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CHARGE CORRELATION EFFECTS IN THE BROADENING OF SPECTRAL LINES FROM HIGHLY CHARGED RADIATORS By JEFFREY MICHAEL WRIGHTON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Jeffrey Michael Wrighton I dedicate this work to my wife, Jennifer Lee Wrighton. ACKNOWLEDGMENTS I would first like to thank Dr. James Dufty, my advisor at the University of Florida, for the guidance he has provided. His patience and dedication during difficult times is truly appreciated, as well as the constant encouragement he has given me. I would also like to acknowledge my previous advisor, the late Dr. C'! i 1! Hooper, Jr. I did not know him nearly long enough, and yet I know I will ahivi remember our meetings, where physics talk was ahivi intermixed with lessons he had learned from life. I also thank Dr. Mark Gunderson and Dr. Donald Haynes, Jr. In recent years, distance has made communication difficult, yet they have ahv taken time to answer questions I have had. I thank the members of my committee, Dr. David Reitze, Dr. David Micha, and Dr. Fredrick Hamann, for the guidance and advice they have given me. iM !1 thanks go to the incredible staff at the Department of Physics. I especially would like to thank Ms. Darlene Latimer, who during my entire stay at the University of Florida has done so much for the students of this Department. I thank my parents, Arthur Wrighton III and Marjorie Wrighton, for instilling in me a love of learning as I grew up. Finally, I thank my wife, Jennifer Wrighton, for her patience and love during these past years. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iv LIST OF FIGURES ..................... .......... vii ABSTRACT ...................... ............ viii CHAPTER 1 INTRODUCTION .................... ....... 1 2 OVERVIEW OF PLASMA SPECTROSCOPY ...... ........ 5 2.1 Experimental Details of ICF ......... ........ ... 7 2.2 Electron and Ion Broadening .......... ....... ... 8 2.3 Evolution of Spectral Line Theory ......... ........ 12 3 LINE SHAPE FORMULATION .......... ............ 14 4 SEMICLASSICAL MOLECULAR DYNAMICS ....... ..... 23 5 LINE SHAPE FORMULA FOR QUASISTATIC IONS ......... 30 6 QUASISTATIC EQUATION OF MOTION ....... ....... 36 6.1 Firstorder Electron Broadening ......... ......... 37 6.2 SecondOrder Static Electron Broadening Term ...... 42 6.3 Dynamic Electron Broadening Term ....... ........ 44 7 CHARGE CORRELATION EFFECTS ON PLASMA PROPERTIES 46 7.1 Plasma Structure ................... ..... 46 7.1.1 Linear Theory ......... ............... 48 7.1.2 Nonlinear model ........... ............ 50 7.2 Screened Ion Field .......... ............... 53 7.3 Ion Microfield Distribution .......... ............ 54 7.4 Constrained Electron Density ............. ... .. .. 55 7.5 Electric Field Covariance .................. .. 56 7.6 Dynam ics .. .. .. ... .. .. .. ... ... .. .. .. .. 57 7.7 Sum m ary .. .. .. ... .. .. .. .. .. .. .. .. .. .. 62 8 INCORPORATING ION DYNAMICS ............. .. 63 9 SUMMARY AND OUTLOOK ................ .... 73 9.1 Static Ion Line Shape Function ........ .......... 75 9.2 Ion Dynamics ................... ..... 77 APPENDIX A HOOPER'S GROUP COMPUTER CODES ................. 80 A.1 Atomic Wavefunctions .................. ... .. 80 A.2 ElectricMicrofield Calculation ............ .. .. .. 80 A.3 ElectronBroadening Operator ....... .. 81 A.3.1 SecondOrder, Full Coulomb, Quantum Theory ...... 82 A.3.2 AllOrder, FullCoulomb, SemiClassical Theory ..... 84 B DOPPLER BROADENING IN THE QUASISTATIC APPROXIMATION 86 C PROJECTION OPERATOR EQUATION OF MOTION ... 89 D AVERAGE DIPOLE EQUATION OF MOTIONQUASISTATIC CASE 91 E CALCULATIONAL DETAILS FOR B(e) ...... ......... 96 E.1 First Order Static Shift B(1) Calculation. ...... .... ... 96 E.2 Constant Perturber Density Assumption .............. 101 F KINETIC EQUATION FOR COUPLED ION DYNAMICS CASE .... 104 F.1 Atomic Liouville Operator ........ .............. 107 F.2 Plasma Liouville Operator ......... ............. 108 F.2.1 Radiator Center of Mass Position .... 109 F.2.2 Radiator Center of Mass Momentum ... 109 F.2.3 Electric Field. .................. ... 110 F.3 Interaction Liouville Operator .... ... 111 F.3.1 Interaction Liouville Operator on D 112 F.3.2 Interaction Liouville Operator on .. 113 F.3.3 SummaryDeterministic Part ..... 114 F.3.4 Dynamics Term .................. ... 115 G THE KELBG AND DEUTSCH POTENTIALS .... 118 G.1 Derivation of the Quantum Potential ..... 118 G.2 Coulomb Potential and Kelbg/Deutsch Results ... 121 REFERENCES .................. ........ .. 122 BIOGRAPHICAL SKETCH ............. .. 126 LIST OF FIGURES Figure page 21 Plasma Parameters ............... .......... 13 71 Plasma ElectronRadiator Pair Correlation Function ... 51 72 Plasma ElectronRadiator Pair Correlation Function ... 52 73 Electric Field Covariance .................. ........ .. 58 74 Normalized Electric Field Autocorrelation ................ .. 60 75 Integral of Electric Field Autocorrelation Function 61 E1 Captured Bound C(! .ige .................. ........ .. 101 E2 Temperature Dependence of B() ................... .. 103 E3 Density Dependence of B(1) .................. ..... .. 103 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARGE CORRELATION EFFECTS IN THE BROADENING OF SPECTRAL LINES FROM HIGHLY CHARGED RADIATORS By Jeffrey Michael Wrighton May 2004 C(! r: James Dufty, C!ii 1! s F. Hooper, Jr. (deceased) Major Department: Physics The theory of spectral line broadening is reformulated to examine the validity of standard approximations for the expected increase in the range of experimental conditions. The new effects studied here are the correlations that exist among the plasma perturbers and with a charged radiator, which are usually neglected or approximated in current theories. The first analysis here assumes quasistatic ions, which is a relevant condition for many lines of experimental interest. The electron broadening operator, or width and shift operator, is calculated to second order in the interaction of the plasma perturbers with the bound electrons of the radiator. All other charge correlations are included without approximations. A semiclassical representation of these results provides the necessary physical quantities required for recently proposed methods of molecular dynamics simulation in plasma spectroscopy. An extension to include the effects of ion dynamics is described. CHAPTER 1 INTRODUCTION The theory of plasma line broadening has been used for many years to study hot, dense plasmas [1, 2]. With advances in experimental equipment, more rigorous theoretical developments are required in order to explore the broader range of state conditions being made accessible. Over time, this has led to the inclusion of effects previously neglected (e.g., ion dyr i' ~[:]) as they became important for specific conditions of higher density and temperature. Therefore, as experimental techniques continue to improve, opportunities for new exploration expand, and continual evaluation of current line broadening theories is needed. One area of the theory that needs evaluation now is the treatment of charge correlations for the spectroscopy of charged radiators. Present theories account for the correlation between the various plasma constituents and between the plasma constituents and the total charge of the radiator in inconsistent vi. Some correlations, such as ionion correlations, are treated quite rigorously through a detailed study of their interactions[4, 5], but an ad hoc screening length is used to account for electronion correlations. Similarly, electron correlations may be treated by assuming independent particles restricted to a Debye sphere[6, 7, 8]. We study the charge correlations for an ionized radiator in a two component fully ionized plasma of ions and electrons. To simplify this first analysis, we focus on those lines and conditions for which the heavy ions behave in a "quasistatic" manner. In this regime the plasma ions and the radiator both have negligible acceleration during the radiation time. The line shape function is formally written in terms of a single operator, the broadening operator, describing all effects of the radiator's environment. The radiator is represented as a point charge plus its residual bound electron distribution. Our main result for this case is an evaluation of the broadening operator that is exact to second order in the interactions of the plasma with the bound electron distribution. All correlations among the plasma constituents are included, as are the correlations induced by the point charge of the radiator. In addition to this general result following from quantum statistical mechanics, several other approaches will be examined. First, the inherent quantum effects of the electronion interactions will be represented by a semiclassical i i ,l! 1. 1 Coulomb interaction[9, 10]. This accounts for diffraction effects at separations of the order of the de Broglie wavelength, and these regularized potentials are well studied for similar plasma conditions in other contexts. A motivation for this representation is to make contact with recent studies of plasma charge correlations using classical molecular dynamics (\!1)) simulations [1]. An outcome of these studies has been the proposal to study plasma spectroscopy by MD as well[12]. Theoretical analyses of the type given here are necessary precursors for any such MD study, so as to identify the appropriate properties to simulate. This is discussed in some detail in C'! plter 4. Another limit for both quantum and semiclassical representations is obtained by approximating the full Coulomb interaction between perturbing electrons and the bound electron distribution by dominant multipoles[6]. It is proposed that monopole and dipole interactions are sufficient following the results of two closely related theoretical formulations for charged radiators developed by the Florida group over the past 10 y, i [7, 8, 13]. The quasistatic ion analysis properly addresses the correlations. However, under some conditions and for some lines, the time for ion motion becomes comparable to times of interest on the line profile. To address this more general case, C! iplter 8 shows how ion dynamics can be incorporated into the theory. Since the ions are generally strongly coupled, the ion dynamics must be described nonperturbatively to avoid inconsistent assumptions about the charge correlations described in the static limit. For practical purposes, following our formal analysis the ion dynamics is modeled using stochastic approximations that allow the inclusion of correlations[14]. The resulting descriptions can then be used, after some modification to allow for the inclusion of experimental effects, to determine characteristics of a broad class of the plasmas, radiators, and lines. This work was begun under the guidance of Professor ('! i i!. F. Hooper, Jr. who led the Florida plasma spectroscopy group for more than 30 years. During that time, a code for predicting spectral line shapes was developed, with continual improvement as the field evolved. It remains one of the only codes general enough to describe multicomponent, multitemperature plasma broadening of 1r ii',  electron atomic lines. In his memory and in an attempt to record the content of that code, we provide an overview of the computer codes developed for this problem at the University of Florida. These programs have been written to calculate line shapes, mainly for nearhydrogenic, highly charged radiators in hot, dense plasmas. Initial work done on the programs focused on accurate calculation of the ion microfield distribution, while more recent work has centered on accurate evaluations of the electron broadening operator. Details are given in Appendix A. The formal analysis given here is an extension of earlier work begun by Dufty and Iglesias for neutral radiators and static ions[15]. Its relevance to highly charged radiators is outlined in the recent thesis of Gunderson[8]. Important differences with the work here are noted in ('! i pter 9. The use of stochastic models for the radiative and transport properties of impurities in plasmas was introduced by Boercker, Iglesias, and Dufty[14]. We hope that a more rigorous treatment of the theory of spectral line broad ening will provide better experimental diagnostics and theoretical understanding. 4 Before developing the theoretical framework, C'! lpter 2 gives an overview of the importance of plasma spectroscopy in general, and gives some physical concepts that characterize the radiative process in a plasma environment. CHAPTER 2 OVERVIEW OF PLASMA SPECTROSCOPY A spectral line shape function I(w) gives the distribution of radiation emitted or absorbed by an atom in the presence of a photon field. For an isolated atom making a simple transition between two atomic states, this is a Lorentzian function with a halfwidth given by the natural broadening that arises from the interaction of the radiating atom or ion with its own radiation field[1]. This width can be given to an order of magnitude by Fnatural ~ a(aZ)2Z2 in Rydberg units, where a is the fine structure constant [16]. It is approximately a factor of a smaller than the fine structure splitting of atomic levels. For cases considered here, this is negligible in the presence of other broadening mechanisms[1]. One of these more important mechanisms is Doppler b ... 1,1~ ii which occurs if the radiator has thermal motion due to its environment[17]. If a Maxwellian distribution for the radiator velocity is assumed, the resulting Doppler profile is IDoppler () exp G 2 (2.1) D7Vp Fdoppler where the Doppler halfwidth is FDoppler { j 1C2 ( 0 (2.2) Here T is the kinetic temperature of the radiator, M is the radiator mass, and wo is the frequency of the unperturbed transition. In C'i lpter 5 we shall show how Doppler broadening is incorporated in a general line shape. The more interesting and informative broadening mechanisms are those due to the direct interaction of the radiator with the other particles of its environment. Through a careful analysis of how these mechanisms determine the shape of I(u), experimental determination of the line shape can provide detailed information about the environment. Plasma spectroscopy is thus a unique tool to serve as a diagnostic and as an exploratory probe of complex environments. Spectroscopic techniques have been used for many years to study the char acteristics of plasmas. The noninvasive character of spectroscopy makes it an ideal tool for the study of plasmas in stellar interiors and in inertial confinement fusion (ICF) experiments, which are the focus of our study. The radiators in these plasmas emit spectral radiation in a manner that is significantly affected by the details of the plasma environment. Each radiating ion has its atomic structure level disturbed by the electric fields of all other plasma constituents. For diagnostic purposes, e.g., in laser produced plasmas, a careful theoretical analysis of a complex line shape often allows reliable determination of key physical properties such as average electron density and the temperatures of the different plasma constituents that could not be obtained otherwise. The accuracy of the calculated line shapes is increased by including more detailed information about the exchange of energy between the radiator and the plasma. The shape of the spectral emission or absorption also depends on the detailed atomic structure of the radiator and the characteristic properties of the plasma constituents (such as proton vs. deuteron concentration). The widths, shifts, merging, and satellites are all features of the spectral line shape that we wish to accurately determine so as to best compare theoretical and experimental results[18]. At the National Laser User Facility (NLUF), part of the Laboratory for Laser Energetics (LLE) located at the University of Rochester in New York, inertial confinement fusions (ICF) experiments are currently being run[19]. Plasma spectroscopy (matching the radiation spectrum with theoretical line shapes) is an important tool to analyze the outcome of these experiments. These spectroscopic techniques will also be important at the National Ignition Facility (NIF) which is currently being made operational at Lawrence Livermore National Laboratory. It is expected that plasmas created at NIF will reach temperatures and densities approximately seven times greater than currently achieved values, which will be conditions beyond the temperature and density of the solar core. (The expected temperature is about 100 million degrees, compared with 15 million at the center of the sun; the expected density is about 300 g/cm3, roughly twice that of the solar core.) The theoretical approximations currently being used for spectroscopy must be r( i, &i1. 1 in the context of these extreme conditions. A primary motivation for our study is the new plasma state conditions and new types of highly charged radiators made possible by such experiments. To better appreciate the complexity of these experiments, and therefore the need for accurate spectroscopic analysis, a brief description is now provided. 2.1 Experimental Details of ICF In ICF experiments, small microspheres with a fill gas are imploded with lasers [13]. In current experiments, these microspheres typically have a diameter of about 1 mm, and consist of a plastic shell of about 20 microns thickness. A fill gas is introduced into the microsphere, which is then placed in the target chamber. Two main classes of experiments are performed: direct drive and indirect drive experiments. In a direct drive ICF experiment, the microsphere is suspended alone in the target chamber[20, 21]. Laser beams strike the sphere from many directions which cause the outer portion of the shell to explode away. The reaction force drives the inner portion of the shell inwards, compressing the fuel gas to form a plasma. In an indirect drive experiment, the microsphere is placed inside a small cylindrical tube (called a hohlraum) which is placed inside the target chamber[21]. The laser beams enter the hohlraum from each end and illuminate the inner walls. These inner walls then radiate, and it is this radiation that leads to the compression of the microsphere in an indirect drive experiment. Using a hohlraum leads to a more even radiation distribution striking the microsphere. This is important since the most common problem is instabilities causing a nonuniform implosion. However, more energy can be delivered to the microsphere in a direct drive experiment. After the fill gas forms a plasma, spectroscopic techniques can be used to determine density and temperature. As an example of typical results, densities of 1.9 x 1024 cm3 were reached at temperatures of 1.15 keV. For ICF experiments, other diagnostic tests can be used (such as neutron emission[22]) and these show agreement with the spectroscopic method to within experimental uncertainties. The choice of the fill gas determines the theoretical approach. We consider two main v v to fill the microsphere: a heavyion dopant and a heavyion fill gas[3, 23]. For example, the microsphere can be filled with a deuterium gas that is doped with a small amount of argon; alternately, the microsphere can be completely filled with argon. The actual choice of radiator (argon, krypton, etc.) depends on the expected experimental results. The plasma will then consist of radiating ions surrounded by free electrons and other ions. The temperature reached by the plasma will determine the relative percentages of different ionization levels of the li, i,. ion radiator. The probability of nonhydrogenic radiators can be accounted for and increases the accuracy of the diagnostics. The radiation is captured by streak cameras and is then analyzed. In the ain 1 ,i the effects of the experimental equipment are also taken into account in the line shape. A set of line shapes is used and a computer code finds the best fit. 2.2 Electron and Ion Broadening Historically, plasma spectroscopic techniques were most important in the study of stellar atmospheres. Stellar atmospheres have electron densities and ionizations that are relatively low; thus pressure broadening by neutral perturbers becomes important [24]. This is essentially collisional bi ...l. 1. 11 and differs considerably from broadening by the long range Coulomb forces of fully ionized matter. The most exciting new developments of the field are therefore in the direction of characterizing the opposite extreme of hot, dense, ionized plasmas. Furthermore, since these conditions support radiators of large charge number, the coupling to this environment can be considerably enhanced over that for neutral radiators. Consequently, the attention here is directed at a careful treatment of interactions of the radiator with the surrounding positive ions and negative electrons as the dominant source of line broadening. A dominant effect of the surrounding charged particles is Stark b .... 1. i.[1], a form of pressure broadening that can be described in terms of the Stark effect. It is also referred to as pressure broadening because it is sensitive to the density of (and thus the pressure from) the surrounding plasma particles. For Stark broadening, the atomic energy levels of the radiator are shifted by the electric fields of the surrounding plasma particles coupling to the radiator dipole. For hydrogenic ions, the bound electron wavefunctions can be solved for exactly using either spherical or parabolic coordinates, and a firstorder perturbation calculation gives the energy shift of a parabolic wavefunction in the presence of an electric field. The energy shift is found to be AE = 3aoenqc/2Z, where ao is the Bohr radius, e is the electron charge, Z is the nuclear charge of the radiator, and the electric field magnitude is e[25]. The numbers n and q are quantum numbers of the wavefunction, and the parabolic coordinate system is oriented along the perturbing electric field. The timescales come into p1l iv here, as the energy shift from the ions changes little during the radiation time, while the energy shift from the electrons' electric field does change. Hence, different theoretical approaches are used for the effects of electrons and ions. As discussed in the next section, the line shape function can be described in terms of a time autocorrelation function for the radiator's dipole moment[1]. A characteristic decay time can be associated with this function to give a radiation time[13]. The characteristic times for various processes can then be estimated and used to determine how they should be handled theoretically. One of the basic assumptions used in the first part of our study is that the perturbing ions are massive enough so that during the radiation time they do not move far enough to alter appreciably their force on the radiator. Consequently, they can be treated as static. In contrast, the electrons, because of their smaller mass, will move a great deal during the radiation time. This means that the effect they have on the atomic energy levels changes during the radiation time, so the electrons require a dynamic treatment. This section provides some orderofmagnitude calculations to check for the validity of these approximations. Several approaches can be taken to estimate the relative effect of perturber motion on the line shape [7]. The first approach is to calculate the relative change in the perturber electric field during the typical radiation time Tr A typical radiation time can be approximated by the experimentally determined line width. One example line shape relevant to our study gives a full width at half maximum (fwhm) of 40 eV. The quantum energytime relation gives a corresponding time of Tr 1017 s, which we use here to estimate the relative field changes. The electric field from a perturber is e = qr/r3 with r being the displacement from the radiator to the perturber. The fractional change in c due to a change in this displacement is then I6e/e = 26r/r a 2vaveTr/ro, where ave = /2kBT/m and the values of m (and possibly T) are different for each species of perturber. Also ro is a characteristic distance from the radiator. For the ions, this characteristic distance can be chosen to be the average ion distance from the radiator ro = ( T,. /3)1 Using typical plasma quantities of temperature of 1000 eV and electron density of 1 x 1024 /cc, the estimated value of 6e/ec for a hydrogen perturber is 6e/e w 0.07, and for argon ions as perturbers 6e/e = 0.01. Clearly, these massive ions can be treated quasistatically for the conditions considered. Using the same density given above for the electrons, this leads to an esti mated fractional change in the electric field of the perturbing electrons of 6e/e w 3. This estimate can be assumed to be too small, since the highly charged positive radiator will tend to pull electrons in closer. Because of the possibility of large field changes, the electrons must be treated with a dynamic theory. An alternate method for determining the need for dynamic theories is by considering the Debye length. An electron separated by more than a Debye length from the radiator has its effect on the radiator screened by that part of the plasma between the two[26]. Using this length and the average velocity of the electron, a characteristic time can be calculated that represents the time that an electron can travel and not be screened by the plasma environment. The Debye length is AD = VkBT/47ne2 and the average thermal velocity was given before. The characteristic time can then be calculated and related to the electron plasma frequency t = AD/Vave = m/8m/ne2 = 1/V2Wp. Therefore, a rough estimate leads to the requirement that the electrons be treated dynamically for energy widths less than the electron plasma frequency. Similar considerations apply to the other perturbing particles of the plasma. The time scales of the plasma under consideration also determine whether Doppler broadening can be calculated separately from Stark broadening. By assuming that it can, the assumption made is that the radiator velocity is constant during the radiation time. The momentum change of the radiator is equal to the impulse delivered to the radiator during some time t, and this impulse can be estimated using characteristic values that were used above. An estimate of the force during this time can be made using the average interparticle spacing of the ions. The resulting change in momentum can be used to determine the fractional change in momentum by using the average momentum determined by the thermal velocity. For the density and temperature given above, these fractional changes are 0.002 and 0.038 for hydrogen ions and hydrogenic argon perturbers. Therefore, neglecting this small acceleration is valid. Then Doppler broadening effects become statistically independent of the Stark broadening by the ions and electrons, as indicated in Chapter 5. 2.3 Evolution of Spectral Line Theory Modern spectral line theory includes many effects, some of which must be dealt with concurrently with the derivation of the line shape. Probably the most important of these effects is the form of the interaction between the plasma elec trons and the radiator. The earlier models of spectral line shapes dealt with neutral radiators[27]. With these plasmas, it was common for the interaction between the radiator and the plasma electrons to be modeled by a dipole interaction. A fundamental change occurred when the plasma experiments reached temperatures high enough to ionize the radiator. Ionized radiators attracted the plasma elec trons, and with the smaller separation, the dipole approximation was no longer adequate[6]. To describe these ionized plasmas, the full Coulomb interaction is required in the theory. When this was done, it was found that line shifts arose from the electronradiator interactions. There are also several effects that are part of spectral line shape analysis (and part of the line shape codes described in Appendix A) that are not incorporated explicitly into our study. One of these effects is spectral line merging[8]. As den sities increase, the positions of neighboring lines move together and merge. The merging of spectral lines at high densities requires a relaxation of the noquenching approximation and a much longer calculation time. Opacity broadening, especially required for stellar applications, has been incorporated into the line shape codes[3]. 109 10 7 r= 0.03 10' SICF r= 0.1 107 r = 0.5 106 I 104 1 1015 1020 1025 no (cm3) Figure 21: Plasma Parameters The temperature and density range of plasmas studied here is compared with other experimental conditions. Parameters used in MD simulations are shown. The observance of satellites required the use of multielectron radiator theories[19]. In addition, the experimental equipment also gives a broadening to the spectral line, which needs to be accounted for to directly compare theoretical and experi mental results. These effects, while important to line shape theory as a whole, can be neglected in our study. Our study will formulate a line shape of a single ionized radiator in a plasma. CHAPTER 3 LINE SHAPE FORMULATION The line shape function I(w) that is the starting point for the theoretical ,in ,i ; here follows directly from an analysis of the radiated power spectrum P(w) of a quantum mechanical system undergoing a transition from initial states a to final states b P() = ( L b de ikq a)2 PaW ab), (3.1) a,b where d = erb is the total dipole moment operator associated with the radiator, with the sum taken over all bound electrons [1, 2, 28]. The position vector rb = b qR is the position relative to the center of mass position qR of the radiator. The vector k in Eq. (3.1) points from the radiator to the detector and has a magnitude of k = w/c. Integrating this vector over all directions captures all of the radiation. The delta function ensures that the transition considered conserves energy. This term contains the quantity jab = (E, Eb)/h, which is the frequency associated with the energy difference between the initial and final state. Finally, this power is found by averaging over the initial states and summing over the final states. The weighting factor pa produces the average over the initial states. For many line shapes, the quantity w4 is approximately constant for the energy range of a line shape[1], and therefore it is convenient to define a line shape I(w) as i(w) = VP() = b de kqR )l 2 6( ab) (3.2) a,b For an equilibrium system, this can be written as an integral of the dipole autocor relation function I(t) 0C I(U)= Re dt (3.3) 0 1(t) Kd. dk(t)jp, (3.4) with the following definitions dk deikqR dk(t) = eiHt/dkeiHtlh (3.5) The brackets in the autocorrelation function denote an equilibrium ensemble average: (X) = Tr peX. (3.6) The density operator p, is from the equilibrium Gibbs ensemble for the radiator and its environment. In this form, the line shape is seen to be determined by the dynamic correlation of the dipole with itself. The decorrelation time Td is therefore a measure of the relevant radiation time or halfwidth of the line shape. The system under consideration here is a fully ionized plasma of point elec trons and ions, plus a single impurity radiator ion of charge number ZR. Overall charge neutrality is assumed. Unlike the plasma electrons and ions, the radiating ion has internal structure and therefore a charge distribution about its center of mass. In the following analysis, the system will be decomposed into two subsys tems, the plasma and atomic systems. The radiator ion will be itself decomposed into two parts, which will be separated into the above groups. The center of mass degrees of freedom of the radiator, which include the kinetic energy of the center of mass and the total charge of the radiator located at the center of mass, will be grouped with the plasma system. The internal degrees of freedom of the radiator, which include the potential energies of the bound electrons with the nucleus and each other and the kinetic energies of the bound electrons about the center of mass, will comprise the atomic system. Because of this grouping, the interaction between the atomic system and the plasma is only due to the distribution of the bound electrons relative to their monopole representation. This is an important point to emphasize here, since the analysis in the next section treats interactions among plasma constituents exactly. That includes exact coupling to the radiator as a monopole. Approximations only occur for coupling to the excess bound electron distribution. The Hamiltonian for this system therefore describes a radiator ion with bound electrons, and also free electrons, free ions, and all interactions among them. The Hamiltonian can be expressed as the sum of a plasma Hamiltonian Hp, an atomic radiator Hamiltonian HR, and an interaction Hamiltonian UpR that describes the interaction between the two systems. H = Hp + HR + Up (3.7) As stated above, a point charge representing the radiator center of mass degrees of freedom will be included in the plasma subsystem. Then, the plasma Hamil tonian Hp contains the kinetic energies of the free electrons, free ions, and the radiator center of mass, as well as all interaction energies between these three components. The result is (N, N17 N, N,' S= K S K( 2) 2 E 2 5 5'(a ,3) (3.8) 17 (a=l a743 q747l a 13=1 In Eq. (3.8), rl represents the type of particle (free electron, ion, or radiator center of mass) and N, is the number of particle of type ty. The first term is the kinetic energy with K,((a) p2/2m,. The second is the interaction between particles of the same type, and the third is the interactions between particles of different types. The interactions V,,' are Coulomb pair interactions. Then Hp describes a two component plasma of electrons and ions, plus a point impurity ion. For the atomic radiator system, the Hamiltonian is written as 2 '2p t 2mHR + (a,/3) + V.(a) (3.9) a a7/3 a=l with nR being the number of electrons bound to the radiator. The three terms in Eq. (3.9) describe the kinetic energy of the bound electrons, the interactions between bound electrons, and the interaction between the bound electrons and the nucleus of the radiator. With the center of mass degrees of freedom absent from these terms, HR is identical to a case of electrons bound to a motionless charged particle, and is therefore a straightforward atomic physics problem. The final part of the Hamiltonian describes the interaction of the atomic and plasma systems. These interactions are Coulomb interactions, and can be written as functions of the position operators of the quantum particles. Because of the mathematical manipulations to be done later, it will be convenient to write these interactions in terms of charge densities. The plasma particle and bound electron charge densities can be defined as: N np p,() = Qe6(x q) Pb = e6(x' r3) (3.10) a=l a=l (In Eq. (3.10), r = {e, i}, indicating the plasma electrons or ions, respectively; in contrast to Eq. (3.8), the point radiator is not included.) With these definitions, the interactions take the form of the above charge densities multiplied by the form of the pair potential for their interaction. For example, the interaction between the free plasma electrons and the atomic system becomes (ectrons) dx dx' p(x (')p ( ) (3.11) The actual form of W(x, x') can be found as follows. Before the introduction of charge densities, the explicit form of the plasma electronatomic system interaction is Ne RR 2 Ne N 2 N e 2 (electrons) e e R Ze N C12 P/ R I q,, qR rTp [q1 qR I aQ qRj (3.12) a=1 \=1 a=1 a=/ The first term describes the interaction between the free and bound electrons, and the second term is the interaction between the free electrons and the radiator nucleus. The interaction between the free electrons and the radiator center of mass was included in the plasma Hamiltonian, so it must be subtracted, giving the last term. The other quantities in the Eq. (3.11), qg, qR, and rp, are the positions of the free electrons, radiator nucleus, and bound electrons respectively. Finally, NR is the nuclear charge and ZR is the radiator charge. Since ZR = NR nR, this can be simplified to Ne R ( 1 /(electrons) = 1 2 1 (3.13) pR (J \ r,3 q (R This shows that the form factor in Eq. (3.11) is V(x, x') = x qR x Ix qRI (3.14) The remaining part of the interaction Hamiltonian is the interaction of the free ions with the atomic system. The ioninteraction Hamiltonian can be written in the same form as above, with the only difference being that the charge of the free ions replacing the charge of the electrons. When these are added together, the total interaction Hamiltonian is UpR = dx dx' pf(x)pb(x') W'(x,x') (3.15) '7 W(x,x') x q 1 (3.16) where the sum over Tl is the sum of plasma species. The form factor for the ionradiator interaction can be simplified by consid ering the various length scales involved in the integration. In the denominators of the form factor, the two vectors of interest are x qR and x'. The first vector represents the separation vector of an ion perturber from the radiator, and the second vector represents the bound electron position relative to the radiator center of mass. The integral over x and x' are formally taken over all space. However, physical considerations limit the effective range of each integral. First, the integral over x' is weighted by the bound electron wave function. Since the wave function has a characteristic length scale given by the Bohr radius, the upper integration limit at infinity can be replaced by a factor on the order of the Bohr radius for that species of radiator. Second, when comparing the ionradiator separation x qR and the relative position of the bound electron x', it is found that the ionradiator sep aration is much larger than the relative bound electron position, or x qR >> x'. To see why this is so, we can use as a characteristic length the ion sphere radius ro. This is the separation of the plasma ions assuming uniform charge density, and is given by 4 / 3 13 rn n = ro = ) (3.17) 3 47n where n is the ion charge density. The probability that the ion will be a certain distance from the radiator can be estimated by the Boltzmann factor. A lower limit on the probability can be found by estimating the Boltzmann factor for the ion at the Bohr radius. This gives exp +ZiZe2 exp ZLZFroP (3.18) where Zi and ZR are the charges of the plasma ion and radiator, ao is the Bohr radius, ro is the ion sphere radius, and F is the electron plasma parameter given by F /3e2/r0. For typical plasmas under consideration, ro/ao is on the order of the radiator charge, and the plasma parameter lies in the range 0.01 $ F $ 0.1. With these values, the exponential factor is small at the Bohr radius, and approaches zero as the exponential of 1/r. Since the radiator typically has a much greater charge than the plasma ions, the average radiatorplasma ion separation is typically larger than the ion sphere radius. The vector x', representing the distance from the bound electron to the radiator center of mass, is limited by the effective cutoff of the bound wavefunction. This distance is on the order of several Bohr radii. The conclusion from this is that in the denominator of the first term of W(x, x'), the vector x qR is much larger than x'. Because of this, the form factor can be expanded in terms of the first quantity. To first order in x', W(x, x') becomes: ( 1 xIx 1 W(x, x')= + + 3 (3.19) x qR x qR x qRj VW(, X') = ( x' (3.20) x qR For the ionradiator interaction, the form factor is now separated. The interaction Hamiltonian for the plasma ions is then, from Eq. (3.13) U[J = [ dx pj(x) (x pb(x')x' (3.21) (x qR)l J S e (ri qR) ] y = E[ d (3.22) ions (ri qR rb where on the last line the quantity in the first brackets was identified as the electric field of the plasma ions at the radiator, and the quantity in the second brackets is the definition for the dipole moment of the radiator. It is important to note here that this ion electric field is the bare Coulomb field. This is a long range force, and it will later be convenient to take advantage of the screening by the plasma electrons to define a modified ion field by including certain averaged plasma electronradiator interactions. We will return to this topic in C'! lpter 5. For the plasma electronradiator interaction, the situation is quite different. The distant electrons will give a contribution similar to Eq. (3.22) due to the electron's electric field. However, the highly charged radiator also pulls the plasma electrons in close, so that the plasma electron charge density is appreciable even within the bound electron orbitals. We will now explore the form factor for these close plasma electrons. To estimate the effect of the close electrons we make the assumption that the plasma electron charge density is spherically symmetric about the radiator, and that the radiator is at the origin. Then Eq. (3.15) for the plasma electrons is Selections 2 x dx p b(x') di W(x, x') (3.23) with dQ being the angular integration over the plasma electron coordinates. The form factor can be expanded in a multiple expansion W(x, x') (1 1 ) 4 Y, (Q) (Q/) _J 1 (3.24) where x> and x< is the greater and lesser, respectively, of {xl, x'}1, and .Y(Q) is a spherical harmonic[29]. If the plasma electron charge distribution is spherically symmetric as assumed, then only the = 0 term in the sum will be nonzero due to the orthonormality of the spherical harmonics. Using Yo0() 1/v4r, the form factor for this case is therefore WV(x, x') x') + x' x) (3.25) x> x x x' x where O(x x') has the value of 1 if x > x' and 0 for x < x'. Since O(x x') + O(x' x) = 1, this factor can be introduced into the last term and the theta functions combined to give W(x, x') = e(x' x) (1 1 e(x' x) (3.26) G x' x xx') This indicates that in addition to the dipole interaction the plasma electronatomic radiator interaction includes a "monc.lp. term from the penetration of the bound charge density by the plasma electrons. In summary, the line shape has been identified in terms of the radiator dipole autocorrelation function. After a separation of center of mass and internal degrees of freedom for the radiator, the dynamics of the dipole is determined from a Hamiltonian of these internal degrees of freedom coupled to a plasma of point charges. This coupling is through a dipole coupling to the ions and electric fields of the plasma charges. In addition there is a coupling to the plasma electrons within the atomic structure of the radiator. CHAPTER 4 SEMICLASSICAL MOLECULAR DYNAMICS In this chapter, the application of molecular dynamics (\!1)) simulations to the line shape problem will be examined. Recall that in C!i lpter 3, an expression for the line shape was expressed in terms of the dipole autocorrelation function as I(t) =(dt dk(t))= ZTrdt iHtdkeH(tt (4.1) where the angle brackets indicate an average over an equilibrium ensemble for the plasma, and Z = Trexp[3H] is the partition function. It is important here to determine where the most difficult part of this calculation lies. To this end, we will approach the calculation of the dipole autocorrelation function by direct solution of the Schr6dinger equation. The Hamiltonian for the system is, from C(i plter 3 H = Hp + H + UpR (4.2) where Hp is the isolated plasma Hamiltonian (including the radiator center of mass), Ha is the Hamiltonian describing the atomic degrees of freedom of the radiator, and UpR is the interaction between the two. Going to the interaction representation[30] defined by eiHt eiH (t) (4.3) gives the autocorrelation function in the form (dt dk(t) Z Tre3Hptd U(t)dkUt (+ i/) (4.4) A basis set {c)} to perform the trace can then be chosen, and interaction picture states can be defined with ',(t) = U(t)la). The correlation function can be calculated by solving the Schrodinger equation H, iUpR(t)) '(t) 0 (4.5) with UpR(t) dx dx' p x(, t)pb(x')W(x, x') (4.6) p(x, t)= eiHtpx)eiH't (4.7) where p,(x, t) is the plasma perturber charge density and pb(X') is the bound charge density. Note that in Eq. (4.6), the only dependence on the plasma degrees of freedom occurs completely within p,(x, t). Once p,(x, t) is specified, the problem represented by Eq. (4.5) is purely an atomic physics problem of the coupling of an atom to an external potential. The difficult n i1 body problem residing in p,(x, t) is therefore of great importance to calculate accurately, which is why MD is relevant here. MD simulations claim to be able to perform an accurate calculation of the plasma charge density with all charge correlations intact. However, there are several difficulties in applying MD to this problem, and this chapter will examine the validity of MD simulations approach. The first topic to study is the fact that MD is inherently a classical method. In MD, Newtonian equations are solved directly to provide particle trajectories [31, 32]. The motion of the particles is then followed for some time interval, and the properties to be calculated are determined by the time dependent coordinates and moment of the particles. The early uses of MD for plasma simulations involved a one component plasma consisting of positive particles, plus a constant negative background for overall charge neutrality. For this case, MD is very useful and clearly applicable, since the repulsive nature of the interparticle interactions keeps the particles far apart. This meant that, with high temperature, the distance between the particles is much larger than the de Broglie wavelength. Therefore a classical description is a good approximation to the one component plasma, since in such a case there is small probability for the particles to be close enough for diffraction effects to be important. For this study, we would like to use MD to study a plasma that would include positive plasma ions, plasma electrons, and a highly charged radiator impurity. One initial problem is the mass difference between the plasma ion and electrons. The electrons require a short characteristic time scale. Any simulation would have to move with time steps less than this time scale, or else the simulation would not capture the effects of the electron motion. However, the ions move much more slowly, and thus have a time scale much larger than that of the electrons. Therefore, the time steps have to not only be small enough to capture the electron motion, but the simulation has to run long enough to capture the effects of the ion motion. This situation requires the recent advances in computer calculation speed to have an effective simulation tool. There is an additional difficulty, however, with the basic validity of the simulation itself. For cases with attractive particles, it is discovered that there are theoretical difficulties with modeling an electronion interaction with a classical interaction energy. For the ions, and especially the radiator impurity, the attractive interactions cause the electrons to approach very closely. Using a bare Coulomb potential for the radiatorelectron interaction causes a singularity to appear in the ensemble. For this classical case, the partition function includes Boltzmann factors with the form e3H ~ = e6/r (4.8) where Z, is the ion charge, and r is the separation from the electron and ion. For small r, this factor diverges, leading to a singularity in the Gibbs ensemble. This reflects the fact that there is no classical limit for a plasma of oppositely charge particles. It has been shown that systems with attractive Coulomb interactions are unstable unless quantum mechanics is used [33]. The result of this is that MD seems to not be a valid tool if bare Coulomb interactions are used for oppositely charged particles, as a collapse of electronion pairs is the inevitable result. Recently a method has been used to resolve this difficulty of using an inher ently classical method to give quantum results. To this end, the potential between electrons and ions is modeled by a form that does not diverge for small distances, but gives the Coulomb potential at larger distances. These are called ;,! i.. 1 Coulomb potentials. To understand this procedure, consider the quantum partition function for two particles [9, 10] Z =Tr e 3H(1,2) dri dr2 rr2 3H(1,2) 1 rr2) (4.9) 2(4.9) where H(1, 2) = H(1) + H(2) + V(1, 2) is the two particle interaction. A classical expression for the partition function is then written down which defines the regularized potential Z Tr e3H(1,2) d drl 2 eU(ir2) (4.10) where A = 2V h2/mkBT. This defines the regularized potential to be exactly U(Irl r2a)  In (A2 rl, r2 e H(1,2) l, r2)) (4.11) This potential resolves the singularity problem, as it is finite even in the limit of Irl T2 i 0. It also agrees with the Coulomb potential for distances farther away from the ion. This characteristic distance is on the order of the de Broglie wavelength AD = 27h/p. The results are somewhat sensitive to the specific distance used, and later in this chapter we will discuss some recent work that shows the optimum distance to use for certain cases. With the regularized potential defined in this way, the expectation value of a quantum operator that depends only on the coordinates of two particles can be given exactly by using this classical potential along with the classical form of the operator (A Tr e3H(,2)A(r, r2) d62Z dr2 e_U(rIr)A(rl, 1 2) (4.12) Because this last equation is exact, the regularized potential must be accounting for all of the quantum effects. Therefore, MD simulations for many particles using these potentials will include quantum effects to some degree, but will also be a welldefined classical problem. At this point, we must examine these results and determine how 'correct' they are, i.e. in what regimes do classical calculations using regularized potentials agree with full quantum calculations. In simulations involving hydrogen plasmas, it is found that good results are obtained for high temperatures. As the temperature lowers, hydrogen atoms appear that are also described well [34, 35]. However, at even lower temperatures, regularized potentials do not adequately describe molecular hydrogen. These results agree with the primary assumption made in using regularized potentials, that quantum effects are most important with pairwise interactions. This holds true for the high temperature plasma and atomic hydrogen systems. At very low temperatures manybody quantum effects become more important. In this study, the system considered is a hot, dense plasma, and MD simulations with regularized potentials would appear to be a good choice for simulations. A form for the regularized potential still has to be found from Eq. (4.11). There are several different methods for evaluating this. In Appendix G one method is shown that results in the Kelbg potential for an arbitrary potential V(ri r2 ) UK(r) dx (/) V(Irl r2 x) (4.13) iXJ * In this form it is straightforward to see that the regularized potential can be interpreted as a smoothing of the bare potential over a volume characterized by the de Broglie wavelength. For the specific case of the Coulomb interactions, the Kelbg potential takes the form [36, 37, 38, 39, 40]: UK(r) V(r) ( e(r 2 + 1 [ erf ()} (4.14) A second form for the regularized potential is the Deutsch potential [41]. This is a phenomenological version that is often used in MD simulations and has the form UD(r) V(r) ( e ) (4.15) Both the Kelbg and Deutsch potentials have similar qualitative features; however, inside the de Broglie wavelength the Kelbg potential is preferable. One additional factor must be evaluated. The plasmas considered here are highly charged. This has the potential for further difficulties as the electron trajectories are pulled closer, and there is the possibility for metastable bound states. More study needs to take place to determine the effects for a given radiator charge, temperature, and electron density. With the use of these regularized potentials, MD simulations can calculate dynamic properties for a plasma. For example, a simple theory might have the radiatorplasma interaction be totally described by the monopole and dipole interaction. In this case, the time dependent electric field of the plasma is required. By using MD to determine the electric field, all correlations between the plasma particles can be retained. The next chapter will look at a specific plasma to determine the properties needed that will give the line shape function. It will be found that some of these properties needed include the constrained average of the electron density, and the electron field autocorrelation function. With statistical mechanics methods, uncontrolled approximations are required to evaluate these 29 properties. MD simulations can be shown to be valid for many cases, and they can provide accurate methods to determine these properties. CHAPTER 5 LINE SHAPE FORMULA FOR QUASISTATIC IONS In C'!I ipter 3, an expression for the line shape function was given, and in ('!C ipter 4, a numerical method to evaluate certain properties important to the line shape was discussed. In this chapter and the next, statistical mechanics will be used to reformulate the problem in terms of those few properties that are most important in determining the line shape function. Recall that in C'! lpter 3, the line shape was written in the form ( (t) K)dk () (5.1) Now an expression for this will be derived that separates the various mechanisms affecting the line shape. Consider first the physical aspects of the plasma particles. The electron and ion components of the plasma affect the dipole in different vv due to their different charges and masses. For positively charged radiators (the only case considered here) the ions are repelled by the radiator and their dominant effect is Stark broadening by their electric field coupled to the dipole of the radiator. In many cases of physical interest, the change in this field is small during the radiation time (relaxation of I(t)). Under such conditions the ions behave as static perturbers and this will be referred to as the quasistatic ion approximation. The quasistatic ion approximation is physically relevant because of the large mass of the plasma ions as the example in ('C! lter 2 illustrates. It is important to note that the quasistatic ion case does not indicate that the ions are motionless, only that the effect of their motion on the forces of interaction is negligible. We want to isolate the effects of the ions from the other processes of the plasma. Shortly we will examine under what conditions this is useful. Going to an interaction representation in which the effect of the ion motion (including the radiator) is separated leads to CiHT CiKTU() (5.2) where U(t) is the desired propagator for the plasma that includes all processes except ion (including radiator ion) motion. This propagator obeys the following equation of motion (O< + iH. + iV({q, + vjt}, {q}, {q}))U(t) = 0 (5.3) We can now examine the conditions under which the ion motion effectively decouples from the other degrees of freedom of the plasma. The important quantity to consider is the fractional change in the ion and electron interaction energy caused by ion motion during the characteristic time under consideration. The fractional change is AV/V, with a differential change given by dV = (VV) (v)dt. With an average speed of vo and a characteristic time r, the conditions required have rvo VV({qiJ, {qeJ) Tvo ( < 1 (5.4) V({qi}, {Jq}) ro Under these conditions, the kinetic energy for the ions commutes with the other terms, and the equation above can be written eiHt eiKteiHt (5.5) where H, = H, + K + Ve({qe}) + IV({qi}, {q,}) + UpR (5.6) The contribution VI~({qj}, {e}, {qb}) denotes the total interaction of both bound and free electrons with the ions. The ionion potential cancels in d(t) for this staticion approximation (but not in the Gibbs density matrix), and H, becomes the Hamiltonian for electrons in the frozen field of the ions. A remark here about the quasistatic case and the absence of certain cor relations is in order. The introduction stated that this work would neglect no correlations. Yet clearly correlations between the radiator center of mass and other plasma particles are absent in going from the general case to the quasistatic ion approximation. There is no discrepancy here, however. It is the time scale of the quasistatic ion approximation, not an approximation made in the calculation of the quasistatic ion case, which causes the lack of these correlations. Therefore there will be no need at the end of the calculation to introduce any theoreti cal device to correct the result. All correlations present in the quasistatic ion approximation are included. We can now decompose the average for the autocorrelation function into a product form, consisting of a static dipole autocorrelation function and a doppler term, resulting in d dk(t)) (d d(t)) eik VRt (=d* d(t)) J dv3/2 e /2ikvt (5.7) = (d d(t)) }(k, t) (5.8) where K(k, t) is the time dependent Doppler line shape. The frequency dependent line shape can be written as a convolution of a doppler line shape and a quasi static ion line shape[42]. I(w) = d'ID ( /')J(/w) (5.9) where 1(w (w ')2 ID () exp (5.10) 2V2 T( 2a2 2 11 /2kBT (5.11) c m, ) and OO J(w) = Re dt ei (d d(t)) (5.12) 7 J 0 The quasistatic ion dipole autocorrelation function is now (d d(t)) = (d(t) d) = Tr Trpd(t) d (5.13) a p and the plasmaatomic Hamiltonian is UpR({qi}, J{qe, qb}) = E({q}, ,{q d)  + d ddx' pe(X)pb(X') (W(x, x') . ) (5.14) This form in Eq. (5.13) for the autocorrelation function still has a difficult ii ii vbody problem in performing the trace over the plasma degrees of freedom. Because the ions are taken to be quasistatic their effect is easier to analyze. First, the large mass of the ions not only justifies the quasistatic approximation but also implies a quasiclassical approximation for the conditions of hot dense matter considered here. This means the ion thermal de Broglie wavelength A 2/h2/mrkBT is short compared to the distance between ions. Therefore, the trace over ion degrees of freedom can be converted to an integral over configurations for the ions. Furthermore, as was discussed in C'! lpter 3, the large separation between the radiator and plasma ions leads to the dominant ionradiator interaction being the coupling of the ion electric field to the radiator dipole. In other words, the only relevant property of the ions is their electric field value at the radiator. Finally, for the quasistatic ions the ion electric field is constant during the radiator time. In performing the integration over ion configurations it is recognized that effect of the plasma ions on the line shape is determined by the value of their instantaneous electric field. Also, many points in phase space correspond to ion arrangements with the same electric field at the radiator. This sil:. 1 the following conceptual method to calculate the trace. For each possible value of the ion electric field, v e, pick out the group of all those configuration points which have their ion microfield equal to e at the radiator. Then average the dipole operator over just that group of points and then perform a weighted average over all possible ion field values. This gives Trpd(t) = dc Trp6(e Es)d(t) (5.15) p J P = dc f (a, )D t) (5.16) which defines a primary quantity to consider, the constrained average dipole operator with an ion electric field constraint Es = e at the radiator D(, t) = Trp(a, e)d(t) (5.17) P p(a, ) =fl(a, )pb( Es) (5.18) f(a, e) = Trp( Es) (5.19) p The quantity Es is the screened ion field at the radiator site. However, at this point in the study the form of the screening is arbitrary. Later in the analysis a particular form will be chosen that simplifies the calculations. With the average over plasma states ahv i constrained, the interaction Hamiltonian can be redefined in such a way that it does not depend on the ion degrees of freedom. The Hamiltonian for this system is H = H+ HP + UpR (5.20) The ions interact with the atomic system solely through a dipole interaction. To explicitly extract these effects, we add and subtract a term based on the screened ion field Es from the Hamiltonian H= Hp+HR+E, d+ UpR Es, d (5.21) Here, the screened ion field Es only depends on the ion coordinates. However, the specific form is still not defined at this point, since the form will be chosen to simplify the analysis later. Now in the calculation of D(e, t), the Hamiltonian alvi, appears in a constrained average, where the trace is restricted by a delta function to only include terms in which the screened ionfield is equal to some given value e. In the quasistatic ion approximation, the plasma dynamics do not affect this restriction, so the Hamiltonian has the following useful form H =Hp+ HR(e) + 6UpR (5.22) where HR(c) = HR + e d is the Hamiltonian for a radiator in an external field e, and UpR = UpR E8 d is the interaction between the plasma and the atomic system in this new Hamiltonian. The new interaction JUpR is given by 6UpR({q,}, {qj, qb}) = (E({q}, {q} ) E({qJ})) d (5.23) + Jdx dx'p(x)pb(x') (W(x,x') x. x' A suitable choice of the screened ion field removes all ion dependence from JUpR and gives a form convenient for analysis. The staticion line shape is then OO J(Uc) 1= Re fdt eiwt de Trf(a, e)D(e, t) d (5.24) 7 JJ a 0 Note that this has the form of an effective atomic physics problem. Here all of the effects from the plasma degrees of freedom have been localized in the quantity D(e, t). Once D(e, t) has been evaluated, the remaining dynamics are that of an atom in an external field e. This problem is well studied, and many results, especially atomic wavefunctions and various matrix elements, are available [17]. The next chapter will evaluate the equation of motion for D(e, t), and use perturbation theory to calculate various plasma effects to second order in the plasmaradiator interaction. CHAPTER 6 QUASISTATIC EQUATION OF MOTION The previous chapter derived an expression for the line shape in which the most difficult many body calculations were contained within the constrained average dipole operator D(e, t). As was mentioned in C'!i pter 5, an equation of motion for D(e, t) will be derived that will give exact dynamics, including all correlation effects, for the case of quasistatic ions. This equation of motion will contain terms that can be identified with various broadening mechanisms for the line shape. The derivation is more straightforward with the introduction of a Liouville op erator [43]. A brief overview is given here, and more details are given in Appendix D. A Liouville operator L can be associated with any Hamiltonian operator H. For some quantity X whose time evolution is determined by the Hamiltonian H, the associated Liouville operator is given by X(t) = eiHt/hX(O)eiHt/ = LtX(O) (6.1) or equivalently LX =[H,X] and + L X =0. (6.2) With the Hamiltonian separated into three terms (see Eq. (5.22)), the Liouville operator associated with the entire system separates into three analogous terms, or L Lp + L(a, c) + 6LR (6.3) Using a projection operator technique detailed in Appendix D and this form of the Liouville operator, the time evolution equation for D(e, t) takes the form + L(a, e) + B(c) D(e, t) + dr M(e; t T)D(e, r) 0 (6.4) 0 The operators B(e) and M(e, t) represent the effects of the coupling of the radiation to the plasma 6LpR. Since the static ion broadening has already been accounted for in L(a, e) it is expected that these are electron broadening terms. However, these electron terms retain their correlations with the ions and a depen dence on the fixed ion value. The operator B(e) is simply the mean value of 6LpR for the constrained plasma B(e)X = Trp(e)6LpRX = Trp() [6UpR, X] (6.5) p P It is seen from the equation of motion that B(e) describes the short time dynamics of D(c, t). The operator M(e; t) describes the dynamical electron broadening and is more complex. Its detailed form is given in Appendix D. These quantities can be expanded in powers of 6LpR as B(c) = B(1(c) + B(2)(c) + M(e,t) = M(2(c,t)+. In the remainder of this chapter these electron broadening operators are brought to a more practical form by an expansion to second order in 6LpR. 6.1 Firstorder Electron Broadening The result for the electron broadening B(e) contains the Liouville operator 6LpR or interaction Hamiltonian 6UpR. In this section, this will be expanded in powers of the interaction energy of the plasma and radiator internal states. As described above, the removal of the monopole and ion dipole terms for JUpR results in a softening of the interaction so that an expansion in powers of JUpR is valid. This will give a form of the first order electron broadening ready for calculation, and also allows us to show that this term arises solely from the effect of the plasma electrons on the radiator. The evaluation of this trace occurring in B(c) is complicated by the presence of 6UpR in the density matrix p(e). However, since the interaction Hamiltonian SUpR itself occurs in the trace, the first order part is found by writing the density matrix to zeroth order in 6UpR. The first order contribution to B(E) is then B (e) = Trp(e)6Lp RX Trp,(e) [6UpR,X] = [(6UpR) ,X] (6.6) p h where (X), Trpp,()X (6.7) P is the constrained plasma average. In this formula, we have pp(C) Ql()pp6( Es) pp = (6.8) Tre 3Hp where pp is the plasma equilibrium density matrix and pp(c) is the constrained density matrix. Up to this point the definition of Es has not been specified. It will now be shown that a suitable definition of Es will remove all ion dependency from the average interaction Hamiltonian. To get this result, the electric field will be defined to be the field of the ions screened by the electrons. First, the form of SUpR is rewritten so that the dipole contribution for distant electrons is also made explicit 6UpR (E E) d f dx dx' pe(X)pb(x') W(xa, x') (x ro) (6.9) where d = Y: ers and E = E + E, = E + dx pe,(x) (X ro) (6.10) / x 3 where Ei is the ion field and E, is the electron field. The constant ro is chosen to be a characteristic size of the bound charge distribution outside of which the dipole interaction is welldefined. The screened ion field E8 is now defined such that this dipole contribution from JUpR gives no contribution to B(1)(e) Trppp()(E E)= Trppp(e)(E, + E E,) (6.11) Trip4(e)(E, + p~TreppE E) (6.12) In the second equality use has been made of the fact that the screened ion field and the total ion field have no dependence on the electron coordinates, so the trace over the electron coordinates can be carried out. Also, pi is the reduced density operator for the ions and pi(c) is the corresponding constrained operator Pi Trepp pi(C) Q1()p6(c E,) (6.13) The definition of Es is now chosen such that this dipole contribution is zero Es E, + p~TrppE, (6.14) This is a primary result of our general analysis that includes all charge correlations. As noted above, the primary effect of the static ions is a Stark broadening of the line. This is described by L(a, c) in Eq. (6.3) above. The field values c are sampled from the probability distribution Q(e). The probability distribution is defined here as an average over the two component plasma, in contrast to current theories which approximate the effect of electrons on the ions using a One Component Plasma (OCP) of ions only. In addition, however, we now see that the choice of microscopic ion field Es whose probability is being computed in Q(c) also must include the effects of the electrons in order that L(a, c) should give all of the ion Stark broadening (i.e., that there be no additional effects from B(1) ()). The cancellation of this additional ion broadening leads to two effects. First, the relevant fields of the ions at the radiator are screened (below it is shown that the second term of Eq. (6.11) provides Debye screening in the weak coupling limit). Secondly, the dipole interaction of the electrons is entirely accounted for by this screening effect. The definition of the screened ion field in this theory includes all plasma correlations. Therefore, the claim that it accounts for all of the dipole interaction between the plasma particles and the internal radiator states is an exact claim for quasistatic ions. The remaining nonzero contributions to B(1) () comes from the second term on the right of Eq. (6.9). This represents the interaction of the perturbing electrons with the bound electron distribution, now with both the monopole and dipole interactions separated out. Using this, the nonzero part of B(1)(e) takes the form B(1c (')X= dx dx' W(x, x') (pe,(x)) [pb(x'),X] (6.15) where ( 1 1 x o( )) (6.16) W(x, x') =( ) (6.16) and (pe(x)), indicates a constrained average of the free electron charge density, and the theta function is defined by 0 ro > x O(x ro) = (6.17) 1 x > ro This form of W(x, x') will be used again for the second order parts of the calculation. However, let us now expand the first term in the form factor W(x, x') using a multiple expansion, and keep just the monopole term. The monopole term is an important piece to study, because in many theories, the assumptions made lead to all other terms going to zero. This term is due solely to the monopole interaction arising from free electrons penetrating the bound electron orbitals (x < x' < ro). The form factor then becomes (including the factor of x2x'2 from the radial integrals) W(x, x') xx'(x x')0(x' x) (6.18) The theta function will change the upper limit on the integration over x. Also, the integration over x' is effectively limited by the bound electron wavefunctions. To represent this, they will be given an upper limit of some characteristic length related to the Bohr radius. The angular integrals can be absorbed in the charge density calculations, giving B((e)X i47 dx' dx (p(x); ) [pb(x'),X] xx'(x x') (6.19) 0 0 (P W(x);e } ( dQ p (p, (x); (6.20) Pb(X) J d6 pb(x') (6.21) For the monopole term of the first order electron broadening, the effects of including all plasma correlations exactly has been localized in the term (pe(x); e)p. This term represents the free electron charge density, averaged over those plasma states in which the ion electric field Es is constrained to have the value e. This free electron charge density is not uniform because the charged radiator at the origin attracts the perturbing electrons. Furthermore, it is not isotropic because of the constrained average: with a specified field e, the ion density will be larger on one side of the radiator than on the other. The fast moving electrons would then be attracted to the more positive side, leading to an .,mmetry in the free electron charge density [44]. The importance of these effects can be explored with a simple estimate calculation. If it is assumed that the perturbing electron charge density pe(x) is constant inside the bound electron wavefunction, pe(x) can be removed from the integral and the integration over x performed, leading to 0 B ( pe(Oe} dx'4 [pb(X'),X] (6.22) 6.2 SecondOrder Static Electron Broadening Term At this point it is useful to more closely examine how the choice of the ion microfield form has affected the physical interpretation of B(e) as a static electron broadening term. Note that initially the plasmaradiator interaction 6UpR represented both electron and ion perturbations, with both the monopole interaction and a ion dipole interaction Es d removed. With the above choice for Es and the neglect of higher order ion multipoles the radiatorplasma coupling energy can be written sUpR = E, d dx dx' p,(x)W(x,X')bX'pb ), (6.23) E, = E, (E,), (6.24) where (E,), is the electron field averaged over the electron degrees of freedom (E,), p7 TrppE, (6.25) Thus, the first term of the interaction is just the fluctuation of the electron dipole interaction whose average value in B(1)(E() is zero by definition. Clearly, 6UpR now represents an electron perturbation of the radiator. Consequently, both B(e) and M(e; t) can now be recognized as electron broadening operators (although still retaining the correlations between electrons and ions). The previous section examined that part of B(e) that was first order in the interaction Liouville operator. The remainder, B(c) B(1) (), can be expanded into a term that is second order in the interaction Liouville operator. The full static broadening term explicitly is B(e)X = Trp'(a, e) 6LRX (6.26) P and the density matrix is given by p(c) = f(a, )pb(c Es), f(a, ) = TrpS(e Es) (6.27) P p =Z e3(HR()+Hp+6UpR) (6.28) where Z is the associated partition function. Define the quantity Ho = HRn() + Hp. Then, to first order in 6UpR, this density matrix is p(e) pp(e) 1 dr (eHO p6R) e'HO Trpp(c)eHo (6UpR) e (6.29) 0 P The first term leads to the first order static shift as described in the previous section. The second term gives a second order contribution of 13 B(2) (eX = dr Trpp()6Upn(ihr) (6UpR (UpR; c) X] (6.30) 0 where UpR(ihr) denotes the Heisenberg operator at an imaginary time. It is expected that the dominant contribution to B(2)(E) should come from the electron dipole interaction of 6UpR. Then B(2)(E) simplifies to 13 B(2)(c)X = dr (6E ( ihr)6Ee; c d(ihr) [d, X] (6.31) 0 This form includes all correlations present in the quasistatic ion case. Also, in this form the atomic and plasma physics have been separated. The electric field autocorrelation function is a plasma physics problem and is solved separately from the atomic dipole operators. 6.3 Dynamic Electron Broadening Term The previous sections have dealt with the term B(e), which describes the static broadening by the plasma electrons. The dynamical electron effects are described by the operator AM(e; t). In Appendix D, this operator is calculated and shown to have the form M(e; t) f'(a)Tr [p(a, e) (6LpR) eQtQ (6LpR)] (6.32) Here Q is a projection operator such that QX gives the deviation of X from its constrained plasma average. QX = X Trp(c)X (6.33) P The previous section accounted for the average value of 6LpR and its time independent fluctuations to second order. What is left is the timedependent fluctuations, which verifies the previous identification of AM(; t) as a dynamic electron broadening term. More physically, AM(; t) describes atomic transitions of the radiator caused by collisions with the plasma electrons. To second order in 6LpR this operator becomes (see Appendix D for details) M (2)(, t) = 6LpRC (LR(e+L) (J6LpR (6LpR; c) P (6.34) This form of the dynamic electron broadening operator is similar to those obtained previously. Typically, however, such results neglect many or all of the electron radiatorion correlations. These correlations enter the broadening term in two places. The definition of Es itself included some of these correlations in that the correlations lead to screening of the ion microfield, and are included only approximately in most previous theories. However, the term above also includes all correlations when performing the trace. Instead of performing the trace over separate ion and electron subsystems, the trace above is over a twocomponent plasma including the radiator monopole. This result can be made more explicit using the form for the perturbation UpR 6E d dx dx' pe(x)W(x,x')pb(x') (6.35) to get the action of M(c, t) on some radiator operator. Clearly, it will be deter mined by the time dependent correlations between 6Ee and pe(x) e(L(x)tM(, xt'X = CPt, e)2 [d, )(t) [d X] [2 b b 1),X]] + dxdx'W(xx')Dox,t,c) ) [dJ(t), [pb(x',X]] Sd Dx (x, t, c) J dx'W(x, x') ) [pb(', t)), [do,X]] where the time dependence of the bound electron density and dipole are given by that for the free radiator in the Stark field c pb(x', t) = (Le ( )tPb (), d (t) = (Lun ()t The plasma properties are contained in the correlation functions S(x, Xl, t' C) P(', t) pe(XI) (pe(ai);E)p) ; e D,(x, t, e) = 6Ec,(t) (pe(X) (pe(X;E));e) ;e Cop(t, e)= (6E,,(t)6Eep; E), CHAPTER 7 CHARGE CORRELATION EFFECTS ON PLASMA PROPERTIES In the previous chapters, a formulation for the spectral line shape has been derived to second order in the plasmaradiator interaction potential, and keeping the charge correlations between the components of the plasma. We now turn to examining the effects that retaining the charge correlations has on various plasma properties. Here the focus is on studying how these properties can be calculated, and giving estimates as to their values. We consider the plasma with the semi classical representation from C!i lpter 4, and use classical statistical mechanics. Quantities of interest include charge densities near the radiator, pair correla tions among perturbing ions and electrons, the screened ion field, the microfield distribution, the reduced density operators for the electrons and ions, and the time correlation functions for the electron charge density and electric field near the radiator. The estimates also provide guidance for more accurate MD simulations as the ultimate test of their range of validity. 7.1 Plasma Structure We will first consider the timeindependent structural properties of the plasma. The plasma is considered to be a twocomponent plasma with ions of charge Z and electrons. In addition, a dilute concentration of another species will be introduced as the radiator. As the equations are derived, the concentration of the radiator will be set to zero, reflecting our model of a single impurity ion radiator. To study the structure of the plasma, an appropriate starting point is the pair correlation function g,,a(r), or the simply related quantity h,a(r) = ga,a(r) 1[45, 46]. The function h,,s(r) is a measure of the total correlation between two particles of type a and /, and has the behavior h,,3 i 0 for large r. The total correlation includes the direct influence from one particle to another, plus the indirect influence caused by a other participating particles. A simpler quantity, which emphasizes the direct influence between particles, is the direct correlation function co,a(r) defined by the OrnsteinZernicke equation[45, 46] h,,a(r) cr,(r) + n7 J dr' (c,,,(r r'\)h,(r') (7.1) where n, is the density of particles of type 7. Another relation in terms of these values can be found from the hypernetted chain (HNC) approximation[45, 46]. This approximation is found to work well for ionic systems. The hypernetted chain approximation results from a specific approximation in terms of ca,p and ha,beta ha,(r) 1 + exp [pVa,(r) + hc,a3(r) c,,(r)] (7.2) or, using the relation h,p(r) = g,a(r) 1 and Eq. (7.1)we get ln g~(r) Va3(r) + n j dr'c( r r' ) (r') 1) (7.3) The two equations Eq. (7.1) and Eq. (7.3) provide the basis for evaluating the pair correlation functions between the various plasma constituents. To proceed, we introduce the "potential of mean force" by defining In9 Sg(r) = 3UaP(r), (7.4) Then Eq. (7.3) immediately becomes a nonlinear integral equation for Up(r) Uc,(r) = V,(r) /31 jn dr'ca ( r r' ) (eU(r') 1) (7.5) These results are quite general. These equations represent the correlation functions for a twocomponent plasma of ions and electrons, with no impurity radiators present. The presence of the monopole of charge number ZR from the radiator does not change these equations for the electron and ion correlations within the plasma away from the radiator because nR = 0. However, we are interested also in the structure of the ion and electron configuration about the radiator. The equations for the densities of electrons and ions about the radiator do couple to these results. We first specialize the above results for the case of a radiator correlated with some other particle. The OrnsteinZernicke equations for the correlation function between the radiator and a particle hcR(r) = ca(r) + Z n J dr'c%( r r' )hR(r') (7.6) and the corresponding HNC equations can be written In gR(r) = /U3R(r), (7.7) UaR(r) VaR(r) 1> / dr'co(I r r' ) (e3U(r') 1) (7.8) Here a and 7 refer to an electron or ion of the two component plasma. The plasma direct correlation function co,(r) is provided independently from the two component plasma HNC equations. 7.1.1 Linear Theory We now turn to an evaluation of the mean potential Uo,R(r). First, we examine the results of the preceding section in the regime of weak coupling. Then the relation of the pair correlation function and the potential of mean force Ua,R from Eq. (7.4) becomes gaR(r) = eU. , 1 /3UR (7.9) Also, when Eq. (7.2) is linearized, the result is Cay V V.1 (7.10) and the linear HNC equations for UJR in Eq. (7.8) become UR(r) V R(r) + n7 J dr'c,(I r r' 1)UR(r') (7.11) In particular, for the electronradiator mean potential, Eq.(7.11) is UeR(r) VeR(r) + ne dr'cee( r r' 1)UeR(r')+ ni dr'cei(I r r' )UR(r') (7.12) Fourier transformation gives (k) ( k) + nie(k)UFR(k) UeR([1 n (k)(713) and the same procedure for the ion radiator mean potential is (R(k) k) + n (k)U(k)4) [1 nijCi(k) Solving these equations gives (feR(k) (1 nih(k)) + niea(k)VR(k)) Ue () = (7.15) (1 necee,,(k) nicii(k) + nnie (c,,ee(k)a(k) Cei(k)Cei(k))) (fR(k) (1 neee(k)) + nei(k)VR(k)) Uin(k~) =(7.16) A dimensionless form for c,, is obtained by scaling the distance with respect to the average electron spacing given by 47rr/3 = 1/ne. Then fe2 Here V* is the dimensionless pair potential and F is the plasma coupling constant taken to be small here. In that case Eq. (7.15) and Eq. (7.16) can be simplified to Vo R( k) UJR(k) = V (k) = 1 na(k) ni(k) (7.17) Here c(k) is the dielectric function for a weakly coupled two component plasma. To be more specific it is necessary to choose the form for the potential VaR(r). A simple and convenient form is the Deutsch potential (see Appendix G) V(r}) Ze ) '(7.18) where Za carries the sign as well as the magnitude of the charge number. The Fourier transforms are V7,(k) 47wZ~Z e2k2 (k2A + )1 (7.19) (k) = 1 + 3F(1 + Zj) (kro)2 (7.20) UCR(k) = 4rZZ~e2 (k2 + ) ((kro)2 + 3F (1 + Z,)) (7.21) and the inverse Fourier transform gives the desired screened potential /3UR(r) ZaZ7F (1 3F (1 + Zi) 62) 1 x 1\ 1 Ic)x (7.22) x = r/ro, 657 = A7/ro (7.23) 7.1.2 Nonlinear model The last subsection calculated the effective potential to determine the charge densities around the radiator at weak coupling gaR(r) 1 /3UaR(r) (7.24) with UaR(r) given by (7.23) from the linearized equations. An estimate for stronger coupling is obtained by exponentiating this result gaR(r) e3UCR(r) (7.25) with the same form for UaR(r). This is an uncontrolled approximation but is confirmed to be qualitatively correct even at strong coupling by comparison with MD simulation. Z=8, r=0.1, 6=.4 C *L A HNC S NLD LD o MD MD 2 0 1 2 3 4 r Figure 7 1: Plasma ElectronRadiator Pair Correlation Function Com parison using the hypernetted chain approximation (HNC), the nonlinear Debye model (NLD), the linear Debye model (LD), and with molecular dynamics (M!1)) results. [47] In Fig.(7.1.2), results for geR(r) are shown arising from HNC, nonlinear Debye, and Debye theories for an idealized plasma of electrons in a uniform positive background. Also results from MD simulations are shown[47]. This figure shows that at Z = 8, the nonlinear Debye and HNC results agree with each other and with MD results. Figure (7.1.2) shows the same quantity for differing values of Z, using the nonlinear Debye model and HNC. With increasing Z, the electron density at the radiator is enhanced. Note that the agreement is much better at lower Z. At Z = 30, there is considerable difference between the two results. However, the functional form of the HNC results, even at this higher Z remains the same as the nonlinear Debye form. If the charge is taken to be a fitting parameter, good agreement is found between the HNC results with Z = 30 and the nonlinear Debye results with Z = 25. 1=0.1 and 8=0.4 Z=1,4,8,20,30 Z=30 10 o HNC NL 0 1 2 3 4 r Figure 7 2: Plasma ElectronRadiator Pair Correlation Function Compar ison of results from the hypernetted chain (HNC) approximation and nonlinear Debye (NLD) model for several radiator charges. 7.2 Screened Ion Field The ion microfield distribution is defined by[44] Q(e) (6( E,)) (27)3 iJ dA e iA.E) (7.26) The brackets denote an equilibrium ensemble average for the plasma consisting of the electrons, ions, and radiator monopole. The screened field is defined by (see Eq. (6.14)) E E + fdppE (7.27) f dFpp where dF, denotes an average over the phase space for the electrons. The momen tum dependence can be integrated out leaving f dql,..dqN eC(ve+Vi++VR)Ee E = Ei + f dql,..dqNee3(Vee+Vi+VeR) SE, + J dqlee (qie) n, (qie {qj}). (7.28) All coordinates have been taken relative to the radiator which can be assumed at the origin. The electron density n, (qle I{qi}) is the number of electrons around the radiator for a given configuration of ions N f dq2e..dqNe6e3(Vee+VKe+Ven) n, (q {qi}J) f dq e .dqNvee,3(ee+V+v) (7.29) If the ionelectron interaction is neglected then the second term of (7.28) van ishes because n, (qle I{qi}) becomes spherically symmetric. This manyparticle correlation function can be obtained from a generalization of Eq. (7.8) ne (qij I{qi) = ne exp (/U (qle I {qj)) (7.30) U(r) Ve(r) + Vie (Ir r,) i, J dr'cee( r r' ) (et ') ) (7.31) J k For weak coupling, we apply an analysis similar to that from above that leads to VeR(k) + VI (k) kE t " Ue(k) e() e(k) = 1 neee(k) (7.32) E, E,j + dqleee (qle) n' (qe) (Ce3U(ljqie ) 1 (7.33) Ue' (r) (2x)3 Idkeikr vs (k) k JU(k(r) de (7.34) Se ( k)() (ql,) nee U, (r) = (27) 3 eikr (7.35) To linear order in UR (r) E, = (e (qj) 3 dqle, (qle) e (qle) Uj (I q1 q)) (7.36) S ees (qi) (7.37) The screened field E, in this weak coupling approximation is the sum of single particle screened fields eis (qi) e (qi) 3 dqleee (qle) We (qle) Ule (Iqie qil) This electron screening of the ion field is enhanced by the presence of the radiator, due to the factor of n' (ql). If in addition n' (ql) is linearized then the usual electron Debye screened field is recovered. 7.3 Ion Microfield Distribution If the screened field is taken to be a sum of single particle fields E, es (qi) (7.38) a then the expression in Eq. (7.26) Q(c) (c Es)) (27)3 dAei ~X eiAE) (7.39) (7 can be written Q(c) (27) / dieX (i (1 + (qa))) (27)3 ie die cG[ (7.40) where S(q) =eie(qi) . (7.41) A functional expansion of G[O] in powers of Q has as its leading term G[QI] J dqi4( (qi) giR(qi) + orderQ2 (7.42) This is the Baranger Mozer approximation for a two component plasma[48, 49, 50]. Note that the screening length for e8 (qi) is determined by the electrons while that for gaR(qi) is for both ions and electrons. 7.4 Constrained Electron Density The average electron density around the radiator, for given electric field value c can be estimated in the following way. Consider again the electron distribution in the presence of a given ion distribution as in (7.31) U (ql {qi}) = VeR (ql) + Vie ( q9e q) in, J dr'ce( r r' ) (e(r') 1). (7.43) The ion electron potential can be written in terms of the ion density SVie (qie qi) J drVe (qie r) i (r). (7.44) Now, consider the ion density to be the average density for the given constraint field c, ni (r)  ni (r,c) = ion density for a given ion field at the radiator satisfying I dr (VVe (0 r)) n (r,c) (7.45) Once the ion density is known the average electron charge density can be deter mined from the solution to (7.43) U (qe  e) = VeR ( ) + J drVi (e r) n (r,e) /3 n dr'cee( r r' 1) (e3U(r') 1 (7.46) This can be solved by the above method of linearization followed by ad hoc exponentiation. The determination of the constrained ion charge density has been discussed by Lado and Dufty, with practical methods available based on the above Baranger Mozer calculation of the microfield distribution[44]. 7.5 Electric Field Covariance The electric field autocorrelation function C(t) = (E E(t)) has a special emphasis in this study. This integral of C(t) is directly related to several transport properties, such as the low velocity stopping power S, the friction coefficient , and the selfdiffusion coefficient D, as well as the impact (fast fluctuation) limit for spectral line broadening by electrons[47]. The transport properties are related by [47] mo 1 S(v) o Z2r4 fdt C(t) (7.47) tip 0 As a first step towards the evaluation of the autocorrelation function, we evaluate the covariance C(O). The electric field covariance in dimensionless form is defined by C(0) (E E) (7.48) C2 where E is the sum of electron microfields at the radiator site. A direct calculation can be performed in terms of the one and two electron charge densities C(0) = 1 dr e(r) ne(r)e(r) + dr' ne(r, r')e(r')] (7.49) C2 \ with f drodrT2 N drNeou nle (di) dN3U = ne9eR (7.50) f dro drNep3 f drodr3 drdeu e(rT, r2) N(N 1) fdro dre3U (7.51) f dro drNCe " However, an expression can be obtained in terms of a mean force field emf defined in terms of the potential of mean force in Eq.(7.4). Writing the covariance as 4 4 C(0) (V, Ui. E) r= (Vo E) (7.52) Ze3 3Ze3 0 7 dr n (r) Vnne(r)) dr ne(r)emf(r) e(r) PZe3 pze e where the last expression defines the mean force field as 1 1 emf(r) V In ne (r,t) VUeR (7.53) pZe Ze Comparing Eq.(7.49) and Eq.(7.52) indicates that the two electron effects have been included in the mean force field. Figure (7.3) shows how the covariance increases monotonically with increasing Z for the idealized one component electron plasma. 7.6 Dynamics We now turn from structural properties to dynamical properties for the plasma electrons in the presence of a highly charged radiator. For the conditions of interest to this study, the plasma electrons are alvi weakly coupled. The nonlinear Vlasov equation[26] is then valid and gives +v V, + (F, + e + F i) V) fe(r, v, t) 0 (7.54) \ L ^ '7'e. 15 10 0 5 10 20 30 40 Z Figure 73: Electric Field Covariance Shown here as a function of Z. Tempera ture is 1000 eV, and density is 1024/cc. where FeR = rVeR (7.55) Fee V r dr' VeR(r r')ne(r',t) (7.56) 7 (7.56) Fe V Jdr' Vi(r r')nr', t) (7.57) and the number density for species a is n, = f dv f,(r, t). The quantities FeR, Fee, and Fe, are the forces calculated from the electronradiator potential, electronelectron correlations, and electronion correlations, respectively. For a hydrogen plasma, the ions will be as weakly coupled as the ions due to the identical charge. The coupling increases with ionic charge, so here we assume that the plasma ions are hydrogen to ensure the Vlasov equation is still valid. An ionic equation analogous to the electron formulation in Eq. (7.54) is thus found. With these two equations and the semiclassical electronradiator interaction, equilibrium solutions to the distribution functions for the electrons and ions can be determined. The equilibrium solutions are equivalent to those of the HNC equations; therefore strong electronradiator coupling is included in this formulation. Time correlation functions can be evaluated in the following manner. The dynamics in this case is determined from the linear Vlasov equation, and the time correlation function can be determined exactly in terms of the initial corre lations (determined from the equilibrium solution), the single particle dynamics arising from potential of mean force for the electron around the radiator, and the dynamical screening due to electron and ion correlations. However, we propose a simpler method based on conclusions drawn from the electric field autocorrelation function C(t) = (E(t) E). MD results for the C(t) expose several relevant qualitative features[47]. Figures (7.3) and (7.4) show an increase of the initial value due to correlations, a decrease in the correlation time, 1.0 MD 0.8 o Theory r=0.1, c=0.25Z 06 Z=4,8,20,30 0.6 o 0.4 Z=4 Z=30 , 0.2 0.2 0.01 0.1 1 cot P Figure 74: Normalized Electric Field Autocorrelation This quantity is nor malized to an initial value of one. The decrease of the correlation time and the increase in anticorrelation with increasing Z is observed. and an increase of anticorrelation, all corresponding to an increase of Z. Figure (7.5) dil'1,iv the total effect on the integral of C(t). The linear Vlasov equation dipl,l these three properties, but neglects the dynamic screening. Thus, for C(t), the features of interest to us can be captured by calculating exactly the initial correlations, and then using a single particle model for the dynamics. We propose that this program can be extended to the plasma dynamics in general and that a single particle model for the dynamics can be used. Then, the initial correlations are calculated exactly from equilibrium conditions, and the 0.8 .8 Area C(t)/C(0) \  Areat C(t) 0.6 S0.4 0.2 E 0.0 10 20 30 40 Z Figure 75: Integral of Electric Field Autocorrelation Function The true and normalized quantities, pi'. il11y relevant because of their relation to transport properties and line shapes, are shown. The conditions are identical to that of Fig. (7.4). dynamics following are that of a single particle in a selfconsistent field, or CAB(t)  Jdr lp/.,.)a(r(t),p(t))b(r,p) (7.58) where r(t) and p(t) are single particle trajectories. 7.7 Summary In this chapter we have made estimates for several important structural and dynamical properties for highly charged plasmas in conditions relevant to this study. By considering the regularized Deutsch potential in the HNC approximation, pair correlation function were determined and found to be in agreement with results from the nonlinear Debye model and from MD simulations. The ion microfield distribution, which has long been a vital part of line shape theories has been reformulated in a manner that includes ion and electron correlations consistently. Finally, the electric field autocorrelation function C(t), related to several interesting transport and radiative properties, is found to have several important qualitative features. The integral of C(t) is modified qualitatively in competing vi as the radiator charge increases. We find that once the initial correlations are dealt with exactly, the subsequent dynamics can be accounted for by using one particle dynamics. This discovery is used as motivation for treating dynamic correlation functions in general by the same procedure. These estimates are intended to complement the results from MD simulations. The MD results are expected to provide information as to the relevant domain of these estimates. Then, once agreement has been reached between our estimates and MD results, we can consider the physical basis behind the models used in this chapter to provide insight into MD results. CHAPTER 8 INCORPORATING ION DYNAMICS The previous chapters have dealt with the calculation of the line shape using the quasistatic ion approximation. This is sufficient for many lines of interest when the time scales for changes in the ion field are larger than the decay time for the dipole autocorrelation function. However, this relation does not ahlvP hold[3]. The ion microfield at the radiator will then change enough during the radiation time to affect the line shape. In this chapter, an analysis accounting for these ion dynamics effects will be outlined and discussed. The starting point once again is the general line shape formula written as a dipole autocorrelation function (Eq. (3.4)) I(t) = Trpdtdk(t) dk = deik (8.1) Recall that in using the quasistatic ion approximation, we extracted the screened ion field as the dominant interaction and separated the trace into separate traces over the atomic and plasma subsystems. Once this separation was effected, the trace over the plasma subsystem was constrained. It was not the entire phase space for the plasma subsystem, but only over a surface of that phase space which corresponded to those plasma states in which the screened ion field at the radiator had some specified value. Now, when considering ion dynamics, we perform the same separation, but we extract not a single property but a set of plasma properties as being most important to treat exactly. The set we choose here is the radiator position qR, radiator momentum p^R, and the screened ion microfield c. Denoting this set by b = {qRpR, p }, we have I(t) f db TrdTreikRpe(b b)d(t) (8.2) a p We have therefore separated the calculation into several steps, with the plasma trace being constrained over surfaces of constant b = {q, PR, e}. We then write the line shape in the form of an atomic physics problem I(t) [ dpR d(p)de Trf(a, )D(k,pR, t) d (8.3) where f(a, b) = Trp 6(b b) = (p)f(a, e) (8.4) P and D(k, pR, ,t) dqReikq D(k, b, t) (8.5) D(k,b,t) fl(a,b)Trpe(b b)d(k,t) (8.6) p is the transform D(k, p, E, t) of the constrained plasma averaged dipole D(k, b, t). In the same manner as before, once we find an expression for D(k, p, E, t) the problem reduces to an atomic physics problem. Again we use the projection operator formalism introduced in Appendix C to derive an equation of motion for D(k, b, t) and its Fourier transform. In Appendix F the calculations and the detailed form are shown for this ion dynamics equation of motion. In this chapter we use those results and the results from the quasistatic ion equation of motion to highlight the differences brought about by the inclusion of ion dynamics. Recall the staticion equation of motion for the constrained plasma averaged dipole operator. This equation was derived using the projection operator technique as outlined in ('i plter 6 and is given by + L(a, e) + B(e) D(, t) + dr M(, t )D(e, r) 0 (8.7) 0 Note here that if the value of the ion microfield were the only degree of freedom to the quasistatic ion plasma, then the integral term would be zero, and the terms in parenthesis would completely specify the dynamics of D(e, ). In this sense, the part in parenthesis can be referred to as the deterministic part of the time evolution equation, since it is completely specified by the ion microfield. Then the integral can be referred to as the nondeterministic part, as it describes the other degrees of freedom of the plasma not accounted for by the ion microfield. (Here the term "nondeterministic p 11 refers to not being determined by the ion microfield, which we chose as the most important plasma information.) In the derivation of an analogous equation for the case when ion dynamics are important, changes to both the deterministic and nondeterministic parts occur. The changes to the deterministic part of the equation will be explored first. In the quasistatic ion approximation, the radiator momentum was, from the definition of quasistatic ions, independent of the ion microfield. The dependence of D(e, t) on the radiator momentum could then be factored out, and thus the Doppler broadening of the line could be handled with a convolution of the Doppler line shape and the stark broadened line shape. The different time scale of the more general ion dynamics case makes this approach invalid due to the requirements of Newtonian mechanics. Consider the radiator momentum. The force on the radiator from the electric field will change the radiator momentum. Also, the momentum propagating the radiator will take the radiator to different spatial positions, leading to a change in the electric field experienced by the radiator. Therefore, even without yet considering the motion of the plasma ions, consideration of the physics from the different time scale for the ion dynamics case leads to a coupling of the radiator momentum and ion microfield. As shown in Appendix F, there are several changes to be made in the projec tion operator technique to derive an equation of motion for the averaged dipole operator. For purposes of the outline here, it will suffice to iv that the constrained averages need to be modified. For the quasistatic ion case, the plasma averages are constrained so that the trace is over those configuration points with a specified value of the ion microfield. The reasoning is that for the quasistatic ion case, the screened ion field is the dominant interaction. Once that is accounted for exactly in the equation of motion, the other effects can be treated with perturbation theory. For the ion dynamics case, the constraint is over a greater set of physical values. We choose this set to be three plasma values: the radiator position qR, the radi ator momentum PR, and the ion microfield e. When these degrees of freedom are chosen and used in the projection operator technique described in Appendix F, the deterministic part of the equation of motion for the average dipole operator changes from the quasistatic form + L(a, e) + B(e) D(e,t) (8.8) to the ion dynamics form + L(a, ) + B() + b D(k, t) (8.9) Here b = qR,pR, ) represents the set of plasma properties chosen, and () is the constrained plasma average of the time derivative of b. Thus, when including ion dynamics, the effect on the deteministic part of the equation of motion is the addition of several terms. The previous definition of the plasma averaged dipole operator was constrained only with the value of the ion microfield. Here, however, the definition includes constraints over the values of all three properties represented by b as indicated in Eq. (8.6). Besides these changes in definition, the deterministic part of the equation of motion includes an entirely new set of terms, given by (b)B(a/0b) acting on the averaged dipole. This form has a straightforward interpretation. The extent to which changes in the degrees of freedom b affect the averaged dipole operator, multiplied by the constrained averaged value of the time rate of change of these degrees of freedom, gives the time rate of change of the averaged dipole operator due to those degrees of freedom. With the calculations performed in Appendix F, these terms can be shown to represent the coupling between the various degrees of freedom represented by b. Explicitly, these terms are ( P V, QRE VP di / (8.10) \ /b 9b m .] ) Op, m ( ). Oj1 The types of terms appear in, for example, the Boltzmann equation and can be thought of in the same manner. The first term on the right hand side relates the change in the averaged dipole due to the radiator momentum; in the context of line shape theory, this term is responsible for doppler broadening. The next two terms arise from the force acting on the radiator from the ion microfield causing the radiator momentum to change. Recall from defining the system Hamiltonian that the charges comprising the radiator were separated, so that that center of mass degrees of freedom were collected in the plasma Hamiltonian, while the internal atomic states of the radiator comprised the radiator Hamiltonian. Classical electrodynamics gives the result that the force on a charge QR in an electric field e is QRE, and the force on a dipole d in an electric field E is (d V)E. That is what these next terms represent. The first is the force on the center of mass degrees of freedom from the electric field, and the second is the force on the atomic states represented by the dipole operator. The total force from these two terms will therefore change the radiator momentum with time. The final term represents the change in the ion microfield value at the radiator due to the momentum of the radiator. The physical content of this term can be understood by the fact that the quantity in angle brackets is related to the gradient of the field. It is instructive to relate this more general case to the previous results for the quasistatic ions. The first term on the right hand side of the above equation arises from qR; the next two terms arise from pR; the last term comes from c. Therefore, for the quasistatic ion case, the first term would still be present as doppler bN ..1. .iii but the time scale defined by the quasistatic ion case would have the other terms equal to zero. This equation shows directly that this gives an upper limit to the ion microfield strength and also the gradient. The repulsion between the radiator and plasma ions is what makes the quasistatic ion case suitable for many situations. Up to this point, the dynamics of the plasma ions have still not been taken into account; all that has been handled is the coupling of the radiator dynamics. Considering points in phase space is worthwhile here. Consider the equation of motion for the constrained average dipole operator with a specific set of values for the plasma properties b. The set of values b picks out a set of phase space points, each of which correspond to a plasma configuration that has the values of b. Now assume that the chosen properties completely specified the plasma, so that the only degrees of freedom to the plasma are b. Suppose one phase point with properties measuring b evolves to a state with properties b'. Then, if these properties completely specified the plasma, all the phase points with b would evolve to some point with values b'. In other words, all the points in the subspace constrained to have values b will move to the subspace constrained to have values b'. It is this sense that the part of the equation of motion above is referred to as the deterministic part. However, the chosen degrees of freedom do not completely specify the plasma, since these other degrees of freedom destroy this simple behavior. The dynamic electron broadening operator described in Chapter 4 is an example of this, and the ion dynamics described in this chapter is another. The properties b were chosen because it is expected that they would dominate the evolution of the phase points, and yet the n ,ivb1ody effects from the plasma electron and ion dynamics will cause some points to end up in the b' subspace that would not be there under the deterministic motion, and some points to not end up in the b' subspace that would have been there. The nondeterministic part of the equation will account for these other degrees of freedom. The dynamical electron broadening term from before will again be used. What is now needed is terms to represent the changing of the ion microfield due to ion dynamics. The ion dynamics terms will first be examined formally for their physical content, and then modeled approximately with a stochastic approach. This is in contrast to a direct ni ,ilbody approach, in which the exact interaction between the plasma ions and the radiator internal states is written, and then approximations are made until a tractable form suitable for calculations is found. The exact form for the equation of motion is + L(a, c) + B(c) + D(k, b, t) t + dr Jdb' M(b,b';t D(k, b', ) 0 (8.11) 0 where M(b, b'; t ) is found in Appendix F to have the form M(b,b';t7)X= f (a,b)Ti,. (b)LU(t r)QL(b')X (8.12) P Here U(t 7) = exp [QL(t 7)], and &(b) = 6(b b) are delta functions that constrain the plasma to have specified values for the set of properties b. The complementary projection operator Q is discussed in Appendix C. As in the quasi static ion approximation, Q projects out deviations from the constrained plasma averaged values. We discuss M(b, b'; t 7) by first analyzing the effects of the Liouville operator L = LR + Lp + 6L, which occurs explicitly twice in the expression for M(b, b'; t r). In Appendix F we follow these effects. The radiator Liouville operator cancels in both places, and several terms result from the remaining parts. One piece is related closely to the dynamic electron broadening term found in the quasistatic ion approximation 3 (b,b',t) f'(a,b)Ti1, ,(b)6LRQe(LP+L)t Q (b')LpR = LpnQe(L+L)tQ (b'6LpR (8.13) The above expression is written to second order in the interaction Liouville oper ator, which is why the density matrix and U(t r) are written to zeroth order in 6LpR. This expression differs from that found in the quasistatic ion approximation in that the constraining delta functions appear twice. The correlation between the free charge densities present through 6L at different times are thus calculated for different possible condition of constrained conditions. That is, in the quasistatic ion approximation, the relationship was between 6LpR at two different times, both of which were constrained to have the same screened ion field. In the ion dynamics case, M(b, b'; t T) is related to the correlation between 6LpR in a plasma state constrained to have the value b, and 6LpR a time t T later in a plasma state con strained to have the value b'. If this change in the value of b during the correlation time is small, then the dynamic electron broadening form found in the quasistatic ion approximation is recovered. Of the remaining terms of M(b, b; t r), the most important is 1 (b,b', t) f (a, b)Trpe (( )(b) ) eQLt ((b')i l) (8.14) where the definition of the Liouville operator as a differential operator has been used. (This operation is shown in Eq. (F.21).) Equation (8.14) actually represent a number of terms, since b is from a set of properties. However, due to our choice of plasma properties b to project onto, this second term can be simplified consider ably. Only the changes in the screened ion field is included. This results from the fact that the time derivative of the momentum is directly related to the electric field force on the monopole, and so projecting onto states in which the electric field is specified account for the collisional variations in the momentum. The resulting expression is 3i (b,b', t) f l(a, b) Tl ()eb (8.15) a' P(815 Still, this expression is complex and rather than analyzing it directly, we use a stochastic approach. The primary feature of Eq. (8.15) is a changing value of e, the screened ion field value at the radiator. For practical purposes this has been modeled by a master equation[43, 51]. A reasonable representation for the ion dynamics term is given by the master equation J de'M (, e', t)Xe) de' ( (e, e')X(e') We', e)X(e)) (8.16) The transition rates W(c, c') set the amplitude and time scale for changes in the field e. For time intervals shorter than this new time the static ion broadening of the previous section occurs. Then this field changes to a new value due to the ion dynamics and the new static ion broadening occurs. This phenomenological picture can be extracted from a reformulation of the exact statistical mechanics motivated by this physical picture. The objective of this section is to give an overview of this "stochastic" description of ion dynamics. In the end, practical approximations are required. However, in the spirit of the inquiry here those approximations are statistical rather than perturbative with respect to any of the charge correlations studied here. A stochastic model is then required for the transition rates. For these tran sition rates, the ion microfield transition will be modeled as a kangaroo process, which is defined by W(e, ')= A(e)B(') (8.17) so that in the transition rate there are no correlations between the initial and final electric fields[51]. With this model, the ion dynamics terms becomes S(e)D(e, t) v(e)D(, t) + v(e) de' Q(e')D(e', t) (8.18) Here, the first term represents those phase points leaving the configuration space with a specific value e due to the ion dynamics. The transition rate specifying the rate of departure due to ion dynamics only depends upon the field strength under consideration. The second term represent points which began at some value e', with probability Q(c'), which end up with field strength e due to the ion dynamics. The frequency v(e) is a parameter of the model, and at this point is arbitrary. With this stochastic model chosen, the formal equation of motion Eq. (8.11) becomes the approximate representation + L(a, e) + B(c) + ( ) D(k, 6, t) t + dT de' M(e,';t T)D(k,b', r) (8.19) 0 v(c) (D(k, t) de Q(e')D(k, b', t) = 0 With this result, we can compare the formalism of our approach with that of previous spectral line theories. With the exception of 6) which couples the components of b, all the terms above appear and have similar physical interpre tations in those previous theories. The more general expressions derived here can therefore be a direct guide into extending the theory to include these charge correlations in regimes where it is necessary to treat them correctly. CHAPTER 9 SUMMARY AND OUTLOOK In this study, we have reexamined the problem of spectral line broadening in plasmas. Our objective has been to explore the charge correlations between the plasma electrons, the plasma ions, and a highly charged impurity ion. To pursue this objective, we therefore were required to treat these correlations consistently and to derive exact expressions to contrast with current semiphenomenological results. Existing models for prediction of spectral line profile have been quite successful in general and it is expected that they should be recovered in many cases (e.g., neutral radiators or radiators with small charge). Two important external influences were felt during this study: previous theoretical work and MD simulations. There exists a large body of theoretical work and computer codes done previously by the plasma spectroscopy group at the University of Florida. These have been developed over several decades, and are believed to provide accurate diagnostics for experimental conditions currently achieved[7, 8]. However, due to the complexity of the analysis, the difficult problem of charge correlations are dealt with in different and possibly inconsistent vi. These correlations are in some cases simply neglected, and in other cases treated by using uncontrolled approximations. Despite the aforementioned success in their use as experimental diagnostics, there are several clear reasons why a more consistent treatment of the charge correlations is desired. The first reason is that by dismissing the correlations or using uncontrolled approximations, the process to extend the theoretical results to new regions of plasma conditions is not made explicit. By deriving results that include these correlations consistently, the ingredients needed in a more general theory are made clear. Rather than replace, our aim is to build upon them and to provide a guide into consistently extending them to treat different plasma conditions. A second reason for a more consistent treatment of charge correlations is related to the purpose of plasma spectroscopy. We have discussed its use as an experimental diagnostic. However, plasma spectroscopy is also a tool for exploring the plasma system. From this point of view, we treat the charge correlations in a consistent manner to give physical insight into the properties of the plasma. Another aspect that has driven our approach is the recent advances in MD simulations. It is no surprise that computing technology has continuously advanced and allows more detailed and longer running simulations. In addition to this, there have been theoretical advances in the classical description of inherently quantum systems through the use of regularized potentials. Thus there is new interest in the simulation of two component plasmas with highly charged radiators, which is the system of interest in this study. Specifically, the application of MD to electrons is an evolving new area. However, the plasma properties to determine through these simulations is not clear without a rigorous treatment of the expression for the line profile. We have determined here the appropriate expressions that treat charge correlations correctly. An example is the presence throughout this study of constrained averages of quantities. In many cases, an average quantity is needed to determine the line shape. This study shows that many of these quantities should not be averaged over all plasma states, but only over a set of states that obey a particular constraint (on the screened ion field, for example). This is a difficult theoretical problem but one that can be studied readily by MD. In addition to our formal expressions, we have provided simple estimates of various structural and dynamical properties of the plasma. These estimates are designed to be used in conjunction with MD simulations. The MD simulations provide the criteria of validity in these estimates, while the estimates provide simple physical insight into the MD results. When calculating these estimates, we followed the constraints brought about by the inherent classical nature of MD simulations. Thus we used regularized semiclassical potentials in our calculations to more closely follow the path taken in performing the simulations. With these potentials, we found that the free charge density did build in the region of the radiator as expected. However, the divergence in the free charge density that would have come about from using the Coulomb potential was avoided. Another example of the physical insight into the plasma processes brought about by our estimates is found in our calculation of dynamical correlation functions. When calculating the plasma correlations functions, it was found that the charge correlations among the plasma electrons and ions was most important in determining the initial value of these correlation functions. Once that initial value was found, the resulting dynamics was mostly due to oneparticle dynamics. More specifically, the primary new contributions of this work are two fold: a detailed derivation of a practical form for the line shape function under conditions of static ions, and an exposition of the more complex structure to be encountered in the treatment of dynamic ions. In the following, the primary results are summa rized in each case with comments on the outlook for their future applications. 9.1 Static Ion Line Shape Function For conditions of static ions the theoretical analysis of the line shape function is simplified in several respects. First, the Doppler broadening is decoupled from the plasma broadening. Second, the primary effect of the ions becomes a statistical distribution of Stark broadening by fields sampled from the microfield distribution Q(e). Our first new contribution appears at this point, with this microfield distribution defined over a two component plasma plus point radiator with all the correlations. Furthermore, the screened ion field E8 is defined in terms of the exact ni liiv1 ody screening by the electrons. Under conditions of weak coupling this field becomes a sum of single particle Debye screened fields and the microfield distribution becomes essentially that for a screened one component plasma as used in current theories. More generally, for highly charged radiators the electron radiator coupling becomes strong and this approximation should be revisited. The general forms given here provide the basis for this study. The remaining broadening due to the electrons was treated to second order in the interaction between the bound state electron distribution and the electrons of the plasma, in the presence of the ions. With the exception of similar early work by Iglesias and Dufty[15] for neutral radiators, this is among the first treatments of electron broadening in a two component plasma. The perturbation expansion is justified since the dominant monopole part of the radiatorplasma interaction has been extracted in the reference state. The broadening operator to first order describes shifts in the spectral line due to electrons penetrating the bound state distribution, discovered recently by the Hooper group[52]. This shift operator has now been described without approximation at this order of perturbation theory and related to the average electron density around the radiator. A new feature is that this average is constrained: the static ion distribution yields a specified value for the field at the radiator. The calculation of the equilibrium electron density around the radiator for large Z is a difficult problem that can be addressed in the semi classical form by the HNC integral equations. The fieldconstrained calculation is a new problem, never addressed before. An approximate means to consider the effects of the constraint is described in C'! lpter 7. The second order contribution includes the dynamical electron broadening and is given in terms of the autocorrelation function for the electron density about the radiator. Again, the average in the correlation function is constrained by the given ion field value. Even without this constraint, the autocorrelation function in the presence of a charge at the origin has been studied only in recent years. These studies have been for the idealized semiclassical case of electrons in a uniform positive background (jellium) with the radiator at the origin. The primary features of the MD study, limited to the electron field autocorrelation function, are captured by a simple mean field model for the dynamics of a single electron around the radiator. We have proposed this model as a practical method for calculating the charge density dynamics as well. In summary, C'! Ilpters 57 give a complete and practical form for the calcula tion of spectral lines under conditions for which the ion motion is negligible. The input for the formulation are electron structure and dynamics around the radia tor that require new theoretical methods for their analysis. First approximations suitable for practical implementation and assessment of correlations have been provided. It is expected that MD simulations of these quantities will be performed soon. 9.2 Ion Dynamics The extension of this work to include ion dynamics is more formal and less complete. It follows the initial work of Boercker, Iglesias, and Dufty[14] who provided the formalism but did not analyze the effects on electron broadening. An important feature of the formalism, discussed at the beginning of ('! Ilpter 8, is the selfconsistent treatment of the radiator center of mass motion and the electric field of the perturbing ions. Previous work in this group has included the stochastic change in the electric field, as described here as well, but neglected the relationship of this field to the radiator motion and the Doppler profile. This consistency problem is a matter of kinematics due to the fact that the rate of change of the center of mass position is proportional to the center of mass momentum, and the change in this momentum is proportional to the total field at the radiator. Thus any formulation of a spectral line shape including ion dynamics must include the deterministic form of the equation given here. The ion Stark broadening is again extracted explicitly as in the static ion case, with the same screened ion field. The first order perturbation term is also similar to that in the static ion result, with the same dependence on the constrained av erage electron distribution around the radiator. The electron broadening operator is now quite different. It still entails an autocorrelation function of the charge density around the radiator, but now with a double constraint: the field and radi ator momentum must have specified values in the initial and final states. If these quantities do not change during the correlation time the previous static ion results are regained. More generally, nothing is known about such correlation functions. Perhaps some guidance is possible from MD simulation. The main conclusion is a warning that when ion dynamics is important, the electron broadening can become quite complex. In addition to the Stark broadening of the ions, a dynamical ion broadening operator is also present. This is essentially another doubly constrained autocor relation function for the rate of change of the field. The effects of this operator have been modeled for plasmas with good success using a master equation. It is proposed here for practical purposes as well. In particular, the Kangaroo process is based on two plasma properties as input, the microfield distribution as its sta tionary state and a jump frequency for the fields. Both can be provided for a two component plasma without compromising the correlations among charges. In summary, this part of the thesis has exposed the important consistency conditions of ion dynamics and provided the beginnings for the controlled analysis necessary for practical model building. With new experiments, containing extreme regimes of temperature, density, and radiator charge, and the recent increasing interest in using MD simulations for attractive, highly charged systems, the problem of treating charge correlations consistently is important. Our study provides practical results to guide MD 79 simulations in the quasistatic ion approximation. For the more general case of ion dynamics, we have provided a foundation for future work to build on. APPENDIX A HOOPER'S GROUP COMPUTER CODES A set of codes has been developed based on recent theories of plasma spec troscopy. There are three main calculations to be done for the calculation of a line shape: the atomic wave functions of the bound radiator electrons, the electric mi crofield due to the screened ions, and the electron broadening term, which accounts for the effects of the free electrons beyond their screening of the ions. Accurate theories to deal with the first two have been in place for a number of years, while recent research has largely focused on the electron broadening term. In addition to the calculation of theoretical line shapes, other codes have been developed which apply experimentally relevant effects, such as doppler bN ,ii.I opacity, and instrumental b .. 1. iii. so that the theoretical results can be most directly compared with experimental line shapes. A.1 Atomic Wavefunctions The atomic wavefunctions to be calculated are for multielectron, ionized radiators [7]. To calculate these, a modified version of a collection of four programs developed by Robert Cowan of Los Alamos National Laboratory is used[17]. The starting point is calculating the oneelectron relativistic radial wavefunction for the electron configuration of interest. The HartreeFock approximation is used for this calculation. Some other needed quantities, such as energy levels, transformation matrices, and dipole and quadrupole interaction strengths, are also calculated and stored at this stage. A.2 ElectricMicrofield Calculation The electric microfield distribution is calculated using the Adjustable Param eter Exponential Approximation, or APEX[53]. This approximation is similar to the exponential approximation used in fluid theory for the pair correlation function. In the present case, the approximated function has the interpretation as a pair correlation function for a fluid with a complex potential energy, which introduces difficulties in a physical understanding. However, the approximation gives good and quick results that agree well with more rigorous theories for stronglycoupled and weaklycoupled plasmas. Once the APEX routine generates a microfield distribution, the .,vmptotic part of the distribution is calculated using a nearestneighbor approximation and the two are matched for a final result. The integral of the fieldconstrained line shape is taken over this distribution for the final result. A.3 ElectronBroadening Operator Much of the recent work in plasma spectroscopy has been focused on accurate evaluations of the effects of the plasma electrons[7, 8]. The codes calculate the electron broadening operator using three different methods. To describe these differing methods, the following terminology is used. The first term is to what order term in perturbation theory that the method uses to evaluate the electron broadening terms. These methods either evaluate these terms to secondorder or to all order. Recall that this study evaluated the electron broadening to second order in the interaction potential 6LpR. The second term is what type of dynamics to use for the electrons. Both quantum mechanics and classical mechanics are used. The final term refers to what type of interaction is calculated between the electrons and the radiator. Recall that in this study, the ionradiator interaction was dealt with as a dipole interaction due to the large repulsion occurring. In the past and especially with neutral radiators, the electronradiator interaction was also considered to be a dipole interaction. Recent theories treat the electron radiator interaction with the full Coulomb expression. With this terminology explained, the three methods are described next. The first method is a secondorder, quantum mechanical dipole theory[54, 55]. In this case, the interaction between the plasma electrons and the radiator is taken to be a pure dipole interaction, in the same way that the plasma ion radiator interaction was handled. The dipole interaction was then expanded using perturbation theory to second order for the result. This type of electron dipole interaction was originally used for neutral radiators, in which case the electrons were not pulled close to the radiator by the radiator charge. The second method is a secondorder, fullCoulomb quantum ti., ii [1, 7]. With this method, the plasma electronradiator interaction is taken to be the Coulomb interaction between the plasma electrons and the radiator. (In the analysis, this interaction may be separated into several parts, such as removing the monopole interaction from the interaction Hamiltonian, but all of the Coulomb interaction is present in the i,: ,1ii See Eq. (3.7) and Eq. (5.22) for how similar modifications were made in this study.) With this full Coulomb interaction, a perturbation expansion is again used to calculate the electron broadening operator to second order in this interaction. A third method is an allorder, fullCoulomb, semiclassical tli., .i [] Here, the same interaction is used as in the previous method. However, the electron broadening operator is calculated to all orders in this interaction. In addition, the plasma electrons are modeled as classical objects with well defined trajectories. The following sections give more details about the full Coulomb theories, as they are most relevant to the type of highly charged radiators considered in this study. A.3.1 SecondOrder, Full Coulomb, Quantum Theory As the plasmas under consideration achieve higher electron densities, the dipole approximation gives inaccurate results for the electron broadening [7]. The effect of the perturbing electrons then must be evaluated using the full Coulomb interaction. The reason that the dipole approximation is still valid for the plasma ions but not the plasma electrons is due to the attraction between the positive radiator and the electrons, which is strong enough to allow the electron perturbers to penetrate the radiator orbitals. This theory was able to calculate the line shifts due to the electrons from a relaxation theory in a consistent manner. In this calculation, the electron broadening operator is split into two parts: a time independent term corresponding to initial correlations that shift the line shape, and a time dependent term that both shifts and broadens the line shape. These two terms are calculated to second order in the radiatorperturber interaction potential. Limiting the calculation to second order places two main constraints on this calculation. First, the calculation is valid in a range that corresponds to less than twice the electron plasma frequency[7]. Second, it is valid only for weak collisions. This weak collision constraint means that the kinetic energy of an average plasma electron must be large enough so that it spends a small amount of time near the radiator so that its momentum change is negligible. The valid temperature region is given by z(z 1) kT> Z( (Ryd) (A.1) which is determined by comparing the perturber's kinetic energy far away from the radiator and at its closest approach to the radiator[7]. The noquenching approximation is applied to the theoretical derivation. This indicates that there are no nonradiative transitions between initial and final states, which is valid when the energy difference is large. The perturbing electrons are described using positive energy Coulomb wavefunctions, and the screening that arises from electron correlations is accounted for by using a cutoff at the Debye length. (This method of using a cutoff is not the only available method, but it allows for convenient comparison with other theories.) The static shift is split into two parts, one that is first order in the interaction potential, and a term that includes all higher order effects. When this is done, it is found that the first order shift arises solely from the monopole interaction of an electron that has penetrated the radiator orbitals. A.3.2 AllOrder, FullCoulomb, SemiClassical Theory When the perturbing electrons undergo strong collisions with the radiator, an allorder classical theory for the electrons can be used [8]. The structure of the derivation is similar to the secondorder theory, in that the radiatorperturbing electron effects are described by an electron broadening operator split into a static part and a dynamic part. The static part describes the timeindependent initial correlations, which is separated into a term that is first order in the interaction potential and a term that contains all higher order effects. The approximations made in this theory are also similar to the second order theory. The noquenching approximation is used, and it is assumed that there are no dynamical correlations between the electrons and the ions. Thus the electrons only affect the ion broadening by screening the ion field. It is also assumed that there is only one strong electron collision during the radiation time, so that the electron broadening effect on a radiator is given by considering only one electron. This approximation would also indicate that there are no electron correlations. To correct for this, a Debye length cutoff is used to account for electron screening. The interaction of a single perturbing electron with the radiator system is handled with a multiple expansion. The (modified) monopole term and the dipole term are dominant here, and angular momentum rules allow only a limited number of nonzero terms. This semiclassical theory treats the perturbing electron as classical objects. This assumption places certain limits on the validity of the theory. There are several main changes needed to evaluate the electron broadening term in this way. First, that part of the density matrix which arises from the perturbing electron Hamiltonian is replaced with its classical analogue. The physical interpretation of this is that those plasma electrons outside the interaction range are in thermal equilibrium and their velocity distribution can be handled using a Maxwell Boltzmann velocity distribution. For this to be valid, it is required that EFermi << kBT (A.2) where EFermi is the plasma Fermi energy given by Fermia { 32e }23 (Ryd) (A.3) which ensures that the degeneracy effects are small for the plasma electrons [8]. A second change is to replace the quantum timedependent electronradiator interaction operator with an interaction term that depends on the coordinates of the perturbing electron as it moves along its path. This hyperbolic path is calculated in a rotating frame. A third change is that the quantum trace must be replaced by a classical trace. Finally, to account for the quantum diffraction effects for distances close to the radiator, a minimum distance cutoff is used. The results from the secondorder and allorder theories are in agreement with each other for current experimental results. APPENDIX B DOPPLER BROADENING IN THE QUASISTATIC APPROXIMATION In C'!i pter 5, it was stated that for the quasistatic ion case, the radiator center of mass degrees of freedom were not coupled to the plasma degrees of freedom, which allows a doppler term to be factored out in the time dependent equation. This section will provide the details of those calculations. At the beginning of C'! lpter 5, the line shape formula was written in the form of I(t) dt dk (t) (B.1) where dk(t) eiHt/hdeikq"eiHt/h (B.2) was defined in Eq. (3.5). The details of the quasistatic ion approximation were then used to separate out the radiator center of mass degrees of freedom from the other plasma degrees of freedom. In that approximation, the motion of the ions in the plasma (and the radiator in particular) was considered to be independent of the rest of the system. Then the density matrix factors into P = Pc.o.m.Pp (B.3) where pc.o.m. depends on the position and momentum of the radiator, and pp depends on all other degrees of freedom. Then Eq. (B.1) can be written I(t) Tr [p..m.ppeik*qdd(t)eik(t)] STr [pc.o.m.ikq Tr d (t)] c.o.m. p = (k, t)Tr [ppd d(t)] (B.4) P In the above, the trace was separated and performed separately over the center of mass degrees of freedom and the degrees of freedom of the rest of the plasma; 4(k, t) describes the effect of the radiator motion on the line shape and is responsible for doppler broadening. We now turn to evaluating this expression, as the quasistatic ion approximation allows K{(k, t) to be considerably simplified. Recalling that the quasistatic ion approximation predicts no acceleration for the ions during the time scale of interest, and also that the ions are treated classically, the position of the radiator center of mass at the time t is qR(t) = (O) + Pt (B.5) where PR is the (constant) momentum of the radiator. Using this in K(k, t) gives S(k, t)= Tr [pc.o.m.ik(pt/m)] [ dp O(p)eik.(pRt/I) (B.6) c.o.m. If the frec q', i,idependent line shape is considered, it is found that the Doppler effect can be incorporated into the line shape through a simple convo lution. Recall that the frc quii'in,dependent line shape is related to the above timedependent line shape through I(w) = Re dt eilI(t) (B.7) We have shown that I(t) is a product of a doppler line shape (k, t) and the dipole autocorrelation function for static ions. Taking the transform converts this product into the following convolution I(w) = dw'ID( o')J(w') (B.8) where ID(w) is the Doppler line shape given by[17] ID(') exp G') (B.9) V2o2 \, 2j2 88 2j.2 /2 (2kBT (B.10) and J(w) is the static ion line shape with J()) = Re dt eit (d d(t)) (B.11) 0 With this the doppler broadening is fully accounted for in the quasistatic ion approximation. APPENDIX C PROJECTION OPERATOR EQUATION OF MOTION In this study a projection operator technique is used to derive an equation of motion for both the quasistatic ion line shape and the more general ion dynamics line shape. The definition of the projection operator is different in each case, yet the general derivation is the same in both cases. Here the general derivation of an equation of motion for an arbitrary operator is presented. The results are later used in Appendix D and Appendix F. We choose an operator P. The actual definition of P will vary for different applications. If we assume that it is a projection operator, then P2 = P. We also define a complementary operator Q 1 P. Note that QP = 0, since QP= (1 ) p p2 =p p (C.1) and also that Q is a projection operator itself (2 = Q). Finally, we assume that for all cases of interest, P(O) = 0 and P aX = jPX. The Liouville equation for some general operator X(t) is + L X(t) 0 (C.2) Acting from the left with P and inserting 1 = P + Q gives P + L (P + Q)X(t) 0 (C.3) Letting the projection operator commute with the derivative and separating the term that includes Q gives S+ PLP PX(t) PLQX(t) (C.4) The procedure is to calculate an expression for QX(t) and substitute it into Eq. (C.4). We follow the same method as above to derive an analogous expression for the complementary projection operator Q, resulting in + QLQ QX(t) QLPX(t) (C.5) or S[QX(t)] = QL [QX(t)] QLPX(t) (C.6) A general solution for an equation of the form 8X = AX + B(t) (C.7) is[43] t X(t) = eAtX(O) + dr eA( ()B() (C.8) 0 A solution for QX(t) is then t QX(t) = eQLtQX(O) dr e L(t) QLPX(t) (C.9) 0 Using this result in Eq. (C.4) gives ( + PLP PX(t) PL eQLtQX(O) dreQL(t7) QLPX(t)) (C.10) 0 The specific form of the projection operator and the details of the system under consideration is used to modify this further. APPENDIX D AVERAGE DIPOLE EQUATION OF MOTIONQUASISTATIC CASE This chapter uses the projection operator results from Appendix C to derive an equation of motion for the time dependent constrained average dipole operator. Here the focus is on finding the result in the quasistatic ion approximation. In Eq. (5.17), the constrained plasma averaged dipole operator was defined as D(, t) Tr [p(a, )d(t)] (D.1) P p(a, c) =fl(a, c)pb(, Es) (D.2) The equation of motion for D(e, t) will be put into the form S+ L(a, ) + B D(, t) + dr M(c, t T)D(, r) 0 (D.3) 0 We define a projection operator P PX Trp(a, e)X (D.4) First note that P satisfies the definition of a projection operator, since P2 p p2X = PX ( Trp(a, c)PX Trp(a, e) PX= PX (D.5) P ) (P In the above, the fact that PX is independent of plasma coordinates (because it is an average over plasma coordinates) was used to bring PX out of the plasma trace. Then, the constrained trace of just the constrained density matrix equals unity, giving the last step. Another consequence is the action of the projection operator on d(t) Pd(t) (Trp(a, e)d(t)} D(E, t) (D.6) \P / We also define the operator Q 1 P. The Liouville equation for d(t) is written as ( +L d(t)= 0 (D.7) The result from Eq. (C.10) is then ( + PLP Pd(t) = PL eQLtQd() J d QL( T)QLPd(r) (D.8) o 0 Using Eq. (D.6), we then have the equation of motion for D(e, t) ( + PLP) D(, t) PL eLQd() J dr eQ(t)QLD(e, t) 0 (D.9) The initial value term for the dipole operator will cancel, since Qd(0) = 0. This is because d(0) does not depend on plasma coordinates, so Qd(0) (1 P)d(0) d(0) Trp(a, )d(0) P Sd(0) Trp(a, e) d(0) d(0) d(0) = 0 (D.10) so the form for the equation of motion is + PLP )D(, t) PL dr eQL(t) QLD(E, r) (D.11) 0 The specific form of the Liouville operator for the quasistatic ion approxima tion can now be used to write the equation of motion in a form in which different plasma effects are separated. Recall that the total Liouville operator was written as L = Lp + LR(E) + 6LpR (D.12) where Lp included the plasma degrees of freedom and their interactions, LR(E) was the Liouville operator for a radiator in an electric field e, and 6LpR is the Liouville 