1!b !

1 b " [b #b #J(b #b )2!b ], ! & ! ! %&& 2 &

(4.79)

1 w " [b !b #J(b #b )2!b ] , 1 2b ! & & ! ! !

b b2 1 1 F"b # !! & ! ! log[b #b #J(b #b )2!b ] & & ! & ! ! 2 2b 2 2 ! 1 b # 1# & J(b #b )2!b . & ! ! 2 b ! It may be interesting to quote explicitly the limiting case b "0, where [79] &

A

B

(4.80)

P

Z(0, b ), dº exp b DTr ºD2 ! !

G

0,

S

A S

B

" 1b #1b 1! 1 !1 log b 1# 1! 1 , ! b 2 b 2 ! 2 ! ! !

b (1 , !

(4.81)

b '1 . !

One may actually show that, in any number of dimensions, a lattice gauge theory with mixed action [117—119] (a trivial generalization of Eq. (4.75)) is solved in the large-N limit in terms of the solution of the corresponding theory with pure Wilson action; Eqs. (4.76) and (4.78) hold as they stand, and 1 w (b )" STr º TD & %&& ! . 1 %&& p b /b , b /0 N

(4.82)

More about the large-N behavior of variant actions can be found in Refs. [120—122]. Different kinds of variant actions have been studied in the large-N limit in Refs. [123—125].

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4.7. Double-scaling limit of the single-link integral In the Introduction, we mentioned that one of the most interesting phenomena related to the large-N limit of matrix models is the appearance of the so-called “double-scaling limit”

G

NPR ,

N2@c1(g !g)"const. , (4.83) # gPg , # where g is a (weak) coupling related to the inverse of b. We already discussed the general physical interpretation of this limit as an alternative description of two-dimensional quantum gravity and its relationship to the theory of random surfaces. Here we only want to consider the double-scaling limit properties for those simple models of unitary matrices that can be reformulated as a singlelink model (cf. Ref. [126]). This specific subject was pioneered by Periwal and Shevitz [127], who discussed the doublescaling limit in models belonging to the class

P

Z " dº exp [Nb Tr V(º#ºs)] , N

(4.84)

where V(º) is a polynomial in º. Because of the invariance of the measure, Eq. (4.84) can be reduced to

P

C

D

Z & d/ DD (e*(1,2, e*(N)D2 exp Nb+ V(2 cos / ) , (4.85) N i i i and solved by the method of orthogonal polynomials. One starts by defining polynomials n~1 P (z)"zn# + a zk , n k,n k/0 that satisfy

Q

(4.86)

A B C A BD

1 dz 1 P (z) P exp NbV z# m z z 2piz n

"h d , n mn

(4.87)

where the integration runs over the unit circle, and moreover obey the recursion relation

AB

1 , P (z)"zP (z)#R znP n`1 n n n z

h n`1"1!R2 , n h n

(4.88)

where R ,a . As a corollary, n 0,n`1 Z JN!< (1!R2 )N~i , N i~1 i and one may show that

Q

(4.89)

C A BD

1 dz exp NbV z# (n#1)(h !h )" n`1 n z 2piz

A BA

1 NbV@ z# z

which in turn leads to a nonlinear functional equation for R . n

B

AB

1 1 1! P (z) P , n`1 n z2 z (4.90)

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The simplest example, corresponding to YM , amounts to choosing V@"1, obtaining 2 (n#1)R2"NbR (R #R )(1!R2) , (4.91) n n n`1 n~1 n and in the large-N limit, setting n"N and R "R, we obtain the limiting form N R2"2bR2(1!R2) , (4.92) showing that b "1 (degeneracy of solution R "0). One may now look for the scaling solution to # # 2 Eq. (4.91) in the form R !R "R "N~kf [No(g !g)], N # N #

1 g" , b

(4.93)

where f 2 is related to the second derivative of the free energy. This is a consistent Ansatz when k"1, o"2 , 3 3 leading to the equation

(4.94)

!2xf#2f 3"f A,

(4.95)

1 "!2(1!R2)(!1!j#3jR2) , b

(4.96)

x"No(g !g) . # In the case V@"1#ju, one finds the equation

which reduces to 1/b"3(1!R4) when j"1. A scaling solution to the corresponding difference 2 4 equation requires k"1 and o"4. When V@"1#j u#j u2, multicriticality sets at j "!3 5 5 1 2 1 7 and j " 1 , and 1/b"10(1!R6), leading to the exponents k"1 and o"6. Rather general results 2 14 7 7 7 can be obtained for an arbitrary order k of the polynomial V: k"1/(2k#1), o"2k/(2k#1), and c"1!6/(k(k#1)). The double-scaling limit can also be studied in the case of the external-field single-link integral [128], and it was found that its critical behavior is simple enough to be identified with that of the k"1 unitary-matrix model. In the language of quantum gravity, the only effect of introducing N2 real parameters A is that of renormalizing the cosmological constant, without changing the ij universality class of the critical point. A few interesting features of the double-scaling limit for the k"1 model are worth a more detailed discussion [21]. In particular let us recall that, according to Eq. (4.94), 2 2 o" " , (4.97) c 3 1 and therefore c "3, implying c"!2. We may now reinterpret the double-scaling limit of matrix 1 models as a finite-size scaling with respect to the “volume” parameter N in a two-dimensional N]N space. As a consequence, we obtain relationships with more conventional critical exponents through the identification c "2l, which in turn by hyperscaling leads to a determination of the 1 specific heat exponent a"2(1!l). Numerically we obtain l"3 and a"!1. The result a"!1 2

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can be easily tested on the solution of the model

G

b2, 1 d2F C(b)" b2 " 2 db2 1, 4

b4b , # b5b , #

(4.98)

with b "1, consistent with a negative critical exponent a"!1. # 2 It is also interesting to find tests for the exponent l, especially in view of the fact that the most direct checks are not possible in absence of a proper definition for the relevant correlation length. Numerical studies have been performed by considering the partition function zero b closest to the 0 transition point b "1, finding that the relationship # 2 Im b JN~1@l 0

(4.99)

is rather well satisfied even for very low values of N; at N55, it is valid within one per mille. Another test concerns the location of the peak in the specific heat in U(N) models, whose position b (N) should approach b with increasing N. Finite-size scaling arguments predict 1%!, # b (N)+b #aN~1@l , 1%!, #

(4.100)

and large-N results are very well fitted by the choice l"3, a+0.60 [129]. 2 4.8. The character expansion and its large-Nlimit: Sº(N) vs. º(N) The general features of the character expansion for lattice spin and gauge models have been extensively discussed by different authors. In particular, Ref. [77], besides offering a general presentation of the issues, presents tables of character coefficients for many interesting groups, including U(R)+SU(R), for the Wilson action. Let us therefore only briefly recall the fundamental points of this approach, which is relevant especially in the analysis of the strong-coupling phase and of the phase transition. In Section 2 we classified the representations and characters of U(N) groups. Because of the orthogonality and completeness relations, every invariant function of » can be decomposed in a generalized Fourier series in the characters of ». Let us now consider for sake of definiteness chiral models with action given by Eq. (2.10); extension to lattice gauge theories is essentially straightforward, at least on a formal level. We can replace the Boltzmann factor corresponding to each lattice link by its character expansion:

G

H

exp MbN Tr[º ºs #º ºs]N" exp N2F(b)+ d zJ (b) s (º ºs ) , x x`k x`k x (r) (r) (r) x x`k (r)

(4.101)

where the sum runs over all the irreducible representations of U(N), F(b) is the free energy of the single-link model

P

1 1 F(b)" log d» exp [Nb Tr(»#»s)]" log detEI (2Nb)E , j~i N2 N2

(4.102)

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and zJ (b) are the character coefficients, defined by orthogonality and representable in terms of (r) single-link integrals as detEI i (2Nb)E j `j~i , (4.103) d zJ (b)"Ss (»)T" (r) (r) (r) detEI (2Nb)E j~i with j defined by Eq. (2.2). We may notice that, for any finite N, zJ (b) are meromorphic functions (r) of b, with no poles on the real axis, which is relevant to the series analysis. However, singularities may develop, as usual, in the large-N limit. Eq. (4.101) and Eq. (4.102) become rapidly useless with growing N. However, an extreme simplification occurs in the large-N limit, owing to the property 1 1 p p (Nb)n``n~[1#O(b2N)] , (4.104) zJ (b)" (l,m) (l,m) n ! n ! (l) (m) ` ~ where n "+ l , n "+ m , and p is the dimension of the representation (l) of the permutation ` ii ~ i i (l) group, which in turn can be computed explicitly as d

d (l,m)

< 1 (l !l #k!j)! k p 1 2 4 " 1yjykys j ; n ! (l , ,l ) <s (l #s!i)! ` i/1 i can be parameterized by

(4.105)

1 1 p p C , (4.106) " (l,m) n ! n ! (l) (m) (l,m) ` ~ where C can be expressed as a finite product: (l,m) s (N!t!i#l )! t (N!s!j#m )! s t (N#1!i!j#l #m )! j < < i < i j , (4.107) C "< (l,m) (N!s!j)! (N!t!i)! (N#1!i!j)! j/1 i/1 j/1 i/1 allowing for a conceptually simple 1/N expansion. These results are complemented with the result d

F(b)"b2#O(b2N`2)

(4.108)

and with the unavoidable large-N constraint b41. 2 The character expansion now proceeds as follows. We notice that, thanks to Eq. (4.104), only a finite number of nontrivial representations contributes to any definite order in the strong-coupling series expansion in powers of b, and each lattice integration variable can appear only once for each link where a nontrivial representation in chosen. A systematic treatment leads to a classification of contributions in terms of paths (surfaces in a gauge theory) along whose non self-interacting sections a particular representation is assigned. Self-intersection points are submitted to constraints deriving from the orthogonality of representations and their composition rules. In the case of chiral models, all relevant assignments can be generated by considering the class of the lattice random paths satisfying a non-backtracking condition [130]. Once all nontrivial configurations are classified and counted, one is left with the task of computing the corresponding group integrals. Only integrations at intersection points are nontrivial, since other integrations follow immediately from the orthogonality relationships. Unfortunately, no special computational simplifications occur in the large-N limit of group integrals.

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Apparently, the character expansion is the most efficient way of computing the strong-coupling expansion of lattice models. In particular, very long strong-coupling series have been obtained in the large-N limit for the free energy, the mass gap, and the two-point Green’s functions of chiral models in two and three dimensions (for the free energy, 18 orders on the square lattice, 26 orders on the honeycomb lattice, and 16 orders on the cubic lattice; for the Green’s functions, 15 orders on the square lattice, 20 orders on the honeycomb lattice, and 14 orders on the cubic lattice). The analysis of these series will be discussed in Section 7. Before leaving the present subsection, we must make a few comments concerning the relationship between SU(N) and U(N) groups. We already made the observation that when NPR there is essentially no difference between SU(N) and U(N) models, at least when considering operators not involving the determinant. In order to explore this relationship more carefully, we may start as usual from the expression of the single-link integral (3.1). Representations of Z(AsA) in the SU(N) case can be obtained [131] in terms of the eigenvalues x of AsA and of h, defined in Eq. (3.23). Introducing the Vandemonde determinant i D(j ,2, j )"< (j !j )"detEji~1E , 1 N j i j j;i one obtains

A

BP

1 N~1 k! < Z(AsA)" 2p N! k/1

A

(4.109)

B

< d/ d + / #Nh i i i i DD(e*(1,2, e*(N)D2

C

D

] exp 2+ Jx cos / , k k D(2Jx ,2, 2Jx ) D(cos / ,2, cos / ) k N 1 N 1

(4.110)

or alternatively

P

A

B

C

D

D(Jx e*(1,2, Jx e*(N) N~1 k! 1 N < d/ d + / #Nh exp 2+ Jx cos / . (4.111) Z(AsA)" < i i k k D(x ,2, x ) 2p 1 N i i k k/1 The only difference between SU(N) and U(N) is due to the presence of the (periodic) delta function d(+ / #Nh), introducing the dependence on h corresponding to the constraint det º"1. i i A formal solution is obtained by expanding in powers of e*Nh:

A

B

N~1 1 = , (4.112) Z(AsA)" + e*NmhdetEzj~1I (2z )E < k! i j~1~@m@ i D(z2,2, z2 ) 1 N k/1 m/~= where z "Jx . Eq. (4.112) in turn leads to the following representation of the free energy for the i i SU(N) single-link model: = F (b, h)"log + A (b) e*Nmh , (4.113) N m,N m/~= where for convenience we have redefined the coupling: bPbe*h. Eq. (4.113) is useful for a large-N mean-field study [112], but it is certainly inconvenient at small N, where more specific integration techniques may be applied.

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We mention that a large-N analysis of Eq. (4.113) for h"0 leads to = F (b, 0)"N2b2#2J (2Nb)!2J (2Nb) J (2Nb)! + kJ2 (2Nb)#O(b3N) . (4.114) N N N~1 N`1 N`k k/1 It is also possible to establish a relationship between SU(N) and U(N) groups at the level of character coefficients. Thanks to the basic relationships s

(º)"(det º)ss 1 2 N(º) , (4.115) j1`s,2, jN`s j , ,j holding in U(N), one may impose the condition det º"1 in the integral representation of the character coefficients and obtain += zJ (r, s) z " 4/~= , (r) += zJ (0, s) 4/~= where, by definition, for U(N) groups

(4.116)

zJ (0, s)"Sdet ºsT,

d zJ (r, s)"Sdet ºss (º)T . (4.117) (r) (r) These relationships are the starting point for a systematic implementation of the corrections due to the SU(N) condition in the 1/N expansion of U(N) models [52,132]. A peculiarity of the SU(N) condition can be observed in the finite-N behavior of the eigenvalue density function o(/, N), which shows a non-monotonic dependence on /, characterized by the presence of N peaks. This is already apparent in the bP0 limit of the single-link integral, where [129] 1 o (/)P , U(N) 2p b?0

A

B

2 1 o (/)P 1#(!1)N`1 cos N/ . SU(N) N 2p b?0

(4.118)

5. Chiral chain models and gauge theories on polyhedra 5.1. Introduction The use of the steepest-descent techniques allows to extend the number of the unitary-matrix models solved in the large-N limit to some few unitary-matrix systems. The interest for few-matrix models may arise for various reasons. Their large-N solutions may represent non-trivial benchmarks for new methods meant to investigate the large-N limit of more complex matrix models, such as QCD. Every matrix system may have a roˆle in the context of two-dimensional quantum gravity; indeed, via the double scaling limit, its critical behavior is connected to two-dimensional models of matter coupled to gravity. Furthermore, every unitary-matrix model can be reinterpreted as the generating functional of a class of integrals over unitary groups, whose knowledge would be very useful for the strong-coupling expansion of many interesting models. This section is dedicated to a class of finite-lattice chiral models termed chain models and defined by the partition function

P

C

D

L L Z " < dº exp Nb + Tr(º ºs #ºsº ) , L i i i`1 i i`1 i/1 i/1 where periodic boundary conditions are imposed: º "º . L`1 1

(5.1)

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Chiral chain models have interesting connections with gauge models. Fixing the gauge A "0, 0 YM on a K]¸ lattice (with free boundary conditions in the direction of size K) becomes 2 equivalent to K decoupled chiral chains of length ¸. Chiral chains with periodic boundary conditions enjoy another interesting equivalence with lattice gauge theories defined on the surface of polyhedra, where a link variable is assigned to each edge and a plaquette to each face. By choosing an appropriate gauge, lattice gauge theories on regular polyhedra like tetrahedron, cube, octahedron, etc., are equivalent respectively to periodic chiral chains with ¸"4, 6, 8, etc. [70]. The thermodynamic properties of chiral chains can be derived by evaluating their partition functions. Free-energy density, internal energy, and specific heat are given respectively by 1 F " log Z , L ¸N2 L

(5.2)

1 ­F L, º " L 2 ­b

(5.3)

­º C "b2 L . L ­b

(5.4)

When ¸PR, Z can be reduced to the partition function of the Gross—Witten single-link model, L and therefore shares the same thermodynamic properties. In particular, the free energy density at N"R is piecewise analytic with a third-order transition at b "1 between the strong-coupling # 2 and weak-coupling domains. Furthermore, the behavior of C around b can be characterized by = # a specific heat critical exponent a"!1. It is easy to see that the ¸"2 chiral chain is also equivalent to the Gross—Witten model, but with b replaced by 2b; therefore b "1 and the critical # 4 properties are the same, e.g., a"!1. 5.2. Saddle-point equation for chiral ¸-chains The strategy used in Refs. [70,133] to compute the N"R solutions for chiral chains with ¸44 begins with group integrations in the partition function (Eq. (5.1)), with the help of the single-link integral, for all º except two. This leads to a representation for Z in the form i L

P

Z " dº d» exp [N2S(L)(º»s)] L %&&

(5.5)

suitable for a large-N steepest-descent analysis. Since the integral depends only on the combination º»s, changing variable to h , e*hj being the eigenvalues of º»s, leads to j

P

(5.6) Z & < dh DD(h ,2, h )D2 exp [N2S(L)(h )] , i 1 N %&& k L i where !n4h 4n, D(h ,2, h )"detED E, D "e*jhk. In the large-N limit, Z is determined by j 1 N jk jk L its stationary configuration, and the distribution of h is specified by a density function o (h), which j L

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is the solution of the equation

P

d/ o (/) cot L

h!/ d # S(L)(h, o )"0 , L 2 dh %&&

(5.7)

with the normalization condition

P

n

o (h) dh"1 . L ~n

(5.8)

For ¸"2, Z is already in the desired form with 2 1 S(2)"2b Tr(º ºs #ºs º ) , %&& 1 2 1 2 N

(5.9)

and the large-N eigenvalue density o (h) of the matrix º ºs satisfies the Gross—Witten equation 2 1 2

P

d/ o (/) cot 2

h!/ !4b sin h"0 , 2

(5.10)

which differs from that of the infinite-chain model only in replacing b by 2b. 5.3. The large-N limit of the three-link chiral chain In the ¸"3 chain model, setting º"º and »"º , S(3) is given by 1 2 %&&

P

exp [N2S(3)]" exp [2Nb Re Tr º»s] dº exp [2Nb Re Tr Aºs ] , %&& 3 3

(5.11)

where A"º#». Recognizing in the r.h.s. of (Eq. (5.11)) a single-link integral, one can deduce that the large-N limit of the spectral density o (h) of the matrix º»s satisfies the equation 3

P

C

D

h!/ 1 sin 1h 2 # "0 , (5.12) 2 2 cos 1h#cos 1/ 2 2 with the normalization condition :o (h) dh"1. In order to find a solution for the above equation, 3 one must distinguish between strong-coupling and weak-coupling regions. In the weak-coupling region the solution of Eq. (5.12) is 2b(sin h#sin 1h)! d/ o (/) cot 3 2

S

C

DC

b h h 1 o (h)" cos 2 cos # 1! 3 n 4 2 3b

S

D

h 1 1@2 2 cos !2 1! 2 3b

(5.13)

for DhD4h "2 arccos #

S

1 1! 3b

(5.14)

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and o (h)"0 for h 4DhD4p. This solution is valid for b5b "1, indicating that a critical point 3 # # 3 exists at b "1. Similarly one can calculate o (h) in the strong-coupling domain b4b # 3 3 # [70,133,134] finding:

A

B

b Jc#J4#c o (h)" y(h)#1! [(y(h)#Jc)(y(h)#J4#c)]1@2 , 3 2p 2

(5.15)

where

S

h y(h)" 4 cos2 #c , 2

(5.16)

and the parameter c is related to b by the equation 1 1#Jc#1c#(1!1Jc)J4#c" . 2 2 b

(5.17)

At b"b , c"0 and therefore # 1 h 3@2 h o (h) " 2 cos cos , 3 #3*5 3p 2 4

A

B

(5.18)

in agreement with the critical limit of the weak-coupling solution (Eq. (5.13)). Since o (p)'0 for b(b and o (p)"0 for b5b , the critical point b can be also seen as the 3 # 3 # # compactification point for the spectral density o (h), similarly to what is observed in the 3 Gross—Witten model. 5.4. The large-N limit of the four-link chiral chain For ¸"4, setting º"º and »"º , S(4) is given by 1 3 %&&

P

P

exp (N2S(4))" dº exp (2Nb Re Tr Aºs ) dº exp (2Nb Re Tr Aºs ) , %&& 2 2 4 4

(5.19)

where again A"º#». The large-N limit of the spectral density o (h) of the matrix º»s must be 4 solution of the equation

P

C

D

h!/ sin 1h 2 # "0 , (5.20) 2 cos 1h#cos 1/ 2 2 satisfying the normalization condition :o (h) dh"1. 4 In order to solve Eq. (5.20) one must again separate weak- and strong-coupling domains. In the weak-coupling region the solution is 4b sin 1h! d/ o (/) cot 2 4

S

2b h h o (h)" sin2 #!sin2 4 p 2 2

for 04h4h 4p , #

o (h)"0 4

for h 4h4p , #

(5.21)

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with h implicitly determined by the normalization condition :h# # o (h) dh"1. The solution # ~h 4 (Eq. (5.21)) is valid for b5b "1p, since the normalization condition can be satisfied only in this # 8 region. 1p is then a point of non-analyticity representing the critical point for the transition from 8 the weak to the strong-coupling domain. In the strong-coupling domain b(b "1p one finds # 8

S

b h o (h)" j!sin2 , 4 2 2

(5.22)

where j is determined by the normalization condition :p o (h) dh"1. The strong- and weak~p 4 coupling expressions of o (h) coincide at b : 4 #

S

h b 1!sin2 . o (h) " 4 #3*5 2 2

(5.23)

Notice that again the critical point b "1p represents the compactification point of the spectral # 8 density o (h); indeed o (p)'0 for b(b , and o (p)"0 for b5b . 4 4 # 4 # 5.5. Critical properties of chiral chain models with L44 In the following we derive the N"R critical behavior of the specific heat in the models with ¸"3, 4, using the exact results of Sections 5.3 and 5.4. From the spectral density o (h), the internal energy can be easily derived by º ":dh o (h) cos h. 3 3 3 One finds that º is continuous at b . In the weak-coupling region b5b "1, 3 # # 3

A

B

1 1 1 3@2 º "b# ! !b 1! , 3 2 8b 3b

A

BS

1 1 C "b2# !b2 1# 3 6b 8

1 1! . 3b

(5.24)

Close to criticality, i.e., for 04b/b !1;1, # 17 1 C " ! (b!b )1@2#O(b!b ) . 3 72 2J3 # #

(5.25)

In the strong-coupling region, one finds 17 1 C " ! (b !b)1@2#O(b !b) 3 72 2J3 # #

(5.26)

for 041!b/b ;1. Then the weak- and strong-coupling expressions of C show that the critical # 3 point b "1 is of the third order, and the critical exponent associated with the specific heat is # 3 a"!1. 2

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In the ¸"4 case, recalling that o (h) is the spectral distribution of º ºs , one writes 4 1 3 1 h h / F " 8b dh o (h) cos ! dh d/ o (h)o (/) log cos #cos 4 4 4 4 4 2 2 2

C P

P

P

A

B

D

3 h!/ ! !log 2b# dh d/ o (h)o (/) log sin2 . 4 4 2 2

(5.27)

Observing that, since o (h) is a solution of the variational equation dF /do "0, the following 4 4 4 relation holds dF ­F 4" 4 , db ­b

(5.28)

one can easily find that

P

1 h º "! # dh o (h) cos . 4 4 8b 2

(5.29)

In this case, the study of the critical behavior around b "1p is slightly subtler, since it requires # 8 the expansion of elliptic integrals F(k) and E(k) around k"1. Approaching criticality from the weak-coupling region, i.e., when bPb`, one obtains # p2 1 p2 C " # ! #O(d2 ) , (5.30) 4 32 8 16 log(4/d ) 8 8 where d2 &b!b , apart from logarithms. For bPb~ 8 # # p2 p2 1 #O(d2) , (5.31) C " # ! 4 4 32 8 16 log(4/d ) 4 where d2&b !b, apart from logarithms. A comparison of Eqs. (5.30) and (5.31) leads to the 4 # conclusion that the phase transition is again of the third order, with a specific heat critical exponent a"0~. In conclusion we have seen that chain models with ¸"2, 3, 4,R have a third-order phase transition at increasing values of the critical coupling, b "1, 1, 1p, 1 respectively, with specific heat # 438 2 critical exponents a"!1, !1, 0~, !1 respectively. It is worth noticing that a increases when 2 ¸ goes from 2 to 4, reaching the limit of a third-order critical behavior, but in the large-¸ limit it returns to a"!1. The critical exponent l, describing the double-scaling behavior for NPR and bPb , can then # be determined by the two-dimensional hyperscaling relationship 2l"2!a. This relation has been proved to hold for the Gross—Witten problem, and therefore for the ¸"2 and ¸"R chain models, where it is related to the equivalence of the corresponding double scaling limit with the continuum limit of a two-dimensional gravity model with central charge c"!2. It is then expected to hold in general for all values of ¸. At ¸"4, the value l"1 has been numerically verified, within a few per cent of uncertainty, by studying the scaling of the specific heat peak position at finite N. Notice that the exponents a"0~, l"1 found for ¸"4 correspond to a central charge c"1.

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5.6. Strong-coupling expansion of chiral chain models Strong-coupling series of the free energy density of chiral chain models can be generated by means of the character expansion, which leads to the result F (b)"F(b)#FI (b) , L L where F(b) is the free energy of the single unitary-matrix model,

(5.32)

1 FI " log+ d2 zL , (5.33) L ¸N2 (r) (r) (r) + denotes the sum over all irreducible representations of U(N), and d and z (b) are the (r) (r) (r) corresponding dimensions and character coefficients. The calculation of the strong-coupling series of F (b) is considerably simplified in the large-N limit, due to the relationships (Eq. (4.108)) and L z (b)"zN bn#O(b2N) , (5.34) (r) (r) where zN is independent of b and n is the order of the representation (r). Explicit expressions for (r) d and zN were reported in Section 4.8. The large-N strong-coupling expansion of FI (b) is actually (r) (r) L a series in bL, i.e., FI "+ c(n, ¸)bnL . (5.35) L n It is important to recall that the large-N character coefficients have jumps and singularities at b"1 [52], and therefore the relevant region for a strong-coupling character expansion is b(1. 2 2 Another interesting aspect of the large-N limit of chain models, studied by Green and Samuel using the strong-coupling character expansion [75], concerns the determinant channel, which should provide an order parameter for the phase transition. The quantity 1 p"! logSdet º ºs T i i`1 N

(5.36)

is non-zero in the strong-coupling domain and zero in weak coupling at N"R. b may then be # evaluated by determining where the strong-coupling evaluation of the order parameter p vanishes. Like the free-energy, p is calculable via a character expansion. Indeed z + d zL~1d Sdet º ºs T" (r) (r) (r) (r,~1) (r,~1) . (5.37) i i`1 + d2 zL (r) (r) (r) Green and Samuel evaluated a few orders of the above character expansion, obtaining estimates of b from the vanishing point of p. Such estimates compare well with the exact results for ¸"3, 4. In # the cases where b is unknown, they found b K0.44 for ¸"5, b K0.47 for ¸"6, etc., with # # # b monotonically approaching the value 1 with increasing ¸. # 2 In order to study the critical behavior of chain models for ¸55, one can also analyze the corresponding strong-coupling series of the free energy Eq. (5.32) [135]. An integral approximant analysis of the strong-coupling series of the specific heat led to the estimates b K0.438 for ¸"5 # and b K0.474 for ¸"6, with small negative a, which could mimic an exponent a"0~. For ¸57 #

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a such strong-coupling analysis would lead to b larger than 1, that is out of the region where # 2 a strong-coupling analysis can be predictive. Therefore something else must occur earlier, breaking the validity of the strong-coupling expansion. An example of this phenomenon is found in the Gross—Witten single-link model (recovered when ¸PR), where the strong-coupling expansion of the N"R free energy is just F(b)"b2, an analytical function without any singularity; therefore, in this model, b "1 cannot be determined from a strong-coupling analysis of the free energy. # 2 From such analysis one may hint at the following possible scenario: as for ¸44, for ¸"5, 6, that is when the estimate of b coming from the above strong-coupling analysis is smaller than 1 # 2 and therefore acceptable. The term FI (b) in Eq. (5.32) should be the one relevant for the critical properties, determining the critical points and giving aO!1 (maybe a"0~ as in the ¸"4 case). For ¸57 the critical point need not be a singular point of the free energy in strong or weak coupling, but just the point where weak-coupling and strong-coupling curves meet each other. This would cause a softer phase transition with a"!1, as for the Gross—Witten single-link problem. We expect b (1 also for ¸57. This scenario is consistent with the results of the analysis of the # 2 character expansion of p, defined in Eq. (5.36). 6. Simplicial chiral models 6.1. Definition of the models Another interesting class of finite-lattice chiral models is obtained by considering the possibility that each of a finite number of unitary matrices may interact in a fully symmetric way with all other matrices, while preserving global chiral invariance; the resulting systems can be described as chiral models on (d!1)-dimensional simplexes, and thus termed “simplicial chiral models” [135,136]. The partition function for such a system is

P

C

D

d d d Z " < dº exp Nb + + Tr(º ºs#º ºs) . (6.1) d i i j j i i/1 j/i`1 i/1 Eq. (6.1) encompasses as special cases a number of models that we have already introduced and solved; in particular, the chiral chains with ¸43 correspond to the simplicial chiral models with d43. One of the most attractive features of these models is their relationship with higher-dimensional systems, with which they share the possibility of high coordination numbers. This relationship becomes exact in the large-d limit, where mean-field results are exact. In the large-N limit and for arbitrary d a saddle-point equation can be derived, whose solution allows the evaluation of the large-N free energy F "(1/N2) log Z d d and of related thermodynamical quantities.

(6.2)

6.2. Saddle-point equation for simplicial chiral models The strategy for the determination of the large-N saddle-point equation is based on the introduction of a single auxiliary variable A (a complex matrix), leading to the decoupling of the

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unitary matrix interaction: Z "ZI /ZI , d d 0 where

(6.3)

P

C

D

d (6.4) ZI " < dº dA exp !Nb Tr AAs#Nb Tr A+ ºs#Nb Tr As+ º !N2bd . i i i d i i i/1 We are now back to the single-link problem and, since we have solved it in Section 3 in terms of the function ¼, whose large-N limit is expressed by Eq. (3.15), we obtain

P

ZI " dA exp [!Nb Tr AAs#Nd¼(b2AAs)!N2bd] . d

(6.5)

It is now convenient to express the result in terms of the eigenvalues x of the Hermitian i semipositive-definite matrix 4bAAs, obtaining

P

C

AB

D

N x ZI " dk(x ) exp ! + x #N d¼ i !N2bd . d i i 4b 4 i The angular integration can be performed, leading to dk(x )"< dx < (x !x )2 . i i i j i i;j The saddle-point equation is therefore

(6.7)

Jr#x 1 (4!d)Jr#x #dJr#x i!d" + j, i 2b x !x N i j iEj subject to the constraint (needed to define r)

G

1 1 + "1 N Jr#x i i r"0

(6.6)

(6.8)

(strong coupling) ; (6.9) (weak coupling) .

The energy º "1­F /­b d 2 d is easily expressed in terms of the eigenvalues:

(6.10)

1 1 (6.11) d(d!1)º " + x !d! . i d 4b2 b i In the large-N limit, after a change of variables to z "Jr#x , we introduce as usual an i i eigenvalue density function o(z), and turn Eq. (6.8) into the integral equation

P

C

D

z d!2 b 2 !d" dz@ o(z@) ! , 2b z!z@ z#z@ a

(6.12)

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subject to the constraints

P P

b

a b a

o(z@) dz@"1 ,

(6.13)

dz@ 41 , z@

(6.14)

o(z@)

with equality holding in strong coupling, where a"Jr. The easiest way of evaluating the free energy F is the integration of the large-N version of Eq. (6.11) with respect to b. $ Very simple solutions are obtained for a few special values of d. When d"0, the problem reduces to a Gaussian integration, and one easily finds that Eq. (6.12) is solved by z J16b!(z2!a2) o(z)" 4pb Jz2!a2

(6.15)

and ZI " exp (N2 log b), independent of a as expected. 0 When d"2 we obtain 1 o (z)" J8b!(z!4b)2, b51 , 8 2 4pb

S

1#6b!z 1 z , o (z)" 4 4pb z!(1!2b)

r(b)"(1!2b)2, b41 , 2

(6.16) (6.17)

and these results are consistent with the reinterpretation of the model as a Gross—Witten oneplaquette system. Notice however that the matrix whose eigenvalue distribution has been evaluated is not the original unitary matrix, and corresponds to a different choice of physical degrees of freedom. This is the reason why, while knowing the solution for the free energy of the d"1 system (trivial, non-interacting) and of the d"3 system (three-link chiral chain), we cannot find easily explicit analytic forms for the corresponding eigenvalue densities. The saddle-point equation (Eq. (6.12)) has been the subject of much study in recent times, because it is related to many different physical problems in the context of double-scaling limit investigations. In particular, in the range of values 04d44, the same equation describes the behavior of O(n) spin models on random surfaces in the range !24n42, with the very simple mapping n"d!2 [137]. In this range, the equation has been solved analytically in Refs. [138] and especially [139] in terms of h-functions. 6.3. The large-N d"4 simplicial chiral model The chiral model on a tetrahedron is the first example within the family of simplicial chiral models which turns out to be really different from all the systems discussed in the previous sections. Explicit solutions were found for both the weak and the strong coupling phases, and they are best expressed in terms of a rescaled variable f"J1!z2/b2

(6.18)

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and of a dynamically determined parameter k"J1!a2/b2 .

(6.19)

The resulting expressions, after defining boN (f) df,o(z) dz, are

C

D

8 Jk2!f2 oN (f)" K(k)!Jk2!f2J1!f2 P(f2, k) , 8 E(k)2 J1!f2

C

(6.20)

D

J1!f2 8 k2 K(k)!Jk2!f2J1!f2P(f2, k) , oN (f)" 4 [E(k)!(1!k2)K(k)]2 Jk2!f2

(6.21)

where K, E and P are the standard elliptic integrals, and 04f4k. The complete solution is obtained by enforcing the normalization condition, which leads to a relationship between b and k, best expressed by the equation

P

1 k (6.22) " df oN (f, k) . b 0 Criticality corresponds to the limit kP1, and it is easy to recognize that both weak and strong coupling results lead in this limit to b "1 and # 4 boN (f)"f log[(1#f)/(1!f)] . (6.23) # Many interesting features of this model in the region around criticality can be studied analytically, and one may recognize that the critical behavior around b "1 corresponds to a limiting case of # 4 a third-order phase transition with critical exponent of the specific heat a"0~. In the doublescaling limit language this would correspond to a model with central charge c"1 and logarithmic deviations from scaling. The critical behavior of the specific heat on both sides of criticality is described by

A

B

­º p2 1 p2#3 C,b2 P ! #O , ­b 12 log(4/k@) log2 k@ 36 k{?0 where k@,J1!k2.

(6.24)

6.4. The large-d limit By introducing a function defined by

P

b o(z@) 1 dz@, f (z) P , (6.25) z!z@ z a @z@?= analytic in the complex z plane with the exception of a cut on the positive real axis in the interval [a, b], we can turn the saddle-point equation (6.12) into the functional equation f (z)"

z !d"2 Re f (z)#(d!2) f (!z) . 2b

(6.26)

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This equation can be the starting point of a systematic 1/d expansion, on whose details we shall not belabor, especially because its convergence for small values of d is very slow. It is however interesting to solve the large-d limit of Eq. (6.26) by the Ansatz o(z)"d(z!zN ) ,

(6.27)

whose substitution into Eq. (6.25) leads to the solution

G

zN "

bd(1#J1!1/bd),

bd51 ,

1,

bd41 .

(6.28)

The large-d limit predicts the location of the critical point b "1/d, and shows complete equiva# lence with the mean-field solution of infinite-volume principal chiral models on a d/2-dimensional hypercubic lattice. The large-d prediction for the nature of criticality is that of a first-order phase transition, with

S

1 1 1 1 º" # 1! ! , bd51 . 2 2 bd 4bd

(6.29)

6.5. The large-N criticality of simplicial models The connection with the double-scaling limit problem naturally leads to the study of the finite-b critical behavior. In the regime 04d44 one is helped by the equivalence with the solved problem of O(n) spin models on a random surface, which allows not only a determination of the critical value (found to satisfy the relationship b d"1), but also an evaluation of the eigenvalue distribu# tion at criticality [137]: 2 ph sinh hu o (z)" cos , # ph 2 cosh u

(6.30)

ph 2 b " tan , # h 2

(6.31)

a "0, #

where h and u are defined by the parametrizations 4 cos2 ph/2,d"1/b , cosh u,b /z . (6.32) # # Unfortunately, the technique that was adopted in order to find the above solution does not apply to the regime d'4, in which case one cannot choose a "0. The saddle-point equation at # criticality can however be solved numerically with very high accuracy, and one finds that the relationship b d"1 (6.33) # is satisfied for all d, thus also matching the large-d predictions. The combinations (a #b )/2 and # # a b admit a 1/d expansion, and the coefficients of the expansion are found numerically to be # # integer numbers up to order d~8. An analysis of criticality for d'4 shows that its description is fully consistent with the existence of a first-order phase transition, with a discontinuity of the internal energy measured by da2/(4(d!1)), again matching with the large-d (mean-field) predictions. #

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6.6. The strong-coupling expansion of simplicial models There is nothing peculiar in performing the strong-coupling expansion of Eq. (6.1). There is however a substantial difference with respect to the case of chiral chains discussed in the previous section: because of the topology of simplexes, the strong-coupling configurations entering the calculation are no longer restricted to simple graphs whose vertices are joined by at most one link, and the full complexity of group integration on arbitrary graphs is now involved [130]. As a consequence, as far as the simplicial models can be solved by different techniques, they may also be used as generating functionals for these more involved group integrals, that enter in an essential way in all strong-coupling calculations in higher-dimensional standard chiral models and lattice gauge theories.

7. Asymptotically free matrix models 7.1. Two-dimensional principal chiral models Two dimensional SU(N)]SU(N) principal chiral models, defined by the action

P

N S" d2x Tr ­ º(x)­ ºs(x) , k k ¹

(7.1)

are the simplest asymptotically free field theories whose large-N limit is a sum over planar diagrams, like four dimensional SU(N) gauge theories. We assume, as usual, that the limit NPR is taken while keeping fixed the (rescaled) coupling b"1/¹, and that the mass gap does not vanish even at N"R. Using the existence of an infinite number of conservation laws and Bethe-Ansatz methods, the on-shell solution of the SU(N)]SU(N) chiral models has been proposed in terms of a factorized S-matrix [140,141]. The analysis of the corresponding bound states leads to the mass spectrum sin(rp/N) M "M , 14r4N!1 , r sin(p/N)

(7.2)

where M is the mass of the r-particle bound state transforming as totally antisymmetric tensors of r rank r. M,M is the mass of the fundamental state determining the Euclidean long-distance 1 exponential behavior of the two-point Green’s function 1 G(x)" STr º(0)º(x)sT . N

(7.3)

The mass-spectrum (7.2) has been verified numerically at N"6 by Monte Carlo simulations [142,143]: Monte Carlo data of the mass ratios M /M and M /M agree with formula (7.2) within 2 3 statistical errors of about one per cent. Concerning the large-N limit of these models, it is important to notice that the S-matrix has a convergent expansion in powers of 1/N, and becomes trivial, i.e., the S-matrix of free particles, in the large-N limit.

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By using Bethe-Ansatz techniques, the mass/K-parameter ratio has also been computed, and the result is [144]

S

M 8p sin(p/N) " , (7.4) K e p/N MS which again enjoys a 1/N expansion with a finite radius of convergence. This exact but nonrigorous result has been substantially confirmed by Monte Carlo simulations at several values of N [132,145], and its large-N limit also by N"R strong-coupling calculations [146,147]. While the on-shell physics of principal chiral models has been substantially solved, exact results of the off-shell physics are still missing, even in the large-N limit. When NPR, principal chiral models should just reproduce a free-field theory in disguise. In other words, a local nonlinear mapping should exist between the Lagrangian fields º and some Gaussian variables [7]. However, the behavior of the two-point Green’s function G(x) of the Lagrangian field shows that such realization of a free-field theory is nontrivial. While at small Euclidean momenta, and therefore at large distance, there is a substantial numerical evidence for an essentially Gaussian behavior of G(x) [132], at short distance renormalization group considerations lead to the asymptotic behavior G(x)&[log(1/xK)]c1@b0 ,

(7.5)

where K is a mass scale, and c /b "2(1!2/N2) P 2 . (7.6) 1 0 N?= b and c are the first coefficients respectively of the b-function and of the anomalous dimension of 0 1 the fundamental field. We recall that a free Gaussian Green’s function behaves like log(1/x). Then at small distance G(x) seems to describe the propagation of a composite object formed by two elementary Gaussian excitations, suggesting an interesting hadronization picture: in the large-N limit, the Lagrangian fields º, playing the roˆle of non-interacting hadrons, are constituted by two confined particles, which appear free in the large momentum limit, due to asymptotic freedom. We must mention that, according to Ref. [148], it is possible to take the large-N limit of principal chiral models in such a way that all the infinite states of the physical mass spectrum lay in a finite range of values. Since these states are equally spaced, the resulting spectrum does not show a mass gap. As a consequence, while exactly solvable, the corresponding theory is not the large-N version of the models whose exact S-matrix has been discussed above. The limit discussed in Ref. [148] cannot be reached by considering lattice versions of the models for larger and larger values of N, and exploring their critical region at fixed values of the conventionally rescaled coupling. Numerical investigations by Monte Carlo simulations of lattice chiral models in the continuum limit show that the conventional large-N limit is rapidly approached, which confirms that the 1/N expansion, were it available, would be an effective predictive tool in the analysis of these models. 7.2. Principal chiral models on the lattice In the persistent absence of an explicit solution, the large-N limit of two-dimensional chiral models has been investigated by applying analytical and numerical methods of lattice field theory,

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such as strong-coupling expansion and Monte Carlo simulations. In the following we describe the main results achieved by these studies. A standard lattice version of the continuum action (7.1) is obtained by introducing a nearestneighbor interaction, according to Eq. (2.10): 1 S "!2Nb + Re Tr[º ºs ], b" . (7.7) L x x`k ¹ x, k SU(N) and U(N) lattice chiral models, obtained by constraining, respectively, º 3SU(N) and x º 3U(N), are expected to have the same large-N limit at fixed b. In the continuum limit bPR, x SU(N) and U(N) lattice actions should describe the same theory even at finite N, since the additional U(1) degrees of freedom of U(N) models should decouple. In other words, the U(N) lattice theory represents a regularization of the SU(N)]SU(N) chiral field theory when restricting ourselves to its SU(N) degrees of freedom, i.e. when considering Green’s functions of the field º x ºK " , x (det º )1@N x

(7.8)

e.g., G(x),(1/N)STr ºK ºK sT , (7.9) 0 x whose large-distance behavior allows to define the fundamental mass M. At finite N, while SU(N) lattice models should not have any singularity at finite b, U(N) lattice models should undergo a phase transition, driven by the U(1) degrees of freedom corresponding to the determinant of º(x). The determinant two-point function G (x),Sdet[ºs(x)º(0)]T1@N (7.10) $ behaves like x~f (b, N) at large x in the weak-coupling region, with f (b, N)&O(1/N), but drops off exponentially in strong-coupling region, where G (x)&e~m$x with [75] $ 1 N! m "!log b# log #O(b2) . (7.11) $ N NN This would indicate the existence of a phase transition at a finite b in U(N) lattice models. Such $ a transition, being driven by U(1) degrees of freedom, should be of the Kosterlitz—Thouless type: the mass propagating in the determinant channel m should vanish at the critical point b and stay zero $ $ for larger b. Hence for b'b this U(1) sector of the theory would decouple from the SU(N) degrees $ of freedom, which alone determine the continuum limit (bPR) of principal chiral models. The large-N limit of principal chiral models has been investigated by Monte Carlo simulations of SU(N) and U(N) models for several large values of N, studying their approach to the N"R limit [132,129]. Many large-N strong-coupling calculations have been performed which allow a direct study of the N"R limit. Within the nearest-neighbor formulation (7.7), the large-N strong-coupling expansion of the free energy has been calculated up to 18th order, and that of the fundamental Green’s function G(x) (defined in Eq. (7.3)) up to 15th order [75,130]. Large-N strong-coupling calculations have been performed also on the honeycomb lattice, within the corresponding

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nearest-neighbor formulation, which is expected to belong to the same class of universality with respect to the critical point b"R. On the honeycomb lattice the free energy has been computed up to O(b26), and G(x) up to O(b20) [130]. Let us define the internal energy density 1 dF E,1!º"1! , 4 db

1 F" log Z , N2»

(7.12)

and the specific heat dE 1 ­2F . C" " b2 d¹ 4 ­b2

(7.13)

Monte Carlo simulations show that SU(N) and U(N) lattice chiral models have a peak in the specific heat which becomes sharper and sharper with increasing N, suggesting the presence of a critical phenomenon for N"R at a finite b . In U(N) models the peak of C is observed in the # region where the determinant degrees of freedom are massive, i.e., for b(b (this feature character$ izes also two-dimensional XY lattice models [149]). An estimate of the critical coupling b has been # obtained by extrapolating the position b (N) of the peak of the specific heat (at infinite volume) 1%!, to NPR using a finite-N scaling Ansatz [129] b (N)Kb #cN~e , (7.14) 1%!, # mimicking a finite-size scaling relationship. The above Ansatz arises from the idea that the parameter N may play a roˆle quite analogous to the volume in the ordinary systems close to the criticality. This idea was already exploited in the study of one-matrix models [21,48,49], where the double scaling limit turns out to be very similar to finite-size scaling in a two-dimensional critical phenomenon. The finite-N scaling ansatz (7.14) has been verified in the similar context of the large-N Gross—Witten phase transition, as mentioned in Section 4.7. Since e is supposed to be a critical exponent associated with the N"R phase transition, it should be the same in the U(N) and SU(N) models. The available U(N) and SU(N) Monte Carlo data (at N"9, 15, 21 for U(N) and N"9, 15, 21, 30 for SU(N)) fit very well the ansatz (7.14), and their extrapolation leads to the estimates b "0.3057(3) and e"1.5(1). The interpretation of the exponent e in this context is still an open # problem. It is worth noticing that the value of the correlation length describing the propagation in the fundamental channel is finite at the phase transition: m(c)K2.8. The existence of this large-N phase transition is confirmed by an analysis of the N"R 18th-order strong-coupling series of the free energy F"2b2#2b4#4b6#19b8#96b10#604b12#4036b14 58471 663184 # b16# b18#O(b20) , 2 3

(7.15)

which shows a second-order critical behavior: C&Db!b D~a , #

(7.16)

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with b "0.3060(4) and a"0.27(3), in agreement with the extrapolation of Monte Carlo data. The # above estimates of b and a are slightly different from those given in Ref. [147]; they are obtained # by a more refined analysis based on integral approximant techniques [150—152] and by the so-called critical point renormalization method [153]. Green and Samuel argued that the large-N phase transition of principal chiral models on the lattice is nothing but the large-N limit of the determinant phase transition present in U(N) lattice models [52,111]. According to this conjecture, b and b should both converge to b in the $ 1%!, # large-N limit, and the order of the determinant phase transition would change from the infinite order of the Kosterlitz—Thouless mechanism to a second order with divergent specific heat. The available Monte Carlo data of U(N) lattice models at large N provide only a partial confirmation of this scenario; one can just get a hint that b (N) is also approaching b with increasing N. The $ # large-N phase transition of the SU(N) models could then be explained by the fact that the large-N limit of the SU(N) theory is the same as the large-N limit of the U(N) theory. The large-N character expansion of the mass m propagating in the determinant channel has $ been calculated up to 6th order in the strong-coupling region, indicating a critical point (determined by the zero of the m series) slightly larger than our determination of b : b (N"R)K0.324 $ # $ [52]. This discrepancy might be explained either by the shortness of the available character expansion of m or by the fact that such a determination of b relies on the absence of singular $ # points before the strong-coupling series of m vanishes, and therefore a non-analyticity at $ b K0.306 would invalidate all strong-coupling predictions for b'b . # # It is worth mentioning another feature of this large-N critical behavior which emerges from a numerical analysis of the phase distribution of the eigenvalues of the link operator ¸"º ºs : (7.17) x x`k the N"R phase transition should be related to the compactification of the eigenvalues of ¸ [129], like the Gross—Witten phase transition. The existence of such a phase transition does not represent an obstruction to the use of strong-coupling expansion for the investigation of the continuum limit. Indeed large-N Monte Carlo data show scaling and asymptotic scaling (in the energy scheme) even for b smaller than the peak of the specific heat, suggesting an effective decoupling of the modes responsible for the large-N phase transition from those determining the physical continuum limit. This fact opens the road to tests of scaling and asymptotic scaling at N"R based only on strong-coupling computations, given that the strong-coupling expansion should converge for b(b . (The strong-coupling # analysis does not show evidence of singularities in the complex b-plane closer to the origin than b .) # In the continuum limit the dimensionless renormalization-group invariant function GI (0; b) A(p; b), GI (p; b)

(7.18)

turns into a function A(y) of the ratio y,p2/M2 only, where M2 ,1/m2 and m is the second G G G G moment correlation length 1 + x2G(x) m2 , x . G 4 + G(x) x

(7.19)

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A(y) can be expanded in powers of y around y"0: = A(y)"1#y# + c yi , (7.20) i i/2 and the coefficients c parameterize the difference from a generalized Gaussian propagator. The i zero y of A(y) closest to the origin is related to the ratio M2/M2 , where M is the fundamental mass; 0 G indeed y "!M2/M2 . M2/M2 is in general different from one; it is one in Gaussian models (i.e. 0 G G when A(y)"1#y). Numerical simulations at large N, which allow an investigation of the region y50, have shown that the large-N limit of the function A(y) is approached rapidly and that its behavior is essentially Gaussian for y[1, indicating that c ;1 in Eq. (7.20) [142]. Important logarithmic corrections to i the Gaussian behavior must eventually appear at sufficiently large momenta, as predicted by simple weak-coupling calculations supplemented by a renormalization group resummation: GI (p)&(log p2)/p2

(7.21)

for p2/M2 <1 and in the large-N limit. G The approximate Gaussian behavior at small momentum is also confirmed by the direct estimate of the ratio M2/M2 obtained by extrapolating Monte Carlo data to N"R. The large-N limit of G the ratio M2/M2 is rapidly approached, already at N"6 within few per mille, leading to the G estimate M2/M2 "0.982(2), which is very close to one [132]. Large-N strong-coupling comG putations of M2/M2 provide a quite stable curve for a large region of values of the correlation G length, which agrees (within about 1%) with the continuum large-N value extrapolated by Monte Carlo data [147]. Monte Carlo simulations at large values of N (N56) also show that asymptotic scaling predictions applied to the fundamental mass are verified within a few per cent at relatively small values of the correlation length (mZ2) and even before the peak of the specific heat in the so-called “energy scheme” [154]; the energy scheme is obtained by replacing ¹ with a new temperature variable ¹ JE. At N"R a test of asymptotic scaling may be performed by using the large-N E strong-coupling series of the fundamental mass. The two-loop renormalization group and a Bethe Ansatz evaluation of the mass/K-parameter ratio [144] lead to the following large-N asymptotic scaling prediction in the b scheme: E p M+16 Jp/e exp K (b ) , 4 E, 2l E

AB

K (b )"J8pb exp (!8pb ) , E E, 2l E E 1 b " . E 8E

(7.22)

Strong-coupling calculations, where the new coupling b is extracted from the strong-coupling E series of E, show asymptotic scaling within about 5% in a relatively large region of values of the correlation length (1.5[m[3) [146,147]. The good behavior of the large-N b-function in the b scheme, and therefore the fact that E physical quantities appear to be smooth functions of the energy, together with the critical behavior

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(7.16), can be explained by the existence of a non-analytical zero at b of the b-function in the # standard scheme: b (¹),a d¹/da&Db!b Da (7.23) L # around b , where a is the critical exponent of the specific heat. This is also confirmed by an analysis # of the strong-coupling series of the magnetic susceptibility s and M2 , which supports the relations G d log s d log M2 G&Db!b D~a & (7.24) # db db in the neighborhood of b , which are consequences of Eq. (7.23) [147]. # The presence of a non-analytical zero (7.23) in the large-N b-function should imply that the asymptotic scaling regime (in the standard temperature variable) should be pushed to very large values of b. On the other hand, this singularity can be eliminated by changing the temperature variable to ¹ , achieving a much faster approach to asymptotic scaling. The improvement obtained E by using ¹ can be observed in perturbation theory. For N'3, the linear correction to the E two-loop relation between the coupling and the K-parameter is considerably smaller in the b scheme. The K-parameter is defined by E 1 1 K" exp ! d¹ , (7.25) a(¹) b(¹)

C P

D

where ¹ is a generic temperature variable (or coupling); at small ¹,

A

BC

D

1 b2!b b 1 0 2 ¹#O(¹2) , (b ¹)~b1@b20 exp ! 1# 1 (7.26) K" 0 b ¹ b3 a(¹) 0 0 where b , b , and b are the first coefficients of the perturbative expansion of the b-function. Unlike 0 1 2 b and b , b depends on the choice of the coupling. The coefficient of the linear correction in 0 1 2 Eq. (7.26) is b2!b b 1 0 2 P !0.00884 b3 0 N?= in the energy scheme, and

(7.27)

b2!b b 1 0 2 P 0.06059 (7.28) b3 0 N?= in the standard scheme. This fact was overlooked in Ref. [132] (cf. [155]). We finally mention that similar results have been obtained for two-dimensional chiral models on the honeycomb lattice by a large-N strong-coupling analysis. In fact an analysis of the 26th-order strong-coupling series of the free energy indicates the presence of a large-N phase transition, with specific heat exponent a+0.17, not far from that found on the square lattice (we have no reasons to expect that the large-N phase transition on the square and honeycomb lattices are in the same universality class). Furthermore, the mass-gap extracted from the 20th-order strong-coupling expansion of G(x) allows to check the corresponding asymptotic scaling predictions in the energy scheme within about 10% [147].

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7.3. The large-N limit of Sº(N) lattice gauge theories An overview of the large-N limit of the continuum formulation of QCD has been already presented in Section 2. In the following we report some results concerning the lattice approach. Gauge models on the lattice have been mostly studied in their Wilson formulation S "Nb + Tr[º (x)º (x#k)ºs(x#l)ºs(x)#h.c.] . W k l k l x,k;l

(7.29)

In view of a large-N analysis one may consider both SU(N) and U(N) models, since they are expected to reproduce the same statistical theory in the limit NPR (at fixed b). As for two-dimensional chiral models, SU(N) and U(N) models should have the same continuum limit for any finite N52. The phase diagram of statistical models defined by the Wilson action has been investigated by standard techniques, i.e., strong-coupling expansion, mean field [77], and Monte Carlo simulations [156—158]. These studies show the presence of a first-order phase transition in SU(N) models for N54, and in U(N) models for any finite N. A first-order phase transition is then expected also in the large-N limit at a finite value of b, which is estimated to be b +0.38 by mean-field calculations # and by extrapolation of Monte Carlo results. A review of these results can be found in Ref. [159]. Some speculations on the large-N phase diagram can be also found in Refs. [28,111]. The roˆle of the determinant of Wilson loops in the phase transition of U(N) gauge models has been investigated in Ref. [111] by strong-coupling character expansion, and in Ref. [160] by Monte Carlo simulations. Large-N mean-field calculations suggest the persistence of a first-order phase transition when an adjoint-representation coupling is added to the Wilson action [116,120]. The first-order phase transition of SU(N) lattice models at N'3 can probably be avoided by choosing appropriate lattice actions closer to the renormalization group trajectory of the continuum limit, as shown in Ref. [161] for SU(5). In U(N) models the use of such improved actions should leave a residual transition, due to the extra U(1) degrees of freedom which should decouple at large b in order to reproduce the physical continuum limit of SU(N) gauge models. It is worth mentioning two studies of confinement properties at large N, obtained essentially by strong-coupling arguments. In Ref. [162], the authors argue that deconfinement of heavy adjoint quarks by color screening is suppressed in the large-N limit. At N"R, the adjoint string tension is expected to be twice the fundamental string tension, as implied by factorization. In Ref. [23], strong-coupling based arguments point out that Wilson loops in O(N), U(N), and Sp(N) lattice gauge theories should have the same large-N limit, and therefore these theories should share the same confinement mechanism. Such results should be taken into account when studying confinement mechanisms. Studies based on Monte Carlo simulations for N'3 have not gone beyond an investigation of the phase diagram, so no results concerning the continuum limit of SU(N) lattice gauge theories with N'3 have been produced. Estimates of the mass of the lightest glueball, obtained by a variational approach within a Hamiltonian lattice formulation, seem to indicate a rapid convergence of the 1/N expansion [163].

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An important breakthrough for the study of the large-N limit of SU(N) gauge theories has been the introduction of the so-called reduced models. A quite complete review on this subject can be found in Ref. [6]. Eguchi and Kawai [63] pointed out that, as a consequence of the large-N factorization, one can construct one-site theories equivalent to lattice YM in the limit NPR. The simplest example is given by the one-site matrix model obtained by replacing all link variables of the standard Wilson formulation with four SU(N) matrices according to the simple rule º (x)Pº . k k This leads to the reduced action

(7.30)

(7.31) S "Nb + Tr[º º ºsºs#h.c.] . EK k l k l k;l Reduced operators, and in particular reduced Wilson loops, can be constructed using the correspondence (Eq. (7.30)). In the large-N limit one can prove that expectation values of reduced Wilson loop operators satisfy the same Schwinger—Dyson equations as those in the Wilson formulation. Assuming that all features of the N"R theory are captured by the Schwinger—Dyson equations of Wilson loops, the reduced model may provide a model equivalent to the standard Wilson theory at N"R. In the proof of this equivalence the residual symmetry of the reduced model º PZ º , Z 3Z , (7.32) k k k k N where Z is the center of the SU(N) group, plays a crucial roˆle. Therefore, the equivalence in the N large-N limit of the Wilson formulation and the reduced model (Eq. (7.31)) is actually valid if the symmetry (Eq. (7.32)) is unbroken. This is verified only in the strong-coupling region; indeed in the weak-coupling region the Z4 symmetry gets spontaneously broken and therefore the equivalence N cannot be extended to weak coupling [164]. In order to avoid this unwanted phenomenon of symmetry breaking and to extend the equivalence to the most interesting region of the continuum limit, modifications of the original Eguchi—Kawai model have been proposed [64,164,165]. The most promising one for numerical simulation is the so-called twisted Eguchi—Kawai (TEK) model [64,165]. Instead of the correspondence Eq. (7.30), the twisted reduction prescription consists in replacing º (x)P¹(x)º ¹(x)s , k k where

(7.33)

¹(x)"< (C )xk k k and C are traceless SU(N) matrices obeying the ’t Hooft algebra k C C "Z C C ; l k kl k l Z is an element of the center of the group Z , kl N 2p Z "exp i n , kl N kl

(7.34)

A

B

(7.35)

(7.36)

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where n is an antisymmetric tensor with n "1 for k(l. C are the matrices implementing the kl kl k translations by one lattice spacing in the k direction (here it is crucial that the fields º are in the k adjoint representation). The twisted reduction applied to the Wilson action leads to the reduced action S "Nb + Tr[Z º º ºsºs#h.c.] . (7.37) TEK kl k l k l k;l The correspondence between correlation functions of the large-N pure gauge theory and those of the reduced twisted model is obtained as follows. Let A[º (x)] be any gauge invariant functional k of the field º (x), then k SA[º (x)]T "SA[¹(x)º ¹(x)s]T . (7.38) k *N/=,YM+ k *N/=,TEK+ Once again the Schwinger—Dyson equations for the reduced Wilson loops, constructed using the correspondence Eq. (7.33), are identical to the loop equations in the Wilson formulation when NPR. The residual symmetry Eq. (7.32), which is again crucial in the proof of the equivalence, should not be broken in the weak-coupling region, and therefore the equivalence should be complete in this case. One can also show that 1. the reduced TEK model is equivalent to the corresponding field theory on a periodic box of size ¸"JN [6]; 2. in the large-N limit finite-N corrections are O(1/N2), just as in the SU(N) lattice gauge theory. Moreover, since N2"¸4, finite-N corrections can be seen as finite-volume corrections. Therefore in twisted reduced models the large-N and thermodynamic limits are connected and approached simultaneously. Monte Carlo studies of twisted reduced models at large N confirm the existence of a first-order phase transition at N"R located at b "0.36(2) [166], which is consistent with the mean-field # prediction b K0.38 [159]. This transition is a bulk transition, and it does not spoil confinement. # The few and relatively old existing Monte Carlo results obtained in the weak-coupling region (cf. e.g. Refs. [166—168]) seem to support a rapid approach to the NPR limit of the physical quantities, and are relatively close to the corresponding results for SU(3) obtained by performing simulations within the Wilson formulation. This would indicate that N"3 is sufficiently large to consider the large-N limit a good approximation of the theory. We mention that hot twisted models can be constructed, which should be equivalent to QCD at finite temperature in the large-N limit (cf. Ref. [6] for details on this subject).

References [1] [2] [3] [4] [5]

H.E. Stanley, Phys. Rev. 176 (1968) 718. G. ’t Hooft, Nucl. Phys. B 72 (1974) 461. S. Coleman, in: A. Zichichi (Ed.), Pointlike structures Inside and Outside Hadrons, Plenum, New York, 1979, p. 11. L.G. Yaffe, Rev. Mod. Phys. 54 (1982) 407. A.A. Migdal, Phys. Rep. 102 (1983) 199.

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[6] S.R. Das, Rev. Mod. Phys. 59 (1987) 235. [7] A.M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, New York, 1988. [8] E. Brezin, S.R. Wadia, The Large N Expansion in Quantum Field Theory and Statistical Physics: From Spin Systems to Two-dimensional Gravity, World Scientific, Singapore, 1994. [9] B. Sakita, Quantum Theory of Many Variable Systems and Fields, World Scientific Lecture Notes in Physics, vol. 1, World Scientific, Singapore, 1980. [10] S.-k. Ma, Rev. Mod. Phys. 45 (1973) 589. [11] S.-k. Ma, J. Math. Phys. 15 (1974) 1866. [12] M. Campostrini, P. Rossi, Riv. Nuovo Cimento 16 (6) (1993) 1. [13] O. Haan, Phys. Lett. B 106 (1981) 207. [14] R. Gopakumar, D.J. Gross, Nucl. Phys. B 451 (1995) 379. [15] M.R. Douglas, Phys. Lett. B 344 (1995) 117. [16] J. Greensite, M.B. Halpern, Nucl. Phys. B 211 (1983) 343. [17] E. Witten, Nucl. Phys. B 149 (1979) 285. [18] A. D’Adda, M. Lu¨scher, P. Di Vecchia, Nucl. Phys. B 146 (1978) 63. [19] S. Nishigaki, T. Yoneya, Nucl. Phys. B 348 (1991) 787. [20] P. Di Vecchia, M. Kato, N. Ohta, Nucl. Phys. B 357 (1991) 495. [21] P.H. Damgaard, U.M. Heller, Nucl. Phys. B 410 (1993) 494. [22] G.M. Cicuta, Lett. Nuovo Cimento 35 (1982) 87. [23] C. Lovelace, Nucl. Phys. B 201 (1982) 333. [24] G. ’t Hooft, Nucl. Phys. B 75 (1974) 461. [25] C.G. Callan, N. Coote, D.J. Gross, Phys. Rev. D 13 (1976) 1649. [26] V.A. Kazakov, Phys. Lett. B 128 (1983) 316. [27] K.H. O’Brien, J. Zuber, Nucl. Phys. B 253 (1985) 621. [28] I.K. Kostov, Nucl. Phys. B 265 [FS15] (1986) 223. [29] D. Weingarten, Phys. Lett. B 90 (1980) 280. [30] D. Weingarten, Phys. Lett. B 90 (1980) 285. [31] G. Veneziano, Nucl. Phys. B 117 (1976) 519. [32] E. Witten, Nucl. Phys. B 160 (1979) 57. [33] S. Coleman, E. Witten, Phys. Rev. Lett. 45 (1980) 100. [34] E. Witten, Nucl. Phys. B 156 (1979) 269. [35] P. DiVecchia, G. Veneziano, Nucl. Phys. B 171 (1980) 253. [36] C. Rosenzweig, J. Schechter, C.G. Trahern, Phys. Rev. D 21 (1980) 3388. [37] P. Nath, R. Arnowitt, Phys. Rev. D 23 (1981) 473. [38] M. Teper, Phys. Lett. B 202 (1988) 553. [39] J. Ambjorn, B. Durhuus, J. Fro¨hlich, Nucl. Phys. B 257 (1985) 433. [40] F. David, Nucl. Phys. B 257 (1985) 45. [41] F. David, Nucl. Phys. B 257 (1985) 543. [42] V.A. Kazakov, Phys. Lett. B 150 (1985) 282. [43] V.A. Kazakov, I.K. Kostov, A.A. Migdal, Phys. Lett. B 157 (1985) 295. [44] V.G. Knizhnik, A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [45] E. Brezin, V.A. Kazakov, Phys. Lett. B 236 (1990) 144. [46] M.R. Douglas, S.H. Shenker, Nucl. Phys. B 335 (1990) 635. [47] D.J. Gross, A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [48] J.W. Carlson, Nucl. Phys. B 248 (1984) 536. [49] E. Brezin, J. Zinn-Justin, Phys. Lett. B 288 (1992) 54. [50] C. Itzykson, J.B. Zuber, J. Math. Phys. 21 (1980) 411. [51] M.L. Mehta, Random Matrices, Academic Press, New York, 1967. [52] F. Green, S. Samuel, Nucl. Phys. B 190 [FS3] (1981) 113. [53] A.A. Migdal, JETP 42 (1975) 743. [54] A. Gonzalez-Arroyo, M. Okawa, Nucl. Phys. B 247 (1984) 104.

P. Rossi et al. / Physics Reports 302 (1998) 143–209

207

[55] Yu.M. Makeenko, A.A. Migdal, Phys. Lett. B 88 (1979) 135. [56] S.R. Wadia, A study of U(N) lattice gauge theory in two dimensions, Chicago preprint EFI Report 79/44, 1979. [57] A. Jevicki, B. Sakita, Nucl. Phys. B 165 (1980) 511. [58] A. Jevicki, B. Sakita, Nucl. Phys. B 185 (1981) 89. [59] G. Marchesini, Nucl. Phys. B 239 (1984) 135. [60] G. Marchesini, E. Onofri, Nucl. Phys. B 249 (1985) 225. [61] V.A. Kazakov, T.A. Kozhamkulov, A.A. Migdal, JETP Lett. 41 (1985) 543. [62] J.P. Rodrigues, Nucl. Phys. B 260 (1985) 350. [63] T. Eguchi, H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. [64] T. Eguchi, R. Nakayama, Phys. Lett. B 122 (1983) 59. [65] A. Gonzales-Arroyo, M. Okawa, Phys. Rev. D 27 (1983) 2397. [66] F.R. Brown, L.G. Yaffe, Nucl. Phys. B 271 (1986) 267. [67] T.A. Dickens, U.J. Lindqwister, W.R. Somsky, L.G. Yaffe, Nucl. Phys. B 309 (1988) 1. [68] R.C. Brower, M. Nauenberg, Nucl. Phys. B 180 [FS2] (1981) 221. [69] E. Brezin, D.J. Gross, Phys. Lett. B 97 (1980) 120. [70] R.C. Brower, P. Rossi, C.-I. Tan, Phys. Rev. D 23 (1981) 942. [71] M. Gaudin, P.A. Mello, J. Phys. G 7 (1981) 1085. [72] S. Samuel, J. Math. Phys. 21 (1980) 2695. [73] I. Bars, J. Math. Phys. 21 (1980) 2678. [74] I. Bars, F. Green, Phys. Rev. D 20 (1979) 3311. [75] F. Green, S. Samuel, Phys. Lett. B 103 (1981) 110. [76] B. Aneva, Y. Brihaye, P. Rossi, Phys. Lett. B 133 (1983) 215. [77] J.-M. Drouffe, J.-B. Zuber, Phys. Rep. 102 (1983) 1. [78] J.B. Kogut, M. Snow, M. Stone, Nucl. Phys. B 200 (1982) 211. [79] Y. Brihaye, P. Rossi, Nucl. Phys. B B 235 [FS11] (1984) 226. [80] Y. Brihaye, A. Taormina, J. de Phys. A 18 (1985) 2385. [81] Y. Brihaye, P. Rossi, Lett. Math. Phys. 8 (1984) 207. [82] V.F. Mu¨ller, W. Ru¨hl, J. Math. Phys. 23 (1982) 2461. [83] V.F. Mu¨ller, W. Ru¨hl, Nucl. Phys. B 210 (1982) 289. [84] H. Hasegawa, S.-K. Yang, Phys. Lett. B 125 (1983) 72. [85] H. Hasegawa, S.-K. Yang, Nucl. Phys. B 225 (1983) 505. [86] D. Weingarten, J. Math. Phys. 19 (1978) 999. [87] B. deWit, G. ’t Hooft, Phys. Lett. B 69 (1977) 61. [88] K.H. O’Brien, J. Zuber, Phys. Lett. B 144 (1984) 407. [89] V.A. Kazakov, JETP 58 (1983) 1096. [90] M.L. Mehta, Comm. Math. Phys. 79 (1981) 327. [91] Harish-Chandra, Am. J. Math. 79 (1957) 87. [92] V.A. Kazakov, A.A. Migdal, Nucl. Phys. B 397 (1993) 214. [93] N. Weiss, in: Proc. Quarks ’94 Conf., 1995, p. 409. [94] D.J. Gross, E. Witten, Phys. Rev. D 21 (1980) 446. [95] P. Rossi, Ann. Phys. (N. Y.) 132 (1981) 463. [96] J. Jurkiewicz, K. Zalewski, Nucl. Phys. B 220 (1983) 167. [97] T. Chen, C.-I. Tan, X. Zheng, Phys. Lett. B 123 (1983) 423. [98] P. Rossi, E. Vicari, Phys. Lett. B 349 (1995) 177. [99] E. Brezin, C. Itzykson, J. Zinn-Justin, J.-B. Zuber, Phys. Lett. B 82 (1979) 442. [100] G. Paffuti, P. Rossi, Phys. Lett. B 92 (1980) 321. [101] J.-M. Drouffe, Phys. Rev. D 18 (1978) 1174. [102] P. Menotti, E. Onofri, Nucl. Phys. B 190 (1981) 288. [103] B. Rusakov, Phys. Lett. B 303 (1993) 95. [104] M.R. Douglas, V.A. Kazakov, Phys. Lett. B 319 (1993) 219.

208 [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154]

P. Rossi et al. / Physics Reports 302 (1998) 143–209 D.J. Gross, W. Taylor, in: M.B. Halpern, G. Rivlis, A. Sevrin (Eds.), Strings ’93, World Scientific, Singapore, 1995, p. 214. E. Barsanti, P. Rossi, Zeit. Phys. C 36 (1987) 143. P. Rossi, unpublished. V.A. Kazakov, I.K. Kostov, Phys. Lett. B 105 (1981) 453. V.A. Kazakov, Nucl. Phys. B 179 (1981) 283. F. Green, S. Samuel, Phys. Lett. B 103 (1981) 48. F. Green, S. Samuel, Nucl. Phys. B 194 (1982) 107. A. Guha, S.-C. Lee, Nucl. Phys. B 240 (1984) 141. P. Rossi, Phys. Lett. B 117 (1982) 72. W. Celmaster, F. Green, Phys. Rev. Lett. 50 (1983) 1556. Y.Y. Goldschmidt, J. Math. Phys. 21 (1980) 1842. T.-l. Chen, C.-I. Tan, X.-t. Zheng, Phys. Lett. B 109 (1982) 383. S. Samuel, Phys. Lett. B 112 (1982) 237. Yu.M. Makeenko, M.I. Polikarpov, Nucl. Phys. B 205 [FS5] (1982) 386. S. Samuel, Phys. Lett. B 122 (1983) 287. M.C. Ogilvie, A. Horowitz, Nucl. Phys. B 215 [FS7] (1983) 249. J. Jurkiewicz, C.P. Korthals Altes, Nucl. Phys. B 242 (1984) 62. J. Jurkiewicz, C.P. Korthals Altes, J.W. Dash, Nucl. Phys. B 233 (1984) 457. J.P. Rodrigues, Phys. Rev. D 26 (1982) 2833. C.B. Lang, P. Salomonson, B.S. Skagerstam, Nucl. Phys. B 190 (1981) 337. S. Samuel, Nucl. Phys. B 205 (1982) 221. K. Demeterfi, C.-I. Tan, Mod. Phys. Lett. A 5 (1990) 1563. V. Periwal, D. Shevitz, Phys. Rev. Lett. 64 (1990) 1326. D.J. Gross, M.J. Newman, Phys. Lett. B 266 (1991) 291. M. Campostrini, P. Rossi, E. Vicari, Phys. Rev. D 52 (1995) 395. M. Campostrini, P. Rossi, E. Vicari, Phys. Rev. D 52 (1995) 358. R.C. Brower, P. Rossi, C.-I. Tan, Nucl. Phys. B 190 (1981) 699. P. Rossi, E. Vicari, Phys. Rev. D 49 (1994) 6072; Phys. Rev. D 50 (1994) 4718 (E). R.C. Brower, P. Rossi, C.-I. Tan, Phys. Rev. D 23 (1981) 953. D. Friedan, Comm. Math. Phys. 78 (1981) 353. R.C. Brower, M. Campostrini, K. Orginos, P. Rossi, C.-I. Tan, E. Vicari, Phys. Rev. D 53 (1996) 3230. P. Rossi, C.-I. Tan, Phys. Rev. D 51 (1995) 7159. M. Gaudin, I. Kostov, Phys. Lett. B 220 (1989) 200. B. Eynard, C. Kristjansen, Nucl. Phys. B 455 (1995) 577. B. Eynard, C. Kristjansen, Nucl. Phys. B 466 (1996) 463. E. Abdalla, M.C.B. Abdalla, A. Lima-Santos, Phys. Lett. B 140 (1984) 71. P.B. Wiegmann, Phys. Lett. B 141 (1984) 217. P. Rossi, E. Vicari, Phys. Rev. D 49 (1994) 1621. I.T. Drummond, R.R. Horgan, Phys. Lett. B 327 (1994) 107. J. Balog, S. Naik, F. Niedermayer, P. Weisz, Phys. Rev. Lett. 69 (1992) 873. S.S. Manna, A.J. Guttmann, B.D. Hughes, Phys. Rev. A 39 (1989) 4337. M. Campostrini, P. Rossi, E. Vicari, Phys. Rev. D 51 (1995) 958. M. Campostrini, P. Rossi, E. Vicari, Phys. Rev. D 52 (1995) 386. V.A. Fateev, V.A. Kazakov, P.B. Wiegmann, Nucl. Phys. B 424 (1994) 505. R. Gupta, C.F. Baillie, Phys. Rev. B 45 (1992) 2883. A.J. Guttmann, G.S. Joyce, J. Phys. A 5 (1972) L81. D.L. Hunter, G.A. Baker Jr., Phys. Rev. B 7 (1972) 3346. M.E. Fisher, H. Au-Yang, J. Phys. A 12 (1979) 1677. D.L. Hunter, G.A. Baker Jr., Phys. Rev. B 49 (1979) 3808. G. Parisi, in: L. Durand, L.G. Pondrom (Eds.), High Energy Physics — 1980, Proc. XXth Conf. on High Energy Physics, AIP, New York, 1980, p. 1531.

P. Rossi et al. / Physics Reports 302 (1998) 143–209

209

[155] P. Rossi, E. Vicari, Erratum: two-dimensional SU(N)]SU(N) chiral models on the lattice. Two-dimensional SU(N)]SU(N) chiral models on the lattice 2. The Green’s function, Phys. Rev. D, submitted. [156] M. Creutz, Phys. Rev. Lett. 46 (1981) 1441. [157] M. Creutz, K.J.M. Moriarty, Phys. Rev. D 25 (1982) 610. [158] K.J.M. Moriarty, S. Samuel, Phys. Rev. D 27 (1983) 982. [159] C. Itzykson, J.-M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge, 1989. [160] K.J.M. Moriarty, S. Samuel, Phys. Lett. B 136 (1984) 55. [161] S. Itoh, Y. Iwasaki, T. Yoshie´, Phys. Rev. Lett. 55 (1985) 273. [162] J. Greensite, M.B. Halpern, Phys. Rev. D 27 (1983) 2545. [163] S.A. Chin, M. Karliner, Phys. Rev. Lett. 58 (1987) 1803. [164] G. Bhanot, U. Heller, H. Neuberger, Phys. Lett. B 115 (1982) 237. [165] A. Gonzalez-Arroyo, M. Okawa, Phys. Lett. B 120 (1983) 174. [166] A. Gonzales-Arroyo, M. Okawa, Phys. Lett. B 133 (1983) 415. [167] K. Fabricius, O. Haan, Phys. Lett. B 139 (1983) 293. [168] O. Haan, K. Meier, Phys. Lett. B 175 (1986) 192.

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Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling Nimrod Moiseyev Department of Chemistry and Minerva Center for Non-linear Physics of Complex Systems, Technion — Israel Institute of Technology, Haifa 32000, Israel Received September 1997; editor: A. Schwimmer

Contents Preface 1. Resonances — Why complex scaling? 1.1. Resonances in full collision experiments 1.2. Lifetime of shape and Feshbach-type resonances from motion of wavepacket calculations 1.3. Resonance lifetime from time-independent calculations of density of states 1.4. Association of the resonance phenomena with the complex poles of the scattering matrix 1.5. The role of complex scaling in the calculations of the resonance poles of the scattering matrix 1.6. Complex scaling and resonances in half collision experiments 1.7. Resonances in multiphoton ionization/dissociation experiments by complex scaling 2. From complex-scaled Hamiltonians to resonance positions and widths 2.1. The complex-scaled Hamiltonian 2.2. The complex ‘‘energy” spectrum — Resonance positions, widths and rotating continua 2.3. Restrictions on the complex-scaling parameter

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2.4. The generalized inner product for complexscaled Hamiltonians 2.5. Do the eigenfunctions of the complexscaled Hamiltonian matrix form a complete basis set? 2.6. The complex analog to the variational principle: The c-variational principle 2.7. The complex analog to the hypervirial theorem 2.8. Cusps, h trajectories and complex-analog Hellmann—Feynman theorem 2.9. The hermitian representation of the complex coordinate method: Upper and lower bounds to the resonance positions and widths 3. Complex scaling of ab initio molecular potential surfaces 4. The complex coordinate scattering theory: from complex-scaled Hamiltonians to partial-widths and cross-sections 4.1. General discussion 4.2. Time-independent Hamiltonians 4.3. Resonance scattering: partial widths 4.4. Time-dependent Hamiltonians by the (t, t@) method References

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QUANTUM THEORY OF RESONANCES: CALCULATING ENERGIES, WIDTHS AND CROSS-SECTIONS BY COMPLEX SCALING

Nimrod MOISEYEV Department of Chemistry and Minerva Center for Non-linear Physics of Complex Systems, Technion — Israel Institute of Technology, Haifa 32000, Israel

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

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Abstract Complex scaling enables one to associate the resonance phenomenon, as it appears in atomic, molecular, nuclear physics and in chemical reactions, with a single square integrable eigenfunction of the complex-scaled Hamiltonian, rather than with a collection of continuum eigenstates of the unscaled hermitian Hamiltonian. In this report, we illustrate the complex-scaling method by giving examples of simple analytically soluble models. We describe the computational algorithms which enable the use of complex scaling for the calculations of the energy positions lifetimes and partial widths of atomic and molecular autoionization resonance states, of small polyatomic molecules and van der Waals molecules in predissociation resonance states, of atoms and molecules which are temporarily trapped on a solid surface and of atoms and molecules which ionized/dissociate when they are exposed to high intensity laser field. We focus on the properties of the complex scaled Hamiltonian and on the extension of theorems and principles, which were originally proved in quantum mechanics for hermitian operators to non-hermitian operators and also on the development of the complex coordinate scattering theory. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 31.15.!p

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Preface In his book on scattering theory Taylor considered the resonances as the “most striking phenomenon in the whole range of scattering experiments”. Resonances are associated with metastable states of a system which has sufficient energy to break up into two or more subsystems. Complex scaling (known also as the complex-coordinate method or as the complex-rotational method) enables one to associate the resonance phenomenon, as it appears in atomic, molecular, nuclear physics and in chemical reactions, with a SINGLE square integrable eigenfunction of the complex-scaled Hamiltonian, rather than with a collection of continuum eigenstates of the unscaled hermitian Hamiltonian. For excellent reviews of the method see Reinhard [23]; Junker [24] and Ho [25]. An updated bibliography is given at the end of the present report. In this report we will illustrate the complex-scaling method by giving examples of simple analytically soluble models which describe scattering of an electron from a negative ion and transition from reactants to products through a potential barrier in three atomic collinear chemical reactions. We will focus on the properties of the complex scaled Hamiltonian and on the extension of theorems and principles, which were originally proved in quantum mechanics for hermitian operators to non-hermitian operators. We will focus also on the development of the complex coordinate scattering theory and on computational algorithms which enable the use of complex scaling for the calculations of the energy positions lifetimes and partial widths of atomic and molecular autoionization resonance states, of small polyatomic molecules and van der Waals molecules in predissociation resonance states, of atoms and molecules which are temporarily trapped on a solid surface and of atoms and molecules which ionized/dissociate when they are exposed to high intensity laser field. The computational algorithms which will be discussed here enabled also the calculations of the rotational vibrational distribution of diatoms obtained in the photodissociation process; for the calculations of specular and non-specular transition probabilities that are obtained in scattering of atoms and molecules from flat and corrugated surfaces; for calculating above-threshold-ionization (ATI), above-threshold-dissociation (ATD) and harmonic generation (HG) spectra of atoms and molecules which interact with a strong laser field. The complex-scaling theory and computational algorithms described in this report also enable the calculations of energies and lifetimes of transition state resonances in reactive scattering collision experiments, cumulative reaction probabilities, and quantum mechanical thermal rate constants for chemical reactions.

1. Resonances — Why complex scaling? 1.1. Resonances in full collision experiments Let us consider an experiment in which a “particle” is scattered from a “target”. The “particle” can be, for example, an electron, an elementary particle, an atom or a molecule and the “target” can be a nucleus, an atom, a molecule, a flat or a corrugated solid surface. In an elastic scattering experiment the energy of the “particle” is conserved. In a non-elastic scattering experiment there is an energy exchange between the “particle” and intrinsic degrees of freedom of the “target”, and the

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Fig. 1. Three different possible scattering orbits.

final energy of the “particle” in its “out” asymptote limit can be smaller or larger than the initial energy of the “particle” in its in asymptote limit. In a reactive scattering experiment the “particle” and the “target” undergo a change during the rearrangement collision and become different species. Even in the simple case of elastic scattering three possibilities may be considered [1]. The first possibility, presented in Fig. 1a, is that of a particle which comes in from infinity, gets trapped by the target and never emerges out of the attractive potential well. As time passes the potential energy of the trapped particle drops down to !R whereas the kinetic energy increases to #R. This “black hole” phenomenon is avoided when the interaction potential at the origin is less attractive than »(r)"!r~2. In order to obtain a free particle in the “in” and “out” asymptotes, »(r) should fall off quicker than r~3 at infinity. A direct scattering event is illustrated in Fig. 1b. In Fig. 1c we illustrate the third possibility, where, due to multiple-scattering events, the particle is temporarily trapped by the target. When the lifetime of the particle—target system in the region of interaction is larger than the collision time in a direct collision process we call the phenomenon a resonance phenomenon. A resonance state is defined as a long-lived state of a system which has sufficient energy to break-up into two or more subsystems. In elastic and inelastic scattering experiments the subsystems are associated with the scattering particle and the target. 1.2. Lifetime of shape and Feshbach-type resonances from motion of wavepacket calculations Probably the most well-known spherically symmetric potential which supports resonances, »(r), is the potential describing decay of radioactive nuclei or of unstable particles [2]. Naturally, we shall refer to the nucleus as the target in this case. An illustrative plot of »(r) in that case is given in Fig. 2 where r is, for example, the distance between an a particle and the nucleus. E and 0 E represent bound and resonance energies, respectively. 1 The lifetimes of the resonance states, DE T, (also known as Gamow or Siegert states) can vary 1 from a few seconds to millions of years (4.5]109 y for the decay of the 238U isotope to Thorium). A very similar potential can describe both the molecular interaction in a diatom for which the total angular momentum, j, is much larger than zero and gives rise to a centrifugal potential barrier [3], and the scattering of an electron from a neutral diatom [4]. In such cases, the lifetime of the metastable resonance states (i.e. predissociation states of rotationally excited diatoms which are

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Fig. 2. Schematic representation of the interaction potential between an a particle and the nucleus. Fig. 3. Gaussian wave packet scattering from a potential barrier. (v(x)"0 if x(6.7n, v"R if x(0, v"1 if x56.7n). (a) the mean energy of the wave packet is below the resonance energy; (b) the mean energy of the wave packet is equal to the resonance energy; (c) the mean energy of the wave packet is above the resonance energy.

formed by the temporary trapping of the two atoms inside the potential well in the scattering experiment, and autoionization states of the negative charged molecular ion) can vary from milliseconds to femtoseconds (about 10~15 s for the autoionization of H~ in its ground state). In all 2 these cases the temporarily trapping of the “particle” inside the potential well is a quantum phenomenon which is known as the tunneling phenomenon. The quantum equations of motion are reduced to the classical one as +P0. While + is taken to zero, the penetration probability of the quantum particle through the potential barrier (into the well or out of it) is reduced and in the limit of +"0 no resonance states will be observed. These kinds of resonances are known in the literature as shape-type resonances. The temporarily trapping of the particle inside the potential well can occur also when the energy of the particle is larger than the height of the potential barrier or even in the absence of a potential barrier (one example is of a Gaussian wave packet scattered from a finite square well for example). In Fig. 3 we show the results obtained for a scattering of a Gaussian wave packet from a potential barrier [5].

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Fig. 4. Schematic representation of two coupled adiabatic potentials.

As one can see from Fig. 3b, unlike the cases where the mean energy of the packet is off resonance, when the mean energy of the Gaussian wave packet is equal to the resonance energy then the wave packet is trapped above the potential barrier for a long period of time q"56.8 s. Any small variation in the resonance energy will reduce the lifetime q. Let us consider a special case where the internal modes of the scattered particle and the internal modes of the target are not coupled to the relative motion between the particle and the target. In this case, when the interaction potential between the “particle” and the “target” is not spherical symmetric it may happen that a bound state of the particle—target system in a fixed orientation (i.e. a square integrable eigenfunction of the one-dimensional time-independent Schro¨dinger equation H(r, h , / ) where h and / are the polar coordinates of the particle relative to the target which are 0 0 0 0 held fixed) is embedded in the continuum of the system in another orientation when h"h and 1 /"/ . The two adiabatic potentials are schematically presented in Fig. 4. 1 Due to the coupling of the r-coordinate with the angles h and / the bound state which is “pushed” up into the continuum becomes a resonance state. These kind of resonances are known as Feshbach-type resonances and can also be obtained in classical calculations. Unlike the shape-type resonances, the lifetime of the Feshbach metastable states get finite values as + is taken to the limit of +"0. A simple illustrative example is of a rigid rotor (stands for a non-vibrating diatomic molecule) scattered from a flat solid surface [6]. The interaction potential between the scattered “particle” (the diatom) and the “target” (the solid surface) depends on the angle, a, between the molecular axis and the normal to the surface and on the distance of the center of mass of the diatom from the surface. If the potential energy terms which couple a and z are neglected then the bound vibrational states of the free-rotor (diatom with the rotational kinetic energy B ( j#1)) are 305 embedded in the continuum of the non-rotating diatom which vibrates above the surface. The potential coupling terms “mix” the bound states of the rotating diatom with the continuum states of the non-rotating molecule and predissociation, which is associated with the resonance phenomenon, takes place. In Fig. 5 the results obtained for a Gaussian wave packet, representing an HD molecule scattered from an Ag(111) surface, are shown [7]. In the off-resonance case, the initially unrotated ( j"0) diatom is excited to its first rotational state, j"1, due to the collision with the flat silver surface. In the resonance case, one can see from Fig. 5b that the HD molecule spends about 0.5 ps close to the Ag(111) surface and as time passes desorption occurs. In the out asymptote limit the probability to observe HD in the j"1 and j"0 rotational states is almost equal.

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Fig. 5. Scattering of Gaussian wave packet which represents an HD molecule from a flat Ag(111) surface: The rotational probability densities o (z, t) as a function of z (!3(z(142 a.u.) are shown, where j is the rotational quantum number. j Each row corresponds to a specific time which is marked on the top right corner (in a.u.). The left column represents o (z, t) (non-rotating HD molecule); the middle column represents o (z, t); and the right column represents o (z, t). (a) 0 1 2 Direct scattering. The initial scattering energy is 18.97 meV. (b) Resonance scattering. The initial scattering energy is 22.69 meV shows very strong trapping at j"2). (c) as in (b) for !3(z(15.

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Fig. 6. The density of states o(E) as function of the energy E. o (E) stands for the box normalized density of states where N o "o(E)!¸/2JE and ¸ is the box size. N

1.3. Resonance lifetime from time-independent calculations of density of states In the above analysis, the lifetime of a resonance state was obtained from time-dependent calculations. It is possible, however, to estimate the resonance lifetime from time-independent calculations. By solving the time-independent Schro¨dinger equation for the potential barrier described in the caption of Fig. 3 the density of states, o, was calculated (note that, when box normalization is used the density of states is obtained by counting number of states with energies between E to E#dE)(8). The results of the time-dependent calculations are presented in Fig. 6. In Fig. 6b the background of the density of states, 1/JE, which results from the use of box normalization condition is removed. The local maxima in the density of states are associated with the resonances. The widths of these Lorenzian peaks, C "*E ; n"1, 2,2, are the inverse n n lifetimes of the resonances states. That is, C "+/q . n n

(1.3.1)

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The value of q obtained from Fig. 6b using Eq. (1.3.1) is in a good agreement with the estimate n of q which was obtained from the time-dependent calculations presented in Fig. 6b. That is n C "1/q "0.0176 a.u. 2 2 1.4. Association of the resonance phenomena with the complex poles of the scattering matrix We will show now that the widths of the shape and Feshbach-type resonance peaks, C , in the n plots of density of states given in Fig. 6, are associated with the poles of the S-matrix. For the sake of simplicity let us assume that » is a short-range potential and »P0 (i.e. the threshold energy E "0) as the “reaction” coordinate rPR. In such a case an eigenfunction of the time5 independent Schro¨dinger equation at rPR is given by /(rPR)"A(k)e~*kr#B(k)e`*krKe~*kr#S(k)e`*kr, E"(+k)2/2k ,

(1.4.1)

The S-matrix is defined as the ratio between the amplitude of the out-going plane wave and the in-coming wave. The S-matrix has a pole in two cases: (1) In the first case when B(k) has a pole. These are “false” poles which are not associated with the resonance phenomenon. “False” poles are independent of the potential, »"jv(r), and exist even when j approaches zero(9). (2) When the amplitude, A(k), of the in-coming wave vanishes. When these poles are concentrated on the positive imaginary axis of k they are associated with bound states, which are not relevant to the dynamics in a scattering process. As will be shown, the poles which are embedded in the fourth-quarter of the complex k-plane (i.e. Re(k)'0, Im(k)(0) are associated with the resonance phenomenon. Near the nth simple isolated pole S(k) can be written as [10,11] S(k)J1/(k!k ) n

(1.4.2)

and dln S(k)/dk"!1/(k!k ) , (1.4.3) n where k is a complex pole such that n Re(k )'0, Im(k )(0 . (1.4.4) n n Considering a closed contour of integration, C, in the complex k-plane then following the residue theorem 1 N" 2p i

Q

­ln S(k) dk , ­k C

(1.4.5)

where N is the number of poles in the fourth-quarter of the complex k-plane. When all the poles of S(k) are embedded in a bounded region in the complex k-plane (i.e. all of them are in a finite distance from the real k-axis as shown in Fig. 7) the closed contour of integration C can be replaced by a contour along the real k-axis (k varies from 0 to R) and, consequently, ­N/­k is defined as ­N/­k"­ln S(k)/2pi­k . Since the density of states o is given by o,dN/dE"­N/­k ­k/­E ,

(1.4.6)

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221

Fig. 7. Schematic representation of a closed contour of integration C in the complex k-plane where “x” denotes the complex poles of the S-matrix.

then 1 ­ln S(E) k ­ln S(k) " . o" 2pi ­E 2pi+2k ­k

(1.4.7)

By substituting Eq. (1.4.3) into Eq. (1.4.7), one can see that when k"Re(k ), the local maxima of n the density of states is obtained for the value of o"o , where .!9 o (k"Re(k ))"(k/2p+2)[!Re(k )Im(k )]~1"[!2p Im(E )]~1 (1.4.8) .!9 n n n n and the complex “energy” is given by (1.4.9) E !E "(+k )2/2k n 5 n with E being the threshold energy. The peaks in the density of states presented in Fig. 6 have 5 a Lorenzian shape. The full-width half-maximum of the nth Lorenzian peak, C , is given by n C "1/po . (1.4.10) n .!9 By substituting Eq. (1.4.8) into Eq. (1.4.10), the connection between the widths of the peaks in o(E), C , and the imaginary part of the complex poles of the S-matrix is established, such that, n C "!2Im(E ) . (1.4.11) n n From this analysis, one can see that the resonance phenomenon as obtained in a scattering experiment is mainly controlled by poles of the scattering matrix. As it will be shown in Section 1.5 the resonance poles are complex eigenvalues of the Hamiltonian, HK (r)/3%4(r)"E /3%4(r), E "e !(i/2)C , (1.4.12) n n n n n n where e is the resonance position above the threshold, C is the width (inverse lifetime) as defined in n n Eqs. (1.4.9) and (1.4.11) and the eigenfunction /3%4(rPR) as given in Eq. (1.4.1) is reduced to n

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exp(ikn) since S(k )"R as one can see from Eq. (1.4.2). Eq. (1.4.12) is the basic equation in the n resonance theory for time-independent Hamiltonians. The resonances are associated with complex eigenvalues of the (unscaled !) Hamiltonian which describes the physical system. Usually, one may expect that the Hamiltonian should be hermitian. That is, in the one-dimensional case

P

P

=

= f (x)HK (x)g(x) dx" g(x)HK *(x) f (x) dx . (1.4.13) ~= ~= By carrying out integration by parts one can see that this equation is satisfied provided

K K

K K

df (x) = dg(x) = !f (x) "0 . (1.4.14) dx dx ~= ~= Consequently, the hermitian property of HK (x) depends on the boundary condition of f (x) and g(x). When, for example, f (x) and g(x) are in the Hilbert space, then f ($R)"0, g($R)"0. Consequently, Eq. (1.4.14) is satisfied and HK (x) is an hermitian operator. In the more general case, Eq. (1.4.14) is satisfied and HK (x) is hermitian when f (x) and g(x) are bounded functions (NOT necessarily bound states). When f (x) and g(x) are bounded functions then D f (x)D(C and 1 Dg(x)D(C where C and C are two finite valued constants (such as the continuum-type functions 2 1 2 for example). When f (x) or g(x) exponentially diverges as /3%4 does, Eq. (1.4.14) is not satisfied and n HK (x) is not hermitian! g(x)

1.5. The role of complex scaling in the calculations of the resonance poles of the scattering matrix Let us now consider the properties of the eigenfunctions associated with these complex eigenvalues of the Hamiltonian. The physical Hamiltonians are hermitian only when they operate on bounded functions (not necessarily square integrable) or, when box normalization is used, on a functional space of all possible square integrable functions (i.e. Hilbert space). Therefore, it is obvious that /3%4 which are associated with complex eigenvalues are not in the hermitian domain of the Hamiltonian (i.e. not in the Hilbert space). For finite range potentials it is clear from Eqs. (1.4.9) and (1.4.12) that the complex wave vector is given by k "Dk De~*rn, where n n u "arctan(C /[2(e !E )]) (1.5.1) n n n 5 with E being the threshold energy. By substituting Eq. (1.4.13) into Eq. (1.4.1), one can see that the 5 resonance eigenfunctions (associated with the poles of the S-matrix) diverge exponentially, /3%4(rPR)"B(k )e`*@kn@%~*rn r"B(k )e*anre`bnrPR , n n n where

(1.5.2)

a "(2k)1@2(e2#(C /2)2)1@4 cos u /+ , n n n n b "a tan e '0 , n n n u "arctan(C /2e ) . n n n The physical interpretation of the divergence property of / is that at r"R one observes the 3%4 “particles” (subsystems as defined above) which were formed an infinitely long time ago [12].

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223

As usual, the solutions of the time-dependent Schro¨dinger equation for time-independent Hamiltonian are given by t3%4(r, t)"/3%4(r)e~*Ent@+ n n and, therefore, the probability density

(1.5.3)

Dt3%4(r, t)D2"D/3%4(r)D2e~Cnt@+ (1.5.4) n n decays to zero as time passes at a constant r. Therefore, the “particles” disappear from any given point in the coordinate space. Due to the exponential divergence of /3%4 the number of particles is conserved only when both the reaction coordinate, r, and the time, t, approach the limit of infinity. Most of the computational algorithms in quantum mechanics have been developed for hermitian operators (as discussed above, the physical Hamiltonians are hermitian only when they operate on bounded functions which get finite values as any point in the coordinate space). For example, variational methods which were successfully used to solve many-body problems in physics and chemistry are not applicable and cannot be used to solve Eq. (1.4.12) even for the one-dimensional case. As we will show, here, an extension of the variational principle and of other well-known theorems in quantum mechanics to non-hermitian operators can be made by carrying out similarity transformations SK which make the resonance functions, /3%4, square integrable functions. That is, (SK HK SK ~1)(SK /3%4)"(e !(i/2)C )(SK /3%4) n n n n such that

(1.5.5)

SK /3%4P0 as rPR (1.5.6) n and SK /3%4 are in the Hilbert space although /3%4 are not. The complex-scaling operator to be defined n below is only one example of a possible similarity transformation for which Eq. (1.5.6) is satisfied [13]. The complex-scaling operator is given by SK "e*hr »@»r

(1.5.7)

such that SK f (r)"f (re*h)

(1.5.8)

for any analytical function f (r). By substituting Eqs. (1.5.7) and (1.5.2) into Eq. (1.5.6) (this is equivalent to scaling r by exp(ih) in Eq. (1.5.2)), one obtains SK t3%4(rPR)"B(k )e`*@kn@%91(*(h~rn)r)"B(k )e*anre~ bnr , n n n where k "Dk De~*rn , n n a "a (cos h!tan u sin h) , n n n b "a (sin h!tan u cos h) , n n n u "arctan (C /2e ) . n n n

(1.5.9)

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One immediately sees from Eq. (1.5.9) that for a sufficiently large value of the rotational angle h the exponential factor b gets a positive value and thereby the scaled resonance wave function becomes n square integrable. In other words, when h5h , #

(1.5.10)

where the critical angle h is given by [14,15] # h "u , arctan C /2(e !E ) # n n n 5

(1.5.11)

with e , C and E being, respectively, the real and imaginary part of the complex energy and the n n 5 threshold energy (E was taken to be zero in our studied example) then, 5 SK /3%4(rPR)P0 . n

(1.5.12)

Note, by passing, that the bound states of the unscaled Hamiltonian are a special case of Eq. (1.5.9) where c"0 (i.e. C "0) and a "iDa D. Therefore, the critical angle h for the bound state is h "0. n n n # # Here we proved that by scaling the “reaction” coordinate the resonance wave function becomes square integrable and, consequently, the number of particles in coordinate space is conserved. Therefore, complex scaling has the advantage of associating the resonance phenomenon with the discrete part of the spectrum of the complex-scaled Hamiltonian. Moreover, the resonance state is associated with a SINGLE square integrable function, rather than with a collection of continuum eigenstates (see the peaks in the density of states presented in Fig. 6) of the unscaled hermitian Hamiltonian. Complex scaling may be viewed as a procedure which “compresses” information about the evolution of a resonance state at infinity into a small well-defined part of space. The tail in spatial space of a single, time-independent, square integrable resonance wave function contains all the information about the system, including information on partial width and on the way in which the system evolves as the separation between the “particle” and the “target” increases to infinity. The complex eigenvalues of the complex-scaled Hamiltonian (see Eq. (1.5.5)) which are associated with the resonance phenomenon are h-independent. The imaginary part of the resonance complex eigenvalues has been shown to be the widths of the Lorentzian peaks, C , in plots of n the probability density of states vs. the energy, and through Eq. (1.3.1) one can associate C with the n rate of decay and with the inverse lifetime of the “particle—target” system. The resonance complexscaled eigenfunctions are, however, h-dependent. We may interpret h as a control parameter which “brings” the information of the decay process from infinity to a finite region in space whose size depends on the value of h. As h is increased, the information about the decay process is compressed into a smaller part of the coordinate space. However, h cannot exceed a critical value as it will be discussed in Section 2. 1.6. Complex scaling and resonances in half collision experiments The resonance wave function which becomes square integrable upon complex scaling can be regarded as the dominant intermediate state in the scattering process. This statement can be explained as follows. The transition probability to get from the in-asymptote to the out-asymptote,

N. Moiseyev / Physics Reports 302 (1998) 211—293

which are denoted by u and u respectively, is given by i f P(E)"DSu D¹K (E)Du TD2 , f i

225

(1.6.1)

where E is the total energy of the particle—target system; the ¹-matrix is defined as usual by ¹"»(x, y, z)#»(x@, y@, z@)G (x@, y@, z@; x, y, z)»(x, y, z) E and the Green operator G "(E!H)~1, provides the probability amplitude to get from (x, y, z) to E (x@, y@, z@) at a given energy E. » is the interaction potential between the free particle and the target. The Su D»Du T term in Eq. (1.6.1) describes the direct scattering events whereas the Su D»G»Du T f i f f term describes the multiple scattering events. The spectral representation of the Green operator is given by D/ TS/ D i , G "+ i (1.6.2) E E!E i i where the intermediate states, D/ T, are the eigenfunctions of the Hamiltonian, H, which describes i the particle—target system. The energy spectrum is discrete due to the use of box normalization. By rotating the contour of integration from the real axis into the complex plane, the intermediate states and energies are replaced by the eigenfunctions and complex eigenvalues of the complexscaled Hamiltonian, respectively. For an isolated resonance E "e !1C , which is sufficiently n n 2 n narrow, i.e. very small C , there is a single dominant term in the series expansion of the Green n operator given in Eq. (1.6.2) when E"e . We shall show that a resonance is considered as n sufficiently narrow when C ;!Im[e exp(!2ih)]. Consequently, when the total energy of the n n particle—target system is about equal to the position of a narrow resonance, Eq. (1.6.1) — the transition probability to get from “reactants” to “products” — is reduced to a Breit—Wigner-type expression [1,16], DSu D»D/3%4TS/3%4D»Du TD2 (C /2) n n i J n P(E)" f . (1.6.3) (E!e )2#(C /2)2 (E!e )2#(C /2)2 n n n n The main conclusion from Eq. (1.6.3) is that when the lifetime of the “particle—target” system is sufficiently large we may say that the “particle” is trapped by the “target” at a single resonance state, /3%4. In such a case, we can distinguish between the mechanism which brings the system to the n resonance state, and the decay process from the metastable resonance state to the products (“out” asymptote in the scattering experiment). The decay process can be considered as a half collision process. A typical half collision process is the decay of an excited state DE T following a laser 1 excitation of a bound state, DE T, to a r esonance state, DE T, (see, for example, Fig. 2). One such 0 1 example is the infra red excitation of a highly rotational excited diatomic molecule from its vibrational ground state to a vibrational state laying just below the top of the centrifugal potential barrier. An example for a Feshbach-type resonance obtained in a half collision experiment is the predissociation of van der Waals complexes [17]. Let us consider, for instance, the predissociation of the van der Waals complex NeICl. The NeICl complex in the B electronic excited state is stable. When the ICl vibration is excited the complex NeICl (B, l"2) gets into a metastable resonance state [18,19]. The lifetime of an excited NeICl molecule is about 2.5 ns and as time passes free ICl

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molecules, in the first excited vibrational state and in rotational states which vary from j"0 to about j"32, are obtained.1 Another example is the autoionization of the helium atom [20] are the two electrons were excited from the doubly occupied 1s orbital to the 2s orbital. The energy of He(2s)2 is smaller than the energy of He`(2s) but greater than the energy of He`(1s). The (2s)2 state is embedded in the continuum of the He`(1s) ion.2 Due to the electronic correlation one electron moves from the 2s orbital to the 1s ground state orbital and thereby proves the energy which is required for the ionization of the second electron: He(2s)2PHe`(1s)#e~2. The kinetic energy of the free Auger electron [21] is about equal to the position of the resonance state above the 1st threshold energy (the real part of the complex resonance eigenvalue), and the rate of decay (inverse lifetime) is the resonance width (the imaginary part of the complex resonance energy). 1.7. Resonances in multiphoton ionization/dissociation experiments by complex scaling In Section 1.5 we proved that the complex-scaled resonance eigenfunctions are square integrable (see also Ref. [22]). The proof based on the assumption that the asymptote of the Siegert resonance wave function is an outgoing plane wave. This assumption holds for finite range potentials for which the cut-off approximation is taken into consideration (i.e. »"0 for r'r ). Balslev and 0 Combes [14] and Simon [15] have shown that also for the infinite range, Coulombic potential the complex-scaled resonance wavefunctions are square integrable (see also the reviews of Reinhardt [23], Junker [24] and Ho [25]). It is entirely clear that the cut-off potential argument cannot be taken into consideration when the interaction between “scattered particle” and the “target” does not vanish as rPR. This is exactly the case when the “scattered particle” is an atom/molecule and the “target” is an electromagnetic field. In this case, the interaction potential is proportional to the dipole moment operator kL . In the interaction of an atom with an dc field the time-independent Hamiltonian is given by HK (r)"HK (r)#e ez , (1.7.1) 0 0 where e is the field intensity, HK is the atomic field-free Hamiltonian and kL "zL . In the interaction 0 0 of atom/molecule with a high intense ac field (where the perturbational methods break down and cannot be used) the time-dependent Hamiltonian is given by H(r, t)"H (r)#e k(r) f (t) (1.7.2) 0 0 when, for example, a monochromatic electromagnetic field is used, then f (t)"cos wt where w is the frequency of the cw laser.

1 The predissociation of NeICl when ICl is in its l"2 vibrational excited state is considered as a Feshbach-type resonance phenomenon since the energy of NeICl (l"2) is smaller than the energy of ICl (l"2, j"0) and hence dissociation due to the shape type resonance mechanism does not occur, i.e., NeICl (l"2);Ne#ICl (l"2, j"0). However, since the energy of NeICl (l"2) is higher than the energy of ICl (l"1; j"0, 1, 2,2, 32 the dissociation occurs due to the coupling between the internal modes of NeICl. 2 The ionization due to the Shape-type resonance mechanism, does not occur, He(2s)2;He`(2s)#e~, since the energy of He(2s)2 is smaller than the energy of He`(2s). The autoionization due to the Feshbach-type resonance mechanism is an example of the electron correlation effect.

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227

Reinhardt, Chu and coworkers studied extensively this problem, and showed numerically that even in these cases, when kPR as xPR, the complex-scaled resonance wave functions are square integrable [26]. Here we shall prove that upon complex scaling of a general time-dependent Hamiltonian (given in the length gauge representation) HK "(pL )2/2m#»(x)#k(x) f (t) , x where »(x)P0 as xPR

(1.7.3)

(1.7.4)

and lim k(x)"e x , (1.7.5) 0 x?= the resonances are isolated from the other states in the continuum and are associated with square integrable functions. The proof will be given in three steps: (I) Transformation of the Hamiltonian from the length-gauge to the momentum gauge By substituting Eq. (1.7.3) into the time-dependent Schro¨dinger equation HK t"i+ ­t/­t

(1.7.6)

where t"e~*k(x):f (t) $t@+/(x, t) ,

(1.7.7)

one can get that

C

D

­/(x, t) (pL !k@(x): f (t) dt)2 x #»(x) /(x, t)"i+ , ­t 2m

(1.7.8)

where k@(x),­k(x)/­x .

(1.7.9)

Using the condition given in Eq. (1.3.1) it is clear that k@(x)Pe 0 as xPR.

(1.7.10)

(II) Transformation of the Hamiltonian from the momentum gauge to the acceleration gauge Following Kramers—Henneberger transformation [27] the complex-scaled wave function /(x, t), /(x, t)"e~*b(t)/I (x, t) , /I (x, t)"e*a(t)pL xs(x, t) ,

(1.7.11)

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where

P P

e2 b(t)" 0 F2(t) dt , 2m+

e a(t)" 0 F(t) dt , m+

P

(1.7.12)

F(t)" f (t) dt is substituted into Eq. (1.7.8). Since we are interested here in the property of the resonance wave functions in its asymptotic limit, we shall assume that k@(x)"e for any value of x. The conclusion 0 obtained on the basis of this assumption holds in the more general case where lim k@(x)"e . As x?= 0 a result one gets that

C

D

(pL )2 ­s(x, t) x #»I (x, t) s(x, t)"i+ , 2m ­t

(1.7.13)

where »I (x, t)"e~*pL a(t)»(x)e`*pL a(t) .

(1.7.14)

By noticing that e`*pL as(x, t)"s(x#a, t)

(1.7.15)

e~*pL a[»(x)s(x#a, t)]"[»(x!a)s(x, t)] .

(1.7.16)

One can get that »I (x, t)"»(x!a)

(1.7.17)

and, therefore,

C

A

PP BD

e (pL )2 x #» x! 0 m+ 2m

­s(x, t) . ­t

s(x, t)"i+

f (t) dt

(1.7.18)

(III) Multiphoton ionization/dissociation resonance wave functions are square integrable From Eq. (1.7.8) one can see that for any given time t, »I "»(x!a(t))P0 as xPR consequently, the coupling of between the atomic/molecular Hamiltonian and the ac field vanishes as xPR and as time passes a free particle (an electron or a molecular fragment) will be obtained. For time periodic Hamiltonians, the resonance Floquet quasi-energy state is given by s (x, t)"e~*E3%4t@+U (x, t) (1.7.19) 3%4 3%4 U (x, t)"U (x, t#2p/w) (1.7.20) 3%4 3%4 and the resonance complex quasi-energy, E , and the time-periodic function U (x, t) are, corres3%4 3%4 pondingly, the eigenvalue and eigenfunction of the Floquet Hamiltonian

A

A

PP BB

­ (pL )2 e !i+ # x #» x! 0 ­t 2m m+

f (t) dt

U (x, t)"E U (x, t) , 3%4 3%4 3%4

(1.7.21)

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229

where E "e!i/2C , (1.7.22) 3%4 = U (x, t)" + e*wntu (x) . (1.7.23) 3%4 n n/~= For the open Floquet channels, n(0, (here we assume that the resonance state is in the n"0 Brillouin zone which implies that !+w(e(0) and within the framework of the cut-off approximation (i.e. »(x'x )"0) 0 u (DxD'x )PJm/+k e*knx (1.7.24) n:0 0 n (+k )2/2m"e!(i/2)C!+wn (1.7.25) n provided n gets a sufficiently large value such that Real (+k )2'0. u becomes square integrable n n when x"x@e*h

(1.7.26)

and x@ is a real variable (for a detailed discussion see Section 1.5). For long-range potential such as the coulombic potential, v(x!a(t)), the complex-scaled resonances are square integrable functions following the Balslev—Combes theorem. From Eqs. (1.7.7), (1.7.11), (1.7.19), (1.7.23), (1.7.24) and (1.7.25) one can see that since the resonance state s(x, t) is a square-integrable function then /(x, t) and, therefore, t(x, t) are the complex-scaled square-integrable resonance wave function. We can summarize the discussion by saying that here we prove that the complex-scaled resonance wave function calculated either in the length-gauge; momentum-gauge, or accelerationgauge representation of the Hamiltonian are square-integrable functions regardless of the fact that k(x) may get an infinite large value as xPR. The proof holds provided that k(x)/x approaches a constant value as xPR. The proof can be extended to time-dependent Hamiltonians which are not necessarily time periodic (this is the case when short pulse high intense lasers are used) by the (t, t@) method recently introduced by Peskin and Moiseyev [28].

2. From complex-scaled Hamiltonians to resonance positions and widths As was discussed in Section 1, the long-lived states of a system which has sufficient energy to break-up into two or more subsystems are associated with the complex poles of the S-matrix. The resonance positions and widths are, respectively, associated with the real and imaginary part of the complex eigenvalues of the time-independent non-scaled [13] Hamiltonian (see Eq. (1.4.12)). The resonance eigenfunctions diverge exponentially as it is shown in Eq. (1.4.13) and therefore they cannot be in the hermitian domain of the Hamiltonian. Consequently, the corresponding resonance eigenfunctions are definitely NOT in the Hilbert space. As it is well known the eigenvalues of hermitian Hamiltonians are real and cannot get complex values. The hermitian property is not only a property of the operator but also of the functions on which the operator is acting (see Section 1.4). As we discussed in Section 1, the resonance eigenfunctions can be taken into the Hilbert space (which contains all possible square integrable functions) by carrying out a “proper”

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Fig. 8. (a) Two possible integration paths, F(x), in the complex coordinate plane (see Eq. (2.1.30) in the text). For j" the conventional complex path is obtained, for j"5 and x "6 a.u. a smooth-exterior-scaling path is obtained. In both two cases h"0.75 rad. (b) The 0 complex » (x) local extremely short-range potential (see Eq. (2.1.21) in the text) which is the coordinate dependent linear factor of the 1 flux-type term in »K (see Eq. (2.1.20)). The path in the complex coordinate plane is F(x) as shown in Fig. 8 for j"5. (c) The complex CAP » (x) potential which vanishes when the physical potential gets non-zero values (see Eq. (2.1.22) in the text) which is the coordinate 2 dependent linear factor of the diffusion-like term in »K (see Eq. (2.1.20)). The path in the complex coordinate plane is F(x) as shown in CAP Fig. 8(a) for j"5.

similarity transformation of the Hamiltonian (see Eqs. (1.5.5), (1.5.6), (1.5.7), (1.5.8), (1.5.9), (1.5.10), (1.5.11) and (1.5.12)). Moiseyev and Hirschfelder [13] suggested the following as “proper” similarity transformations: SK & exp[!hf 1@2(r)(­/­r) f 1@2(r)] ,

(2.0.1)

where f (r) can be any function for which f (r)/rP1 as rPR. In the conventional complex scaling, as first proposed by Hartree during the Second World War [29] and more recently by Balslev and Combes [14] and by Simon [15], f (r)"r. In Fig. 8a we represent several illustrative contours, SK r, in the complex coordinate space.

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The exterior complex scaling [30—32]

G

if r(r , 0 (2.0.2) exp[!h(r!r )­/­r if r5r 0 0 was suggested by Simon [32] to avoid the intrinsic non-analyticities of the potential, V, by passing on the right-hand side of the singularity points of V. We shall represent here several properties of the complex-scaled Hamiltonians. SK "

1

2.1. The complex scaled Hamiltonian 2.1.1. “Conventional” complex scaling For the sake of clarity, and without loss of generality, let us describe the one-dimensional complex-scaled Hamiltonian. When the potential is dilution analytic, the complex-scaled Hamiltonian (in dimensionless units) H "S~1(h)HK SK (h) h (where SK is the complex scaling operator as defined in Eq. (1.5.7)) is given by e~2*h ­2 H "! #»(x e*h) h 2 ­x@2

(2.1.1)

(2.1.2)

where !R4x@4R

(2.1.3)

and the complex rotational coordinate x (shown in Fig. 8a) is given by x"x@e*h .

(2.1.4)

If »(x@) gets a minimal value at x@"x , one may wish (to achieve a rapid numerical convergence 0 in the solution of the time-independent Schro¨dinger equation) to define x as x"(x@!x )e*h#x . 0 0

(2.1.5)

2.1.2. “Exterior” complex scaling For non-dilution analytical potential one may use the exterior complex-scaling operator which is defined in Eq. (2.0.2). The complex rotational coordinate x (shown in Fig. 8a) in this case is given by

G

x"

x@ e*hx@

if x@(x , 0 if x@5x . 0

(2.1.6)

Consequently, ­/­x"g(x@)­/­x@ ,

(2.1.7)

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where

G

1, ­x@ g(x@), " ­x e~*h,

x@(x , 0 x@5x . 0

(2.1.8)

Therefore, g@,­g/­x"(e~*h!1)d(x@!x ) 0

(2.1.9)

and ­2/­x2"g2 ­2/­x@2#g ) g@ ­/­x@ "g2(x@)­2/­x@2#e~*h(e~*h!1)d(x@!x )­/­x@ . (2.1.10) 0 From Eqs. (2.1.6), (2.1.7), (2.1.8), (2.1.9) and (2.1.10) one can see that the complex exterior-scaled Hamiltonian is given by

A

B

[g(x@)]2 ­2 ­ e~2*h !e~*h H "! ! d(x@!x ) #»(x) , h 0 2 ­x@ ­x@2 2

(2.1.11)

where g(x@) and x are defined above. By carrying out integration by parts it has been shown by Rom and Moiseyev [33] that the complex exterior-scaled kinetic matrix is given by

T K K U P A BA B P A BA B

1 1 x0 d/ d/ ­2 i j dx@ ! / / " 2 i ­x2 j 2 ­x@ dx@ 0 e~2*h = d/ d/ 1 d/ (x@) i j dx@! / (x ) j # ) (1!e~*h) dx@ dx@ 2 2 i 0 dx@ 0 0 x i x{/x when / (x@) and / (x@) are real square integrable basis functions. i j

K

(2.1.12)

2.1.3. “Smooth-exterior” complex scaling The Moiseyev—Hirschfelder [13] generalization of the complex coordinate method associated the resonance-poles of the S-matrix, E"E !iE , with the h-independent complex eigenvalues r i of H K : H K W"EW ,

(2.1.13)

where H K "!+2/2M ­2/­z2#»(z)

(2.1.14)

and z"F(x) is a path in the complex coordinate plane such that z"F(x)Px exp(ih) as xPR .

(2.1.15)

In the spirit of Simon’s proposition to avoid the need to carry out analytical continuation of the potential term in the Hamiltonian into the complex coordinate plane [30—32] Rom, Engdahl and Moiseyev first proposed to define a smooth-exterior-scaling path which is defined as [34] f (x)"­F/­x"1#(exp(ih)!1)g(x) ,

(2.1.16)

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where g(x) is varied from 0 to 1 value around the point x"x . If »(x5x )"0 one can use the 0 0 unscaled potential »(x) rather than using the complex potential »(z). Note however, that the path which defined in Eq. (2.1.16) is very general and is not necessarily limited to the case where »(z)"»(x) or when »(z) & »(x). It is easy to see that since ­/­z"f ~1(x) ­/­x ,

(2.1.17)

­2/­z2"!f ~3(x) ­f (x)/­x ­/­x#f ~2(x)­2/­x2 .

(2.1.18)

then

Consequently, the smooth-exterior-scaled Hamiltonian as derived by Moiseyev is given by [35] H K "!+2/2M ­2/­x2#»[F(x)]#»K , CAP where »K "1» (x) ­/­x#» (x)­2/­x2 CAP 2 1 2

(2.1.19)

(2.1.20)

and » (x)"+2/Mf 3(x) ­f (x)/­x , 1 » (x)"(+2/2M)(1!f ~2(x)) . 2 The volume element is given by dz"f (x) dx .

(2.1.21) (2.1.22)

(2.1.23)

As usual one can transform the Hamiltonian in order to simplify the expression of the volume element to be equal to dz"dx by defining a new function U, W(x)"f ~1@2U(x) .

(2.1.24)

Such that H K "!(+2/2M)+ 2#»[F(x)] , f f where

+ 2"f `1@2(x)(­2/­ z2) f ~1@2(x) . f After some algebraic derivations one gets that U(x) is an eigenfunction of H K , f (f) , H K "!+2/2M ­2/­2x#»[F(x)]#»K CAP f where

(2.1.25)

(2.1.26)

(2.1.27) »K (f) "» (x)#» (x)­/­x#» (x)­2/­x2 . CAP 0 1 2 The functions » (x) and » (x) are as defined in Eq. (2.1.21) and Eq. (2.1.22) where » (x) is given by 1 2 0 (2.1.28) » (x)"+2/4M f ~3(x)­2f/­x2!(5+2/8M) f ~4(x)(­f/­x)2 . 0

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As one can see the price one pays for the simplification of the expression of the volume element is in the inclusion of an extra term » in the complex absorbing potential (CAP) and in doubling the 0 weight of the flux-type term. Without loss of generality, let us define a specific family of integration paths in the complex coordinate plane by defining g(x) in Eq. (2.1.16) as [144] g(x)"1#0.5(tanh(j(x!x ))!tanh(j(x#x ))) . 0 0 By carrying out integration over g(x) the complex paths, F(x), are obtained,

(2.1.29)

F(x)"x#(exp(ih)!1)[x!x #(1/2j)ln(1#exp(!2j(x!x )))] 0 0 !ln(1#exp(!2j(x#x ))) . (2.1.30) 0 Illustrative examples for different possible integration paths are given in Fig. 8a for j"0, 3, 5 and x "6. For j"0, the usual complex coordinate path, z"x exp(ih), is obtained. For large 0 values of j the smooth-exterior-scaling path is obtained. At the limit of jPR the exterior scaling path [32] is obtained and

G

x

if !x 4x4x 0 0 if x'x . (2.1.31) z" (x!x )e*h#x 0 0 0 (x#x )e*h!x if x(!x 0 0 0 The CAP terms can be calculated using the above expression for g(x) and the following analytical expressions for the first and second derivatives of f (x): ­f/­x"0.5 j(exp(ih)!1) (cosh2(j(x#x ))!cosh2(j(x!x ))) 0 0

(2.1.32)

and ­2f/­x2"j2(exp(ih)!1)(tanh(j(x#x ))(1!tanh2(j(x#x ))) 0 0 ! tanh(j(x!x ))(1!tanh2(j(x!x ))2)) . (2.1.33) 0 0 In Fig. 8b, c illustrative examples for » and » (x) are given for j"5. As one can see from 1 2 Fig. 8b » (x) is an extremely short-range potential for the chosen value of j. Since » (x) looks like 1 1 a delta function and since the flux operator is defined as id(x!x )­/­x we refer to the correspond0 ing first term in the CAP which is defined in Eq. (2.1.20) as a flux-type operator. The second term is a kinetic-type operator which describes the diffusion at x&x . As one can see from Fig. 8c » (x) 0 2 vanishes when !x (x(x . The value of x can be chosen such that only within this interval of 0 0 0 x the physical potential gets non-zero values and vanishes elsewhere. It should be stressed that in the calculations the potential »(x) can remain unscaled due to the properties of the chosen integration path, F(x), (see Fig. 8a for j"5). We can summarize it by saying that the use of the universal Flux-diffusion-type CAP which is constructed from the » and » functions which are presented in Fig. 8b, c, enables us to obtain in 1 2 a very high accuracy many resonances (regardless to their widths and being isolated or overlapping resonances) from a single diagonalization of a complex non-symmetric matrix. Exactly as in the conventional complex scaling. The use of the fact that for some cases we can choose a path in the complex coordinate plane that leaves the physical potential unscaled, i.e. »(F(x))"»(x), enables

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one to construct the unscaled Hamiltonian matrix elements by conventional computational algorithms and by using available program packages (for example, the Gaussian code by which the neutral and ionize polyatomic molecular Hamiltonians can be constructed). The complex smoothexterior scaling Hamiltonian is obtained by adding to the unscaled Hamiltonian matrix a matrix which represents the universal »K (using the same basis set which has been used to represent HK ). CAP 2.1.4. Comment on the “smooth-exterior” complex scaling and other approaches using “complex-absorbing-potentials” In multi-channel problems, one can define an non-local, energy-dependent operator, the socalled optical potential, that enables the calculation of cross sections for a subset of channels. All effects caused by the excluded channels are accounted for by the optical potential. “Optical potentials” has been used in nuclear physics a long time ago [36]. This type of optical potentials should be distinguished from negative-imaginary-short-range potentials that are added to the Hamiltonian in order to impose absorbing boundary conditions which provide outgoing waves in the asymptotic limit. In the literature this type of potentials are sometimes also called optical potentials [37]. To avoid confusion we do not use the term optical potential here but adopt the more suitable expression CAPs, complex-absorbing-potentials, as proposed by Riss and Meyer [38] in order to name the artificial potentials introduced to impose absorbing boundary conditions. In molecular physics the use of CAPS avoids artificial reflections which result from the use of the finite basis/grid approximation [39] and allows simulations of large scale strongly coupled scattering problems (such as in four-center reactions) involving millions of basis functions [40]. In optical simulations the Maxwell equation is solved by using CAPs (see, for example, Refs. [16, 41]) to design wave guides which have specific properties. More recently, Neuhauser presented a new highly accurate and anomaly free time-independent approach to reactive scattering based on the use of very short-range imaginary potentials [42]. In a one-dimensional simulation the CAPs covered only two grid points! Unlike other methods such as the complex-coordinate method which stays on a solid mathematical ground given by Balslev and Combes [14] and Simon [15], the use of CAPs was based on intuition and numerical experience. It has been proved that poles of the scattering matrix are also the eigenvalues of the complex-scaled Hamiltonian, but it has not been proved that they are the eigenvalues of the Hamiltonian which is perturbed by a CAP. Rom et al. [43] have shown that the use of a CAP is similar to the use of the exterior scaling and more precise to the use of the smooth-exterior-scaling method which was formulated by Rom et al. [34]. Riss and Meyer [38] addressed themselves also to the question of under what conditions the resonances obtained by the use of CAPs are indeed the poles of the scattering matrix. Their strategy and derivation was as follows: the CAPs would provide the exact poles of the S-matrix if they would be reflection-free potentials. The reflection can be made smaller than any given limit, if the CAP is made weak and long enough; moreover, they showed that one may minimize the reflections of a short-range CAP by adding specific energy-dependent terms to the effective Hamiltonian which includes the CAP; from the requirement that the CAP in the free-reflection modified Hamiltonian will be energy linearly dependent they obtained a new effective Hamiltonian which consists of smooth-exteriorscaled kinetic operator and a potential term which is vanished (unlike the usual used CAPs) when the potential of interaction, », is vanished as well; Their so-called TCAP-method [44] is, in fact,

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very similar to the smooth exterior complex scaling [34] except that they add an extra local complex potential which is problem-dependent. The strategy by Moiseyev [35] is in reversed to the one by Reiss and Meyer. Reiss and Meyer started from the Hamiltonian perturbed by a CAP and end up with a complex-scaled-type operator. We get out from the Moiseyev—Hirschfelder generalized representation of the complex coordinate method and end up with the non-scaled Hamiltonian perturbed by a CAP which is problem independent. As we show above this universal energy-independent CAP is a linear combination of Flux- and diffusion-type operators. In order to avoid confusion we should stress that the Flux CAP derived here is not related to the use of reactive flux and CAPs made recently by Ja¨kel and Meyer in the formulation of a new modified flux operator formalism for the calculations of state-to-state transition probabilities [45]. It is also different from the Bloch flux-type CAP derived by Lipkin, Bra¨ndas and Moiseyev [46] from the exterior complex scaling approach [32]. The main difference between the Bloch-type CAP derived by Lipkin and coworkers and the CAP operator derived here, is in the fact that the Bloch operator is energy-dependent whereas the CAP presented below (see Eq. (2.1.20)) is energy-independent. Therefore, the use of the Bloch-type CAP enables the calculations of the resonances one by one and by the use of an iterative numerical procedure [8], whereas the use of the CAP derived here enables one to get at once many resonances from a single diagonalization of a complex non-hermitian matrix. 2.2. The complex **energy++ spectrum — Resonance positions, widths and rotating continua 2.2.1. General discussion Following the Balslev—Combes theorem [14], the complex eigenvalues (defined as the complex “energy” spectrum) of the complex scaled Hamiltonian, SK (h)HK (x)SK ~1(h), which are associated with BOUNDED not necessarily square integrable eigenfunctions are given in Fig. 9. We have already shown in Section 1 that bound states and resonance states are associated with the discrete part of the energy spectrum. Both the bound states and the resonance states are associated with square integrable functions and consequently belong to the Hilbert space. For a simple proof see Eqs. (1.5.7), (1.5.8), (1.5.9), (1.5.10), (1.5.11) and (1.5.12). The continuum, unlike the bound and resonance states, is affected by the value of the rotational angle h. Following the Balslev and Combes theorem, the scattering states are rotated into the lower-half of the complex energy plane by the angle 2h. It is easy to prove this theorem for short-range potentials. In such cases, the scattering states have the asymptotic behavior given by /4#!55(rPR)"A(k)e~*kr#B(k)e`*kr

(2.2.1)

where E"(+k)2/2m .

(2.2.2)

The energy gets any real positive value (provided that the threshold energy is taken as zero). The complex-scaled scattering states are given by /4#!55(r@e*h) P A(k)e~*k%*hr{#B(k)e`*k%*hr{ . r{?=

(2.2.3)

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237

From Eq. (2.2.3) one can see that for real values of k (i.e. real positive eigenvalues E) /4#!55 diverges as r@PR and h(n since the real part of the exponential factor is positive. That is, Re(!ike*h)"k sin h'0 .

(2.2.4)

Fig. 9. (a) Schematic representation of the eigenvalues (i.e. “energy spectrum”) of the complex coordinate plane scaled Hamiltonian, H , according to the theorem of Balslev and Combes. (b) The complex eigenvalues of the complex-scaled h Hamiltonian which describes the interaction of helium atom with the corrugated Cu(117) surface as calculated by Engdahl et al. [138]. The first three full circles below the first threshold stand for the vibrational bound states of helium adsorbed on the surface. The other full circles stands for the selective adsorption resonances. (c) The complex eigenvalues of the complex scaled Hamiltonian which describes the reaction, NeICl(B, l"2)PNe#ICl(B, l"1) as calculated by Lipkin et al. [19,139]. (d) The complex eigenvalues of the complex-scaled DVR (discrete variable representation) HCO(J"0) Hamiltonian which are associated with the HCO molecular predissociation resonances as calculated by Ryaboy and Moiseyev [94]. (e) The eigenvalues of the complex-scaled Hamiltonian HD/Ag(111) which are associated with square integrable and continuum-type bounded eigenfunctions (Peskin and Moiseyev in Refs. [6,108]). (f ) The complex eigenvalues of the complex-scaled Floquet operator, !i+ ­2/­t2!1­2/­x2#»(x, t), were »(x, t) is a time 2 periodic driven Rosen—Morse potential, as calculated by Moiseyev and Korsch [128]. The full circles which are laying close to the real axis are the quasi-energy resonances which are defined modulo of +u, where u"1 is taken as the laser frequency.

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Fig. 9. Continued

The ONLY bounded non-divergent (not square integrable) functions are obtained when k gets complex values, k"DkDe~*h

(2.2.5)

and therefore (when the threshold is taken as the zero reference energy) E (h)"DEDe~2*h; DEDe[0, R] 4#!55

(2.2.6)

where E is defined in Eq. (2.2.2). Consequently, the complex-scaled scattering eigenfunctions have exactly the SAME asymptotical behavior as the unscaled states, but are associated, however, with a continuum which is rotated into the lower-half part of the complex energy plane by the angle 2h. The rotating one-dimensional continuum is a “white” continuum in the sense that its density of states is varied monotonously as the energy, DE (h)D, is increased. When the continuum-type wave functions are 4#!55 box normalized, the density of states of the rotating one-dimensional continuum is expected to be proportional to DE (h)D~1@2. The deviation from this behavior due to the use of a finite box size in 4#!55 the normalization procedure has been discussed by Alon and Moiseyev [8]. The schematic representation of the eigenvalues of the complex-scaled Hamiltonian given in Fig. 9a describes the “energy” spectrum of potentials which support shape-type resonances (see, for example, Fig. 2). In the case of potentials that support Feshbach-type resonances (see, for example, Fig. 4) the continuum spectrum splits into several branches which come out of the threshold energies. See, for example, the eigenvalues spectrum of the complex scaled Hamiltonian, presented in Fig. 9b—f, which describes the interaction of helium with Cu(117); dissociation of van der Waals complex NeICl and of HCO; interaction of HD molecules with Ag(111) surface and multiphoton ionization/dissociation of a model Hamiltonian. The threshold energies are the bound state energies of the “subsystems” (e.g. the rotational energy levels of the free HD molecule) which are obtained due to the decay process (e.g.

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predesorption in our illustrative example) of the system (e.g. HD adsorbed on Ag(111)) which was prepared in a metastable (resonance) state. In the case of 3D potential energy surfaces it is sufficient to scale only the reaction coordinate. We (Lipkin, Ryaboy and Moiseyev) developed a code that enables the calculations of the predissociation resonances of any three atomic molecular system by complex scaling the distance of the dissociative atom from the center of mass of the diatom. In Fig. 9d we show, for example, thousands of HCO resonances (HCOPH#CO) which were all obtained from a single diagonalization of the complex-scaled Hamiltonian.

2.2.2. Resonance transition states for analytical models By using complex-scaling arguments, the h-independent complex eigenvalues which are associated with shape-type resonances can be immediately obtained from the bound state spectrum of analytically solvable model Hamiltonians. Let us assume that for a given system an analytical expression of the bound state discrete spectrum E(n, j), as a function of the quantum energy level n and as a function of the potential strength parameter j V(r)"jv(r) is known. For positive values of j, »(r) has a potential well which supports the bound states. There are two cases where complex energy spectrum can be immediately obtained from the known analytical expression of the discrete bound state spectrum. In the first case, a finite number of bound states associated with real eigenvalues are obtained when n(n . By substituting values for n which are larger than 0 n , E(n'n , j) gets complex values which are associated with the resonance states. In the second 0 0 class of potentials there are either finite or infinite number of bound states (i.e. n "R). By making 0 the transformation of jP!j the up-side-down potential !»(r) has a potential barrier. If the new potential !»(r) supports resonance states then E(n, !j) would be a COMPLEX function of the quantum number n. The real and the imaginary parts of E(n, !j) are, respectively, the resonance positions and widths. Here we used indirectly the fact that upon complex scaling the resonance eigenfunctions become square integrable. Let us denote by /(E, j, r) a solution of the time-independent Schro¨dinger equation with the original potential well »(r). The eigenfunction / reduces to a square integrable function if we let the eigenvalue E get only discrete values assigned by the quantum number n, E "E(n, j), n"1, 2,2, n . In the first case mentioned above, the n 0 /(E,#j, r) solutions reduce to square integrable functions when E gets only discrete COMPLEX values, E "E(n,#j), n"n #1, n #2,2, by taking a complex contour r"DrD exp(ih) in the n 0 0 coordinate space. In the second case, the /(E,!j, r) solutions reduces to a product of an exponential decaying function and a polynomial (and thereby to a square integrable function) when E gets only discrete COMPLEX values, E "E(n,!j) by taking the inverse sign of j and by n choosing a complex contour r"D rD exp(ih) in the coordinate space. The main conclusion from this analysis is that there is a SINGLE expression for the discrete energy spectrum of a given model Hamiltonian. By varying either the quantum number n or one of the REAL potential parameters (j in our above discussion), the REAL energies (associated with bound states) and the COMPLEX “energies” (ONLY when resonance states do exist in the studied class of potentials) are obtained. To illustrate this use of the complex coordinate method we shall look at several simple analytically solvable model Hamiltonians. We will show here for several model Hamiltonians that the expressions of the resonance positions and widths obtained from the requirement that the resonance complex scaled solutions

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are square integrable are identical to the expressions obtained from the bound state energy spectra when the simple procedures proposed above are used. 2.2.2.1. ¹he parabolic potential barrier. The eigenfunctions of the well-known harmonic oscillator potential »(x)"kx2/2 are product of an exponential decaying function and the Kummer functions, M(E/(a/4!2E/(4+Jk/m), a/2, +Jk/mx2), where a"1 for even functions and to a"3 for odd functions. The Kummer functions reduce to Hermit polynomials when a/4!E/(2+Jk/m)"!n .

(2.2.7)

A parabolic potential barrier is obtained when kP!k. When x is scaled by the complex factor exp(ih) and from the requirement that M(E/(a/4!2E/(4+J!k/m), a/2, +J!k/mx2 exp(2ih)/+) would be an n-degree finite polynomial, we get that complex-scaled square integrable eigenfunctions of the complex-scaled Hamiltonian, HK "!e~2*h+2/2m d2/dx2!e`2*h(k/2)x2

(2.2.8)

are associated with the negative “purely” imaginary eigenvalues given by a/4!(E /2+)(!k/m)1@2"!n , n where for k'0 (!k/m)`1@2"!i(k/m)`1@2"iu .

(2.2.9)

(2.2.10)

Consequently, or

E "!i+u((4n#a)/2) n

E "!i+u(n#1/2), n"0, 1, 2,2 . (2.2.11) n Using the terminology given in Section 1 we can say that in this case the resonance positions (i.e. Re(E )) are all equal to zero, (note that the top of the barrier is located at »"0) whereas the n resonance widths are given by C "2+u(n#1/2), n"0, 1, 22 . (2.2.12) n For a detailed study of this problem see Atabek et al. [47]. In chemical reactions the potential barrier between the reactants and products is often described by a parabolic potential barrier. In such cases, the resonances are transition states associated with complex-scaled square integrable wavefunctions which are located inside the potential barrier. The lifetime, q, of these transition states is associated with the inverse resonance widths, q(transition states)"[2u(n#1/2)]~1 .

(2.2.13)

2.2.2.2. A model Hamiltonian of an electron scattering from a negative ion. One of the examples which were given by Landau and Lifshitz [48] for analytically soluble problems is of a particle moving in a central symmetric field with potential energy »(r)"j(c/r2!1/r). The solution of the time-independent Schro¨dinger equation is given by t"R(r)½ (h, /) , lm

(2.2.14)

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241

Fig. 10. The potential »(r)"l2/r!c/r2 as function of the radius r.

where ½ (h, /) are the spherical harmonics and R(r) is obtained by solving the following radial lm Schro¨dinger equation,

C

A

B

D

+2 d2R(r) 2 dR(r) 2k # # E! l(l#1) !»(r) R(r)"0 . r dr +2 2kr2 dr

(2.2.15)

The solution which is associated with negative energy levels and is finite for r"0 is given by R(r)"rs exp([!2kE]1@2r/+)F(s#1!j[k/(!2E)]1@2/+, 2s#2, 2(!2kE)1@2r/+) ,

(2.2.16)

where s(s#1)"l(l#1)#2kjc/+2

(2.2.17)

The confluent hypergeometric function F reduces to a polynomial of degree n (i.e. square integrable function) when n obeys the equation n"jJ(k/!2E)/+!s!1

(2.2.18)

and gets positive integer (or zero) value. Consequently, the bound states energy spectrum is given by E "!(2j2m/+2)[2n#1#M(2l#1)2#8mjc/+2N1@2]~2 . n By making the transformation of jP!j

(2.2.19)

(2.2.20)

we get the up-side-down potential which is presented in Fig. 10. This potential is a model for an electron approaching a negative ion (such as H~ for example). Resonances and bound states are associated with the complex-scaled square integrable eigenfunctions R(r). Therefore by scaling the r coordinate by a complex factor exp(ih) the confluent

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hypergeometric function F, F(s#1#jJ(k/!2E)/+, 2s#2, 2J(!2kE)re*h/+)

(2.2.21)

reduces to a polynomial of degree n"0, 1, 2, 32 when n"!jJ(k/!2E)/+!s!1

(2.2.22)

and s is defined as in Eq. (2.2.17). Note that Eq. (2.2.22) is obtained also from Eq. (2.2.18) by making the transformation jP!j. The corresponding h-independent resonances and bound states discrete spectrum is obtained by substituting Eq. (2.2.17) into Eq. (2.2.22) or by replacing j by !j in Eq. (2.2.19). For j"1 we obtain the complex energy levels which are given by 2j2m E "# [(2n#1)#iM8mc/+2!(2l#1)2N1@2]~2 n +2 2j2m (2n#1)2!D!2i(1n#1)D1@2 " , +2 [(2n#1)2!D]2#4(1n#1)2D

(2.2.23)

where D,8mc/+2!(2l#1)2 . From Eq. (2.2.23), one can see that shape-type resonances are obtained (i.e. Im(E )(0) when the n attractive part of the potential is strong enough such that C'0 and c'+2(l#1/2)2/2m, l"0, 1, 2 .

(2.2.24)

The resonance eigenvalues and eigenfunctions for the special case of l"0 were first introduced by Doolen [49], and for the more generalized case where lO0 by Junker [24]. Note by passing that in a very similar way the resonance energies of a barrier potential given by »(r)"$A/r2!Br2 can be obtained by making the transformation AP!A and BP!B in the analytical expression of the bound spectrum of !»(r) given by Landau and Lifshitz [48]. 2.2.2.3. ¹he Eckart potential — a model for A#BC reactive system. The transition from reactants to products in an elementary chemical reaction characterized by the presence of a potential barrier. The symmetric potential barrier presented in Fig. 11 describes the potential energy of H#H 2 (within the framework of the Born—Oppenheimer approximation) as a function of the reaction coordinate x. The reaction coordinate is defined as the minimum energy path from the reactants’ “valley” to that of the products. For non-homonuclear chemical reactions the potential barrier is non-symmetric. The Eckart potential is probably the most commonly used model for the description of a chemical reaction potential barrier. It has been shown on the basis of semiclassical transition theory formulated by Miller [50] that the cumulative reaction probability as a function of the translational energy, N(E), can be calculated only from the contributions of the Siegert poles of the scattering matrix even when N(E) is structureless [50,51]. Following the strategy to obtain analytical expressions for the resonance positions and widths (i.e. complex Siegert eigenvalues of

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243

Fig. 11. Non-symmetric Eckart potential energy curve.

the complex-scaled Hamiltonian) described above, Ryaboy and Moiseyev [52] obtained the resonance energy levels of a particle moving in the symmetric and non-symmetric Eckart potential barrier. (a) Symmetric potential barrier Here we follow Landau and Lifshitz [53] solution for the Rosen—Morse model Hamiltonian [54] which supports bound states. Note that V(Rosen—Morse)"!V(Eckart). The Rosen—Morse model Hamiltonian is given by HK (x)"!+2/2m d2/dx2#» /[cosh2(x/a)] . 0

(2.2.25)

By making the transformation » P!» we get the up-side-down Rosen—Morse potential which 0 0 is the symmetric Eckart potential barrier. By choosing in Eq. (2.2.25), a complex contour, x"x@ exp(ih), where x@ 3 (!R,#R), and sufficiently large value of h, the resonance eigenfunctions become square integrable. The wave functions t(x) are proportional to hypergeometric functions F, t(x) & F[e!s, e#s#1, e#1, x] ,

(2.2.26)

where e"(a/+)J!2mE

(2.2.27)

s(s#1)"!2m» a2/+2 . 0

(2.2.28)

and

In order to get square integrable functions, F should be truncated to a finite order n-degree polynomial by following the requirement of: e!s"n"0, 1, 2,2

(2.2.29)

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Fig. 12. The Siegert transition-state resonance eigenvalues and 2h rotated continuum. Fig. 13. Symmetric Eckart potential »(x) (solid lines) and five Siegert eigenvalues Re t (x); n"0!4. Two states closest n to the reaction threshold E"0 continuum eigenfunctions are given by dashed lines for comparison.

Therefore, E"!(+2/8ma2)[!(1#2n)#J1!8m» a2/+2]2 0 "(+2/8ma2)[(1#2n)!iJ8m» a2/+2!1]2 . 0 Eq. (2.2.30) can be rewritten E "1/a[u /2!i(n#1/2)]2 , n s where a"2ma2/+2 and u "J4a» !1 . s 0

(2.2.30)

(2.2.31)

(2.2.32)

Note that here we choose the positive root of !1, that is, J!1"#i. See, for example, the Siegert eigenvalues presented in Fig. 12. As one can see the Siegert states can be counted by their nodes, in the same way as bound state wave functions. The complex-scaled resonance wave functions are localized within the potential barrier region. The two closest eigenfunctions to the reaction threshold, E"0, continuum are given in Fig. 13 for comparison. (b) Non-symmetric potential barrier The non-symmetric Eckart potential is given by »(x)"!» cosh2 kMtanh[(x!ka)/a]#tanh(k)N2#» e~2k , (2.2.33) 0 0 where the potential barrier height is » e~2k and the two threshold energies are 0 and 0 !2» sinh(2k), respectively. When k"0 Eq. (2.2.33) reduces to the symmetric potential barrier 0 that appear in Eq. (2.2.26). Similarly to the derivation of the Siegert eigenvalues for the symmetric

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245

potential we shall follow here the analytical solution presented by Morse and Feshbach [55]: »(Morse!Feshbach)"!»(non-symmetric Eckart) and obtain the complex Siegert eigenvalues which are given by b2 2 E "!2b #[b #i(n#1/2)]2# "(z#!b /z)2 , n 2 1 2 [b #i(n#1/2)]2 1 where

(2.2.34)

a"2ma2/+ , e"aE , l"a» , 0 (2.2.35) b "(l cosh2k!1/4)1@2 , 1 b "(l sinh 2k)/2 , 2 z"#b #i(n#1/2) . 1 2.2.2.4. Resonances of non-dilation-analytical potential by complex translation. This is an example for the case where the complex resonance eigenvalues are obtained by letting the quantum number n in the expression for the discrete bound state spectrum to exceed a certain value n . The 0 Natanzon model Hamiltonian [56] is given by HK "!d2/dx2#»(x) ,

(2.2.36)

where »(x)"!j2l(l#1)(1!y2)#(1!y2)(1!j2)[5(1!j2)y4!(7!j2)y2#2]/4

C

A BD

1 i#cy 1#y x" ln #icln 2j2 i!cy 1!y

"Re(x) ,

(2.2.37)

c"(j2!1)1@2 , and !R(x(#R as !1(y(1

(2.2.38)

j and l are dimensionless parameters describing the shape and number of bound states. The potential shown in Fig. 14 (for the values of j"26, l"5.5) supports six bound states and resonances which result from the tunneling through the two symmetrical potential barriers. There are four singularities at y"$1 and y"$i/c. As pointed out by Certain and coworkers [57], these singularities avoid a straightforward use of the complex-scaling procedure. As discussed in Section 1.5, the resonance eigenfunctions diverge asymptotically but upon scaling the coordinate x by exp(ih), t(xe*h)P0 as xP$R

(2.2.39)

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Fig. 14. The Natanzon non-analytical potential.

when h'h and the critical angle h is given by Eq. (1.5.11). Since in the studied case even the # # critical angle h associated with the longest lived resonance is beyond the angle h"tan~1(1/c) at # which the singularity occurs, h 'tan~1(1/c) , (2.2.40) # the complex eigenvectors obtained for h'h do not have the characteristic resonance asymp# totical behavior of a purely out-going plane wave. In order to avoid singularity points and to obtain the resonance complex eigenvalues by bound state techniques one can use the exteriorscaling procedure described above, or, as it will be shown here, to translate into the complex coordinate plane, rather than dilate, a transformation of the variable y. The transformation of the variable used by Ginnochio [58], cos a"jy/[1#(j2!1)y2]1@2

(2.2.41)

reduces the original Schro¨dinger equation for the Natanzon potential to the Gegenbauer equation,

C

A

B A

BD

d2 1 2 k2!1 4 ! # a#k# ! da2 2 sin2a

f (cos a)"0 ,

(2.2.42)

where k"(!e)1@2/j2 , a"[1#k2(1!j2)#l(l#1)]1@2!k!1 4 2

(2.2.43)

and

A

t"

B

j2#(1!j2) cos2 a 1@4 f (cos a) . sin2a

(2.2.44)

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247

The Gegenbauer function f (cos(a)) is square integrable if the parameter defined in Eq. (2.2.43) gets non-negative integer values. a"0, 1, 2,2, n .

(2.2.45)

From Eqs. (2.2.43) and (2.2.45) we get E "!j2(l#1)2#(j2!2)(n#1)2#(2n#1)[j2(l#1)2#(1!j2)(n#1)2]1@2 , n 2 2 2 2 where

(2.2.46)

n4n (2.2.47) 0 and n is given by 0 n "j(l#1)/(j2!2)1@2!1 . (2.2.48) 0 2 2 This is exactly the algebraic form of the eigenenergies obtained by Ginocchio for bound states. However, following the above analysis, this expression is valid for resonances as well when n'n . (2.2.49) 0 As was shown by Lipkin and Moiseyev [59], the complex translated (not scaled!) eigenfunction t(a#ih) is a square integrable function when n'n . 0 Consequently, the complex resonance eigenvalues are given by E"!j2(l#1)2#(j2!2)(n#1)2 2 2 (2.2.50) !i(2n#1)Dj2(l#1)2#(1!j2)(n#1)2D1@2, n'n . 2 2 0 The same analytical expression for the resonances was obtained also by Ginocchio and by Alhassid et al. [60]. 2.3. Restrictions on the complex-scaling parameter As it was pointed out in Section 1, the resonances are associated with square integrable eigenfunctions of the complex-scaled Hamiltonian, provided the rotational angle h is larger than a critical value, h , given in Eq. (1.5.11), where the complex-scaling parameter is given by exp(ih). # The value of the critical angle depend on the ratio between the resonance width and the resonance position. Here we shall address the question of what are the upper bounds to the rotational angle h. We shall distinguish between two cases. In the first one the exterior scaling procedure is used to avoid complex scaling of the SHORT range potential. That is, H"!1d2/dr2#»(r) when r(r 2 0 and since »(r'r )"0, then 0 e~2*h d2/dr2 when r'r . H"! 0 2

(2.3.1)

(2.3.2)

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From Eq. (1.5.9), one can see that the scaled resonance function is square integrable when the exponential factor b"Dk D sin(h!/ ) is positive. Consequently, n n 0((h!/ )(n . (2.3.3) n By substituting / "h as defined in Eq. (1.5.11), one immediately sees that n # h (h(h #n . (2.3.4) # # In the second case the “reaction” coordinate is scaled by a complex factor, exp(ih), over the entire available space. Therefore, the upper limit of the rotational angle h depends on the analytical properties of the potential. The upper limit of h results from the inapplicability of the Balslev—Combes theorem to the bound states (rather than to resonances) when h exceeds a given value which is potential dependent. Namely, when h gets beyond a critical value the bound states are not associated with real eigenvalues as they should. Following the work of Moiseyev and Katriel [61] let us consider for example the bound class of potentials »(x)"x2m ,

(2.3.5)

where m"1, 2,2 .

(2.3.6)

The asymptotic behavior of the eigenfunctions of the Hamiltonian !1d2/dx2#»(x) 2 can be easily shown to be (note that for large values of x v(x)!EKv(x) and Ht"0).

(2.3.7)

/ (x) P x~m@2 exp[$21@2(m#1)~1xm`1] . (2.3.8) B x?= Here / are square integrable functions but / are not. Note that the square integrable solutions, ~ ` / , are associated with the discrete bound states, whereas the non-square integrable solutions, / , ~ ` are not associated with resonances (which do not exist in this case) and are known in scattering theory as anti-bound or virtual states. Upon complex scaling, xPx exp(ih), and the scaled, / (x e*h) or / (x e*h), are respectively square integrable when ` ~ Re exp[i(m#1)h]"cos[(m#1)h] (2.3.9) is negative or positive. The critical angles, for which cos[(m#1)h]"0 and the eigenfunctions “jump” from one type of boundary condition (e.g. / P0) to another type (e.g. / P0) are: ` ~ h(j)"n/(m#1)( j#1), j"0, 1, 2,2 . (2.3.10) # 2 Writing the complex-scaled Hamiltonian in the form

C

D

1 d2 HK (h)"e~2*h ! #e2*h(m`1)x2m , 2 dx2

(2.3.11)

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249

we note that for exp[2ih(m#1)]"1, i.e. h,h "pj/(m#1), j

j"0, 1,2

(2.3.12)

exists HK (h )"e~2*hjHK (h"0) j and, therefore,

(2.3.13)

E(j)"e~*2jp@(m`1)+iE (h"0), j"0, 1, 2,2, (2.3.14) n n where E(j) are the eigenvalues of the complex-scaled BOUND Hamiltonian, (i.e. no continuum n spectrum), and E (h"0) are the eigenvalues of the unscaled Hamiltonian H(x). n Within the range (2.3.15) h(j)(h(h(j`1) , # # the spurious complex energy E(j) as given by Eq. (2.3.14) is h-independent. The changes in the phase n occur abruptly at the critical angles h(j). ¹he energy spectrum of the bound states remain on the real # energy axis only when h(h(0). Atabeck and Lefebvre [62] numerically observed a similar phenom# enon. That is, by rotating the coordinate beyond its critical upper limit, complex eigenvalues of exponential BOUND potential were obtained. This numerical observation can be understood on the basis of the analytical results obtained for the xn (note that here n"2m and Eq. (2.3.14) is valid also for odd values of n). The dominant term, m, in the expansion of the exponential potential = »(x)"Ae`jx"A + (jx)n/n! n/0 in the vicinity of x"x 0 ­ln[(jx )n/n!] 0 Kln(jx )!ln(n)"0 . 0 ­n

(2.3.16)

(2.3.17)

That is, for j"1 nKx . (2.3.18) 0 Let x be the classical turning point, determined by 0 (2.3.19) »(x )"Aejx0"E . 0 Consequently, at the classical turning point the exponential can be approximately represented as »(x)"x2m; m"x /2. Using Eqs. (2.3.14) and (2.3.15) for n"x and m"n/2 we get 0 0 (2.3.20) E(j)"e~(4jp@x0`1)* E (h"0), j"0, 1, 2,2 . n n and

A B

arctan

4nj Im E n K , j"0, 1, 2,2 x #1 Re E 0 n

(2.3.21)

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when x is scaled by a complex factor exp(ih), p/(x #1)(2j#1)(h(p/(x #1)(2j#2) . (2.3.22) 0 0 ¹herefore the Balslev—Combes theorem is satisfied (i.e. bound states are associated with real eigenvalues even when the Hamiltonian is complex scaled) only with h(p/(x #1). A similar analysis for 0 estimating the upper limits of the rotational angle for other analytical given potentials can be carried out. 2.4. The generalized inner product for complex-scaled Hamiltonians 2.4.1. The c-product for time-independent Hamiltonians The inner product is defined in quantum mechanics as the scalar product, that is,

P

S f DgT"

f *g dq

(2.4.1) !-- 41!#% where f, g are functions in the Hilbert space. When f and g are exponentially divergent functions — as the resonance Siegert eigenfunctions of HK are — they are NOT in the Hilbert space, and they are NOT in the hermitian domain of HK . By scaling the coordinates in the time-independent Schro¨dinger equation by a complex factor, the exponentially divergent resonance states become square integrable. However, in spite of the fact that the complex-scaled resonance states are in the generalized Hilbert space, they are not associated with an hermitian operator (HK (xe*h) is non-hermitian) and as we shall show in this section the scalar hermitian product given in Eq. (2.4.1) should be replaced by a generalized inner product (the so-called c-product). In the c-product, the complex conjugate of the terms in the function which are complex not as a result of the complex scaling are taken. In the c-product first the complex conjugate of the function is taken and only then the coordinates or the momentum are analytically continued into the complex plan. If, for example, f *"exp(!ikx) when k is real, then the complex scaled f * is given by exp(!ik e`*hx). Here e*h can be interpreted as the scaling factor of x or (when h(0), as the analytically continuation of k into the complex plan such that (+k e*h)/(2k) is the complex resonance energy and exp(#ik e*hx) is the exponentially diverged Siegert state. The c-product as it is described below facilitates the discussion of theorems for the complex scaled (non-hermitian) operator H(h) which are analogous to those for H, and enables the derivation of the complex-coordinate scattering theory for time-independent and for time-dependent Hamiltonians which are discussed in Section 3. Only when the c-product rather than the metric scalar product is used in the calculation of the spectral representation of the Green operator, a remarkable agreement between the theoretical and experimental non-specular scattering intensities of helium atom scattered from corrugated Cu(115) surface were obtained. Moiseyev, Certain and Weinhold [63] proposed a c-product where ON¸½ the complex conjugate of the terms in the **bra++ state, which are complex regardless of the use of the complex scaling, should be taken. This definition is in the spirit of the usual understanding of analytical continuation. One does not take the complex conjugate of the terms in the wavefunction which become complex only due to the rotating of the coordinate into the complex plane. If for example, the eigenfunctions of the unscaled Hamiltonian are real and the COMPLEX eigenfunctions of the complex-scaled

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Hamiltonian are expanded in terms of REAL functions, then the inner product is given by

P

| f Dg},( f Dg),S f *DgT"

fg dq . (2.4.2) !-- 41!#% For different approaches see “The Letropet Symposium View on a Generalized Inner Product” [64]. In order to get a generalized definition of the inner product that should be used, let us represent the differential time-independent Schro¨dinger equation as a matrix eigenvalue problem, HwR"E wR , (2.4.3) i i i where wR is the vector representation of the “ket” state. Without loss of generality, we assume that H is a symmetry adapted Hamiltonian matrix and the eigenvalue spectrum is non-degenerate. The inner product is required to satisfy the conditions, ((wL)5DwR)"d . (2.4.4) i j ij Thus, the set of right and left eigenvectors MwR, wLN forms a complete set of orthonormal functions. i j The right-hand eigenvectors of H are defined as the “ket” states. In order to define the “bra” states the left-hand eigenvectors of the same H should be calculated: (wL)5H"EI (wL)5 . j j j By taking the transpose of Eq. (2.4.5), one gets

(2.4.5)

H5wL"EI wL . j j j Since the eigenvalues of H5 are the eigenvalues of H,

(2.4.6)

E "EI j j

(2.4.7)

then H5wL"E wL . (2.4.8) j j j From Eq. (2.4.8), one can see that wL are the right-hand-side eigenvectors of the transposed of j the Hamiltonian matrix H. Therefore, the inner product is the product of the eigenvectors of the Hamiltonian matrix, H, and the eigenvector of its transpose, H5, associated with the same eigenvalue. When the Hamiltonian matrix is hermitian H5"H* .

(2.4.9)

Consequently, (2.4.10) wL"(wR)* j j and the inner product is the ordinary metric scalar product. When the Hamiltonian matrix is complex and symmetric (for example, when the Hamiltonian becomes complex only upon complex scaling, and real basis functions are used to construct the

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Hamiltonian matrix), then (2.4.11)

H5"H and therefore the two column vectors are identical

wL"wR . (2.4.12) j j In the general case, the Hamiltonian matrix is complex because of the use of complex-scaled coordinate, and because of the use of complex functions as a basis set. In such a case, the right-hand side eigenvectors of the Hamiltonian matrix and of its transposed should be calculated. If A¸¸ the right-hand side eigenvectors of H are known, then the left-hand side eigenvectors (i.e. eigenvector matrix of the transposed Hamiltonian) are obtained by calculating the transpose of the inverse of the eigenvector matrix of H, (wL, wL,2)"[(wR, wR,2)~1]5 . (2.4.13) 1 2 1 2 A complex basis function is a natural choice, for example, when the energies and lifetimes of three atomic molecules are calculated. To conserve the total angular momentum J, and its projection M along the laboratory z-axis, the eigenfunction of the complex-scaled Hamiltonian are expanded in terms of the yJ, lM basis functions, given by j, l j YJ, lM(c)" + (2.4.14) + (ljmlm DljJM)½ j(RK )½l l(rˆ ) j, ,m j j, m l l m /~ mj/~j when J"M"0 then l"j and YJ, lM"½ (cos c) (2.4.15) j, j, 0 and cos c"RK ) rL where R and r are, respectively, defined as the distance from the dissociative atom C to the center of mass of the diatom AB and the internuclear distance of the diatom AB; (l j ml m Dl j J M) are the Clebsh—Gordon coefficients and ½ is the spherical harmonic. When real j j,0 basis functions u (R) and s (r) are used, the Hamiltonian matrix elements given by n m (2.4.16) [H(h)] "Su s YJMl DHK (Re*h, r, c)Du s YJMl T (n{, m{, j{, l{), (n, m, j, l) n{ m{ j{ { n m j, are complex even for the case where h"0 (i.e. no rotation of the coordinates to the complex plane) and J O 0. Here the basis functions are defined as S/ (rL , RK ),Du s YJMl T (2.4.17) k n m j, when the index k stands for the collective indices Mn, m, j, lN. The complex-scaled square-integrable eigenfunctions of H(h) are orthonormal, |tLDtR},(tLDtR)"e*h j i j i when tR is given by i tR"+ (C) / (RK , rL ) i k, i k k

P

tLtR dq"d j i ij !-- 41!#%

(2.4.18)

(2.4.19)

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and tL is given by j tL"+ (C~1)5 /*(RK , rL ) , (2.4.20) j k, j k k where C is the eigenvector matrix of the complex-scaled Hamiltonian matrix H(h). A knowledge of the eigenfunctions of the complex-scaled Hamiltonian enables us to calculate the uncertainty in measuring different dynamical quantities. The COMPLEX expectation value of a given quantity is given by expressions bi-linear in tR and tL, : tLOK (h)tR dq |tLDOK (h)DtR},(tLDOK (h)DtR)" !-- 41!#% , (2.4.21) tLtR dq : !-- 41!#% where OK is the COMPLEX-scaled operator associated with the measured dynamical quantity. If OK is, for example, the complex-scaled Hamiltonian, H(h), the expectation value is a complex number, E"e!(i/2)C. The real part of E, e, is the average measured energy of the system in its metastable state, whereas the imaginary part, C is the uncertainty of e, even when the standard deviation of H(h) is zero (i.e. |H(h)2}!|H(h)}2"0). Similarly, the imaginary part of (tLDOK (h)DtR) can be interpreted as the uncertainty of measuring the quantity of OK when the system is prepared in a resonance (i.e. metastable) state. 2.4.2. The c-product for time periodic Hamiltonians The interaction of atoms and molecules in CW lasers is described by time-periodic Hamiltonians. However, even in short pulse experiments the time periodic behavior of the Hamiltonian may be assumed when the field oscillates more than 50—100 times during the pulse “lifetime” [65]. When the Hamiltonian is time periodic HK (x, t)"HK (x, t#¹) ,

(2.4.22)

the solutions of the complex-scaled time-dependent Schro¨dinger equation ­t (x, t) HK (xe*h, t)t (x, t)"i+ h h ­t

(2.4.23)

for a given initial state t (x, 0) can be described as a linear combination of the quasi-energy h solutions (known also as the Floquet states). The quasi-energy complex-scaled states are given by t (x, t),tQE(x, t) , h a tQE(x, t)"e~*Eat@+/h(x, t) , a a where

(2.4.24)

/h(x, t)"/h(x, t#¹) (2.4.25) a a (when box normalization is used then the spectrum is discrete and a"1, 2,2). For dissociative/ionizing systems the quasi-energies get complex values and the resonance quasi-energies, E , a are h-independent: E "e !(i/2)C . a a a

(2.4.26)

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The periodic functions, /h, and the quasi-energies, E , are respectively defined as the eigenfunctions a a and eigenvalues of the complex-scaled Floquet operator, H K /h"E /h , h a a a where

(2.4.27)

­ H K "!i+ #HK (xe*h, t) . h ­t

(2.4.28)

The complex-scaled time periodic eigenfunctions /h can be expanded in a Fourier basis set, a = /h(x, t)" + uh (x)e*unt (2.4.29) a n, a n/~= where u"2p/¹ and uh are the components of the right eigenvectors of the Floquet Hamiltonian n, a matrix [66,128], Hh(x)uh(x)"E uh(x) a a a

(2.4.30)

and

P P

A

B

­ 1 T e~*wn{t !i+ #H(xe*h, t) e*wnt dt [Hh(x)] " n{, n ¹ ­t 0 1 T HK (xe*h, t)e*w(n~n{)t dt#+ud . " n, n{ ¹ 0 The time-dependent inner product should satisfy the following condition: |tL (QE)DtR(QE)},(tL (QE)DtR(QE)) a{ a a{ a 1 T dt " dxtL, QE(xe*h, t) ) tR, QE(xe*h, t)"d a{ a a, a{ ¹ 0 !-- 41!#% Therefore (see Eqs. (2.4.24) and (2.4.29)), the right-hand quasi-energy state is given by

P P

= tR, QE,e~*Eat@+ + uh (x)e`*wnt, a"1, 2,2 , n, a a n/~= whereas the left-hand quasi-energy state is given by

(2.4.31)

(2.4.32)

(2.4.33)

= (2.4.34) tL, QE"e`*Ea{t@+ + uh, L (x)e~*wnt, a@"1, 2,2 , n, a{ a{ n/~= where uh, L are the components of the left eigenvectors of Hh(x). It should be stressed here that one n, a should NOT take the complex conjugate of the complex quasi-energies E . Only when real basis a functions, Ms (x), i"1,2, NN are used to diagonalize the Floquet Hamiltonian matrix Hh(x) i and the Hamiltonian has a time reversal symmetry, then uh,L"uh . The uh (x) functions n,a n,a n, a

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(components of the vector uh), which are used in Eq. (2.4.33), are given by a uh (x)"+ CR (h)s (x) , n, a (n,i),a i i where CR are the components of the right eigenvectors of Hh, (n, i), a HhCR(h)"E CR(h) , a a a where

P

(H)h " k{,k

dqs* (x)[H K (xe*h)] s (x) . i{ n{, n i

(2.4.35)

(2.4.36)

(2.4.37)

!-- 41!#% The row and column indices, k and k@, are associated, respectively, with (n, i) and (n@, i@). n defines the nth Fourier basis function and i defines the ith complex basis function s (x). The uh (x) functions i n, a{ which are used in Eq. (2.4.34) are given by (2.4.38) uh (a)"+ CL (h)s*(x) , i n, a{ (n,i),a i where MCL N are the components of the left eigenvectors of Hh (as defined in Eq. (2.4.37)) (n,i),a (Hh)5CL (h)"E CL (h) . (2.4.39) a{ a{ a{ For example, in the study of multiphoton processes in time-dependent ionizing/dissociative systems, the use of the inner product as defined in Eqs. (2.4.33), (2.4.34), (2.4.35), (2.4.36), (2.4.37), (2.4.38) and (2.4.39) is the key point in the evaluation of accurate time-independent expressions for the probability to obtain high harmonics [65]. The probability to obtain high harmonics, i.e. radiation with the frequency X"kw, is associated with the Fourier transform of the timedependent dipole moment [67],

KP

K

1 T X 2 e~* tD(t) dt . ¹ 0 When the system is “trapped” in a long lived resonance quasi-energy state p(X)J

th "e~*E3%4t@+/h (x, t) 3%4 3%4

(2.4.40)

(2.4.41)

and E "e !(i/2)C . 3%4 3%4 3%4 The complex expectation value of the dipole moment is given by

P P

1 T dt tL,h(x, t)xe*htR,h(x, t) dx , D(t)" 3%4 3%4 ¹ 0 !-- 41!#% where

P P

1 T dxtL, h(x, t)tR, h(x, t)"1 dt 3%4 3%4 ¹ !-- 41!#% 0

(2.4.42)

(2.4.43)

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Using the c-product as defined above in Eqs. (2.4.33) and (2.4.34) Ben-Tal et al. [65] derived the time-independent expression for the probability of obtaining radiation with the frequency X,

K

P

K

= 2 p(X"kw)" + (2.4.44) uh,L (x)xuh (x) dx n`k, 3%4 k, 3%4 n/~= where uh (x) and uh (x) are the Fourier components of /h (x, t) as defined in Eqs. (2.4.41) n`k, 3%4 k, 3%4 3%4 and (2.4.38). 2.4.3. The c-product for a general time-dependent Hamiltonian The solution of the complex-scaled time-dependent Schro¨dinger equation for any time-dependent Hamiltonian (not necessarily a time periodic one) can be obtained as [68] t (x, t)"t (x, t@, t)D , h h t{/t where and

(2.4.45)

t (x, t@, t)"e~*@+ HK (%*hx, t{)(t~t0)t (x, t@, t ) h h 0

(2.4.46)

t (x, t , t ),t(xe*h, t ) (2.4.47) h 0 0 0 is the initial given state, and H K (xe*h, t@) and H K L(xe*h, t@) are the right and left Floquet-type operators H K (xe*h, t@)"HK (xe*h, t@)!i+ ­/­t@ ,

H K L(xe*h#i+ ­/­t@)

(2.4.48)

The time variable t@ acts as an additional coordinate in a generalized Hilbert space introduced by Howland [69,70]. This space contains all possible square integrable functions of x and t@, where box normalization is used for x and for t@ (0(t@(¹). The eigenfunctions of H(xe*h, t) forms a complete set for the generalized Hilbert space (see the discussion in Section 2.4.4). The inner product in the generalized Hilbert space is defined as

P P

1 T dt@ dx/L(xe*h, t@)/R(xe*h, t@)"d , |/ D/ },(/ D/ )" i j ij i j i j ¹ 0 !-- 41!#% where

(2.4.49)

H K (xe*h, t@)/R(e*hx, t@)"Eh/R(e*hx, t@) , j j j

(2.4.50) H K L(xe*h, t@)/L(e*hx, t@)"Eh/L(e*hx, t@) . i i i Assuming that the eigenfunctions of the complex-scaled Floquet Hamiltonian form a complete set, the unit operator is given by + D/R}|/LD"1ª . (2.4.51) j j j By substituting Eq. (2.4.51) into Eq. (2.4.46), one gets that the solution of the time-dependent Hamiltonian (which describes ionizing or dissociative systems) for the initial state t(x, t ) is given 0 by t (x, t)"+ e~*Ehjt|/LDt(t )}/R(xe*h, t) , h j 0 j j

(2.4.52)

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257

where

P P

1 T dt@ |/LDt(t )}" dx/L(xe*h, t@)t(xe*h, t ) . (2.4.53) j 0 j 0 ¹ 0 !-- 41!#% When complex basis functions s (x) and Fourier basis set, exp(2pti/¹), are used to diagonalize i H(xe*h, $t@) then /L and /R are calculated as described in Eqs. (2.4.29), (2.4.35), (2.4.36), (2.4.37), j j (2.4.38) and (2.4.39). Two cases may be considered. In the first case, the Hamiltonians are NOT time periodic. In such a case box normalization for x and t@ implies that t(xe*h, t@, t) is a square integrable function of x and t@. Long time (i.e. t) propagation requires the use of a large box (i.e. ¹ should get sufficient large value). In the second case time periodic boundary conditions are required although the Hamiltonians are NOT time periodic, t(xe*h, t@, t)"t(xe*h, t@#¹, t) , where 04t@4¹ . The physical solution is obtained when t@"t mod ¹ . Solving the complex-scaled time-dependent Schro¨dinger equation for time-dependent Hamiltonians by the (t, t@) method enables the evaluation of time-independent state-to-state transition probabilities, and the calculation of the time evolution operator as for time-independent Hamiltonians (provided the propagated wave packet remains square integrable for t4¹ for ionizing/dissociative systems [55]). 2.4.4. The turn-over rule If HK (h"0) is a real and self-adjoint operator, then the complex-scaled Hamiltonian is given by HK "SK ~1(h)HK SK (h) h and obeys

(2.4.54)

HK `"HK * . (2.4.55) h h If f and g are two complex square integrable basis functions (regardless of the fact that hO0) then, (2.4.56) S f DHK gT"SH`f DgT"SHK * f DgT"Sg*DHK f *T . h h h h However, if f and g are complex functions ONLY due to the fact that hO0 as we shall discuss below (see Section 2.4.1 on the c-product) ( f DHK g)"S f *DHK gT"S(HK `f *)DgT"S(HK * f *)DgT"(HK f Dg) . (2.4.57) h h h h h Therefore, HK is symmetric and satisfies the turn over rule with respect to the complex product h ( f DH g)"(H f Dg)"(gDH f ) . (2.4.58) h h h

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2.5. Do the eigenfunctions of the complex-scaled Hamiltonian matrix form a complete basis set? On the basis of Motzkin and Taussky results [71], Moiseyev and Friedland [72] proved that for very special values of the complex-scaling parameter, g"exp(ih), (where h may get complex values) an incomplete spectrum of N-dimensional Hamiltonian matrix is obtained. The spectrum is incomplete in the sense that the eigenfunctions of the complex-scaled Hamiltonian matrix do not form a complete basis set and the number of non-linear independent eigenvectors of the nonhermitian matrix is smaller than N. Note, however, that any infinitesimally small variation in h will make the spectrum to be a complete one. Within the framework of the finite basis set approximation the time-independent Schro¨dinger equation is given by H(h)wR"E wR and H5(h)wL"E wL , i i i i i i where the complex-scaled Hamiltonian matrix is given by

(2.5.1)

(H(h)) "Ss DHK (xe*h)Ds T, (i, j)"1, 2,2, N (2.5.2) ij i j and MSs TN are square integrable basis functions. In Section 2.4 we assumed that the eigenfunction i of H(h) forms a complete set. That is, there are N normalizeable independent eigenvectors (wLDwR)"d , (i, j)"1, 2,2, N . i j ij Since the complex-scaled Hamiltonian matrix is not hermitian and

(2.5.3)

H (h)OH* (h) (2.5.4) ij ji it may happen that the N]N Hamiltonian matrix H(h) has N!1 or even less linear independent eige:nvectors for special values of h. In such a case the eigenvectors of H(h) do NOT form a complete set for the N-dimensional vectors. For example [72], the complex symmetric 3]3 matrix

A

6

e*h

!1

H(h)" e*h

5

2eih

!1 2e*h

1

B

(2.5.5)

has only two distinct eigenvectors when h"p/2,

A B AB 1

E "6, wR"wL" 1 1 1

i/3

,

(2.5.6)

!1/3 0

E "3, wL"wR" 1 . 2 2 2 i

(2.5.7)

It should be stressed here that although the 3]3 matrix given in Eq. (2.5.5) has only 2 eigenvalues there is no degeneracies in the spectrum. Degeneracy implies two non-linear-dependent eigenvectors which are associated either with E"3 or with E"6. As one can see from Eq. (2.5.6) and

N. Moiseyev / Physics Reports 302 (1998) 211—293

259

Eq. (2.5.7) there are no such degenerated states. Since the matrix is complex and symmetric we use here the c-product rather than the usual scalar product. The fact that for h"p/2 the spectrum of H(h) is incomplete is reflected in the fact that the second eigenvector w is not c-normalizable and 2 (wLDwR)"0 (2.5.8) 2 2 w and E are considered, respectively, as a defective eigenvector and eigenvalue of the matrix H. 2 2 In such a case E is NOT a simple pole of the resolvent (H!E1)~1. A defective eigenvalue and 2 eigenvector is obtained at a branch point h "p/2, where b (2.5.9) E PE 2 3 and w and w are coalesced, 2 3 w Pw (2.5.10) 2 3 as hPh . (2.5.11) " We can summarize it by saying that the eigenvalues E (h) are analytical in h except for a finite i number of complex values of the rotational angle h "h ,2, h , which are the branch points for " 1 q some eigenvalues. Therefore, some E (h) do not have a Taylor expansion in g"exp(ih) around the i branch points g "exp(ih ). However, E (h) does have an expansion in (g!g )1@p (known as b " i " Puiseux series): = E (h)" + a (g!g )k@p#E (h ) . (2.5.12) i ik " i " k/0 Here N5p52 is an integer which implies the coalescence of p eigenvectors of H as hPh . " Usually, h gets complex value and, consequently, the absolute value of the complex scaling factor " g"exp(ih) is not equal to unity, i.e. DgDO1. For a computational method to calculate the branch points h for complex-scaled Hamiltonian matrices see Moiseyev and Certain in Ref. [73]. In " Fig. 15 the complex eigenvalues of a finite Hamiltonian matrix for 1S resonance of Helium as a function of the complex-scaling parameter g are presented [72], showing a branch point in the complex “energy” plane at Dg D"1.0837, cosDh D"!0.996432. " " 2.6. The complex analog to the variational principle: ¹he c-variational principle The complex analog to the variational principle provides the formal justification to the use of computational techniques that originally were developed for bound states in the calculation of the resonance position and widths by the complex coordinate method. As discussed in Section 2.6 one can assume that the eigenfunctions of HK (h) form a complete c-normalizable (see Section 2.4) set. In such a case it is easy to verify that the Rayleigh quotient EM h"(/DHK D/)/(/D/) h

(2.6.1)

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Fig. 15. 0-trajectory energy spectrum of the complex scaled Hamiltonian of helium which was calculated with 36 Hylleeraas-type basis functions. The black dot denotes the branch point in which the spectrum of the 36]36 Hamiltonian matrix is incomplete.

provides a stationary approximation to the true complex eigenvalue Eh when / is a c-normalizable k (i.e. (/D/)"1) approximation that is close to the eigenfunction of HK , t ; that is, h k /"t #­(e) implies Eh"Eh#­(e2) . k k

(2.6.2)

This complex variational principle [63] is, however, a stationary principle rather than an upper or lower bound for either the real or imaginary part of the complex eigenvalue. As noted above, even this stationary property fails if the eigenfunctions are not c-normalizable (i.e. c-normalization implies that (t Dt )O0 for any value of k). k k ¸inear c-variational calculations. Let us expand the trial functions / by N orthonormal basis functions, Ms , i"1,2, NN; that is, i N D/T" + CRs , j j j/1 N S/D" + CLs . i i i/1

(2.6.3)

By substituting Eq. (2.6.3) into Eq. (2.6.1), one obtains + CLCR(H (h)!EM d )"0 , i j ij ij i, j

(2.6.4)

where H (h)"Ss DHK Ds T . ij i h j

(2.6.5)

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261

On the basis of the c-variational principle given in Eqs. (2.6.1) and (2.6.2) and proved by Moiseyev et al. [63], we require that ­EM /­CL"0 for i"1, 2,2, N . i Consequently,

(2.6.6)

N + CR(H (h)!EM d )"0 j ij ij j/1

(2.6.7)

H(h)CR"EM CR .

(2.6.8)

or

Here we proved that the solutions of the matrix eigenvalue problem are stationary solutions in the complex variational space, and as NPR the exact solution of the time-independent complexscaled Schro¨dinger equation would be obtained. This theorem is the ground for the c-variational calculations by which the autoionization, predissociation and other type resonance positions and widths (inverse lifetimes) were obtained. For the generalized variational theorem (not necessarily linear variational space) see Moiseyev in Ref. [74]. 2.7. The complex analog to the hypervirial theorem From the turn-over rule Eq. (2.5.5) for the complex-scaled Hamiltonian, it follows immediately that commutators of HK and any given operator KK have vanishing c-expectation values in any h eigenstate t , k (t D[HK , KK ]t )"0 . (2.7.1) k h k This complex analog to the hypervirial theorem [63] holds for a wide class of operators KK , whether hermitian or not. In particular, if KK is chosen to be r ) e (or a sum of such terms in a many-particle system), and if the potential » is a homogeneous function of coordinates of degree m, Eq. (2.7.1) reduces to the complex virial theorem (KK is the kinetic energy operator) h (t DD¹ Dt )"(m/2)(t D»K Dt ) (2.7.2) k h k k h k relating the (complex) kinetic and potential “energies”. For example, when atomic autoionizing resonances are studied then m"1(40). For alternative proofs of the complex analog to the virial theorem see Froelich and Brandas [75], Wingler and Yarris [76], and Moiseyev [77]. 2.8. Cusps, h trajectories and complex-analog Hellmann—Feynman theorem The Hellmann—Feynman theorem [78] can be extended to the complex-scaled non-hermitian Hamiltonians [63]. Suppose U depends on a variational parameter C (e.g. C can be taken as the i linear variational parameters as described in Section 2.6). Variationally optimal values of these parameters must satisfy the relation ­EM /­C "0, i

i"1, 2,2, N ,

(2.8.1)

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where (/DHK D/) h EM " (/D/)

(2.8.2)

and H contains an embedded parameter f, h H "H (f) . (2.8.3) h h For example, f can be the scaling factor, g"exp(ih), or a physical parameter such as a nuclear charge, intramolecular distance in the Born—Oppenheimer Hamiltonian, etc. However, for c-normalized / (i.e. (/D/)"1),

AK KB

A BA B

dEM M ­EM ­H ­C h/ #+ j . " / (2.8.4) df ­f ­C ­f j j/1 Thus, if / is variationally optimal for any given value of f Eqs. (2.3.1) and (2.6.6) is satisfied and therefore Eq. (2.8.4) reduces to

AK KB

dEM /df" /

­H h/ . ­f

(2.8.4a)

This is the complex form of the Hellmann—Feynman theorem. When f"exp (ih) the requirement of dEM /dh"0 (the resonance solution satisfies this condition following the Balslev—Combes theorem) leads to another derivation of the virial theorem Eq. (2.8.2). Within the framework of the finite basis set approximation (C are the linear variational parameters) the resonance stationary condition

K K dEM dh

"0 h015 (f,h in Eq. (2.8.4)) leads to an iterative procedure for calculating the resonances:

(2.8.5)

H "e~2*hT#V , (H ) "Ss DH(xe*h)Ds T h h h ij i j H CR"EM CR, H5 CL"EM CL, (2.8.5a) h k k k h k k k (CL)5(­V /­h)CR h k. e~2*h" k (2.8.6) 2i(CL)5TCR k k Note that following the Cauchy—Reimann conditions h for which Eqs. (2.8.5) and (2.8.6) are 015 satisfied will usually get a complex value. Real (h ) should satisfy the conditions discussed in 015 Section 2.3. The Im(h ) can get either negative or positive values, that is, the absolute value of the 015 complex-scaling parameter, g"exp(ih ), can be either smaller or larger than unity. For atomic 015 coulombic potentials Eq. (2.8.5) reduces to the c-virial theorem. Moiseyev, Certain and Weinhold used this iterative algorithm for calculating the autoionization resonances of Helium [63]. Moiseyev and Corcoran [79] succeeded to avoid numerical difficulties in the first studied molecular autoionization resonances in H~ and H due to the fact that Im h was found to be negative in 2 2 015 their calculations.

N. Moiseyev / Physics Reports 302 (1998) 211—293

263

Cusps and h-trajectories. Doolen et al. [80] proposed a graphical method of solving Eq. (2.8.4) (known as h-trajectory), in which EM (h) is plotted as function of Re(h), holding Im(h) fixed. The stationary points along the trajectories which provide the minimal (not necessarily zero value) of DdEM /dhD have been assumed to correspond to resonances [81]. Moiseyev et al. [82] proved that the stationary solution where dEM /dh"0 at h"h is associated with a cusp in h-trajectory plot. The 015 proof is as follows: in the neighborhood of a stationary point the complex energy, EM , can be expanded in powers of (g!g ), (Puiseux expansion). Where the first two terms are 015 EM "E #a(g!g )k#2 (2.8.8) 0 015 g"e*h"ae*hR, g "e*h015"a e*h(0)R 015 015 h"h #ih , h "h(0)#ih(0) (2.8.9) R I 015 R I (0) a"e~hI, a "e~h I 015 and k is a positive rational number. For small enough (g!g ) along the h-trajectory 0 g"a e*hR (2.8.10) 015 and, therefore, we can write g!g "ig x , 015 015 where

(2.8.11)

x"h !h(0) . R R Consequently, for x50 (i.e. approaching the cusp from “above”)

(2.8.12)

EM "E #a(ig )kDxDk#2 ` 0 015 while, for x40 (i.e. approaching the cusp from “below”)

(2.8.13)

x"!DxD"DxDe*n

(2.8.14)

and EM "E #a(ig )kDxDke*nk2 . (2.8.15) ~ 0 015 One thus sees that a cusp exists between the two branches EM and EM at the stationary point, h"h , ` ~ 015 with a cusp angle pk. A smooth curve (cusp angle"n) will be observed at the stationary point when k is an odd integer (3, 5, 7,2). A schematic representation of such an example is given in Fig. 16a, where it is seen that the h -trajectory “slows down” at the stationary point, but yet there is no cusp. It should be R remembered, however, that the h -trajectories for non optimum Dg D (i.e. non-optimal value of R 015 Im(h )) are smooth also. This case (i.e. no cusp) has been observed numerically by McCurdy [83] 015 in calculations of shape-type resonances of the v"r2 exp (!0.61r) potential. A zero cusp angle will be observed at the stationary point when k is an even integer (2, 4, 6, 2). This case which is shown schematically in Fig. 16b is the most common case. For example, see the zero-angle cusps obtained in the calculations of H (1pu)2 and He(2s)2 resonances [79,84]. 2

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Fig. 16. h trajectory when a is held fixed at a"a . The arrows show the motion of complex “energies” as h is varied. R 015 The open circles denote the stationary solutions that are obtained from g"h "h(0) , (a) k"3, 5, 7,2 (b) k"2, 4, 6,2 R R (c) k"3/2, 7/22.

An interesting case occurs for rational k"n/m'1, where n and m are coprime, since here the cusp angle is neither zero nor n. A fractional n is the result of the coalescence of at least m eigenvalues (and also the corresponding eigenvectors!) of the complex Hamiltonian matrix at the stationary point h"h . As discussed in Section 2.5, EM (h ) in such a case is a defective eigenvalue. 015 015 This case which is shown schematically in Fig. 16c, for k"3, apparently has not been observed 2 so far in resonance calculations. Presumably, this is due to the very special conditions at which a defective eigenvalue is obtained. This cusp conditions has been used also by Peskin and Moiseyev [85] for calculating the optimal scattering transition probability amplitude, ¹ (h). The optimal condition 4#!55 ­¹ 4#!55 "0 (2.8.16) ­h 015 h has been found to be the key point in the successful application of the complex-coordinate scattering theory (Section 3) to long-range potentials.

K

K

2.9. The hermitian representation of the complex coordinate method: ºpper and lower bounds to the resonance positions and widths The c-variational principle presented in Section 2.6 is a stationary principle rather than an upper or lower bound for either the real or imaginary part of the complex eigenvalue. On the basis of the hermitian representation of the complex coordinate method developed by Moiseyev [86], it has been proved that DE(exact)!EM D5j ,

(2.9.1)

where j2 is the lowest real and positive eigenvalue of the complex, yet hermitian, operator H K 2 H2(EM , EM *)/"j2/

(2.9.2)

H2,(HK !EM )*(HK !EM ) h h

(2.9.3)

N. Moiseyev / Physics Reports 302 (1998) 211—293

265

and EM ,eN !(i/2)CM .

(2.9.4)

If / is a trial given function, then by using the Hellman—Feynman theorem, one can show that the minimal expectation value, S/DH2D/T/S/D/T, is obtained for EM "S/DHK D/T/S/D/T . (2.9.5) h By carrying out an iterative calculation, EM as obtained from Eq. (2.9.5) is substituted in Eq. (2.9.2) to get a new estimate for /. Here we assume that the ground state of the hermitian operator H2 can be accurately obtained from numerical computations (see, for example, Refs. [64—66,87—89]). Eq. (2.9.1) can be used to obtain an error estimate of EM obtained from c-variational calculations: (/DH D/) |/DH D/} h h , . EM " |/D/} (/D/)

(2.9.6)

Note that in Eq. (2.9.6) the c-product is used rather than the scaler product which has been used in Eq. (2.9.5). In such a case by combining Eq. (2.9.1) with the bounds to resonance eigenvalues evaluated by Davidson et al. [90] it is obtained that E (exact) is embedded in an annular ring centered at EM , where the inner and outer radii are, respectively, the lowest eigenvalue of H2(EM , EM *) and the complex-variance of H2(EM , EM *). Dp D4DE(exact)!EM D4j C

(2.9.7)

and Dp D4p DS/D/TD1@2 (2.9.8) C H provided DS/Dt(exact)TD2'1, and S/D/T"1. p and p are, respectively, the complex and C H 2 Hilbert-space variances given by [90—92] (/D(H !EM )2D/) h p2" , C (/D/) S/DH2(EM , EM *)D/T p2 " . H S/D/T

(2.9.9) (2.9.10)

Note that / is a complex normalized function (/D/)"1, and therefore S/D/T'1. An illustrative numerical example for the model Hamiltonian H"!1 d2/dx2#(x2/2!0.8)e~0.1x2#0.8 2 is presented in Fig. 17. Proof of Eq. (2.9.1): DE(exact)!EM D5j

(2.9.11)

HK t"E(exact)t (2.9.12) h H2(EM , EM *) is an hermitian operator and j2 is its lowest, real, and positive eigenvalue. Therefore, following the conventional variational principle: StDH2(EM , EM *)DtT 5j2 . StDtT

(2.9.13)

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Fig. 17. Bounds of the estimate complex-coordinate resonance eigenvalues EM , obtained for N even-parity harmonic oscillator basis functions. EM are indicated by the signs “#” and the exact value E by a dot. The dashed areas give an optimal estimate of the resonance location. The upper and lower bounds of the estimate shape-type complex-coordinate resonance eigenvalue, obtained for N"2, 3, 4 even-parity harmonic oscillator basis functions [64].

Since, StDH2DtT"SsDsT ,

(2.9.14)

where s"(H !EM )t"(E(exact)!EM )t , h therefore, StDH2DtT"DE(exact)!EM D2StDtT .

(2.9.15)

(2.9.16)

By substituting Eq. (2.9.16) into Eq. (2.9.13), one gets that DE(exact)!EM D25j2 .

(2.9.17)

Proof of Eq. (2.9.7): DE(exact)!EM D4Dp D C Here we shall use the Lanczos recursion procedure [93] to construct a three-diagonal complexscaled Hamiltonian matrix, a 0 b H" 0 h

A

b 0 a b 1 1 , b a b 1 2 2 } } }

B

(2.9.18)

where s ,/ is a given wave function which describes well enough the exact eigenfunction 0 t(exact), and a ,EM "(/DHK D/) , 0 h

(2.9.19)

N. Moiseyev / Physics Reports 302 (1998) 211—293

b ,p2"(s Ds )1@2 , 0 C 1 1 s "(HK !a )/ . 1 h 0 0 Similarly for n51,

267

(2.9.20) (2.9.21)

a "(s DHK Ds ) , (2.9.22) n n h n s "(HK !a )s !b s , (2.9.23) n`1 h n n n~1 n~1 b "(s Ds )1@2 . (2.9.24) n n`1 n`1 By diagonalizing the three-diagonal Hamiltonian matrix H the exact eigenfunction is obtained h = t(exact)" + C s (x) (2.9.25) n n n/0 where C is an eigenvector of H , h H C"E(exact)C (2.9.26) h and s are the Lanczos recursive functions n (s Ds )"d (2.9.27) n n{ n, n{ as defined in Eq. (2.9.23). By substituting Eq. (2.9.18) into Eq. (2.9.26), one gets a C #b C "EM (exact)C . (2.9.28) 0 0 0 1 0 Since a "EM (see Eq. (2.9.19)) and b "p2 (Eq. (2.9.20)) then from Eq. (2.9.28) one immediately 0 0 C obtains that DE(exact)!EM D2"Dp D2DC /C D2 . C 1 0 If s "/ is the dominant function in the Lanczos basis set expansion then DC D2'1, 0 0 2 DC /C D2(1 1 0 and, consequently, DE(exact)!EM D24Dp D2 . C

(2.9.29)

(2.9.30)

(2.9.31)

3. Complex scaling of ab initio molecular potential surfaces When the potential »(x) is known analytically its analytic continuation, »(x exp(ih)), is also known analytically. The problem is, however, that the molecular potentials are calculated on a finite grid in the conventional unscaled coordinate space. To overcome this difficulty several numerical procedures were proposed (see Section 2.2 in Ref. [94]). Moiseyev and Corcoran [95] studied the H~ and excited H autoionization resonances by using an analytic continuation of the 2 2 Hamiltonian matrix elements to the complex plane rather than of the Hamiltonian operator. Therefore, one can fit an ab initio potential to a given order polynomial, calculate analytically the potential matrix elements as function of the basis set parameters (such as box-size or scale

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parameter), and finally carry out analytic continuation of the potential matrix elements into the complex plane. This approach was used by Lipkin and coworkers [96] in their study of resonances for piecewise potentials and more recently in the study of certain atmospheric conditions by which downwind at the mountains is formed [97]. A preferable method, however, is the one for which there is no need to fit the ab initio potential to any finite order polynomial or to any other analytic form in order to calculate the complex-scaled potential energy matrix elements. Lipkin et al. [98] suggested to carry out similarity transformation of the hermitian Hamiltonian matrix where the transformation matrix is an overlap between the complex scaled and the unsealed basis functions. More recently, Ryaboy and Moiseyev [94] used another approach. The y calculated the HCO and DCO predissociation resonances by carry out numerical integration along the real axis when the basis functions were scaled by exp(!ih) whereas the ab initio potential remained unscaled. All the methods described above, use basis functions in the reaction coordinate which is scaled by a complex factor. Starting from the theoretical work of Moiseyev and Hirschfelder [13] Mandelshtam and Moiseyev [99] developed a method which avoid the need of using the basis set approach along the reaction coordinate. Namely the complex-scaled potential can be generated on the same grid as »(x exp(ih))"+ S »(x ) , (3.1) n n, k k k where the matrix S represents the complex-scaling operator (conventional scaling or smooth exterior scaling) which is given by SK "e~*h@2ehWK ,

(3.2)

where ¼ K is given by ¼ K e**f1@2(x)(»@»x)f1@2(x)+ ,

(3.3)

and f (x) is a smooth function of x. When f (x)"x the x coordinate is rotated by h into the complex plane as in the conventional complex scaling [so-called the CCM (complex coordinate method)]. To illustrate how the complex-scaling operator SK can be defined on a grid we shall consider here the case of f (x)"x. In such a case ¼ K is a hermitian unbounded operator, ¼ K "(xPK #PK x)/2 , (3.4) x x where P "i ­/­x is identical with the momentum operator. x The difficulty to carry out such a transformation can be illustrated by analyzing the Fourier transform of »(x ): n »(x)"+ C exp(2pnxi/¸) , (3.5) n n where ¸ is the size of the grid. From the first glance the complex-scaled potential can simply be given by `= »(xe*h)" + C exp(2pnx exp(ih)i/¸) , n n/~=

(3.6)

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269

However, this expansion will not be stable because of the exponentially diverging terms when nxP#R. The choice of basis functions is crucial when a finite basis set approach is taken to represent the complex scaling operator SK "exp(h¼ K ). It is easy to see it when SK is introduced by using the spectral representation of ¼ K . That is, N S "e~*h@2 + ehjiC(n)*C(k) , (3.7) n, k i i */1 where C and j are, respectively, the eigenvectors and eigenvalues of the matrix W which represent i i ¼ K . For example, if one uses sin(pnx/¸) basis functions (i.e. particle-in-a-box basis functions) infinitely many terms should be taken to correctly describe eigenfunctions of ¼ K since the frequency of their oscillations is continuously changing from infinity (at small x) to zero (at large x). An alternative seemingly more stable expansion could be obtained if local basis functions are used. Such local basis functions are the sinc-DVR basis functions (see Refs. [8—10,100—102]) defined as / (x)"(1/JD) sinc[p(x/D!n)] . n In this basis the off-diagonal matrix elements of ¼ K can be approximated as i(n#m) , ¼ "S/ D¼ K D/ T+(!1)(n~m) n, m n m (n!m)

(3.8)

(3.9)

and the diagonal matrix elements are ¼ "0 . n, n Even though the sinc-DVR basis functions are an approximation for the Dirac d-functions, d(x!nD)+/ (x) , (3.10) n they are not quite localized in space leading to a very slow decay of the off-diagonal matrix elements of ¼ K . On the other hand, the operator ¼ K is local and therefore its grid representation could be made local by damping the elements ¼ with large Dn!mD, n, m i(n#m) e~p(n~m)2 . (3.11) ¼ +(!1)(n~m) n, m (n!m) Such an ansatz which has been checked numerically [99] can also be understood by analyzing the relation:

P

P

K Dx@Tf (x@)+ dx@SxD¼ K Dx@Te~(p@D)(x~x{)2f (x@) . [¼ K f ](x)" dx@SxD¼

(3.12)

For a sufficiently small p this is a good approximation due to the local nature of the operator ¼ K . The Mandelshtam—Moiseyev method [99] has been used by Narevicius and Moiseyev [103] in the calculations of thousands of broad and overlapping ArHCl resonances for 3D potential energy

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surface which has been calculated on a finite grid. In a very similar way, one can use the smooth-exterior scaling approach. It should be stressed that it is sufficient to scale the reaction coordinate and there is no need to scale by a complex factor all degrees of freedom. This method enables the study of the dynamics and resonance phenomena in polyatomic molecular systems for which the potential energy surface is available only on a grid.

4. The complex coordinate scattering theory: From complex-scaled Hamiltonians to partial-widths and cross sections 4.1. General discussion The complex coordinate time-independent scattering theory [104—112] enables one to calculate cross sections, partial widths and state-to-state transition probabilities even for time-dependent Hamiltonians. There are three key points in the derivation of the complex-coordinate scattering theory: (1) The state-to-state, / P/ , transition probability amplitude matrix elements, i f ¹"S/ D»(1#GK »)D/ T, are calculated when the integration is carried out along a complex f i contour in the coordinate space. The complex spatial contour is obtained by scaling the cartesian coordinates by a complex factor exp(ih). Scaling the “reaction coordinate” only by a complex factor, where the other coordinates remain unscaled, has numerical advantages. The “reaction coordinate” may be considered as the distance between the center of mass of the scattered particle and the center of mass of the target. (2) The Green operator, GK , is defined as (E!H )~1 when H is the complex-scaled Hamiltonian h h of the system. Consequently, the energy contour of integration becomes complex when GK in the spectral representation is calculated. Since this is a crucial point in the derivation of the complex coordinate scattering method, we shall discuss it with some more details: The Green operator GK (E) provides the probability amplitude to get from x to x@ at a given energy E. GK in the spectral representation is given by

P

t (x@)t (x) E{ , G (x@, x)" dE@o(E@) E{ E E!E@

(4.1.1)

where t are the eigenfunctions of the full Hamiltonian and o(E@) is the density of states. As one E{ can see from Eq. (4.1.1) G has a branch cut along the real energy axis. Consequently, large E numerical errors are expected in a brute force application of Eq. (4.1.1), due to the vanishing of the energy denominator. This is a very serious technical problem, which can be avoided by rotating the energy contour of integration from the real axis into the complex plane. The energy contour of integration becomes complex when the relevant spatial contour is rotated into the complex plane, as discussed in Sections 1 and 2. The wave functions t and the energy E@ in Eq. (4.1.1) are E{ replaced by the eigenfunctions and eigenvalues of the complex-scaled Hamiltonian H . h The eigenfunctions of the complex-scaled Hamiltonian are associated with the discrete resonance spectrum, E3%4"e !(i/2)C , where t3%4; n"1, 2,2 are square integrable functions, and with n n n n

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271

the rotating continua, t jh where Ej "e exp(!2ih)#E5)3%4, e is varied from 0 to R and E h j E5)3%4; j"1, 2,2 are the (real) threshold energies. The complex-scaled Green operator in the j spectral representation is given by (see Refs. [2—6,105—109], Rescigno and Reinhardt [113], Johnson and Reinhardt [117], Rescigno and McCurdy [118], and Froelich and coworkers [119]),

P

t3%4(x@)t3%4(x) t (j) (x@)t (j)(x) n Eh . G (E)" + n # + dEj o(E(j)) E h h h h (j) E!E3%4 E!E n n(3%4) j(5)3%4) h

(4.1.2)

Here we assume that the wave functions are complex only resulting from the complex scaling. When the eigenfunctions of the complex-scaled Hamiltonian are variationally obtained with complex basis functions one should use the left and the right eigenstates of H as defined in h Section 2. o(E(j)) in Eq. (4.1.2) is the density of states. Due to the complex scaling the resonances are h isolated and separated from the other states in the continuum. Consequently, o(E(j)) does NOT h depend on the potential interaction and depends ONLY on the dimension of the studied problem. For a d-dimensional continuum o is proportional to e(d@2~1) where e,DEj !E5)3%4D. When box h j normalization is used the integrals in Eq. (4.1.1) and in Eq. (4.1.2) are replaced by summations over the discrete quasi-continuum and, respectively, o(E@) and o(E(j)) are replaced by 1. h (3) The transition probability amplitude, ¹(E), gets complex values and is in principle hindependent. In the numerical calculations when finite grid or finite basis set methods are used ¹(E) is h-dependent. The optimal values of ¹ are associated with the stationary solutions for which d¹/dh"0. Since ¹ is complex h should be allowed to get complex values and to be optimized to satisfy the two Cauchy—Riemann conditions:

K

K

K

K

­Re ¹(h) ­Re h

"0 and h015

K

­Im ¹(h) ­Re h

K

K

­Im ¹(h) ­Im h

h015

or ­Re ¹(h) ­Im (h)

"0 and h015

K

"0 .

(4.1.3)

h015

The fact that the optimal rotational angle h may get complex values can be introduced as the 015 result of using a complex scaling factor g"e*h"ae*h{ ,

(4.1.4)

where a and h@ are real and a is not necessarily equal to one. The optimal values of a and h@ are those for which ­¹/­a"0 and ­¹/­h@"0. Peskin and Moiseyev [111] have shown that the optimization of the complex scaling factor n (i.e. h@"h , a"a ) enables the application of the complex 015 015 coordinate scattering theory to long range potentials, and avoids the serious singularities and convergence problems due to the exponential divergence of the complex scaled “in”-asymptote D/ T i or “out”-asymptote S/ D which were explored by Rescigno and Reinhardt [111], Baumel et al. f [114], Johnson and Reinhardt [115], by Rescigno and McCurdy [116], and by McCurdy et al. [120].

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4.2. Time-independent Hamiltonians The probability to get from the initial state / to the final one / is given by the S-matrix element, i f P "DS D2 . (4.2.1) f0i f, i In the case of a one-dimensional “reaction coordinate” the scattering matrix is given by S "1!¹ (4.2.2) f, i f, i if the scattering process is an elastic one and if the sign of the wave vector in the reaction coordinate is unchanged during the collision; i.e. k "k . In non-elastic scattering and in elastic scattering i f when k "!k (as in the case of atom (molecule)/surface scattering), i f S "¹ , (4.2.3) f, i f, i ¹ is the energy normalized complex-scaled ¹ matrix element f, i ¹ (E)"(1/+)|/ D»#»GK »D/ } , (4.2.4) f, i f E i where »K (x@, r)"» describes the interaction potential of the scattered particle and the target; MxN are the internal (or target) coordinates; r is the reaction coordinate and GK "(E!H)~1. The E Hamiltonian is given by HK "HK #»K 0 where

(4.2.5)

HK "!(+2/2k)D #hK (x) , (4.2.6) 0 r D stands for the Laplacian operator with respect to the coordinates of the scattered particle and r hK (x) is the Hamiltonian of the target. In this step of the representation we will carry out an analytical continuation of H, », / , / and H and GK by scaling the reaction coordinate by i f 0 a complex factor. That is, r"r@e*h

(4.2.9)

where r@ gets real values only and, » "»(r@e*h, x) , h H "HK (r@e*h, x) , (4.2.10) h GK "1/(E!HK ) . h h The energy normalized initial and final states are eigenfunctions of the complex-scaled H Hamil0 tonian

S

k D/ }" e~*k*%*hr{ ) s i(x) i m +2Dk De*h i

(4.2.11)

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and

S

k e*kf%*hr{ ) s* f(x) , (4.2.12) |/ D" f m +2Dk De~*h f where s are given by m s (x) , hK (x)s i(x)"E5)3%4 mi mi m (4.2.13) hK (x)s f(x)"E5)3%4 s (x) . m mf mf If the reaction coordinate is radial, !2i sin(kehr@) replace the exp(!ike*hr@) in Eqs. (4.2.11) and (4.2.12). The initial and final wave vectors are defined such that the total energy of the “particle—target” systems is conserved: (+k )2 (+k )2 E" i #E5)3%4 " f #E5)3%4 . i m mf 2k 2k

(4.2.14)

The right and left eigenfunctions of the complex scaled Hamiltonian can be obtained by the diagonalization of the Hamiltonian matrix H when Ms N are used as a basis function: h m WR, h"+ /R, h (r@)s (x) , n m, n m m

(4.2.15)

WL, h"+ /L, h (r@)s (x) n m, n m m when

H /R, h"Eh/R, h , h n n n (4.2.16) H5 /L, h"Eh/L, h , h n n n [H ] "Ss DHK (x, r@e*h)Ds T . h m{, m m{ m By using the complex eigenvalues and eigenfunctions given in Eqs. (4.2.15) and (4.2.16) to construct the complex-scaled Green operator (see Eq. (4.1.2)), the complex coordinate ¹-matrix element is obtained (See Eq. (4.2.4)) k ¹h (E)" f, i +2JDk DDk D i f

PP

dr@dMxNs* f(x)e~*kfr{»(r@, x)s i(x)e~*kir{ m m

CPP

# e*h+ (E!Eh)~1] n n

CPP

]

dr@dxs* f(x)e~*kf%*hr{»(e*hr@, x)tR, h(r@, x) m n

D

dr@dxs i(x)e~*k*%*hr{»(e*hr@, x)tL, h(r@, x) , m n

D (4.2.17)

where the summation + is over all resonances and rotated continua solutions (which are n associated with a discrete spectrum due to the use of finite number of basis functions). Since the integral in the first term is unaffected by the value of h (provided »(r@, x) is an analytical function) we choose here to carry this integral along the real r"r@-axis.

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Note by passing that Eq. (4.2.17) can be simplified if the exterior scaling procedure rather than the commonly used complex scaling is used. If r"r@ when r@4r ; r"(r@!r )e*h#r when r@'r 0 0 0 0 and »(r@'r , x)"0, then the initial and final states and the interaction potential » remain 0 unscaled. The only complex functions used in this case in Eq. (4.2.17) are the left and the right eigenfunctions of the exterior-complex-scaled Hamiltonian H , and the only complex numbers in h Eq. (4.2.17) are the corresponding complex eigenvalues of H . h Eq. (4.2.17) provides an expression for the ¹-matrix element that does not suffer from the numerical disadvantages of ¹(h"0): The continuous spectrum of HK is discretizised and the complex eigenvalues, Eh, avoid the vanishing of the energy denominators in the Green operator (see n Eq. (4.1.1)), such that the energy integral in Eq. (4.1.2) for summation in Eq. (4.2.17) converges. Another benefit is the inclusion of the resonance states (poles of the S-matrix) into the spectrum of HK , where one resonance eigenstate represents a large number of scattering eigenstates of the non-scaled Hamiltonian. The price we pay for these benefits is the need to confirm the stability of the calculated complex ¹-matrix in terms of the size of the basis set and the need to look for the optimal value of the scaling parameter g"exp(ih), for which

K

K

­¹h f, i ­Re(h)

"0 and

K

K

­¹h f, i ­Im(h)

"0 . (4.2.18) h015 h015 The complex coordinate method for time-independent Hamiltonians has been successfully used to calculate the specular and non-specular intensities of gas atom and molecules scattered from corrugated and smooth solid surfaces. See the remarkable agreement between the experimental results obtained for helium diffraction from Cu(115) which is shown in Fig. 18a and the theoretical (no fitting parameters) results presented in Fig. 18b which we obtained by using Eq. (4.2.30) for calculating the cross sections. The method is suitable also for the calculations of state-to-state transition probabilities in chemical reactions resulting from reactive scattering experiments. 4.2.1. The complex coordinate scattering theory and the Kohn variational principle On the basis of the c-variational principle as proved in Section 2, the eigenfunctions of the complex-scaled Hamiltonian can be expanded in terms of N square integrable basis functions, / , n N N Dth}" + Ch U , |thD" + Dh U (4.2.19) a n, a n a n, a n n/1 n/1 The linear variational parameters C and D are, respectively, the right and left eigenvectors of the complex-scaled Hamiltonian matrix H , h H C h"EhC h , h a a a (4.2.20) H 5 Dh"EhDh , h a a a where [Hh] "SU DHK (r@e*h)DU T . h n{, n n{ n Since the Hamiltonian is non-hermitian (due to the complex scaling) the c-product should be used as an inner product rather than the usual scalar product. Consequently, D5C"1 .

(4.2.21)

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Fig. 18. (a) The experimental non-specular intensity as measured by Perreau and Lapujoulade [140], in scattering of He from Cu(115) corrugated non-symmetrical surface. (b) The theoretical transition probabilities as a function of the incident angle c as calculated by Peskin and Moiseyev [109] from Eq. (4.2.30) in the text.

The spectral representation of the Green operator in terms of the variational eigenfunctions of the complex-scaled Hamiltonian matrix Dth} |thD a G "+ a E (E!Eh) a a

(4.2.22)

is valid if and only if the eigenfunctions of HK form a complete set. As discussed in Section 2, for any h given finite number of basis functions, N, one can always find well-defined complex values of h for which the spectrum is incomplete. In such cases, the number of the non-linear dependent (i.e. “orthogonal”) eigenvectors of H is smaller than N. Since under very small variation of h the h spectrum becomes complete this possibility can be ignored (see the relevant discussion on the generalized inner product in Section 2). By substituting Eqs. (4.2.19), (4.2.20), (4.2.21) and (4.2.22) into Eq. (4.2.4), one can get ¹"S/ D»K D/ T#e`2*h(m )5(CA~1D5)m , f i f i

(4.2.23)

where A is defined as A "(E!Eh)d a, a{ a a, a{

(4.2.24)

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and [m ] "|/h D»K (r@e*h)DU } , f n f n [m ] "|U DVK (r@e*h)D/h} . i n n i

(4.2.25)

Since A"D5[E1!H(h)]C

(4.2.26)

then, by using Eqs. (4.2.22), (4.2.23) and (4.2.26), we get ¹"S/ D»K D/ T#e`2*h(m )5[E1!H(h)]~1m . f i f i Let us denote the matrix elements of [E1!H(h)]~1

(4.2.27)

(4.2.28)

by M[E1!H(h)]~1N "[|U DE!HK (r@e*h)DU }]~1 . n, n{ n n{ Then, by substituting Eq. (4.2.29) into Eq. (4.2.28) we obtain

(4.2.29)

¹"S/ D»K D/ T f i #e2*h + |/h D»K (r@e*h)DU }[|U DE!HK (r@e*h)DU }]~1|U D»K (r@e*h)D/h} . (4.2.30) f n n n{ n{ i n, n{ This is exactly the result obtained from the Kohn ¹-matrix variational principle by Nuttall and Cohen [104]. Here we followed Peskin and Moiseyev’s [110] proof that the complex coordinate scattering theory gives the Kohn variational solution for the ¹ matrix when the “reaction” coordinate is analytically continued into the complex plane. Eq. (4.2.30) can be obtained from the following expression: ¹"S/ D»K D/ T#e2*h[|/h D»K (r@e*h)Ds} f i f # |s@D»K (r@e*h)D/h}!|s@DE!HK (r@e*h)Ds}] , (4.2.31) i where s and s@ are square integrable functions (note that the c-product is used rather than the “usual” inner product, as discussed in Section 2) in the space spanned by the MU ; n"1,2NN basis n functions and are written as N N s" + a U , s@" + b U . n n n n n/1 n/1 By substituting s and s@ in Eq. (4.2.31) and by the requirement of ­¹/­a "0, n one can get that

­¹/­b "0; n"1,2, N , n

N a " + [|U DE!HK DU }]~1|U D»K D/h} , n n h j j h i j/1 N b " + [|U DE!HK DU }]~1|/h D»K DU } . n n h j f h j j/1

(4.2.32)

(4.2.33)

(4.2.34)

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By substituting Eqs. (4.2.32) and (4.2.34) into Eq. (4.2.33) the complex transition probability amplitude given in Eq. (4.2.30) is obtained. Another variational basis set method which satisfies the Kohn variational principle was developed by Miller and coworkers [121,122]. 4.3. Resonance scattering: Partial widths Partial widths represent the probability per unit time of getting a specific reaction product in a well-defined quantum state in a full scattering or half-collision experiments. The resonancescattering-theory enables the calculations of partial widths from the “tail” (i.e. asymptote) of a single, time-independent, square integrable resonance wave function. This method has been used, so far, in the calculations of the rotational distribution of a diatom in a specular and non-specular scattering of the diatom from a solid surface; the rotational distribution of a diatom obtained in a photodissociation of a van der Waals complex; and of the probability for ionizing an atom or dissociate a diatom resulting from the absorption of n-photons in the presence of very strong electromagnetic fields. Within the framework of the resonance-scattering-theory, as developed by Moiseyev and Peskin [105], one assumes that the dynamics of the studied system is controlled by a single intermediate narrow resonance state, and C "D¹h (E)D2 , f0i f, i where the complex scaled ¹-matrix element is approximately given by ¹h (E)"|/h D» #» G (E)» D/h} f, i f h h h h i

where

|/h D» Dth }|th D» D/h} 3%4 h i , K f h R%4 E!E 3%4

(4.3.1)

(4.3.2)

H th "E th , h 3%4 3%4 3%4 (4.3.3) E "E !(i/2)C . 3%4 r This approximation holds provided that the contribution of the direct scattering event to the cross section is small relative to the contribution of the multipole-scattering events (i.e. the »G » terms), E provided that EKE and provided that the resonance is narrow and is located in the complex r energy plane much closer to the real energy axis than the rotating continuum, that is, 2(E !E5)3%4)tan(2h)
/h i

Q

tPR

th P /h 3%4 f

(4.3.5)

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or |th }"(1#G (E)» )D/h} , 3%4 h h i (4.3.6) |th D"|/h D(» G (E)#1) . 3%4 f h h Eqs. (4.3.5) and (4.3.6), however, cannot be satisfied since / and / are associated with “in” and i f “out” going plane wave with a real momentum (+k ) and real (+k ). From the asymptotic behavior i f of the resonance wave function given in Eq. (1.5.2) one can see that Eqs. (4.3.5) and (4.3.6) can be satisfied if, and only if, (+k ) and (+k ) will be complex as required in Eq. (1.4.9). Consequently, the i f assumption that the dynamics is controlled by a single complex-scaled resonance state lead us to postulate that in the complex-coordinate resonance scattering theory the complex energy is conserved and we introduce the complex momentum for the final state which is determined by [105] (+k(m))2/2k"E !(i/2)C!E5)3%4 . f r m From the same initial state several different final states can be obtained. That is,

G

D/ (k(m))T,D/ T , f f m D/ TP D/ (k(m{))T,D/ T , i f f m{ D/ (k(m{{))T,D/ T . f f m{{ From Eqs. (4.3.8), (4.3.1) and (4.3.2) one gets the branching ratio

K

(4.3.7)

(4.3.8)

K

C |/h D»(r@e*h)Dth } 2 m" m 3%4 . (4.3.9) C |/h D»(r@e*h)Dth } m{ m{ 3%4 Eq. (4.3.9) is the expression proposed by Noro and Taylor [123] for the calculation of branching ratios. We omit here the label “i” from C which is defined in Eq. (4.3.1) (“f ” is associated with f 0i the final complex momentum (+k(m))) since the derivation of Eq. (4.3.9) from Eq. (4.3.2) shows that f during the scattering process the initial state has been “forgotten”. This is probably true in a half-collision process when the system is initially prepared in a resonance state. In the radial case r@3[0, R], |/h(m)D"0 for r@"0 and yet

S

k |/h DP e~*k(m)f r{%*h as rPR . m +2k(m) f Therefore, in the radial case

(4.3.10)

S

k |/h D,/L (re*h)"!2i sin(k(m)r@e*h) (4.3.11) m m f +2k(m) f and k(m) is defined as in Eq. (4.3.7). Eq. (4.3.9) can be rewritten by carrying out integration by parts f as described by Moiseyev and Peskin in Ref. [105] to provide another formula for the branching ratio,

K

K

dUh (r@) d/L (r@e*h) 2 /L (r@e*h) m !Uh (r@) m m m C dr dr m " lim C dUh (r@) d/L (r@e*h) m{ r{?= /L (r@e*h) m{ !Uh (r@) m{ m{ m{ dr dr

(4.3.12)

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/L and /L are the divergent complex-scaled incoming plane waves (defined in Eq. (4.3.11)) with m m{ a complex momentum defined in Eq. (4.3.7). Uh and Uh are the right square integrable functions m m{ obtained from the variational solution of the complex-scaled Schro¨dinger equation and are associated with resonance complex eigenvalue E !(i/2)C. See Eq. (4.3.3). That is, r th "+ s (x)Uh (r@) , (4.3.13) 3%4 m m m where Ms N are the bound state eigenfunctions (associated with the eigenvalues E5)3%4) of the HK (x) m m 0 Hamiltonian given in Eq. (4.2.6). The Moiseyev—Peskin expression [105] for the partial widths given in Eq. (4.3.12) appears to be more compact (in dimensionality) than the first one [Noro—Taylor expression [123] given in Eq. (4.3.9)] and thus avoids the need to integrate over the target coordinates. By noticing that as r@PR |Uh (r)DPa /L (r@e*h) , m m m DUh (r)}Pa /R (r@e*h) , m m m where /L is defined in Eq. (4.3.10) and /R is defined as m m k e`*k(m)f r@e*h as r@PR , /R (re*h)P m +2k(m) f such that

S

(4.3.14)

(4.3.15)

k . /L ) /R " m m +2k(m) f Eq. (4.3.12) is reduced to Peskin—Moiseyev—Lefebvre formula for the branching ratios [124]: C /C "Da /a D2 , (4.3.16) m m{ m m{ where Da D2 provides the probability to decay into the open channel m. By taking into consideration m Eq. (4.3.14) and the fact that Uh are energy normalized functions, one can see that Da D2 is the m m probability flux of Uh (r@) as r@PR and provides the number of particles at the quantum state m m detected at r"R per unit time per unit area. a can be easily obtained by carrying out an m asymptotic analysis of the resonance wave function,

A

B

Uh (r@) m . (4.3.17) a "lim m /R (r@e*h) r?= m By substituting Eq. (4.3.16), Eq. (4.3.17) into Eq. (4.3.15) the branching ratio as proposed by Peskin et al. [124] is obtained

K

K

(k(m{))1@2Uh (r) m{ (m) *h 2 C m e*(k f~k f)r% m " lim f (4.3.18) (k(m))1@2Uh (r) C { m{ (m ) r?= f In this section three different formulas for the branching ratio are given in Eqs. (4.3.9), (4.3.12) and (4.3.18). As shown here, following Moiseyev and Peskin proof [105], the three formulas are related to one another in a simple manner. Consequently, in the limit of infinite-basis-set variational calculations the three formulas should provide the same results for the partial widths. The

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Fig. 19. (a) The experimental rotational distribution of ICL as measured by Lester and co-workers [141], in photodissociation of NeICl. (b) The theoretical rotational distribution of ICL as calculated from the tail of the complex-scaled square integrable NeICl resonance wave function by Lipkin et al. [19,139], in photodissociation of NeICl.

complex-coordinate-resonance-scattering approach has been successfully used in the calculation of partial widths in photodissociation of van der Waals molecules. In such experiments the partial widths provide the rotational distribution of the diatomic molecule obtained by exciting the vibrational mode of the diatom in the atomic van der Waals complex. In Fig. 19 we show a comparison between the experimental and the theoretical (obtained by calculating the partial widths from the asymptotic analysis of a single square integrable complex-scaled resonance wave function) rotational distribution of ICl which is obtained when NeICl complex is vibrationaly excited. 4.4. Time-dependent Hamiltonians by the (t, t@) method Before describing the (t, t@) method and the complex coordinate time-independent scattering theory for time-dependent Hamiltonians, we should mention the fact that quasi-energy resonance positions and widths have been calculated by complex-scaling long time before the development of the (t, t@) method. Reinhardt and coworkers [125], and Howland [126] studied the ac Stark Hamiltonians in the dipole approximation, k x cos(ut). Chu [127] has extended it to molecular 0 time-dependent Hamiltonians in the dipole approximation. Moiseyev and Korsch [128] extended

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281

Fig. 20. The complex eigenvalues of the complex scaled time evolution operator of H` in time periodic field (wavelength 2 of 329.7 nm and laser intensity of 2.5]1013 W cm~2) as calculated by Moiseyev et al. [142]. The almost degenerated spirals are the two rotating continua and their ‘rate’ of convergence to the unit circle is h-dependent. The 9 isolated circles are associated with the multiphoton quasi-energy resonances and are stable to small variation of the scaling angle h.

the work of Chu and Reinhardt to any periodic field. Since the real part of the quasi-energies (eigen values of the complex-scaled Floquet operator which will be defined later in Eq. (4.4.1)) are defined modulo of +u, it is more natural to represent them as the eigenvalues of the complex-scaled time evolution operator, j"exp(!iE t/+). See, for example, Fig. 20 where the complex eigenvalues of QE the time evolution operator are shown for the case where the bound ground electronic H` potential energy curve is coupled to the repulsive first excited electronic curve via cw laser. 2 The time-independent scattering theory has been developed for time periodic Hamiltonians by Peskin and Moiseyev [111]. As they have shown [112], it is possible to extend the timeindependent scattering theory to general time-dependent cases where the Hamiltonians are not necessarily time periodic. This scattering theory enables the derivation of time-independent expression for state-to-state transition probabilities. The time-independent scattering theory for time-dependent Hamiltonians enables one to propagate analytically a given initial state to t"R. For example, the above-threshold-ionization (ATI), the “breathing” above-threshold-dissociation of Cl~, and the high harmonic generation spectra of 2 Xe model Hamiltonian were accurately obtained by the time-independent scattering theory which will be described below [112,129]. The time-independent scattering theory for time-dependent Hamiltonians is based on the assumption that as time passes a free particle (e.g. an electron or a molecular species) is obtained [112]. This assumption provides two kinds of limitations on the time-dependent Hamiltonians: (1) The “bare” (i.e. field-free) atomic or molecular potential should be a non-coulombic potential. It can be a finite, short- or long-range potential. This is definitely not a serious restriction in the study of dissociation of molecules where the atom—atom potential interaction decays as R~3 when R is the distance between the two atoms. Moreover, due to the correlation effect of the inner shell electrons, the effective potential which describes the interaction of the ionizing electron and the ion

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is often taken as a short-range potential. In the study of multiphoton ionization phenomena in hydrogenic like atoms the time-independent scattering theory can be used only within the framework of the cut-off approximation where the potential is finite and »(x'x )"0. 0 (2) The interaction of the atom/molecule with the time-dependent field, »(x, t), should vanish as xPR. This condition is satisfied if the acceleration gauge is used rather than the length gauge (see the discussion in Section 1.7). 4.4.1. State-to-state transition probabilities and time evolution operator by the complex scaled (t, t@) method Let us define H(x, t@) as H(x, t@)"!i+ ­/­t@#H(x, t@) ,

(4.4.1)

where H(x, t@) is the given time-dependent Hamiltonian and !i+­/­t@ can be described as the momentum operator of the “photon” particle. Here t@ acts as an additional coordinate. Using box normalization condition t@e[0, ¹], the inner product of the functions f (x, t@) and g(x, t@) in the generalized Hilbert space is defined by

P P

1 T = dt@ dxf *(x, t@)g(x, t@) . | f Dg}" (4.4.2) ¹ 0 ~= For ionizing or dissociative systems complex scaling is used and f *(x, t@) is replaced by f (x@e`*h,!t@) and g(x, t@) by g(x@e`*h, t@) assuming g(x@, 0) and f (x@, 0) as real functions. The reason for replacing the complex conjugate by t@P!t@ is due to the fact that we wish to take the complex conjugate of the terms in the function which are complex NOT resulting from the use of complex scaling. Upon complex scaling x"x@e*h is substituted in the Hamiltonian when x@ is varied along the real axis. The solutions of the time-dependent Schro¨dinger equation H(x, t)W(x, t)"i+

­W(x, t) ­t

(4.4.3)

for the initial state t(x, t ) is given by 0 1 T W(x, t)"/(x, t@, t)D " dt@/(x, t@, t)d(t@!t) , t{/t ¹ 0 where /(x, t@, t) is obtained by solving the following equation

P

­/(x, t@, t) H K (x, t@)/(x, t@, t)"i+ ­t

(4.4.4)

(4.4.5)

or /(x, t@, t)"e~*HK (x, t{)(t~t0)@+/(x, t@, t ) . 0 For time periodic Hamiltonians we use the periodic boundary condition W(x, t)"W(x, t,#¹) , t@"t mod ¹ .

(4.4.6)

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It is an excellent approach when, for example, the time-dependent Hamiltonian describes the interaction of an atom or molecule with a pulse laser whose pulse envelope evolves oscillates over 50—100 optical cycles [129]. This periodic boundary condition can be used, however, even for non-periodic Hamiltonians. By substituting Eq. (4.4.6) into Eq. (4.4.4) and by noticing that at t"t , /(x, t@, t )"t(x, t ), the 0 0 0 time evolution wave function is given by t(x, t)"[e~*HK (x, t{)(t~t0)@+t(x, t )] 0 t{/t

P

1 T dt@d(t@!t)[e~*H(x, t{)(t~t0)@+t(x, t0)] . " ¹ 0

(4.4.7)

The solution of the time-dependent Schro¨dinger equation for a general time-dependent Hamiltonian is obtained from Eq. (4.4.7) by calculating the time evolution operator exp(!iH K t) where H K (x, t@) is a Floquet-type operator and 04t@4¹ acts like an additional coordinate. Since H K is t-independent, the need to calculate time-ordering operators (such as first-, second- or high-order terms in the Magnus series expansion) is avoided. From Eq. (4.4.7), one can see that the time evolution operator is given by

P

(4.4.8)

P

(4.4.9)

1 T dt@d(t@!t)e~*HK (x, t{)(t~t0)@+ UK (tDt )" 0 ¹ 0 and 1 T dt@d(t@!t ) e~*HK (x, t{)(t0~t)@+ . UK (t Dt)" 0 0 ¹ 0 Since

lim UK (tDt )W(x, t )"lim UK (tDt )t (x, t ) , (4.4.10) 0 0 0 0 f 0 t?= t?= where t(x, t ) is the solution of the time-dependent Schro¨dinger equation given in Eq. (4.4.3) and 0 the final energy normalized state is given by t (x, t)"UK (tDt )s (x, t ) f 0 0 f 0

S

"e~*HK 0(x, t{)(t~t0)@+

S

k k e*kfx"e~*Ef(t~t0)@+ e*kfx , +2k +2k f f

(4.4.11)

where H K is the free particle Hamiltonian which consists of the kinetic operator (pL )2/2k of the free 0 x ionized/dissociated particle and the operator pL of the momentum of the “photon”. t{ H K (x, t@)"!+2/2k ­2/­x2!i+ ­/­t@ , 0

(4.4.12)

E "(+k )2/(2k) . f f

(4.4.13)

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By substituting Eq. (4.4.11) into Eq. (4.4.10) the solution of the time-dependent Schro¨dinger equation which provides as tPR the final state t (x, t) is obtained: f W(x, 0)"lim UK (0Dt)UK (tD0)t (x, 0) 0 f t?= 1 T dt@d(t@)e*H(x, t{)t@+e~*HK 0(x, t{)t@+t (x, 0) "lim (4.4.14) f ¹ 0 t?= 1 T = ­ dt@d(t@) 1# dt e*H(x, t{)t@+e~*H0(x, t{)t@+ t (x, 0) , (4.4.15) " f ­t ¹ 0 0 t "t"0 is taken as the time the atomic/molecular system interacts with the time-dependent field. 0 Illustrative schematic representation is given by

P

A

P

A P

t"t 0 t (x, t) describes i the initial state of

BA t Pt 0

B

t"0 W(x, 0) describes the “particles” (e.g.

the system before the P electrons/atoms/molecules) interaction with the time-dependent field

interacting with the electromagnetic field (laser)

B

A

tPR

P

t"#R

B

t (x, t) describes . f free “particles”

When a sudden interaction of the system with the fields occurs we may assume that t "0. 0 Following a similar derivation made by Taylor for time-independent Hamiltonians, one can get that [112]

P

1 T W(x, 0)" dt@d(t@)(1#G f(x, t@)»(x, t@))t (x, 0) , E f ¹ 0 where

(4.4.16)

»(x, t@)"H K (x, t@)!H K (x, t@) , (4.4.17) 0 1 G f(x, t@)" . (4.4.18) E E !H K (x, t@) f The state-to-state [t (x, t )Pt (x, t) as tPR] transition probability, P(E ), is given by the i 0 f f overlap integral between the initial time-independent scattering state t (x, t "0) and the solution i 0 of the full time-dependent problem, t(x, 0), which provides as tPR the final state t (x, t): f P(E )"DSt (0)DW(0)TD2 . (4.4.19) f i 4.4.2. Above-threshold-ionization (A¹I) or above-threshold-dissociation (A¹D) spectra In multiphoton ionization/dissociation experiments the initial scattering state, t (x, 0), is the i bound state of the field-free atomic/molecular Hamiltonian, t (x, 0)"W (x) . i "

(4.4.20)

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By substituting Eq. (4.4.11) and Eqs. (4.4.16) and (4.4.17) into Eq. (4.4.19) one gets the timeindependent expression for the state-to-state transition probability P(E )"D|U (t"0)D1#G f(x, t@)»(x, t@)DU (t"0)}D2 , f f i E where

(4.4.21)

= |U (t"0)D"W (x)d(t@)" + e*wnt{t (x) , i " " n/~=

(4.4.22)

S

k DU (t"0)}" e*kfx f +2k f

(4.4.23)

w"2p/¹ .

(4.4.24)

and

Eq. (4.4.21) provides the probability to detect a free electron (in ATI experiment) or a free atom/molecules (in ATD experiment) with a kinetic every (+k )2/2k. As it was observed in f experiments and also in theoretical calculations (see Ref. [111] and references therein) the ATI spectra of atoms consists of a sequence of isolated peaks, separated by one photon energy. 4.4.3. Harmonic generation spectra by the complex-scaled (t, t@) method Kulander [131] associated the harmonic generation spectra with the Fourier transform of the time-dependent dipole moment,

P

p (X)" HG

=

dte~*XtSt (t)DxDt (t)T " "

(4.4.25) 0 where t (t) is the time-dependent solution obtained from Eq. (4.4.7) which stands for the initial " bound state of the field-free Hamiltonian, t(x, t )"W (x). By substituting Eq. (4.4.7) into 0 " Eq. (4.4.25) when the eigenfunctions of H(x, t@) are used as basis functions, the time-independent formula for the harmonic generation (HG) spectra has been derived:

K

K

= = p (X)" + + dX a{ a + + (W Ds (t"0)) ) (s (t"0)DW ) HG ,*(e ~e )@ `mw+ " a{ a " m/0 a, a{ n/~= ]D(u DxDu )D2 , (4.4.26) a{, n~m a, n where w is defined as in Eq. (4.4.24); Ms (x, t@)N and Me N are, respectively, the eigenfunctions and a a eigenvalues of the complex-scaled Floquet-type operator H(x, t@); Mu N are the Fourier compoa, n nents of s (x, t@); and (2D2) stands for the c-product. a and a@ get discrete values when box a normalization is used for x. The high harmonics for which X"mw; m"1, 3, 5,2 (for the conditions at which only odd harmonics are obtained see Ref. [127]) Eq. (4.4.26) is reduced to

K

K

= 2 = p (X"mw)" + + dX + (t Ds (t"0))2(u DxDu ) . HG , mw " a a, n~m a, n m/0 a n/~=

(4.4.27)

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Here we assume that the t and s are complex functions only due to the use of complex scaling, " a x"x@exp(ih), where x@ is a real variable. Usually, t is a real function and therefore " (t Ds (t"0))"St Ds T. " a " a 4.4.4. The resonance approach As in the resonance scattering theory described in Section 3 we assume here that the dynamics of the field-atom/molecule system is controlled by a single intermediate narrow resonance state. The use of complex scaling enables one to associate the metastable state of the time-periodic Hamiltonian (which describes the semiclassical interaction between radiation and matter) with a single square-integrable eigenfunction, W (x, t)"e~*E3%4ts (x, t), x"x@e*h (4.4.28) 3%4 3%4 where s and E are correspondingly the eigenfunction and the h-independent complex eigen3%4 3%4 value of the complex scaled Floquet operator H K (x, t). Note by passing that usually we used different notation for the complex-scaled eigenfunction. That is, th (x@, t) instead of t (x, t) or 3%4 3%4 sh (x@, t) instead of s (x, t). Since s (x, t) is time periodic s can be written as 3%4 3%4 3%4 3%4 = s (x, t)" + e*wntu3%4(x) (4.4.29) 3%4 n n/~= when for the open channels, n(0, (here we assume that the resonance is located at the n"0 Brillouin zone and !+w(Re(E )(0 while the threshold of the field-free potential is taken as 3%4 zero reference energy).

S

k u3%4(x)Pa e*knx as xPR n n +2k n

(4.4.30)

(+k )2 n "E #+wn 3%4 2k

(4.4.31)

and

4.4.4.1. Harmonic generation spectra within the framework of the resonance approximation. Within the framework of the resonance approximation Eq. (4.4.27) can be simplified by assuming that the dynamics is controlled by a single intermediate narrow resonance state. When a is associated with 0 the longest-lived resonance state, then Eq. (4.4.27) is reduced to Ben-Tal et al. [130] expression for HG, spectra

K

K

= 2 p3%4 (X"mu)" + (u 0, n~mDxDu 0, n) , HG a a n/~= where (2D2) stands for the c-product. If real basis set are used then

(4.4.32)

(u 0, n~mDxDu 0, n)"Su*0, nDxDu 0, nT , (4.4.33) a a a a where S2D2T stands as usual for the scalar product. Note that Eq. (4.4.27) is reduced to Eq. (4.4.32) when Tt Ds 0(t"0)TK1. " a

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287

Fig. 21. The experimental harmonic generation spectra of helium in 5 eV KrF laser (full circles) as measured by Sarukura and co-workers [143], and the theoretical ab initio spectra as calculated by Moiseyev and Weinhold (Ref. [135]) from a single complex-scaled resonance Floquet state.

By analyzing the symmetry properties of p3%4 (X) Ben-Tal et al. [133] provided a non perturbative HG proof for field-free Hamiltonians which support discrete and continuum spectra, which shows that under specific well-defined symmetric conditions only odd harmonics are obtained. Moiseyev et al. [134] have shown that for mononuclear diatomic molecule only odd harmonics are obtained regardless of the symmetry properties of the electronic potential surfaces. The fact that a single complex-scaled Floquet state describes the atom/molecule even in high intensity fields has been used by Moiseyev and Weinhold [135] in their calculations of the high harmonic generation spectra of He in KrF laser. The remarkable agreement of the theoretical HG spectra with the experimental results even for the very high harmonics first has been shown that high harmonics can be obtained by neutral atoms. See the experimental vs. the theoretical harmonic generation spectra of helium in Fig. 21. This is due to the dynamical electronic correlation which “push” apart the two electrons to the opposite sides of the nucleus, thus “decreasing” the nuclear charge and thereby avoid ionization. 4.4.4.2. ¹he A¹I/A¹D spectra within the framework of the resonance approximation. As Moiseyev et al. [134,132] pointed out Da D2 which are given in Eq. (4.4.30) are the partial widths, C , which n n provide the probability for ionization/dissociation resulting from absorbing n number of photons and, therefore, +2Dk D n lim Du3%4(x@e*h)e~*knx{%*hD2 . C" n n k x?=

(4.4.34)

Within the framework of the resonance scattering theory two assumptions can be made to simplify the expressions of the state-to-state transition probability (e.g. the expressions of the abovethreshold-ionization (ATI), the above-threshold-dissociation (ATD) and the high harmonic generation spectra):

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1. The Green operator in the spectral representation is given by DW (t)}|W (t)D Ds (t)}|s (t)D 3%4 " 3%4 3%4 . GK fK 3%4 E E !E E !E f 3%4 f 3%4

(4.4.35)

2. The projection of the resonance state on the bound initial state is almost time-independent, |W Ds (t)}"|W Du3%4}K1 . (4.4.36) " 3%4 " 0 By substituting Eqs. (4.4.34), (4.4.35) and (4.4.36) into Eq. (4.4.21) Bensch, Korsch and Moiseyev received a simple expression for the isolation peaks in the ATI/ATD spectra [137]: C n P(E )K + (4.4.37) f DE !(E #+wn)D2 3%4 n:0 f which explains the experimental evidence that the isolated peaks in the ATI/ATD spectra (e.g. of Xe as shown in Ref. [136] are separated by +w; all of them have the same width (i.e. C"2Im(E )); 3%4 and their relative heights are the branching ratio C /C , as obtained from Eq. (4.4.34) by carryn n{ ing out calculations which combine Floquet theory and complex scaling. As an example we represent in Fig. 22 the ATI spectra for a driven Rosen—Morse model Hamiltonian as obtained by

Fig. 22. The ATI spectra for a driven Rosen—Morse model Hamiltonian as obtained by Bench et al. [137] by using the complex scaling approach for calculating the quasi-energy resonance position, width and partial widths. Fig. 23. The “breathing ATI spectra” of Cl~ in ArF excimer-laser as calculated by Peskin and Moiseyev [112] using the complex coordinate time-independent scattering theory for time-dependent Hamiltonians.

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Bench et al. [137] by using the complex scaling approach for calculating the quasi-energy resonance position, width and partial widths. In Fig. 23 The “breathing ATI spectra” of Cl~ in ArF excimer-laser field as calculated by Peskin and Moiseyev [112] using the complex coordinate time-independent scattering theory for timedependent Hamiltonians they developed. The “breathing ATI spectra” describes a phenomenon where the atoms are stabilized (i.e. the quasi-energy resonance lifetime is increased) at high intensity fields.

Acknowledgements My work on complex scaling started when I was still a graduate student and extended the virial theorem to resonance poles of the scattering matrix (complicated proof which is different from the simple proof given later with Weinhold and Certain from Wisconsin). Thanks to Prof. Gabi Kventzel with whom we shared an office at that time; he went through my derivation and encouraged me by citing Landau — the result looks simple enough to be true. Special thanks are due to Prof. Frank Weinhold and Prof. Phillip Certain with whom we get together into the subject of complex scaling. In particular, it is a pleasure to thank Phil Certain for twenty fruitful and most enjoyable years of closed collaboration. His wisdom, patience and friendship were always a source of support and encouragement. It is also a great pleasure to thank the graduate students who participate in the development of the method and its application to many different subjects. Thanks to Dr. Nurit-Nuphar Lipkin, Dr. Nir Ben-Tal, Dr. Uri Peskin, Dr. Naomi Rom, Mr. Ofir Alon, Mr. Vitali Averbukh, Mr. Ilya Vorobechick and Mr. Edi Narievichus for the fun we had doing science together. My colleagues and friends Prof. Roland Lefebvre and Dr. Victor Ryaboy should be acknowledged for many years of collaboration with applications of the complex-scaling method to many different problems in Physics and Chemistry. The US—Israel Binational Science Foundation, The Israeli Academy of Sciences and the Foundation of Promotion Research at the Technion should be acknowledged for many years of support. Last but not the least, I wish to thank Mrs. Charlotte Diament who not only shaped and typed this report and most of my papers, proposals and correspondence during the last decade but also took care of most of the administration work of the scientific meetings we organized.

References [1] J.R. Taylor, Scattering Theory, Wiley, New York, 1972. [2] G.A. Gamow, Zs. f. Phys. 51, 204; 52 (1928) 510; R.W. Gurney, E.U. Condon, Phys. Rev. 33 (1929) 127. [3] See for example A. Schutte, D. Bassi, F. Tommasini, G. Scoles, W.C. Stawlley, A. Niehaus, D.R. Herschbach, J. Chem. Phys. 62 (1975) 600; J. Chem. Phys. 63 (1975) 3081. [4] I. Eliezer, H.S. Taylor, J.K. Williams, J. Chem. Phys. 47 (1967) 2165. For a study by complex scaling see: N. Moiseyev, C.T. Corcoran, Phys. Rev. A 20 (1979) 814.

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[5] O. Alon, M.Sc. Thesis, Technion — Israel Institute of Technology. [6] For experimental and theoretical study of scattering of MD molecule from Ag(111) and Pt(111) surfaces see: J.P. Cowin, C.F. Yu, S.J. Siebener, J.E. Hurst, J. Chem. Phys. 75 (1981) 1033; C.F. Yu, C.S. Hogg, J.P. Cowin, K.B. Whaley, J.C. Light, S.J. Sibener, Isr. J. Chem. 22 (1982) 305; C.F. Yu, K.B. Whaley, C.S. Hogg, S.J. Sibener, P.R.L. 51 (1983) 2210; K.B. Whaley, J.C. Light, J. Chem. Phys. 81 (1984) 2144. C.F. Yu, K.B. Whaley, C.S. Hogg, S.J. Sibener, J. Chem. Phys. 83 (1985) 4217; K.B. Whaley, C.F. Yu, C.S. Hogg, J.C. Light, S.J. Sibener, ibid. 83 (1985) 4235; R. Schinke, Surf. Sci. 127 (1983) 283. For a study by Complex Scaling see: N. Moiseyev, T. Maniv, R. Elber, R.B. Gerber, Mol. Phys. 55 (1985) 1369; U. Peskin, N. Moiseyev, J. Chem. Phys. 96 (1992) 2347. [7] U. Peskin, D.Sc. Thesis, Technion — Israel Institute of Technology (1993). [8] O. Alon, N. Moiseyev, Phys. Rev. A. 46 (1992) 3807. [9] H.J. Korsch, R. Mo¨hlenkamp, H.-D. Meyer, J. Phys. B 17 (1984) 2955 and references therein. [10] A.I. Baz’, Ya.B. Zel’dovich, A.M. Perelomov, Scattering, Reactions and Decay in Nonrelativistic Quantum Mechanics; Israel Program for Scientific translation, Jerusalem, 1969. [11] V.I. Kukulin, V.M. KrasrOpol’sky, J. Hora´cek, Theory of Resonances: Principles and Applications, Kluwer Academic Publishers, Prague, 1989. [12] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965. [13] N. Moiseyev, J.O. Hirschfelder, J. Chem. Phys. 88 (1988) 1063. [14] E. Balslev, J.M. Combes, Commun. Math. Phys. 22 (1971) 280. [15] B. Simon, Commun. Math. Phys. 27 (1992) 1; Ann. Math. 97 (1973) 247. [16] For the proof that upon complex scaling DSu D»D/3%4TD2JC /2 see N. Moiseyev, U. Peskin, Phys. Rev. A 42 (1990) f n n 225. [17] D.H. Levy, Adv. Chem. Phys. 47 (1981) 363; J.A. Beswick, J. Jortner, Adv. Chem. Phys. 47 (1981) 363; K.C. Janda, Adv. Chem. Phys. 60 (1985) 201. [18] J.M. Skene, J.C. Drobits, M.I. Lester, J. Chem. Phys. 85 (1986) 2329. [19] O. Roncero, J.A. Beswick, N. Halberstadt, P. Villarreal, G. Delgado-Barrico, J. Chem. Phys. 92 (1990) 3348; for a study by complex scaling see: N. Lipkin, N. Moiseyev, C. Leforestier, J. Chem. Phys. 98 (1993) 1888. [20] A. Tempkin, Autoionization, Mono Book Corp., 1966. [21] P. Auger, Compt. Rend. 180 (1925) 1935; J. Phys. Rad. 6 (1925) 205. [22] N. Moiseyev, Isr. J. Chem. 31 (1991) 311. [23] W.P. Reinhardt, Annu. Rev. Phys. Chem. 33 (1982) 223. [24] B.R. Junker, Adv. At. Mol. Phys. 18 (1982) 107. [25] Y.K. Ho, Phys. Rep. C 99 (1983) 1. [26] Ref. [23]; S.-I. Chu, Adv. At. Mol. Phys. 21 (1985) 197; S.-I. Chu, Adv. Chem. Phys. 73 (1989) 739. [27] W.C. Henneberger, Phys. Rev. Lett. 21 (1958) 838; H.A. Kramers, Collected Scientific Papers, North-Holland, Amsterdam, 1956, p. 272. [28] U. Peskin, N. Moiseyev, J. Chem. Phys. 99 (1993) 4590. [29] See J.N.L. Connor, J. Chem. Phys. 78 (1983) 6161 and references therein. [30] B. Gyarmati, R.T. Vertse, Nucl. Phys. A 160 (1971) 523. [31] C.A. Nicolaides, D.R. Beck, Phys. Lett. A 65 (1978) 11. [32] B. Simon, Phys. Lett. A 71 (1979) 211. [33] N. Rom, N. Moiseyev, J. Chem. Phys. 99 (1993) 7703. [34] N. Rom, E. Engdahl, N. Moiseyev, J. Chem. Phys. 93 (1990) 3413. [35] N. Moiseyev, J. Phys. B (1997). [36] R.R. Johnson, Multiple Scattering Corrections to the Nuclear Optical Potentials, 1961. [37] See for example, G. Jolicard, E.J. Austin, Chem. Phys. Lett. 121 (1985) 106; D. Neuhauser, M. Baer, J. Chem. Phys. 90 (1989) 4351; ibid. 91 (1989) 4651; G. Jolicard, G. Billing, J. Chem. Phys. 97 (1992) 997; J. Perie, G. Jolicard, J.P. Killingbeck, J. Chem. Phys. 98 (1993) 6344. [38] U.V. Riss, H.-D. Meyer, J. Phys. B. 26 (1993) 4503. [39] See for example: C. Leforestier, R.E. Wyatt, J. Chem. Phys. 78 (1983) 2334; D. Neuhauser, R.S. Judson, D.J. Kouri, D.A. Kliner, N. Schefer, O. Adelman, R.N. Zare, Science 257 (1992) 522. [40] U. Manthe, T. Seideman, W.H. Miller, J. Chem. Phys. 99 (1993) 10 078; D. Neuhauser, ibid. 100 (1994) 9272; D.H. Zhang, J.Z.H. Zhang, ibid. 101 (1994) 1146.

N. Moiseyev / Physics Reports 302 (1998) 211—293

291

[41] R. Mittra, H.M. Ramahi, A. Khebir, A. Kouki, IEEE Transact. Magnetics, 25 (1989) 3034; J.A. Kong, M.A. Morgan (Eds.), in Differential Methods in Electromagnetic Scattering, Elsevier, New York, 1989, ch. 4, pp. 133—173. [42] D. Neuhauser, J. Chem. Phys. 103 (1995) 8513. [43] N. Rom, N. Lipkin, N. Moiseyev, Chem. Phys. Lett. 151 (1991) 199. [44] U.V. Riss, H.-D. Meyer, J. Phys. B 28 (1995) 1475. [45] A. Jaeckle, H.-D. Meyer, J. Chem. Phys. 105 (1996) 6778. [46] N. Lipkin, E. Brandas, N. Moiseyev, Phys. Rev. A 40 (1989) 549. [47] O. Atabek, R. Lefebvre, M. Garcia Sucre, J. Gomez-Llorente, H. Taylor, Int. J. Quantum Chem. 15 (1991) 211. [48] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon, Oxford 1965. Problem 3 in Section 36. [49] G. Doolen, Int. J. Quantum Chem. 14 (1978) 523. [50] W.H. Miller, Fraday Discuss. Chem. Soc. 62 (1977) 40. [51] T. Seideman, W.H. Miller, J. Chem. Phys. 95 (1991) 1768. [52] V. Ryaboy, N. Moiseyev, J. Chem. Phys. 98 (1993) 9618. [53] Ref. [126], Problem 4 in Section 23. [54] N. Rosen, P.N. Morse, Phys. Rev. 42 (1932) 210. [55] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1983. [56] G.A. Natanzon, Vestn. Lening. Univ. 10 (1971) 22; Teor. Mat. Fiz. 38 (1979) 146. [57] D.T. Colbert, R. Mayrhofer, P.R. Certain, Phys. Rev. A 33 (1986) 3560. [58] J.N. Ginocchio, Ann. Phys. 152 (1984) 203. [59] N. Lipkin, N. Moiseyev, Phys. Rev. A 36 (1987) 4076. [60] Y. Alhassid, F. Iachello, R.D. Levine, Phys. Rev. Lett. 54 (1985) 1746. [61] N. Moiseyev, J. Katriel, Chem. Phys. Lett. 105 (1984) 194. [62] O. Atabek, R. Lefebvre, Nuovo Cimento 76B (1983) 176; Chem. Phys. Lett. 84 (1981) 233. [63] N. Moiseyev, P.R. Certain, F. Weinhold 36 (1978) 1613. [64] E. Bra¨ndas, N. Elander (Eds), The Letropet Symposium View on a Generalized Inner Product, Lecture Notes in Physics, vol. 325, Springer, Berlin, 1989. [65] N. Ben-Tal, N. Moiseyev, R. Kosloff, C. Cerjan, J. Phys. B 26 (1993) 1445. [66] See for example: N. Moiseyev, J.H. Korsch, Isr. J. Chem. 30 (1990) 107. [67] K.C. Kulander, Phys. Rev. A 35 (1987) 445. [68] U. Peskin, N. Moiseyev, J. Chem. Phys. (1993). [69] J.S. Howland, Math. Ann. 207 (1974) 315; Indiana Univ. Math. J. 28 (1971) 471. [70] H.L. Cycon, R.G. Forese, W. Kirsch, B. Simon, Schro¨dinger Operators, Springer, Berlin, 1987, p. 144. [71] S.T. Motzkin, O. Taussky, Trans. Ann. Math. Soc. 73 (1952) 108. [72] N. Moiseyev, S. Friedland, J. Chem. Phys. 22 (1980) 618. [73] N. Moiseyev, P.R. Certain, Mol. Phys. 37 (1979) 1621. [74] N. Moiseyev, Mol. Phys. 47 (1982) 585. [75] E. Bra¨ndas, P. Froelich, Phys. Rev. A 16 (1977) 2207. [76] P. Winkler, R. Yaris, J. Phys. B 11 (1978) 1475. [77] N. Moiseyev, Phys. Rev. A 24 (1981) 2824. [78] H. Hellmann, Einfu¨hrung in die Quantumchemie (F. Deutsche), 1937; R.P. Feynman, Phys. Rev. 56 (1939) 340. [79] N. Moiseyev, C.T. Corcoran, Phys. Rev. A 20 (1979) 814. [80] G.D. Doulen, J. Nuttal, R.W. Stagat, Phys. Rev. A 10 (1974) 1612. [81] See for example Int. J. Quantum Chem. 14 (1978). [82] N. Moiseyev, S. Friedland, P.R. Certain, 74 (1981) 4739. [83] C.W. McCurdy (private communication). See also footnote 7 in Ref. [80]. [84] N. Moiseyev, P.R. Certain, F. Weinhold, Int. J. Quantum Chem. 14 (1978) 727. [85] U. Peskin, N. Moiseyev, J. Chem. Phys. 97 (1992) 6443. [86] N. Moiseyev, Chem. Phys. Lett. 99 (1983) 364. [87] N. Moiseyev, P. Froelich, E. Watkins, J. Chem. Phys. 80 (1985) 3623. [88] P. Froelich, N. Moiseyev, J. Chem. Phys. 81 (1988) 1352. [89] E. Engdahl, N. Moiseyev, J. Chem. Phys. 84 (1986) 1379.

292 [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125]

[126] [127] [128] [129] [130] [131] [132] [133] [134] [135]

N. Moiseyev / Physics Reports 302 (1998) 211—293 E.R. Davidson, E. Engdahl, N. Moiseyev, Phys. Rev. A 33 (1986) 2436. P. Froelich, E.R. Davidson, E. Bra¨ndas, Phys. Rev. A 28 (1983) 2641. N. Moiseyev, F. Weinhold, Int. J. Quantum Chem. 17 (1980) 1201. C. Lanczos, J. Res. Natl. Bur. Std. 45 (1950) 255. On the complex resonance eigenvalues by the Lanczos recursion method see K.F. Milfeld, N. Moiseyev, Chem. Phys. Lett. 130 (1986) 145. V. Ryaboy, N. Moiseyev, J. Chem. Phys. 103 (1995) 4061. N. Moiseyev, C.T. Corcoran, Phys. Rev. A 20 (1979) 814. N. Lipkin, N. Moiseyev, J. Katriel, Chem. Phys. Lett. 147 (1988) 603. D.S. Tannhauser, N. Moiseyev, Meteorol. Zeitschrift 4 (1995) 257. N. Lipkin, N. Moiseyev, C. Leforestier, J. Chem. Phys. 98 (1993) 1888. V. Mandelshtam, N. Moiseyev, J. Chem. Phys. 104 (1996) 6192. D.T. Colbert, W.H. Miller, J. Chem. Phys. 96 (1992) 1982. G.C. Groenenboom, D.T. Colbert, J. Chem. Phys. 99 (1993) 9681. C. Schwartz, J. Math. Phys. 26 (1985) 411. E. Narevicius, N. Moiseyev, Chem. Phys. Lett. 287 (1998) 250; Mol. Phys. (1998), in press. J. Nuttal, H.L. Cohen, Phys. Rev. 188 (1969) 1542. N. Moiseyev, U. Peskin, Phys. Rev. A 42 (1990) 255. E. Engdahl, N. Moiseyev, T. Maniv, J. Chem. Phys. 94 (1991) 1636. E. Engdahl, T. Maniv, N. Moiseyev, J. Chem. Phys. 94 (1991) 6330. U. Peskin, N. Moiseyev, J. Chem. Phys. 96 (1992) 2347. U. Peskin, N. Moiseyev, J. Chem. Phys. 97 (1992) 2804. U. Peskin, N. Moiseyev, J. Chem. Phys. 97 (1992) 6443. U. Peskin, N. Moiseyev, Phys. Rev. A 49 (1994) 3712. U. Peskin, N. Moiseyev, J. Chem. Phys. 99 (1993) 4590; see also Comments At. Mol. Phys. 31 (1995) 87. T.N. Rescigno, W.P. Reinhardt, Phys. Rev. A 8 (1973) 2828; 10 (1974) 158. R.T. Baumel, M.C. Crocker, J. Nuttall, Phys. Rev. A 12 (1975) 486. B.R. Johnson, W.P. Reinhardt, Phys. Rev. A 29 (1984) 2933. T.N. Rescigno, C.W. McCurdy, Phys. Rev. A 31 (1985) 624. B.R. Johnson, W.P. Reinhardt, Phys. Rev. A 28 (1983) 1930. T.N. Rescigno, C.W. McCurdy, Phys. Rev. A 37 (1985) 624. P. Froelich, A. Flores-Riveros, W. Wegric, J. Chem. Phys. 82 (1985) 2305. C.W. McCurdy, T.N. Rescigno, B.I. Schneider, Phys. Rev. A 36 (1987) 2061. W.H. Miller, J. Op de Haar, J. Chem. Phys. 86 (1988) 6233. J.Z.H. Zhang, S.-I. Chu, W.H. Miller, J. Chem. Phys. 88 (1988) 6233. T. Noro, H.S. Taylor, J. Phys. B 13 (1990) L377. U. Peskin, N. Moiseyev, R. Lefebvre, J. Chem. Phys. 92 (1990) 2902. W.P. Reinhardt, Int. J. Quantum Chem. S 10 (1976) 359; S.-I. Chu, W.P. Reinhardt, Phys. Rev. Lett. 39 (1977) 1195; C. Cerjan, W.P. Reinhardt, J. Avron, J. Phys. B 11 (1978) 2201; C. Cerjan, R. Hedges, C. Holt, W.P. Reinhardt, K. Scheibner, J. Wendoloski, Int. J. Quantum Chem. 14 (1978) 393; C.R. Holt, M.G. Raymer, W.P. Reinhardt, ibid. 27 (1983) 2971. J.S. Howland, J. Math. Phys. 24 (1985) 1240. S.-I. Chu, J. Chem. Phys. 75 (1981) 2215; T.-S. Ho, S.-I. Chu, O. Laughlin, ibid. 81 (1984) 788; S.-I. Chu, J. Cooper, Phys. Rev. A 32 (1985) 2769. N. Moiseyev, H.J. Korsch, Phys. Rev. A 41 (1990) 498. N. Moiseyev, M. Chrysos, O. Atabek, R. Lefebvre, J. Phys. B 28 (1995) 2007; N. Moiseyev, M. Chrysos, R. Lefebvre, J. Phys. B 28 (1995) 2599. N. Ben-Tal, N. Moiseyev, R. Kosloff, C. Cerjan, J. Phys. B 26 (1993) 1445. K.C. Kulander, Phys. Rev. A 35 (1987) 445. F. Bensch, H.J. Korsch, N. Moiseyev, J. Phys. B 24 (1991) 1321. N. Ben-Tal, N. Moiseyev, A. Beswick, J. Phys. B 26 (1993) 3017. N. Moiseyev, F. Bensch, H.J. Korsch, Phys. Rev. A 42 (1990) 4045. N. Moiseyev, F. Weinhold, Phys. Rev. Lett. 78 (1997) 2100.

N. Moiseyev / Physics Reports 302 (1998) 211—293

293

[136] P. Agostini, F. Fabre, G. Mainfray, G. Petite, N.K. Raham, Phys. Rev. Lett. 42 (1979) 1127; R.R. Freeman, P.H. Bucksbaum, H. Milchberg, S. Darak, P. Schumacher, M.E. Gensic, Phys. Rev. Lett. 59 (1987) 1092. [137] F. Bensch, H.J. Korsch, N. Moiseyev, Phys. Rev. A 43 (1991) 5145. [138] E. Engdahl, T. Maniv, N. Moiseyev, J. Chem. Phys. 88 (1988) 5864. [139] N. Lipkin, N. Moiseyev, C. Leforestier, J. Chem. Phys. 98 (1993) 1888. [140] J. Perreau, J. Lapujoulade, Surf. Sci. 122 (1982) 341. [141] J.M. Skene, J.C. Drobits, M.I. Lester, J. Chem. Phys. 85 (1986) 2329. [142] N. Moiseyev, M. Chrysos, R. Lefebvre, J. Phys. B 28 (1995) 2599. [143] N. Sarukura et al., Phys. Rev. A (1991) 1669. [144] N. Moiseyev, J. Phys. B 32 (1998), in press.