Dense plasmas, screened interactions, and atomic ionization

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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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.

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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]

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