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- Permanent Link:
- http://ufdc.ufl.edu/AA00003303/00001
## Material Information- Title:
- Higher order microfield effects on spectral line broadening in dense plasmas
- Creator:
- Kilcrease, David Parker, 1952-
- Publication Date:
- 1991
- Language:
- English
- Physical Description:
- vii, 154 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Approximation ( jstor )
Atomic physics ( jstor ) Electric fields ( jstor ) Electrons ( jstor ) Energy ( jstor ) Ions ( jstor ) Line spectra ( jstor ) Plasmas ( jstor ) Quadrupoles ( jstor ) Radiators ( jstor ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1991.
- Bibliography:
- Includes bibliographical references (leaves 149-153).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by David Parker Kilcrease.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 001689199 ( ALEPH )
AJA1235 ( NOTIS ) 25138674 ( OCLC )
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HIGHER ORDER MICROFIELD EFFECTS ON SPECTRAL LINE BROADENING IN DENSE PLASMAS By DAVID PARKER KILCREASE 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 1991 All we know of the truth is that the absolute truth, such as it is, is beyond our reach. Nicholas of Cusa, 1401-1464 A.D. ACKNOWLEDGEMENTS I am extremely grateful to all of the people who have helped me in my scientific career; there are too many to enumerate them all. I am also most grateful to the taxpayers for making it possible for me to participate in the great enterprise of scientific research. TABLE OF CONTENTS page ACKNOWLEDGEMENTS ........................................... iii ABSTRACT ......................................................... vi CHAPTERS I. INTRODUCTION ................................................1 1.1 The Big Picture ................................................1 1.2 Focus of This Work ........................................... 3 1.3 Outline ................................................ ...... 5 II. THEORETICAL BACKGROUND TO THE LINE BROADENING PROBLEM............................... 8 2.1 Dense Plasma Fundamentals...................................9 2.2 Historical Background........................................ 17 2.3 Theoretical Formulation....................................... 18 III. HIGHER ORDER FIELD EFFECTS .......................... 39 3.1 The lon-Quadrupole Effect ...................................39 3.2 Atomic Data by Perturbation Theory Solution ...............54 3.3 Atomic Data by Numerical Solution ..........................70 3.4 Electron Delocalization and Field Ionization................. 78 IV. RESULTS AND DISCUSSION .................................89 4.1 The Lyman a Line...........................................91 4.2 The Lyman P Line...................... ........... ....... 92 4.3 Conclusions ........................... .................. 99 APPENDICES A. SOME EXPRESSIONS IN USEFUL UNITS ...................123 B. THE APEX CONDITIONAL DISTRIBUTION FUNCTION ....125 C. PARABOLIC COORDINATES ...............................130 D. IMPORTANCE OF THE ION-QUADRUPOLE EFFECT.......132 E. FINE STRUCTURE CORRECTIONS .........................138 F. COMPUTER CODE DOCUMENTATION.....................144 REFERENCES ..................................................... 149 BIOGRAPHICAL SKETCH ...................................... 154 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 HIGHER ORDER MICROFIELD EFFECTS ON SPECTRAL LINE BROADENING IN DENSE PLASMAS By David Parker Kilcrease May 1991 Chairman: Charles F. Hooper, Jr. Major Department: Physics Radiating atoms in dense plasmas are highly affected by their local plasma environment. We examine the effect of this environment on hydrogenic radi- ators by using an atomic wave function basis set that is itself a function of the plasma electric microfield for the calculation of spectral line profiles. Our theoretical development includes the effect of static ion perturbers up to the quadrupole term in the multiple expansion of the radiator-perturbing ion in- teraction, as well as dynamic electron broadening up to second order in the radiator-perturbing electron interaction. Effects due to the presence of the uni- form ion-electric field are treated exactly within this framework, and the field gradient is then treated as a perturbation by inclusion of the ion-quadrupole term. We employ a basis set for the representation of the upper state of the transition that includes states of principal quantum number n as well as n+1 to allow for the field mixing of these two manifolds. The ion-quadrupole effect is treated in a new way that includes ion correlations by using a renormalized vi independent-particle model for the perturbing ions. We also present a pre- liminary assessment of the importance of field ionization on the spectral line shapes. Our results are compared with previously reported calculations using field independent matrix elements, neglect of possible field ionization and the independent-particle model of the ion-quadrupole effect. Hence, we are able to identify the most important effects that arise from the use of the field depen- dent basis set. These comparisons also allow us to examine the importance of ion correlations when determining the ion-quadrupole effect. CHAPTER I INTRODUCTION In this dissertation we, that is you and I, will be examining the theory of spectral line radiation from highly ionized atoms in dense plasmas. Before we go into this in detail it will be helpful and illuminating to stand back and look at the relevance of this from a wider perspective; to take a look at the big picture. 1.1 The Big Picture In general, a plasma can be defined as a statistical system containing mo- bile charges.1 We will be concerned with a subcategory of this definition and will limit ourselves to discussing electrically neutral plasmas associated with hot highly ionized atoms. This is no great limitation since over 99 percent of the known universe is covered in this category.2 With the aid of sophisti- cated astronomical detection equipment, most of what we see of the universe is electromagnetic radiation emanating from a variety of objects ranging from ordinary stars to extremely luminescent quasistellar objects located at the edge of the known universe. The matter in all of these entities exists almost entirely in the plasma state; plasmas whose constituents range from ordinary ionized atoms to exotic matter-antimatter electron pairs. The radiation reaching the earth from these distant bodies runs the gamut from radio waves, through the visible spectrum and into the realm of X-rays and 7-rays. It is easy to see that an understanding of the behavior of these plasmas can and will give us great insight into the structure and workings of the universe in which we live. 2 Of more mundane concerns are the plasmas encountered here on earth. These are evident in such natural phenomena as lightning and fire. Man- made plasmas are also numerous in example, ranging from ordinary fluorescent lighting to terrifying large scale thermonuclear explosions that result from the fusion of hydrogen isotopes. Since the early 1950s, one of the primary goals of plasma physics has been to bring under control these tremendous quantities of energy liberated by thermonuclear fusion. When this goal is achieved, we will have succeeded in obtaining access to a nearly unlimited supply of relatively clean and safe energy that will last into the foreseeable future.3 Due to the extremes of physical conditions under which many plasmas ex- ist and due to their often ephemeral nature, making direct measurements of the composition, density, temperature and other physical parameters that are necessary for a clear understanding of their inner workings is often impossi- ble. What is needed is an indirect nonintrusive probe of the plasma interior. The radiation emitted or absorbed by these plasmas is the ideal answer to this problem. Isolated, partially ionized atoms can emit or absorb radiation that results from atomic electrons passing from one energy level to another. This radiation is emitted or absorbed in discrete amounts leading to distinct and identifiable sharp spectral lines. These lines have a slight width, called the natural width, due to quantum fluctuations in the atom's own electromagnetic field. As long as the atoms remain isolated, the spectral lines remain rela- tively sharp. When linked to their plasma environment, however, the situation changes radically. The amounts of energy emitted or absorbed by the atoms are now affected by the plasma, often revealing pertinent information about this environment. Thus we may explore the atom's surroundings indirectly through changes in the profile of its spectral lines. This technique is employed 3 in many areas of science to reveal information about the internal structure of matter in a noninvasive way by examining emitted or absorbed radiation. Some examples of this application are nuclear magnetic resonance (NMR) and M6ssbauer spectroscopy.4 In this dissertation we will be concerned with spec- tral lines of emitted radiation from plasmas although the formalism applies equally well to spectral lines originating from the absorption of radiation. The line shapes will reveal information about the density and temperature in the vicinity of the radiating atoms, regardless of whether they are located in a simple laboratory experiment or in a distant star. 1.2 Focus of This Work The main focus of this work will be on the shapes of line spectra from dense, high temperature plasmas consisting of atoms of moderate atomic num- ber, Z. These terms "dense" and "high" are quite relative and historically have meant the densest and hottest plasmas obtainable in laboratory experiments for a particular element at that particular point in time. As an example, in the mid 1960's these terms referred to electron number densities, ne, of around 1017 cm-3 and temperatures corresponding to a few eV for elements such as hydrogen and helium.5 Today, high density refers to ne = 1024 to 1025 cm-3 and temperatures corresponding to 600 to 1000 eV for elements such as argon and krypton with Z = 18 and 36 respectively.6,7,8,9,10 One method of creat- ing these dense plasma conditions in the laboratory is by laser driven inertial confinement. Current experiments using this method have created conditions of many times solid density for periods of a few hundred picoseconds.9 The mechanism of compression employs high intensity laser light to symmetrically irradiate spherical targets containing the material under investigation.11 When 4 this energy is rapidly deposited in the outer layers of the target (mostly by in- verse Bremsstrahlung), these layers are quickly vaporized and blown outwards away from the target. This propels the remainder of the target material in- ward. This is in close analogy to the way a rocket works by combusting fuel in an engine and expelling it in one direction in order to be propelled in the other direction. If our laser irradiation is sufficiently symmetrical, the target will be reduced in size many times as it is compressed, and resulting shock waves will heat the core to a high temperature. This compression and heating can produce the plasmas of high density and temperature that we wish to examine in this dissertation. Approximate descriptions of spectral line shapes emitted from dense, hot plasmas are usually derived from theories that use an atomic physics descrip- tion of atoms that are isolated from their environment. In reality, the atomic wave functions for the radiating atoms can be strongly perturbed by surround- ing charged particles, and in extreme cases it is not realistic to think of the atoms as separate entities at all but rather as clusters of ions surrounded by clouds of electrons. Radiation from atoms in this highly interacting environ- ment will run the gamut from continuum to discrete. Therefore, in order to construct a tractable method for the calculation of spectral line shapes, we must often, somewhat arbitrarily, distinguish between the radiator and its per- turbers. However, the more influence of the surrounding particles that we can include in the description of our radiating atoms at the outset of our develop- ment, the better off we will be when considering further perturbations. Our goal in this work will be to include some of the higher order effects of the plasma microfield on atomic spectral line emission. To this end, we will use for our zero-order atomic wave functions, numerical solutions to Schr6dinger's 5 equation for an atom in a uniform electric field. These field dependent solutions are supplemented by solutions obtained from a systematic perturbation theory treatment for small field values. The zero order atomic Hamiltonian will thus contain the perturbing ion-dipole interaction and higher order corrections will describe nonuniformities in the plasma electric field. We will include the first of these higher order corrections, the ion-quadrupole term, to account for the effects of the field gradient.12'13,14,15 Griem16 has pointed out as early as 1954 the possible importance of this field dependence, as well as the ion quadrupole effect, as a source of spectral line asymmetry. The plasma electric microfield can produce additional effects on the radi- ating atoms beyond those given by the simple field dependence of their atomic wave functions and energy levels. Atomic bound state energies which were dis- crete in the absence of the plasma electric field now become resonance states with finite widths. The formerly bound electrons are now free to quantum mechanically tunnel through the potential barrier created by the electric field. Although atomic electrons that lie deep in the radiator potential well are little affected by the plasma electric field, high lying electron levels are perturbed to the point where they are, for all practical purposes, no longer bound to the atom. This field ionization phenomenon could have important consequences for the relative populations of the radiator excited states. 1.3 Outline Chapter 2 focuses on the development of a general plasma line broaden- ing theory and the accompanying approximations. We discuss fundamental length and time scales for the interactions of the two main components of the plasma with the radiator: the highly charged ions and the unbound plasma 6 electrons. We also discuss under what conditions and where in the line profile the phenomenon of perturber motion will be important (both ion and elec- tron). Next follows a discussion of the strength and nature of the various interactions between the plasma particles. From this consideration, we decide which interactions are important and which can be ignored. A brief account of the historical development of the main ideas of spectral line broadening theory is presented. We find the historical roots for the ideas of Stark, Doppler and electron broadening in the late 1800's and early 1900's. Next follows a detailed derivation of a calculable form of the line shape function. We first present a general formalism appropriate for multi-electron radiators that includes field dependent wave functions, an exact treatment of the perturbing ion radiator-dipole interaction, field dependent NLTE level pop- ulations and field dependent electron broadening. This general formalism is then restricted to the case of hydrogenic radiators with no lower state broad- ening. We also discuss a method for approximately including field induced resonance widths in the line shape. Chapter 2 ends with a brief note about other line broadening phenomenon that may need to be included before com- parisons with experiment can be carried out. Chapter 3 begins with the development of a theory for the approximate evaluation of the ion-quadrupole effect. We present approximations for the probability function for the field gradient term as well as an approximation for the average field gradient constrained to have a given field value. These ap- proximations include ion-ion correlations and represent an improvement over previous theories. We then compare our calculation of this field gradient with previous calculations and simple approximations. Next, we discuss the methods 7 of calculation of field dependent wave functions and matrix elements by pertur- bation theory and by direct numerical solution of Schr6dinger's equation. The numerical solution technique is also used to calculate resonance widths that give an approximate measure of the lifetimes of the field dependent resonance states produced by the presence of the plasma microfield. These lifetimes are then used to study the effect of the resulting possible level depletion on the line shape. We discuss a simple population kinetics model for this purpose. In chapter 4, we present some theoretical spectral line shapes resulting from our work. We calculate line shapes for the La and Lp lines of hydrogenic argon at kT = 800 eV and for an electron density range of 1 x 1024 to 1 x 1025 cm-3. We then examine the importance of the various effects studied in this dissertation on the spectral line shape, width and asymmetry. We close with some conclusions and suggestions for further work. Appendices discuss several useful topics and give some details of calcula- tions presented in the main text. We have a discussion of physical units, some details of the model used to calculate the average field gradient required for the treatment of the ion-quadrupole effect, a discussion of parabolic coordinates and the details of the perturbation theory calculation of the field dependent fine structure corrections to the radiator Hamiltonian. CHAPTER II THEORETICAL BACKGROUND TO THE LINE BROADENING PROBLEM We will now develop a theory17,18 for the description of spectral line shapes in dense plasmas. The physical conditions we will be interested in considering will be pertinent to inertial confinement fusion plasmas. These plasmas will be at, or near solid density and at relatively hot temperatures (i.e. hot enough that the plasma is not quantum mechanically degenerate). We will also be interested in systems where the plasma ions have much greater mass than the electrons. This difference will allow us to consider different domains: one defined by the fast moving or dynamic electrons and one appropriate for the description of slow moving or quasi-static ions. This will allow us to proceed with the development of a simplified formalism. The physical systems and conditions that we will examine in this disser- tation are restricted to argon radiators immersed in a pure argon plasma at a temperature corresponding to 800 eV and an electron number density rang- ing from 1024 cm-3 to 1025cm-3. These densities are high enough to bring out the higher order field effects but not high enough for the plasma to be in the degenerate electron regime. In local thermodynamic equilibrium (LTE), the line shape is not very sensitive to small variations in temperature. Con- sequently, we will not examine the temperature dependence of the line shape. These physical conditions are also relevant to current experiments9 that employ argon radiators. 9 2.1 Dense Plasma Fundamentals In dense plasmas there are several fundamental length and time scales that are important for understanding the interactions among the plasma particles. Since our plasmas are dynamic systems, the particles are in constant motion and travel, on the average, a characteristic distance in a given relevant time interval. This fact allows us to establish some relations between length and time scales. The use of these scales will allow us to classify the various plasma particles into categories that are governed by interactions of differing strengths. This will be important when deciding what approximations are suitable and consistent in our description of the plasma. Without approximations we would be hopelessly lost in a morass of ~ 1023 coupled equations of motion. A fate, preferably, to be avoided. We will now discuss some relevant length scales1,19,2 (see Appendix A). The ion-sphere radius ro is an estimate of the average distance between two ions of species j. It is given by ro, = ( ) 1, (2.1) where nj is the number density of species j. Another important quantity is the Debye length AD. It is an indication of the distance from an ion beyond which screening of its charge becomes significant. It is given by ( kT 1/2 (2.2) AD, = (4Z2e2n (2.2) where k is Boltzmann's constant, T is the temperature and Ze is the charge of species j that forms the screening cloud. For more than one component, the 10 resultant screening length is AX2 = Ej AX2 where the sum is over the different species. The thermal de Broglie wavelength A gives an estimate of the quantum mechanical wave nature of the plasma particles. It is given by A 27rh2 1/2 (2.3) A = mkT ' where m is the particle mass. Except for the factor of 27 this is just the quantum mechanical wave length of plasma particles possessing average kinetic energy kT. It gives us an estimate of the spatial extent of the particle wave packet. The radial dimension of an atomic ion can be estimated by considering the radius of the Bohr orbit for the most probable quantum state, and writing the radius as, rn = aon2/Z ; (2.4) n is the principal quantum number for the atomic ion, ao is the radius of the first Bohr orbit for hydrogen and Z is the nuclear charge. In order to relate time scales to characteristic lengths we need an estimate of the velocity for particles of species j. Using the magnitude of the most probable velocity for classical motion, we have Vmp,j = (2kT/mj)1/2 (2.5) We have now defined the basic quantities which will allow us to make estimates of several important time scales of interest. Additionally, the length 11 scales will be used in determining appropriate scale factors for perturbation expansions. We will now look at the time scales relevant to motion of the plasma parti- cles, specifically the perturbing electrons and ions. It will be shown in section 2.3 that the line shape function I(Aw) can be represented by the transform of a dipole autocorrelation function C(t) such that I(Aw) = f exp(iAw t)C(t)dt (2.6) where Aw is the separation from line center and C(t) is a decreasing function of t. When t > 1/Aw, the exponential begins to oscillate rapidly causing the contribution to the integral from the integrand in that region to be small. So, for a given value of Aw, the integral is determined, for the most part, by C(t) such that 0 < t < 1/Aw. Hence, we can define the time interval of interest for the line shape at the distance from line center Aw as r = 1/Aw. If the duration of a perturber's collision with the radiator is significantly greater than the time of interest 7 corresponding to the part of the spectral line Aw, we may regard the perturber as being stationary during the time of interest and treat its perturbation as static. This is known as the quasi-static approximation. If the duration of collision does not meet this criterion, we must regard its perturbation as dynamic and explicitly take its time dependence into account. For a particular separation from line center Aw, the effect of perturbing ion motion will be negligibly small if17 F(t) F(t) << Ai(F)|, (2.7) where F(t) is the perturbing ion electric field strength at the radiator, F(t) is its time derivative and Awi, (F) represents the separation from line center, Aw, 12 due to the static linear Stark effect. Note that Awi,f(F) = wi,f(F) wi,f(0), where i and f refer to the initial and final state of the transition, respectively, and wi,f is the transition energy wi,f = wi-wf. The linear Stark shifted energy for level i is wi(O) + 3niqihF/me. We use the parabolic representation (see Appendix C) and take F = Zie/r2 where r is the distance from the radiator to the perturbing ion, and obtain on substitution into Eq. (2.7) Aw = 3>n-nq (2.8) 2Zmer2 r where ni,f and qg,f are the parabolic quantum numbers for the initial and final states and v is the perturbing ion velocity. Eliminating r from both sides of the inequality gives 2mev2Z Aw > 7. (2.9) 3hZi(niqi nfqf) We can use the most probable velocity as an estimate for v to obtain 4mekTZ Aw > Z Awi (2.10) phZi(niqi nfqf) where we have used the perturbing ion-radiator reduced mass p in the expres- sion for v because we are only interested in the relative motion. Here, we have defined Awi as the characteristic shift from line center due to ion motion. The criterion for the quasistatic ion approximation to be valid is thus: Aw > Awi. For the Lyman a line in pure argon plasma with kT = 800 eV, qi = 1 and qf = 0 we have hAwi = 0.044 eV. For higher series members hAwi will be less. We conclude that estimated shifts due to ion dynamics will be much smaller than the estimated shifts due to the static-ion Stark effect. The latter are on the order of tens to hundreds of electron volts for the physical conditions we 13 are interested in in this work. Consequently, we can confidently use the qua- sistatic approximation for the perturbing ions. For low mass perturbing ions the situation is quite different. In the case of proton perturbers hAwi = 16 eV and ion motion will become a significant component of the line width. The story for the perturbing electrons is, however, another matter. Due to their much lighter mass, the electrons will be traveling much faster when they have the same kinetic energy; their dynamic influence on the radiator will need to be considered. If we treat the plasma ions as static, the plasma electrons will form shielding charge clouds around them, screening the ionic charge at a distance of roughly the Debye length.20 This is given by AD,e = (kT/4re2ne)1/2. If we regard the time of interest for the interaction of a perturbing electron with the screened radiator as roughly the time it takes to cross the Debye sphere radius AD,e, we have -1 _Ve ,mp,e Te -= ,Ve De (2.11) e D,e AD,e where we have estimated Ve by its most probable value. This gives us e-1 = V/2p,e (2.12) where Wp,e is the electron plasma frequency given by S 47rnee2) 1/2 (2.13) p,e = me (2.13) The portion of the line profile that corresponds to electron dynamics is then given by Aw < Wp,e , (2.14) 14 with a quasistatic electron approximation being applicable for Aw > Wp,e. For our physical conditions wp,e is in the neighborhood of 10-150 eV. Examination of the statics and dynamics of the perturbing ions and electrons allows us to conclude that the quasistatic approximation for ions is good throughout the line profile except at the very center for Aw less than 1 eV while electrons behave dynamically inside wp,e. This will allow us to separate the problem of plasma perturbations of the radiator into two regimes coupled only by the static plasma microfield: that of purely static screened ions and purely dynamic electrons. This will greatly simplify the line shape theory that we wish to formulate. For our plasma particles to behave non-degenerately we require that the interparticle spacing be much greater than the quantum mechanical wavelength of the moving particles. For the plasma electrons this condition is Ae < ro,e (2.15) This requirement, except for a slight difference in the numerical constant, is the same as EF/kT < 1, where the electron Fermi energy, EF, is given by EF = (37r2ne)2/3 (2.16) 2me For our plasma at kTe = 800 eV and ne = 1 x 1025 cm-3 the above ratio is 0.21. However, for a density of ne = 1 x 1026 cm-3 the ratio becomes 0.98 and it is no longer a reasonable approximation to treat the plasma electrons non-degenerately. We will take an electron density upper bound of ne = 1 x 1025 cm-3 to avoid the problem of electron degeneracy in this work. The much heavier plasma ions, on the other hand, will behave classically under any conditions we will be considering here. 15 To judge the importance of correlations between particles in our system we can look at the Coulomb coupling constant P. This is the ratio of a particle's potential energy of interaction with a neighboring particle to its kinetic energy. If r > 1 then interactions with other particles will dominate ideal gas behavior. If F < 1 then the kinetic energy will be most important in describing the particle behavior. First we look at the coupling between electrons. For the potential energy we use the potential produced by an electron at the average interparticle spacing ro,e. For the kinetic energy we use kT to give 2 Fe,e = (2.17) ro,e6kT For our highest density, ne = 1 x 1025 cm-3, this gives Fe,e = 0.063. We con- clude that the plasma electrons are weakly coupled to each other and can be treated, in a first approximation, as an ideal Boltzmann electron gas. Correla- tions between electrons will be treated in an approximate way.21 For the ions we use ro,i and obtain r,= T (2.18) r,,ikT ' but since ro,i = Zl/3ro,e we have ri,i = Z5/3Fe,e. For our fully stripped argon plasma, Z=18 and for our highest density we have Pi,i = 7.8. Clearly, then, we must not ignore ion-ion correlations. In particular the calculation of the ion microfield will require a careful treatment of these correlations. For ion-electron correlations we take as the interparticle spacing ro = (ro,e + ro,i)/2 to give Ze2 Fe,i = oT (2.19) rokT 16 Noting that ro = (Z1/3 + l)ro,e/2, we can write 2Z Fe,i = Fe,e (2.20) (Z/^3 + 1) For ne = 1 x 1025 cm-3 this gives Fe,i = 0.62. So ion-electron coupling is of order one and, although not as strong as ion-ion correlations, will need to be taken into account. This will be accomplished by a mean field approximation using screened ion potentials. In order to examine the effect of ion radiator interactions it will be use- ful to make a multiple expansion of the radiator-perturbing ion interaction potential. To do this we define an expansion parameter 6, where 6 = ra (2.21) ro,i Here, ra = (n|lr I|n) ~ aon2/Z is the most probable orbital radius for a radiator in the state with principal quantum number n and nuclear charge Z. So /4 1/3aon2n 1/3 6= (471/3 an2n3 (2.22) 3\3/ Z4/3 This is 6 = 8.531 x 10-9n2nel/3/Z4/3 with ne given in cm-3. At ne = 1 x 1025 cm-3 for n = 2 we have 6 = 0.16 and for n = 3 we have 6 = 0.35. The dipole term of the multiple expansion is second order in 6 and the quadrupole term is third order in 6. For n = 3 this gives 62 = 0.12 and 63 = 0.043. So our expansion is quite legitimate. 17 2.2 Historical Background We will now take a brief look at the historical development of spectral line shape theory with the goal of identifying the origin of the main ideas of line broadening.22 In 1889, Lord Rayleigh23 correctly explained the broadening of spectral lines due to the Doppler effect. He used the not completely accepted idea at that time, of a collection of randomly moving atomic radiators whose velocity was governed by Maxwell's distribution. Interest was such that by 1895 A. A. Michelson24 was able to summarize the possible mechanisms of line broadening as he saw them. Among the proposed mechanisms were the following: 1) The change in the emitted radiator frequency due to neighboring molecules. In those days the term "molecule" was used to refer to any atomic sized entity;22 2) Doppler broadening due to radiator motion; 3) Broadening of radiator emission lines due to radiation interrupting colli- sions. Michelson developed a theory for this effect and showed that the resulting line shape was that of a Lorentzian. These early theories consid- ered the radiators to be classical oscillators. We will see that collisional broadening is often proportional to the density, and hence, to the pressure, assuming the temperature is held constant. This is where the terminology pressure broadening originated; 4) The natural line width due to radiation damping which was viewed as the leaking away of the oscillator's energy to the environment. In 1901 C. Godfrey25 combined Doppler and collisional broadening in a unified description. At that time these two effects were viewed as the most sig- nificant causes of observed line broadening. With the advent of the quantum 18 theory the time was ripe for a reworking of the old classical line broadening theory. In 1924, W. Lenz26 applied some elements of the new quantum theory to collisional broadening but still considered the radiator as a classical oscil- lator. Returning to the problem in 193327, Lenz incorporated line shifts and asymmetries into his collisional line broadening theory by using finite collision times. Previously all radiation interrupting collisions had been assumed to occur instantaneously. Michelson's point about the radiation from atoms being affected by atoms in their vicinity was taken up by Holtsmark28,29 in 1919 and developed into the theory of statistical Stark broadening. Holtsmark considered the effect of a particular configuration of static perturbers surrounding the radiator on the shift of the radiator's characteristic frequency. He then averaged over the possible configurations to produce a broadened line profile. If the perturbers are electric monopoles this theory is referred to as Stark broadening theory after an idea originally suggested by J. Stark30 in 1914. Holtsmark also developed similar theories for perturbing dipoles and quadrupoles. This brings us to a point in the historical development of line broadening theory where we see the emergence of the primary theoretical points of view as to the physical causes of line broadening in plasmas. These are Doppler broadening, collisional broadening and statistical Stark broadening. Many fur- ther developments and refinements17,18 have occurred since these "early days" of line broadening physics but we will not follow the historical development further. We will jump to the point of present day developments. 19 2.3 Theoretical Formulation In order to examine the line radiation from a plasma, we must look at the power emitted by dipole radiation from an emitter immersed in the plasma. At the outset we assume that the plasma consists of ionic emitters surrounded by a plasma consisting of perturbing ions and electrons. This will allow us to model the plasma line radiation by examining the radiation from one of the emitters and averaging over all possible configurations of the surrounding plasma. This is equivalent to looking at a large number of plasma radiators in differing environmental conditions. The resulting power radiated by dipole radiation from an emitter immersed in a plasma, as a function of the frequency, is given by31,32 4w4 P(w) = c3 I(w) (2.23) where I(w), the line shape function, is defined by I(w) b 6(w wab) (blde-ifja) 2Pa (2.24) a,b Here, a and b refer to the initial and final states of the total radiator-plasma system, respectively. The energy difference between these initial and final states can be written as wab = (Ea Eb)/h. The dipole moment of the radiator is given by d, and the population of the initial states is given by Pa which is an eigenvalue of the density matrix p, where p = Ei pi ii) (i. The radiator center-of-mass coordinate is given by R and the wave vector of the emitted radiation is given by k where k = w/c. The eigenvectors ii) are eigenfunctions of the system Hamiltonian H. Free-free and free-bound transitions will not be examined in this dissertation so we consider only dipole radiation from atomic ions. 20 Since a similar formalism can be used to describe absorption line profiles, we will concern ourselves only with the line shape function I(w). Our system will consist of plasma electrons, ions and the radiator which will always be referred to in this section by the subscripts "e", "i", and "r", respectively. For simplicity, we also set h = 1 for the remainder of this section. We can express the Dirac delta function as an integral representation that will allow us to incorporate the time development of the system into the dipole moment operators (i.e. the Heisenberg representation). If we work with this representation, we have the advantage that all of our approximations must be made explicitly on the system Hamiltonian which will be contained in the expression for I(w). This gives (w) = L dei(W-(W-Wb))t (blde-'k2la) Pa a,b (2.25) 1 roo = Re dt eiw(t) 7r 0 where 0(t) = S (al eikRdlb) (bi eiwtde-ikRe- pa Ia) a,b (2.26) = (al d(0) b) (b d-k(t)p la) a,b Here we have used the fact that p Ia) = Pa la) and e-iHt Ia) e-iwat la) and the definition of the time development of the dipole and radiator position operator, d-k(t) = eiHtd(0) e-i'Re-iHt (2.27) It follows that dk = dei~'. We will consider our sum over states a and b to now be a sum over a complete set of states and replace the sum with a trace. 21 We are free to do this at the outset and only look at the elements of the sum that correspond to the spectral transitions we are interested in. Now we have (t) = Trr,e,i [4k(O) k(t)p (2.28) (2.28) = (k(0) d-k(t) and thus 1 -. - I(w) = -Re dteit(dk(0) dk(t)) (2.29) 7r Jo0 To address the problem of radiator motion, we will assume that the radiator velocity' is statistically independent of perturbing ion and electron effects as well as of the internal state of the radiator. Consequently, we can factor the radiator translational motion from the density matrix. This gives P = PtrPr,i,e (2.30) Here, ptr is the density matrix for the radiator translational motion. Assuming ideal gas behavior for the radiators, we have Smr 3/2 _mr,2 Ptr = ( e 2T (2.31) where mr is the radiator mass and v is its velocity. We may now separate out the radiator's translational motion to give I(w) = -Re o dt eit I dptreiE )) ( d(t)) (2.32) If we assume that the radiator's motion is constant during the time of interest, then we can write (2.33) Ri(t) = R- + Vt This gives I(w) = -Re dt edt d!ptre-~7kt( d (t)) (2.34) The integral over v may be performed to give 0 1Reodt it- ( 2kt2 I(w) = Re d eit e ~- (t)). (2.35) 7r Jo It can be shown with the aid of the Fourier convolution theorem33 that this expression is equivalent to I(w) = J 'd ID( w ') Is(w') (2.36) loo where ID(W w') is the Doppler distribution given by ID(w w') exp (2.37) where34 2a2 = 2 (2.38) C2 \ mr / Here, mr is the mass of the radiator and c is the speed of light. The line shape function for the static radiator is given by 1s( 0 I(w) = -Re dt eiwt. d(t)) (2.39) This result is the usual method for treating the Doppler effect but we see that it follows, with suitable approximations, directly from the expression for the power emitted by a radiator. The interaction of radiator motion with plasma perturbations can be treated32,35 as an extension of the above devel- opment by not making the approximation: R(t) = R + vt. We will employ 23 the Doppler convolution given by Eq. (2.36) to account for radiator motion and continue with the development of IS(w). We will drop the subscript from Is(w) for convenience in what follows. This gives 1 (00 I(w) = 1Re dt ei (d(0) d(t)) (2.40) 7r 10 We see that the line shape problem is reduced to the calculation of the system's dipole-dipole autocorrelation function. This is the starting point for some numerical simulation approaches.36 We, however, will continue with the formal development of the theory along the general lines of the work of Smith and Hooper.37 We will see below that it is possible to approximately factor the density matrix into product form so that P = PiPePr (2.41) Here the individual density matrices are for the three subsystems: the plasma ions, the plasma electrons and the radiator. We will examine the LTE case where the density matrix becomes the simple Boltzmann factor p = e-H /Tr e-fH Consider our system Hamiltonian H = H + HO + HO + Vi,r + Vi,e + Vr,e. (2.42) Here the HO's refer to the kinetic and potential energies of the respective subsystems so that Hi = Ti + Vi and H0 = Te + Ve. The Vj,'s refer to the potential energy of interaction between the subsystems j and k. We make the quasistatic approximation for the ions so we assume that they do not move appreciatively during the atomic radiation time. The use of the quasistatic 24 approximation allows the interaction of perturbing ions with the radiator to be statistically screened by the rapidly moving plasma electrons. Consequently, we will replace Vi,e +Vi with a screened ionic potential and as an approximation obtain V' Vi + Vi,e. (2.43) This gives us the effective ionic Hamiltonian HP' = Ti + V'. What precise form this screened potential should take has been the subject of considerable work.38,39,40 Here, we will use the lowest order screening approximation for V': the Debye-Hiickel potential.20 In order to make the density matrix factor- ization, we will neglect Vr,e, but only in the density matrix. This amounts to neglecting radiator-perturbing electron correlations at the initial time. This is a reasonable approximation as long as kT > Vr,e. Now we have H w H?' + HO + HO + Vi,r. (2.44) The ion-radiator interaction, Vi,r, will be expressed as a multiple expansion which we will truncate. For the density matrix we will keep only terms through the dipole. The monopole term has only ion coordinates and is incorporated into HP' to give H!. The dipole term contains both ion and radiator coor- dinates but we will use a mathematical device to eliminate those of the ions and combine it with HO to give HrI(i. Here f is the plasma microfield to be discussed below. If we assume that the remaining effective Hamiltonians for the three subsystems commute amongst themselves the density matrix is now factorable with the Hamiltonian H w H, + Hr(e + Hf. We will see what these terms are in detail shortly. 25 Turning from the approximate form for the Hamiltonian in the density matrix, we consider the terms in the Hamiltonian of the dipole moment time development operator. We make a multiple expansion41 of the radiator-ion interaction 1 .. i .Ei(0) Vr, = x(0) d. E() Q. +.... (2.45) i,J The electric fields and potentials refer to those produced by the plasma ions at the radiator which we take to be at the origin. The effective charge of the radiator is given by X = Z a where Z is the radiator charge and a is the number of bound radiator electrons. The radiator dipole moment operator is given by d and the components of the radiator quadrupole moment tensor are given by Qij. One effect of the quadrupole term is the production of an asymmetry in the line by intensification of the blue portion of the line relative to the red.12 The octupole term is fourth order in the expansion parameter 6 compared to the third order of the quadrupole term. For the special case of hydrogen-like ions, Joyce42 has shown that the octupole term, to first order in perturbation theory, produces symmetric shifts of the hydrogenic energy levels and thus will not contribute significantly to the asymmetry of the spectral line. Since we are using a basis set for this calculation that is made up of eigenfunctions of the radiator Hamiltonian that includes the field dependent dipole term, we take the quadrupole term as the first order correction to this Hamiltonian, and neglect all higher order terms. We will symbolically represent all field gradient terms 9Ei(0)/zxj by the generic term Env = QpEV. Whenever we use this term we will mean all rele- vant partial derivatives. Our device for removing the ion coordinates from the interaction potential Vi,r involves the introduction of a delta function in the 26 manner shown below. This will alow us to separate the ion coordinates from the radiator-perturbing electron subsystem and proceed with the development as shown below. Introducing the delta functions into Eq. (2.28) gives (t) = dJ de, Trr,e,i [pib(- E)6(epv EP,)O() d(t)pre] (2.46) Here, the integration variable epv is used in the same way as Etv above. The time dependence of the radiator dipole operator is given in terms of the total system Hamiltonian H. The ion coordinates in H that are not removed by the delta function will commute with d(0) in the static ion approximation and cancel.43,44 This leaves the Hamiltonian He,r that has only radiator and perturbing electron coordinates. This is illustrated by d(t) = eiHe d(O)e-iHer, = eiLt(0) (2.47) where Her = H0 + Ve,r + Hr(e Qi ,j (2.48) ij J and L is the Liouville superoperator expressed in terms of a commutator acting on an arbitrary operator O as LO =[Her, O] (2.49) We have now successfully separated the system ion coordinates from the radiator and perturbing electron subsystems. This is expressed by I(w) = d deyW(e e^)J(, (2.50) where W(f, eiV) is the ion probability function, and J(w, ew,) = -Re dt eiwt ((0) d(t))e,r 7r JO 1 = -RIm Tre,r [d- Re,r()(pe,rd] We have defined the resolvent operator as /0OO Re,r(w) -i I dt ewt-iLt = [w L]- (2.52) Re and Im refer to the real and imaginary parts respectively. The ion proba- bility function W(f, e1v) is given by W(, ejv) = Tri [,PI(- E)6(el Epv)] (2.53) ( ( E)b(eyv E}))i. The field E and field gradient Eu, are sums of contributions from all of the individual system ions. We will return to the evaluation of W(C, eCv) in chapter 3. Recalling that pe,r = Pepr, we can perform the trace over plasma electron coordinates to give J(w, ) = -Im Trr [d- (Rer(w))e(Prd) (2.54) We see that the effects due to the perturbing electrons have been isolated in Rr(w) = (Rer(w))e. Note that L = L(,7 te). Next, we proceed with the evaluation of the operator Rr(w). We now introduce a coupling constant A as an expansion parameter. Write L = Lo + ALI, (2.55) where L corresponds to the Hamiltonian: He,r = H + AVi = H, + A (Ve,r Qi (2.56) The Hr, corresponds to Lo and Ve,r + V9 corresponds to LI. Here, Qij is a function of the radiator's electron coordinates (for single or multielectron radiators), Ve,r is the radiator and plasma-electron interaction, Vi is the ion- quadrupole interaction, and Hr = Hr(,)+H0 is the combined non-interacting radiator plasma-electron Hamiltonian containing the ion-dipole term. We will now implicitly define an electron broadening operator M(w), by setting Rr(w) = [w Lr(4 ev) M(w)]-1 (2.57) where Lr(E, eyv) is the Liouville operator corresponds to the Hamiltonian: Hr(F, epv) = Hr(,) Qijiej (2.58) i,j Here, Hr(E, is the Hamiltonian of the field-free radiator with the ion-dipole term, and we have used the quadrupole approximation for the remainder of the interaction of the plasma ions with the radiator. Specifically, Hr() = H0 d1. (2.59) Note that Hr(i) = Hr(fE0). So, using Eqs. (2.52),(2.55) and (2.57), we can write, [w- Lr(f ec) M(w)] = ([w Lo ALI-)e (2.60) Next, we expand M(w) in powers of the coupling constant, expand both sides of Eq. (2.60), equate coefficients of equal powers of A, and thus identify the terms of the M(w) expansion. 29 The right-hand side of Eq. (2.60) can be expanded in a Lippmann-Schwinger expansion and the left-hand side can be expanded in a Taylor series in A. First, we will expand the right-hand side. For simplicity in notation we will denote the full resolvent operator by Re,r(w) R, and define a zero order resolvent operator as R [w Lo]1 (2.61) We can construct an iterative equation for R by noting that R R = RORO R-1] R (2.62) or explicitly, 1 1 1 1 S- [w L (w L)] (2.63) -L w Lo w L w L This can be written equivalently as, 1 1 1 1 = + L LI (2.64) L w w-L w-L or simply as just R = Ro + RoLIR (2.65) This is the Dyson equation for this system in Liouville space. This implicit equation for R can be iterated to give a series expansion for R in powers of A: 00 R = A"Ro(LIRo)n (2.66) n=0 To second order in A we have R = Ro + ARoLIRo + A2RoLiRoLRo (2.67) In order to examine what (. *)e does to this, look at Rr(w) = (R)e (2.68) To second order in the interaction Liouville operator this is Rr(w) = (Ro)e + A(ROLIRo)e + A2(RLLIRoLJRo)e (2.69) Here Ro is a function of Lr and Le, the zero-order-radiator and perturbing- electron Liouville operators with Lr corresponding to Hr(e) and Le to H,0. Tighe44 has shown that when L0 operates on objects of the form B = peb, where b operates on radiator coordinates only, then (Ln)e = L (2.70) On expanding Ro this leads to (Ro)e = [w Lr]-1 RO(w) (2.71) For combinations of Ro with a general operator in Liouville space Tighe44 has shown that (Ro0)e = RO()e, (2.72) and (Ro)e = (C)R . (2.73) 31 These identities allow us to simplify the expression for the resolvent. We obtain (R)e = R' + ARO(LI),eR0 + X2RO(LIRoL)eR0 (2.74) For the left-hand side of Eq. (2.60) we postulated an equivalent form for the resolvent that contains all of the perturbing electron statistics in an implicitly defined operator M(A, w) which is analytic in the coupling constant A. Restating this we have (R)e = [w Lr(E e y) M(A, )] -1 (2.75) where Lr is defined above. Expanding M(A, w) in powers of A we have M(A,w) = M(0)(w) + AM(1)(w) + AM(2)(w) + (2.76) To expand the right-hand side of Eq. (2.75) about A = 0 we will need the following easily proved operator identity for some operator A in Liouville space: BA-1 dA = -A-l A- (2.77) Expanding A-1(A) about A = 0 we have A-(A) = A-(0) + A -() 2 2A1 (0) + O(3) (2.78) OA 2 + )2 In this case we have A(A) = [w Lr(, 0) ALr,i(euv) M(A,w)] (2.79) with Lri(ep) corresponding to VQ. Using the identity of Eq. (2.77), along rith, with OA( = Lr,i(e,) M(1)(w) (2.80) 02A(0) a2 = -2M(2)(w) (2.81) and A-(0) = w- Lr(, 0) M()(w) (2.82) we can evaluate the expansion of A-1 about A = 0. We obtain (R)e = A- = A-(0) + AA-1(0) [Lr,i(ev) + M()(w)] A-(0) A2A-1(0) [Lr,i(ep) + M(l)(w)] A-(0) [Lr,i(e ) + M(1)(w) A-1(0) A-1()M(2)(w)A-1(0)} + O(A3) . (2.83) Comparison of the two expansions for (R)e, term by term in powers of A up to second order, allows us to make the following identifications for the terms in the expansion of M(A, w): A(0) term M()(w) = 0, (2.84) A(1) term M )(w) = (Lr,e)e (2.85) A(2) term M(2)() = (LIRoLI) (Li)eRO(LI)e . (2.86) This last expression for M(2)(w) can be further simplified if we expand LI. Recall that LI = Le,r + Li,r with Le,r corresponding to Ve,r and Li,r to V. This gives M(2)(w) = (Lr,eRoLr,e)e (Lr,e)eRO(Lr,e)e (2.87) + Lr,i((Ro R))Lr,e)e + (Lr,e(R0 R ))eLr,i - With the use of Eqs. (2.72) and (2.73) the last two terms cancel to give M(2)(w) = (Lr,eRoLr,e)e (Lr,e)eRO(Lr,e)e (2.88) If the dipole approximation is made for Ve,r, we will find for the corre- sponding Liouville operator, (Le,r)e = 0. This is easy to see by noting that in order to evaluate this expression we need to calculate (Ee)e which must average to zero since there is no preferred direction in the plasma. This holds true for the case of hydrogenic or multielectron radiators. The first surviving term in the M(w) expansion will then be to second order. So to lowest order in A, M(w) = M(2)(w)= (Le,rR(w)Le,r)e (2.89) where Ro(w) = [w Lo]-1. This gives us M(w) = -i dteit (Le,re-iLotLe,r)e (2.90) This operator is tetradic37 because of the use of the Liouville operators and, when evaluated as a matrix element, will give45 M(w)ab,ab, = Ib [6 a daa" daa',G(Awal.b) ta" + baa' E db'b" db"biG(- ab") (2.91) b" aa, dbb{G(Aab,) + G(-AWab)1 where the first term represents electron broadening of the upper state, the second term represents electron broadening of the lower state and the third term represents interference effects. The many-body electron broadening function G(Awab) is given by G(Awab) = dteiAwbt( e Ee(t))e (2.92) with Ee(t) = eiHOt e(O)e-iHt and Awab = w (wa wb). Thus the elec- tron perturber effects are given in terms of an autocorrelation function for the plasma electric field at the radiator due to the plasma electrons. We have calculated only the imaginary part of M(w) which corresponds to line width operator. The real part corresponds to the line shift operator and yields only a small shift44 which is ignored here. In our approach to this calculation we approximate the effects of corre- lations among the perturbing electrons by the use of G(Aw) = G(wp) for |Aw| < wp, where wp is the electron plasma frequency. This procedure has been shown to be a good approximation by a detailed treatment of dynamic electron correlations.21 This handling of electron correlations allows us to use single particle Coulomb waves in taking the trace over perturbing electron co- ordinates. The Coulomb waves are single particle solutions to Schr6dinger's equation with a central 1/r potential with effective radiator charge X, where X = Z aON; aoN being the number of bound radiator electrons. G(Aw) is 35 then evaluated using the method of Tighe and Hooper.46 Other theories47 also exist for the evaluation of G(Aw). It is instructive to point out, however, that we can relate this electric field autocorrelation function to the charge density autocorrelation function48 which gives us, (jEe e(t))e_/dJ3x f3 'f (e e(t))e = dx d 2X (p(x)p(, t))e (2.93) This illustrates the relationship of the plasma charge-density fluctuations to the spectral line-broadening problem. Note that this charge density correlation function is not translationally invariant for ionic radiators as is sometimes assumed.49 Translational invariance would imply a uniform perturbing electron gas; for highly charged radiators, this assumption would ignore aspects of the radiator-perturbing electron correlations. Altogether, our approach leads to a line shape function of the following form: I(w) = de depvW(-, e)J(w, ) ,v) where J(w, C, efp) = ImTrr ( [w Lr(, e-7 ) M(w)] prd} (2.94) or in matrix element form 1 -1 (. J(w, E ey) = Im padb'a' 'dab [w- Lr( Ep) M(W)] ab,ab (2.95) aba'b' where a and b refer to initial and final radiator states respectively. 36 The above equations for the line intensity I(w) are valid for the general case of multielectron radiators with combined upper and lower state electron broadening effects and Stark shifts. In this work we will be considering hy- drogenic ion resonance lines which have the ground state as the lower state. For hydrogenic ions the ground state is not subject to electron broadening and only very weakly affected by the plasma electric microfield. Therefore we will ignore lower state broadening and interference effects. This will leave only the upper state term of the three terms in the matrix element expression for the electron broadening operator M(w) given by Eq. (2.91). This gives M(W)aa, = -- aa,, daalG(Awa,,f) (2.96) a"l where subscript f refers to the single lower radiator state. Since lower state ion produced Stark shifts are unimportant here, the ra- diator Liouville operator Lr(E, epv) in Eq. (2.95) can be evaluated to give 1 -1 J(w, E, e,) = --Im padaf dfa, [ (Hr(e ~e) f) M(w)] , aa' (2.97) where wy is the lower state energy of the radiator. We note that since there is only one lower state the sum over b is absent. For our case, the dipole and quadrupole operators become single electron quantities with d= -er (2.98) Qi,j = -e(3xiXj r2 i,j) 37 Here r and xi refer to radiator electron coordinates. The corresponding mul- tielectron quantities are just sums of terms similar to these for each radiator electron. Any intrinsic line width, such as the resonance width induced by the pres- ence of an electric field (see Chapter 3), can be approximately included in the line profile formalism by replacing the affected level energy E by a complex resonance energy50 E. For level j this gives Ej Ej i (2.99) where Ej is the energy at the center of the resonance and the presence of irj/2 assumes a Lorentzian shape for the resonance. Fr is the full width at half maximum of resonance j. We have chosen the minus sign in Eq. (2.99) to be consistent with the fact that the electron broadening operator M(w) is proportional to -i. A choice of Ej iY would correspond to a proportionality factor of +i. The use of this complex energy is only an approximation since we are attempting to describe the time dependent behavior of a resonant state in a stationary state formalism. We use this procedure to approximately include the field induced resonance width for the upper states of the Lyman series in hydrogenic argon. When considering the comparison of theoretical line profiles to experimen- tal results, it is important to remember the factor w4 in the relation for the total power, P(w) = Const. w41(w). If the radiation energy is high enough and the line shape is broad enough, this factor will not be approximately constant across the line shape and I(w) must be redefined to include it. Otherwise an 38 additional source of line asymmetry will be neglected. For experimental com- parison, it is also important to include instrumental broadening5 as well as opacity effects51 if significant. CHAPTER III HIGHER ORDER FIELD EFFECTS In the proceeding chapter we have reviewed some fundamentals of dense plasmas and formulated a theory of spectral line shapes. We will now examine some of the processes and effects that can make important contributions to these line shapes in the high density regime. 3.1 The lon-Quadrupole Effect Previously we introduced the ion coordinate joint probability function W(, e/,V) to describe the joint probability distribution for the microfield f to- gether with the field gradients ev. Recall that for notational convenience, ep, represents all of the field gradient terms. Writing out this probability function explicitly we have W(, epv) =( 6(E- E)6(ez Ey)6(eyy Eyy)6(ezz Ezz) x 6(exy Ezy)6(eyz Eyz)S(ezx Ezz) )i (3.1) =- e- #b(- E)6(e Exx)... d3Nr where < ... >i is the ensemble average over ion coordinates, Zc is the configu- rational partition function for the ion subsystem, N is the number of ions, and V is the potential energy of the ion subsystem in the presence of the charged radiator. The quantities E and Eij are the many particle electric field and field gradients at the radiator and are functions of the perturbing ion coordinates. Using an integral representation of the field gradient delta function, 6(e Exz) = d az e-io,"(e"-E, ) (3.2) gives W(f, =)- (2r) Zc d'a e-i(O'e'+--') d3Nre b (- )ei(-,,E+-) (2ir)6Zc d 3r 6(E-- E)eV v d6e-i(UzE:+'-) (eie) = Q(2iJI d6ae-'i(Ox,, e +oU acyu+w )(eie)e (21r)6 J = (1(P(l'|,E) , (3.3) where 0 = ozxxEx + ayyEyy + azzEzz + aZyExy + ayzEyz + azxEz (3.4) We have defined the conditional averaged quantity (eiE) f d3Nr e-3V(g- .)eie (e d3N, (3.5) f- d3Nre-(Vt6(-- E) This is the ensemble average of eie where all included ion configurations are constrained to have the microfield value f. It is normalized to give one if 0 = 0. By defining the average in this way we have retained the benefits of using the ordinary plasma microfield function Q(e) to describe the uniform field at the radiator and we are now free to approximate the higher order field gradients if we choose. The function P(E|e t) is the conditional probability for ey given g. We take the field to be in the z direction for simplicity and without loss of generality. 41 Direct evaluation of this constrained average is quite difficult if we wish to retain all of the ion interactions. Consequently, we will make a simple approx- imation to this function in order to proceed. We will expand the exponential in a cumulant expansion52 by taking (eie), = exp En (3.6) n=l1 where C1 = (0)c C2 = ((O (),)2), (3.7) C3 = ((6 (e),)3) . Higher order terms are more complicated. In terms of the expansion parameter 6, which we introduced in the multiple expansion (see Eq. (2.22)), we have C1 ~ 63 and C2 ~ 66. We see this by noting that 0 goes as aEi/Oxj which goes as 1/r3 and 1/r is of order 6. For our highest density of ne = 1 x 1025 cm-3, this gives C1 ~ 0.043 and C2 ~ 0.0018 for the n = 3 level. The C2 term, which is actually the variance of the quantity Theta, is therefore in the neighborhood of 4% of the lower order term C1. Here, we will keep only the C1 term in our approximation. This gives us (eie) z ei(e) (3.8) Inserting this into Eq. (3.3) gives W(f (2_)6 Q( / df d e-i(Ezz +-)ei(e) = Q(Ce6(ez (Ezx)c)6(eyy (Eyy),)... (3.9) = Q(e1E6(ep (ESv)c) . 42 This gives us for the approximation of the conditional probability: P( epv) 6(epv (Epv),) (3.10) We have developed a systematic expansion for the conditional probability func- tion and have examined the magnitude of the expansion terms. We can identify the first term of this expansion as the approximation used by Joyce.12 Return- ing to the expression for the spectral line intensity, Eq. (2.50) now gives I(w) = df de, Q(e~P(f e,,)J(w, f," ev) (3.11) The microfield function Q(g) does not depend on the direction of but only on its magnitude, e. We can therefore preform the angular integration and define a new microfield distribution function as P(e) = 47re2Q(e (3.12) Now, the intensity becomes I(w) = J de de, P(e)P(epv)J(w, e, e). (3.13) Using our approximation for P(e lepv) we can carry out the epv integration to obtain I(w) = de P(e)J(w, e, (Eg,)) (3.14) The field gradient terms in J(w, e, e~,) have now been replaced with their constrained averages that depend only on e so we can write J(w, e, (Epv)E) = J'(w, e), where the prime indicates the replacement of the field gradient by its constrained average. For the remainder of this work, we will drop the 43 prime with the understanding that we actually mean J'(w, e). Our use of the approximation for P(ejEpv) has lead to a significant simplification of the line shape formula that still retains the approximate effect of the field gradients. In order to proceed we will need to evaluate this constrained average of the field gradient. From the ensemble of all possible ion configurations in the presence of the radiator we choose only those configurations that give a certain field magnitude e at the origin. If we average a quantity F over this subset of configurations, we have constructed the constrained field average that we introduced above. We denote this average by (F) = 6(-E)) (3.15) Q() (3.15) where Q(e) is the plasma microfield function and (* *) is the complete ion subsystem ensemble average. This average, as we have said, is such that for F = 1, (F), = 1. We will show that it is possible to relate this constrained average to a functional derivative of the plasma microfield function. This will allow us to evaluate the constrained average in terms of available simplified models for the microfield function Q(E). For our purposes we will take F to be an additive function of the ion coordinates, F = f(r) = d f(f)n(f) (3.16) where n(rf-) = r(f-i). (3.17) i 44 Note that f drn(r) = Ni, the total number of ions in the system. So we can now write (F)e = 1 df df(-)(n(fr)6(- E)) (3.18) = Jdff(Pr)nig(;E , where we have defined nig(r; (E (3.19) On expanding the delta function we have nig( j = (2 ) e-i"(n(reiE) (3.20) We would like to relate the average on the right-hand side of Eq. (3.20) to a functional derivative of the plasma microfield function. To do this we introduce G(A), the generating function of the microfield distribution, where G(A) = In (X) = In(eiE) (3.21) If we define a function 4, such that = i E(r) (3.22) then G(X) becomes a functional of or G(A) = G[] (3.23) Now we can cast G(A) in the form of a functional derivative, where G(A) = dl OG) aJo 3(1 / Of b~6G[4b ] = f ddl drO 6G[ (3.24) = dl driil*- E() E G[ 0 (il. E(-) where 1= l\ and A is the unit vector in the direction of A. Alternately, from the definition of G(X) in Eq. (3.21), we have Sa In W(1 G(X) f dl1 (i -eig) = dl (3.25) 0 (eif) dl (eiff Note that we can write, N i E d = I di. E(F b (6- r-) i=1 (3.26) = dr-i .* (r-n(). Using Eqs. (3.25) with (3.26) gives ( n( -eil-"E) In Q(A) = dl dFl I d i- ( (3.27) o' (- {eil-E) Comparison of Eqs. (3.24) and (3.27) gives us the connection between the constrained average considered here, (F)E, and a functional derivative of the microfield generating functional G[O]: (n(r-)eil E) 6G[i] = (3.28) (eil-E) 6(il E,(r)) which on substitution into Eq. (3.20) above, gives 1g/ dX ei= e- At() (G (3.29) nig( Q (2) 3 E(i Er-)) Substitution into Eq. (3.18) above gives us the relationship we want: (F) = df(r) e-iX.Q(A) G[] (3.30) This explicitly shows that we can relate the constrained average to a functional derivative of a microfield generating function G(A) = In Q(A). We are free to use whatever functional model we choose for Q(X). As an aid in evaluating these constrained averages we define the transform of g(r'; ( by 1 6G[#] F; 6A l (3.31) ni b(iA E(F)) From Eqs. (3.20) and (3.28) we see that the transforms are related by g(1; 0 = e-ilX (X~)(A ; X). (3.32) Thus a knowledge of W(Fr; X) obtained from Q(X) will lead to g(r; e) through the transform and then to (F)e through Eq. (3.18). To evaluate the ion-quadrupole term recall that the perturbed radiator Hamiltonian is Hr(E, (EV)c) = Hr(e) E Qij(Eij) (3.33) ij where the ion-quadrupole term is just Vr,i. To evaluate the (Eij), terms note that when the angular integration are performed, the o, or azimuthal, integrals lead explicitly to42 (Exy)E = (Eyz)e = (Ezx)c = 0 (3.34) Together with the property cjEi = OiEj, this leads to the simplification SQi,j(Eij) = c Qii(Eii) (3.35) i,j i With the field in the z direction there is no difference between the x and y directions so (Exz), = (Eyy),. Additionally, with the tracelessness of the quadrupole tensor41 we have SQi,j(Eij), = Qz ((Ezz) (EZx),) (E) 1 (3.36) If the ion interaction potentials are screened we have V E 5 0 due to the continuous charge distribution. However, the form of the multiple expansion we have employed made use of V E = 0 at the origin which is not strictly true for our screened ion potentials. We reconcile these two facts by noting that if we restrict the screening charge density of the perturbing ions to lie outside the Bohr orbit of the radiator's bound electron, we will have V E = 0 at the origin. Tests of the computation of the field gradient term show that its value upon taking V E = 0 is approximately 20% less than that obtained on taking V E 5 0 at a field value corresponding to the maximum of the microfield probability function at ne = 1 x 1025 cm-3 and kT = 800 eV. The difference is less for lower densities. In light of the uncertain nature of the actual form of the ion potential, this does not seem unreasonable. Consequently, we will take V E = 0 in the evaluation of the field gradient term in agreement with the form of our multiple expansion. This amounts to requiring that we have 48 no penetration of the radiator by the perturbing ions or their screening charge density. For the ion-quadrupole term this gives us 1 Vr,i= --1Qzz(Ezz), (3.37) with the radiator Hamiltonian given by Hr(e) + Vr,i. To evaluate Vr,i we must calculate the constrained average term (Ezz)E, and to do this we must pick a model for the microfield function Q(e). In this dissertation we choose the adjustable parameter exponential approx- imation (APEX)53,54,55,56 for the microfield. It has the advantage of incorpo- rating correlations between the perturbing ions while retaining the functional simplicity of the independent particle (IP) model42 of the microfield. Joyce42'12 used the IP model to evaluate the ion-quadrupole effect which we will see leads to an overestimate of its magnitude due to the neglect of the inter-ion correla- tions. We will compare these two models along with a nearest neighbor (NN) approximation for the field gradient. The APEX model for the microfield is based on the independent particle model but with an effective inverse screening length, a. The field is effectively renormalized in this way by requiring that two conditions54 be met: the so called second moment condition and the local field condition (see Appendix B). The transformed conditional distribution function W(rf; A) for the APEX microfield distribution is given by (see Appendix B) (; A) = g(r)ei *(r) (3.38) where g(r) is the equilibrium radial distribution function and E*(r-) is the APEX renormalized field. This field is given by =*(r) = E(r(1 + ar)e-ar, (3.39) where a is determined by the second moment of Q(E), and E(f) is the screened field produced by the plasma ions. We calculate the radial distribution function g(r) by integral equation methods due to Rogers.39 Lado and Dufty57 have shown that this method of calculating g(r'; e) by using the APEX microfield underestimates the inter-ion correlations in some cases. In particular, for the case of a hydrogen plasma with Iii = 10 at a field value of e/eo = 0.4 (eo = e/r2,i), the APEX calculation underestimates the maximum value of g(r'; ) by a factor of two. We retain the APEX model, however, because of its simplicity and significant improvement over the IP model of Joyce12,42 which ignores ion-ion correlations entirely. With the APEX model for g(F; e) obtained from Eqs. (3.32) and (3.38) we can evaluate the constrained average by using Eq. (3.18) to obtain (F), = J dFf(P)g(r)Q( *(r-F)), (3.40) where it is understood that the microfield distributions, Q(e), are in the APEX approximation. For the ion-quadrupole effect, the constrained average we need to evaluate is (Ezz),. In particular, we have F = Ezz = zz(ri). (3.41) In general, the one perturber derivative term is given by EU(r) = Ze 1 + r) ( 3, (3.42) r3 A r so for our case we have ezz(r) = Ze ~ + 1 3 cos2 ) .(3.43) The microfield function in the APEX approximation is54 QAPEX() = i dkk sin(ke)en hl(k) (3.44) with hl(k) = 47 drr2g(r) (j0 (kE*(r)) 1), (3.45) nj E*(r) where jo(x) = sin(x)/x is the zero order spherical Bessel function of the first kind. From now on when we speak of Q(e) we will implicitly assume the APEX approximation and suppress the subscript. Using this approximation we can evaluate Eq. (3.40) for the case of F = Ezz to obtain Zeni dkk2enooh(k) OO -r/A (Ezz) e d= kk2e () drr2g(r) 0 (1-r/A)I(r). (3.46) The angular integral IQ has been evaluated by Joyce42 and is given by (r) d(1 3cos2O)j (k ?- *(r) (3.47) = -8rj2(ke)j2 (kE*(r)) , with dQ = dpd(cos 9). To evaluate (Ezz)E numerically, we cast its variables in unitless form by taking S= E/Eo S= Eok (3.48) a = ro,i/A x = r/ro,i where Eo = (3.49) 0,i We obtain (Ez = -6 d2 en hl( )j2(k ) fo dxg(x) -(1 + ax)j2 (k*()) foi JO" dkken'hi(k) sin(ki) (3.50) with hl(k) = 00 dxx2g(x) ) J( *(x)) 1) (3.51) ni ( E*(x) ( In this form, the equation for (Ezz)c can be directly evaluated numerically. Following Joyce42 we implement this function in the line shape calculation by the use of a Pad6 approximate. For a particular case, (Ezz)e is calculated for a mesh of i values. The Pad6 function is then fit to the calculated values by minimizing the squared difference between the two while varying the coeffi- cients of the Pade function. The resulting approximation then yields (Ezz)f for any value of the scaled field up to i = 20 with an accuracy of about 5% compared with the values calculated from Eq. (3.50). This is quite sufficient for calculating most line shapes. If more accurate values of the field gradient are required, the directly calculated values can be used. 52 We would like to examine the behavior of the field gradient term (Ezz)f in the limit of large and small values of e. For e --, 0 it is easy to show that the field dependence follows the limit of j2(ke) to lowest order in e. This will give lim(Ezz)= = -C2 (3.52) where C is a constant that depends on ne, T and Zi. Thus, the zero field limit is zero. This is easy to see physically. If e = 0 there will be no preferred direction about the radiator and the fluctuations in the field gradient Ezz will sum to zero upon averaging over the ensemble of ion configurations. The e -- oo limit is somewhat more complicated so we will examine the behavior of a simple nearest neighbor model for (Ezz)E. The large field behavior of the microfield at a charged radiator is governed by a single nearest neighbor ion.58,59 For the strongest fields, the nearest neighbor will be significantly inside the screening length AD,e so screening will be negligible. To approximate (Ezz)c let the field gradient at the radiator be that produced by a single ion located at the distance from the radiator sufficient to give a field value at the radiator of e. This gives us the nearest neighbor approximation of (Ezz}),N = -) /2 (3.53) for large e. In Fig. (3.1) we compare three different models for (Ezz)e. For purposes of comparison, we take (e/ro,i3) = 1. We look at the APEX model which includes ion-ion correlations, the independent particle model of Joyce42 which ignores ion-ion correlations and the nearest neighbor model for large fields as discussed above. We see that the APEX model gives a field gradient whose magnitude is consistently smaller than that of the other two. This is to be expected since the 53 ion-ion correlations tend to cause the plasma ions to repel each other and thus to be less likely to converge on the radiator and create a large gradient. The ion-ion correlations will, in general, lessen plasma ion microfields by causing a rough ordering of plasma ions. The nearest neighbor model produces the largest gradient due to its assumption of all charge located on the z axis. It also includes no correlation effects. The IP model of Joyce falls in between because it includes radiator-ion correlations but no ion-ion correlations. In Fig. (3.2) we examine the behavior of the APEX model for the field gra- dient at several values of the electron number density. For ne = 1 x 1024 cm-3 the average Ezz gradient's magnitude is less than that of the nearest neighbor model as was pointed out above. As the density increases the APEX field gradient approaches that of the nearest neighbor model for all relevant val- ues of the field (in this case we refer to a range of field values of e/eo = 0 to 5). For a particular density, we can see that the gradient also approaches the nearest neighbor value as the field increases. This behavior can be explained by recalling that the presence of large fields at the radiator is generally due to the location of a nearest neighbor perturbing ion close to the radiator. The large field behavior at constant density should be due to a nearest neighbor ion and ion-ion correlations play little role in the interaction of two nearest neighbors at these densities. The probability is small for having a positively charged perturbing ion close to the positively charged radiator. It is even less probable to have two ions close to a positively charged radiator; and we need two for ion-ion correlations to be important. The approach of (Ezz), toward the nearest neighbor value as density increases should be due, simply, to the ions being forced closer to the radiator. With decreased interparticle spacing the field gradient naturally increases. 54 3.2 Atomic Data by Perturbation Theory Solution To evaluate the radiator subsystem matrix elements necessary for the cal- culation of the spectral line shape I(w), we are free to use any complete set of basis states that spans the Hilbert space of the system. A true representation of an operator in this basis will thus consist of a matrix whose dimension is the same as the number of basis vectors necessary to span the particular Hilbert space. For our case this dimension is a nondenumerable infinity. However, a reasonable representation of the operator can be obtained by using a truncated set of basis vectors. For weak fields, a reasonable representation of the Hamil- tonian for a radiator in the state with principal quantum number n can be obtained by using eigenvectors of the field-free radiator Hamiltonian for states n and n + 1 as basis vectors.60 For stronger fields, the need to include other states, including positive energy states, will be necessary. In fact, the reso- nance nature of the strong-field states can not be described without including positive energy states to account for the unbound character of field dependent solutions that allow for the possibility of tunneling. In order to include these effects we will employ an alternative method using a basis set that consists of the eigenvectors of a radiator Hamiltonian that includes the ion-dipole term from the perturbing-ion multiple expansion. This Hamiltonian is Hr() = H0 d- t. (3.54) Hr(c) is diagonal in this field dependent basis set. By using this Hamiltonian, we will automatically incorporate the characteristics of the resonate nature of the states into our basis. Specifically, we will use Hr(e) as our zero order radiator Hamiltonian. The first order correction to this will be the next term in the perturbing-ion multiple expansion, the ion-quadrupole term Vr,i. 55 To calculate our field dependent basis we solve Schr6dinger's equation in the position representation with the Hamiltonian Hr(e). The equation is Hr(e)(r-) = E(r (3.55) The natural coordinates to use for this Hamiltonian are parabolic coordinates.61,62 The wave equation will then be separable. Assuming the plasma microfield to be in the z direction, we write Eq. (3.55) in atomic units (see Appendix A) and scaled to Z = 1 as V2 + + 2E 2Fz (r) = 0, (3.56) where F is the magnitude of the plasma ion microfield. In parabolic coordinates (see Appendix C), Schr6dinger's equation becomes 4 9 f89O 4 8 (89 1 92 +r 0 \" U + 9++rr 0 ^ +a S+ 7 a 9 C + 777 a7 C (W2 (3.57) + (+4 +2E-F( 1-7) 0=0. We exploit the separability of the equation by looking for a solution of the form: e-imW (V) = f(A0)g()--e- (3.58) v'27 We define new scaled variables and transform the two resulting separated equa- tions. We take x -/n y /n (3.59) A (1/4)n3F , and obtain the system of equations 56 {d d m2 2} + -Ex AX2 + A f(x) = 0 , dz dz Tx 4x 2 2 Ey + Ay2 +B g(y) = 0 T- ~ -JU A+B=n. (3.62) Field-Free Solution For the field-free case we take A = 0 and obtain the solutions (in Z = 1 atomic units) 1 EO=- 2n2 m+l Ao = nl + 2 m+l B0 = n2 + -+ 2 (3.63) with (3.64) n=nl + n2 +m + l . Here nl and n2 are the integer valued parabolic quantum numbers.61 The field-free eigenfunctions are fo(x) = u(ni,m, x) where go(y) = u(n2, m, y). vTm/2 u(n1,m,) = e-VXm -/2 -2m() [(nl + m)!]3/2 n" (3.65) (3.66) Lm (x) is an associated Laguerre polynomial given by63 and Wdy dY with (3.60) (3.61) L(x) = [(n( + m)!]2 (-x)P j- (m + p)! (nl p)! p! p=O0 For our case m is restricted to integers for which m > 0. Other definitions of Lm (x) are quite common.33 Field Dependent Solution In order to solve Eqs. (3.60) and (3.61) for A 0 0 we will expand the unknowns f(x), g(y), A, B and E in powers of A. The zero order terms will be given by the A = 0 results above. This perturbation expansion technique closely follows the work of Hoe et al.6465 The expansions are 2P>1 A = ApAP p>0 B = Z BpAP (3.68) p>0 f(x) = AP ah u(n1 + h,m,z) p>O h>-ni g(y) = AP bp u(n2 + h,m,y). p>0 h>-n2 The sums for index h are over all possible values of nI and n2. Upon evaluation of the coefficients ah and b^, however, we will find that all terms in the sum are zero except for max(-nl, -2p) < h < 2p. This is also true for the sum involving n2. We immediately see that because of the condition A + B = n, we have A0 + B0 = n (3.69) and Bp = -Ap for p > 0. (3.70) The other coefficients in the expansion can be obtained by solving for the p'th coefficients in terms of the p- 1 and lower coefficients. Thus, a knowledge of the zero order terms will allow us to construct all higher order terms iteratively. To accomplish this we substitute the expansions for f(x) and g(y) into Eqs. (3.60) and (3.61), multiply by u(x) and u(y) and integrate over x and y, respectively. For the x coordinate we obtain Sdx u(n + k, m, x) {Ho(x) + A + x(1 + 2n2E) Ax2 f(x) = 0 (3.71) where d d m2 x Ho(x) = -xz 4 4 (3.72) dx dx 42 4 Following Hoe et al64,65 we call Ho(x) the field-free effective Hamiltonian for the x coordinate. It gives Ho(x)u(n1 + h, m, x) = -Aou(n1 + h, m, x) (3.73) with m+l A0 = n1 + h + 2 (3.74) Carrying out the integration gives 2p a AP y 2 ap dxu(n + k,m,x)Ho(x)u(n1 +h,m,x) p>O maz(-nl,-2p) (3.75) + A6k,h + (1/4)(1 + 2n2E) (x)k,h A (x2)k,h} 0 where (X")k,h j dxxnu(nl + k,m,x)u(ni + h,m,x) (3.76) and 00 Sdxu(ni + k,m, x)u(ni + h,m, ) = 6k,h. (3.77) Upon expanding E and A in powers of A, we see that the p'th coefficient is given by p-i 2p p-i ka = Ap-i a x)k,ha -1 + (x)k,h E vp- i=0 h>max(-nl,-2p) i=0 (3.78) If the maximum value of the sum index is less than the starting value of the index, then the sum is defined to be zero. We will return to the special case of k = 0 below. Likewise, we obtain for the y coordinate p-1 2p p-1 kbk Bp6 + ()k,hh-1 (Y)k,h E p-ibh i=0 h>maz(-n2,-2p) i=O (3.79) For p = 0 we must have ah = bh = 0,h. (3.80) To obtain the energy coefficients up we take the k = 0 case of Eqs. (3.78) and (3.79). We obtain two equations for Ap. Equating these and solving for Vp gives i = 1 Z [(1O + 0,h] S=1 (3.81) The sum over h is indexed from max(-nl, -2p) or max(-n2, -2p) to 2p as appropriate. We still need to evaluate the coefficients ak and bp for k = 0. To accomplish this note that Sdxf2(x) = 1 (3.82) and 0 dyg2(y) = 1. (3.83) Evaluating these by using the series expansions for f(x) and g(y) and exam- ining the coefficients of a particular term of order AP will give 2p p-1 a -- a ah h (3.84) h>maz(-ni,-2p) i=1 and 2p p-1 b = bb, _, (3.85) h>maz(-n2,-2p) i=1 We now have the coefficients for calculating the energy E and the components of the wavefunctions f(x) and g(y). To evaluate these we need the integrals (x)k,h and (z2)k,h as well as those for (Y)k,h and (y2)k,h. These can be straight- forwardly evaluated to give (x)k,h = (2nl + 2k + m + 1)6k,h /(nl + k + l)(nI + k + m + 1)6k+1,h (3.86) v/(nl + k)(ni + k + m)k-,h, and (x2)k,h = {6(n1 + k)(nl + k + m + 1) + (m + 1)(m + 2)} Sk,h 2(ni + k + 1)(nj + k + m + 1)(2ni + 2k + m + 2)6k+l,h 2/(ni + k)(n1 + k + m)(2n, + 2k + m)6k-_,h + V/(nl + k + 2)(ni + k + m + 1)(ni + k + 1)(ni + k + m + 2)k+2,h + /(nl + k)(ni + k + m 1)(ni + k + m)(ni + k 1)bk-2,h (3.87) The relations for (Y)k,h and (y2)k,h are obtained from these formulae by taking x -+ y and ni -- n2. Normalization In order to compute matrix elements from the field dependent wave func- tions given in Eq. (3.58), we also need to determine their normalization. We take the normalization constant to be N(ni,n2, m, A) =- dxdy (x + y)[f(x)g(y)]2 (3.88) Expanding N(nl, n2, m, A) in powers of A we have N(n, n2, m, A) = E NpAP (3.89) p>0 To evaluate the coefficients Np, we expand the functions f(x) and g(y) in powers of A, matching terms on the right- and left-hand sides to obtain Np = E E [a ()h,k + bhb-i(Y)h,k] (3.90) i=O h,k The second sum is indexed from max(-nl, -2p) or max(-n2,-2p) to 2p as appropriate. Note that N = n (3.91) Dipole Matrix Elements Now that we have developed the formulae for the perturbation solution of the Schridinger equation for a radiator in the presence of a uniform electric field, we can go on to calculate the dipole and quadrupole matrix elements needed for the calculation of the spectral line shape I(w). Since we know the wave functions only in terms of their perturbation expansions, it will be necessary to develop perturbation expansions for the matrix elements as well. We will follow the calculation of the atomic matrix element of the coor- dinate z as an example. The matrix elements for x and y, as well as the quadrupole matrix elements are calculated in the same way and only the re- sults are presented here. Further details concerning the technique can be found in the references of Hoe et al.64,65 We wish to calculate the atomic matrix element ( /(n1, n', m', A')|z(n1, n2, m, A)) . The normalized wave function is given by 0 = 0(nl,n2, m,A) = 2 jr(n, A (gnmA() em (3.92) V2vrN(n1, n2, m, A) so the matrix element of the z coordinate of the radiator's bound electron is 1 roo oo 2r I i dx 7)* (n, n,2, m, m'')(nl 2, m,) (ol zd1) ( 27 A) =-j d dT dp (2 ^2)4,*(nl ,, n', m ', X)(ni, n2, m, X) _'n dr dJ77 ( 2 _- ,72) 8 0/0 where As a notation x fn,,'A,(l/n')fnlmA(l/n)gn's,:,(i/n')g,,,mA( /n) a =s iN(n'1nilm'')N(nln2mA) . nal simplification, take ZninM ( |z) . In order to evaluate b, consider AP = N(n', n'2, m', ')N(n,n2, m,A) n3F) = N1 "n'3) p'>0 p>0 = C EQ QpF '+P P p'>0op>O E QiQp-iFP p>0 i=0 p>O We have defined -(n3)P Qp - 4Q (3.93) (3.94) (3.95) (3.96) (3.97) and p WVp ZQ Qp-i . i=O (3.98) This gives us an expansion for N in powers of the field F. Next, we use this to obtain the expansion of $. Since 4 = vl/, we have (3.99) If we expand 4 and AN in powers of A and multiply it out, we obtain P P iFi F = F NpFP p>0 i=O p>O P A/= pZip-i i=0 p-1 = Eip-i + 240P* i=l1 Note that p = 0 gives 0 = v/= No Solving for 4p gives for p 1 , Ap E&1 ^p-i pD P=1 i P-i P- 2n t or in terms of the more basic quantities p p-1 P (n13)(n3)-N'iNpi E ip- i=0 i=1 (3.100) (3.101) (3.102) (3.103) (3.104) P = n2n2 1P~2--1 1"n n for p > 1. For the p = 0 case, we have 0 = t22 (3.105) n/ In/ I To proceed with the evaluation of Zn2m return to Eq. (3.93). We can separate the functions of the two coordinates to obtain Z 2' = F(2)G() F(0)G(2)} (3.106) where F(a) d (ifnimA,(/nflm)fnImA( /n) (3.107) and G(a) j d .Trgnm'A.,(7/n')gng2mn(l/n). (3.108) The orthogonality condition will not apply since we are dealing with integrals of the form fo" d( f'((/n')f((/n) and in general n' 5 n. To evaluate Eqs. (3.107) and (3.108), we expand their left-hand sides in powers of F to obtain F(a) = F"a) FP p>0 p>G. (3.109) G(o) = G( ) FP p>O Expanding the right-hand sides of the integrals and rearranging as a series in increasing powers of F allows us to equate terms of like powers in F from both sides to obtain F( ) E 1 '3'n"( -' -. Fa) h ni3in3(p-i) a i ak h,k i=0 ) ) 3i3(p-i) bbk =4P E k,h n h,k i=0 where the sums over h and k are as before, and where ,(a =h d "u(ni + k, m, /n) u(n'l + h, m', /n') , au(2 + km, ) u( +) h, /) kh =- dr q"u(n2 + k, m, n/n) u(n' + h, m', r)/n) . JO (3.110) (3.111) (3.111) Ih can be evaluated from the definition of u(nI, m, ?'/n) to obtain I(l) ( 2nn' ~)+1 kh = +'n) J(nl + k)!(n1 + k + m)!(n' + h)('1 + h + m)! 2n m' nx+k nI+h nn+n',-, q=O s=O n-2n' \n+n'j q --2n n+n/ (3.112) (1 + -T- +q+s)! (m + q)!(n1 + k q)!q!(m' + s)!(n' + h s)!s! For J( just take n1 -+ n2 in Eq. (3.112). To evaluate 'nn2 we use the expansions for F(a) and G(a). We obtain T n' nm' nin2lm -I- (3.113) and ( +2n' ' n + n/ with Tln:m = T TpFP, (3.114) p>0 where Tp 1F, (2)G(. o 2'.) (3.115) i=0 We can also introduce the expansion Zn n2m = Z F Z P. (3.116) P>O Rewriting Eq. (3.113) as Tfim' _y- 4Zfi 1 -2 Tn1n2m n2m (3.117) and using the expansions for each side gives P Tp= ZiP-i. (3.118) i=0 Solving this for Zp gives Zp= T- p Zi0p-i (3.119) If p = 0 we take the sum to be zero. This expression gives the terms in the expansion for (O'l z \1i) given by Eq. (3.116). We have therefore successfully calculated the perturbation expansion for the atomic matrix element of the radiator's z coordinate. We have followed the calculation of the z matrix element in detail in order to demonstrate the perturbation theory method. For the remaining matrix elements we present only the final results. The matrix elements for x and y 68 are calculated by the same procedure except for the angular integrals which we evaluate explicitly. For x = v/7 cos V we have f27 dqp cos ve-i(m'-m)p = 2 r 0 2 [ + '(3.120) 2x (A) 2 m,m' For y = V/ sin W we have 2 dp sin e-i(m'-m)" = [m+1,m -l,m] S2i m, m l (3.121) 27A (-) i2 m,m * The factor of 27r will cancel with a similar factor from the wave function nor- malization. The x matrix element is given by Xnm = XpF, (3.122) p>0 where Xp = Rp i- Xip- (3.123) x0 with S- a (3/2)G(1/2) + 1/2)G(3/2) (3.124) i=0 The y matrix element is given by Y-m = -i Yp FP, (3.125) p>O where Yp i (3.126) with Kp m,m 1 +/2)G (3.127) P p-i p- i=0 Quadrupole Matrix Elements The last atomic matrix element we need is the quadrupole moment tensor component Qzz. Its operator in terms of radiator electron coordinates is given by (taking e = 1) zz = (3z2 r2) 1 2 (3.128) = 2 2+ 2 4r). The matrix elements of Qzz are then given by (Qzzn = EQpFP, (3.129) p>0 where O S, EYip QOQ p-i S-= (3.130) with Sp = m,m (F )G(O) + F(O)G 3F)G(1) 3F(I)G-)i i=0 (3.131) The calculation of Qzz goes beyond the work of Hoe et al64,65 but the method is the same and is as outlined above in the calculation of the z matrix element. 70 For the perturbation theory calculations in this work, we will go to sixth order, taking p = 6. This will be more than sufficient for our purposes. 3.3 Atomic Data by Numerical Solution Next, we give a qualitative discussion of the numerical solution of Schr6dinger's equation for a hydrogenic radiator in a uniform electric field. We have already developed the perturbation solution useful for relatively small field values. Di- rect numerical solution is applicable, in principal, for fields of any magnitude, though, in practice, it turns out to be inconvenient for very small field val- ues because it is difficult to numerically handle the resultant extremely sharp resonances. We will discuss this point further below. Consequently, the pertur- bation and numerical solution techniques are complementary and additionally should serve as consistency checks since they should produce matching results for a suitable intermediate range of fields. For large field values, the resonance nature of the radiator states becomes an essential part of their description and the perturbation theory used here is no longer useful for finding the field depen- dent eigenvalues and eigenfunctions of Schrodinger's equation. We also discuss the calculation of the width of these resonances and the numerical evaluation of the necessary atomic matrix elements. The presence of the uniform field at the radiator changes the very nature of bound atomic states by turning each discrete energy level into a shape reso- nance; in other words, there will be a solution for a particular set of quantum numbers that includes a continuum of energy values.66,67 For a field strength given by A, the states can be characterized uniquely by the quantum numbers ml, A and the energy E. The quantum numbers mi and A remain discrete but E now has a continuous spectrum. For weak field values, the density of 71 states is greatest for a radiator electron with an energy that closely corre- sponds to the discrete states given by the perturbation theory solution to the problem. As the field value increases, however, the density of states broad- ens and the probability of finding the electron with an energy more widely spaced from the resonance center increases. This is a result of the field in- duced broadening of the resonance. The width, F, of the resonance is also roughly inversely proportional to the lifetime of the electron inside the radia- tor. This is most easily seen from the time-energy uncertainty principle but also follows from a detailed WKB treatment of resonance decay.68 In fact, the WKB treatment68,69 of the field ionization problem leads to a probability of finding the quasibound electron inside the potential barrier given by the time dependence factor e-t. Thus, when t = 1/P the probability has decreased from one at t = 0 to 1/e = 0.37. The presence of the field lowers the Coulomb potential along the upfield direction thus producing a potential well defined by a finite-size potential barrier. The electron is then able to tunnel out of the radiator. Indeed, in the presence of the field, there are now no truly bound states in the sense of the electron being permanently associated with a partic- ular atom baring radiative or collisional ionization. However, if the lifetime of the electron inside the radiator potential barrier is long compared to the time of interest for the radiator, it is effectively bound to the radiator. For even larger field values, the plasma electric field potential will exceed the attractive central Coulomb potential; the energy level will be above the top of the poten- tial barrier. In this case, if the energy level is close to the top of the barrier, the electron can still spend a significant amount of time in the vicinity of the radiator nucleus and will retain some of the character of a bound state. As the field is increased further this bound character will gradually be reduced. This 72 phenomenon results in the smooth broadening of the resonances as the field increases until they overlap. At that point the resonances have merged into a relatively smooth background continuum. We need to numerically solve the system of coupled second order ordinary differential equations (ODE's) given by Eqs. (3.60) and (3.61). In order to em- ploy the solutions of these equations for the calculation of the matrix elements and line shapes, and since the eigenvalues are not discrete, we will use the Breit-Wigner62 Lorentzian parameterization of resonances and take the eigen- functions and eigenvalues at the center of each resonance as a representative value over the entire resonance. As long as the resonance is fairly narrow and remains distinct, this is a reasonable approximation. It breaks down for strong fields where the resonance becomes strongly asymmetric. Fig. (3.3) gives an example of this phenomenon. The reason for this asymmetry lies in the shape of the potential barrier; as the energy level goes higher it sees a thiner potential barrier. The thiner the potential barrier, the shorter the lifetime of the state and the greater the uncertainty in its energy. Consequently, the broader width on the high energy side of the resonance is due to the greater uncertainty in the energy value at that point. These resonance asymmetries have been inves- tigated experimentally by Harmin.70 We do not need to describe this behavior in detail since its contribution to the line shape is greatly attenuated due to the low probability of occurrence of the relevant high field values as reflected in a small value of the microfield probability function. We employ the numerical solutions of Eqs. (3.60) and (3.61) by R. Mancini and C. Hooper71 who follow the method of E. Luc-Koenig and A. Bachelier.66,67 73 We give here a general qualitative procedural description of the solution tech- nique (see the above references for further details). The solutions for the equa- tion in the scaled variable x are effectively bound and, therefore for large x are exponentially damped. The number of nodes in the wavefunction's x com- ponent is given by the parabolic quantum number nl. The solutions for the equation in the scaled variable y are effectively unbound; that is, the solutions for large y outside the potential well are oscillatory in nature. On the other hand, for x and y -- 0 we have f(0) = g(0) = 0. To solve the the two equations, Eq. (3.60) and Eq. (3.61) consistently, first, we solve the effectively bound equation in x for a particular case given by specific values of F, E, nl and ml. This will give a value for the constant A by imposing the exponentially damped behavior of F(x) for large x (this is an eigenvalue problem for A). We use this to determine the constant B and proceed with the solution of the effectively unbound equation in y. We change Eqs. (3.60) and (3.61) into a form more convenient for numerical solution by eliminating the first order derivative with the transformations '(xZ) = V f() (3.132) and g(y) = vg(y) (3.133) Our two equations will then be transform to d2 ( + Tz() =0 (3.134) dx2 and d2 g(y) dy2 + TyG(y) = 0 ,(3.135) where A n2 m2 -1 T = -+ E Ax (3.136) x 2 4x2 and B n2 m2 -1 Ty = E-2 +Ay. (3.137) y 2 4y2 The motivation for making this transformation is that now we have to deal with second order ODE's with missing first derivatives which can be efficiently integrated using the Numerov algorithm. To solve Eq. (3.134) for an energy value E, a scaled field value A, a value for the z component of the orbital angular momentum ml and a value for the parabolic quantum number nl, we pick a trial value for A. We solve the differential equation starting at x = 0 using a power series expansion; this is used to initialize an outward numerical integration using a fourth order Numerov algorithm.66,72 This numerical integration is continued up to the outer turning point of the effective potential barrier. (If the energy level is above the top of the potential barrier so that there is no outward turning point, we use the maximum of the potential barrier.) For suitable large x values, an exponentially damped asymptotic solution is used to begin an inward numerical integration, again using the Numerov method. The two solutions overlap at the outer turning point of the potential barrier (or the potential barrier maximum if the energy level is above it). At the meeting point, we match F(x) by adjusting a multiplicative constant, and from the matching 75 condition for d.(x)/dx we compute a correction to our initial guess for A. This defines an iterative procedure which is continued until the initial guess for A agrees within a given tolerance with the value found from matching the derivatives.73 To solve Eq. (3.135), the unbound equation in the scaled variable y, we note that once we have the value of the constant A we have B = n A. This gives us E, A, nl, ml, A and B; therefore, it is not necessary to make any initial guesses. We start the outward integration as we have done for the variable x using a power series solution, and then switch to the numerical integration using the Numerov method as before. Since the constant B has already been determined, we continue the integration outward to large y beyond the effective potential barrier where we fit a large-y analytic oscillatory asymptotic solution to the numerical results. This will give the value of the amplitude of g(y) for large y. By studying the distribution of amplitudes with respect to the energy, the width and center-location of the resonance can be estimated. Since the values of the energy can have a continuous spectrum, we have the normalization condition per unit energy given by d dpj df d77 ( + r) m*,A,E Cm,A,,E = mmIA,A,6(E E') . (3.138) it can be shown62 this is equivalent to the condition Iout = 1/2r, where Iout is the outward flux in the r7 direction at large 77. The outward flux lout can be given in terms of the outgoing wave ,out where we have decomposed the stationary wavefunctions into outgoing and incoming components in the r di- rection. 76 The density of states can be obtained by studying the behavior of the electron probability density near the nucleus for the component of the wave- function associated with a constant value of the outward flux. As stated before, we can approximate the density of states for a particular resonance using the Breit-Wigner parameterization.62 We obtain 1r2 D ( (E) = m (Er) E ) (3.139) n 1 n i (E Er)2 + 'ir2 where Er corresponds to the energy value at the maximum of the density of states of the resonance, and F is the full resonance width at half maximum. This is a reasonable parameterization as long as the resonance remains distinct. The average lifetime rres, of the resonance can be associated with the resonance width through65 Tres ; h/F (3.140) This is just a manifestation of the time-energy uncertainty principal. Once we have numerically evaluated F(x) and g(y), we are free to calculate the matrix elements needed for the line shape. By straight forward numerical integration we evaluate matrix elements for x, z and Qzz. We cut-off the in- tegration over the scaled variable y at the outer turning point of the effective potential barrier for energy values below the maximum of the potential bar- rier. (For energies greater then the maximum of the potential barrier, we stop the integration at the value of y corresponding to the maximum of the poten- tial barrier.) Due to their resonance nature, the wavefunctions extend overall space but we are only interested in the portion located near the radiator. The numerical values of the matrix elements are not significantly sensitive to small changes in the location of this integration cut-off. Since we now have properly 77 normalized wavefunctions, we can evaluate the field dependent matrix element of a real operator 0 between two resonance states denoted by subscripts "i" and "j" by integrating over the energy range of each resonance. The square of the matrix element for an operator 0 can be written as, IOi,j(F)I = dEi dEj I ( EF) ((Ei, F)1)I2 (3.141) For narrow resonances we can approximate this results by O i ( ) 2 (2)2 (i (Er,i,F) O1 j (Er,j,F)) (3.142) where we have used the value of the eigenfunctions at resonance center. This is a reasonable approximation because the main field dependence of the wave functions, for narrow resonances, is contained in the normalization factor which is a function of the square root of the density of states. Our evaluation of the square of the matrix element allows us to integrate over the density of states to obtain the 7r7i/2 factors. Since these factors are always positive and real, the sign (and any factors of i) of Oij is reliably given by the evaluation of (4i(Ei,F) 0 ICj(Ej,F)) at Ei = Er,i. Fig. 3.4 gives an example of the calculation of atomic matrix elements by both perturbation theory and direct numerical solution of Schr6dinger's equation. We see that the two solutions join smoothly for intermediate field values and diverge for larger field values as the accuracy of the perturbation theory solution breaks down. As the field F goes to zero, Eq. (3.141) will reduce to the usual no field limit where O,j(0) = ( 0 ) (3.143) Here, i = u(ni, ml, )u(n2, m, ) (3.144) which, as expected, is just the solution to Schr6dinger's equation for the field free case. Here, the state "i" is specified by E, nl, ml and F. Since for a solution with a discrete values of E there is a unique value of the principal quantum number n, we can obtain n2 through the relation n = nl+n2+(ml+1 and our limiting state can be characterized by ni, n2 and ml as expected. 3.4 Electron Delocalization and Field Ionization We have been using, as a model for the radiator-plasma interaction, the picture of a radiator in the presence of a uniform plasma microfield produced by the averaging of the fields from all the plasma particles. We have added the ion-quadrupole correction to account for a relatively small field gradient at the radiator. This picture is only a model, of course. As the plasma density increases, it becomes unreasonable to ignore the presence of highly charged nuclei close to the radiator. This fact creates a qualitatively different picture. In the uniform field case, a quasibound electron will tunnel out of the radiator and continue to accelerate toward the field source at infinity. In reality, large fields are produced primarily by nearby charges. The quasibound electron will then orbit about the ensuing multicenter ionic potential in a molecular orbital74 instead of tunneling into the interparticle space; the electrons will become delocalized. A complication to this molecular orbital picture is the large number of free plasma electrons in the internuclear volume. It is unrea- sonable to assume that the quasibound electrons will not strongly interact with these free electrons as they resonate between various nuclei. Perhaps a more accurate picture for these conditions would be the self consistent treatment of 79 both bound and free electrons in the presence of the multicenter potentials. Some work has been done along these lines from an analytic point of view by Rogers,75 and from a numerical simulation point of view by Younger et al.76 Additionally, a model consisting of a cluster of nuclei with bound electrons has been considered by Collins and Merts77 but for Te = 0. Much work remains to be done before these models are capable of accurately treating spectral line transitions for plasmas of high density and temperature. In addition to the question of whether field induced ionization or resonance exchange of electrons between nuclei is a better description of what strong fields do to quasibound electrons, there is the question of the level broadening produced by these two mechanisms. The field ionization picture contains level broadening due to the resonance character of the energy levels in the presence of the nearly uniform electric field. This is most easily seen from the uncertainty principle where the width is related to the resonance lifetime by AE hi/rres. In the molecular orbital picture, an electron bound to two nuclei will still have a discrete energy. However, as the number of nuclei in the molecule or cluster increases, splitting is induced. If each level is m-fold degenerate and there are N nuclei in the cluster, the splitting will be on the order of mN-fold.77 This large degree of splitting will be qualitatively similar to the intrinsic level broadening produced in the field ionization case. We see, therefore, that the level broadening from the two different models should produce similar results under similar conditions, at least as far as the level width is concerned. At this point, it appears that the molecular orbital picture may be the most capable of describing our dense plasma since it is a truly many-body theory. However, there is the complicating factor of the intercollision of free and bound electrons. As the bound electrons exchange between the nuclei, 80 they will be subject to collisions with the free plasma electrons. If they are then thermalized into the plasma, the situation will be more like the field ionization case. A realistic picture probably lies somewhere between the two over simplified models. In the molecular orbital model, radiator levels are not depopulated. On the other hand, in the field ionization model, all levels will be depopulated on time scales relevant to radiative decay, for sufficiently strong fields. If the actual phenomenon lies somewhere between these two pictures, then the resultant effect will range somewhere between no change in the line shape due to no level depopulation, to a substantial decrease in line radiation for large fields due to level depletion from field ionization. The levels will not be completely depopulated, of course, due to collisional and radiative repopulation. What effect could these two models have on the spectral line shape? In order to estimate the importance of this possible ionization effect on level populations, we will examine the limiting case of field ionization due to tunneling with no reverse process to repopulate the radiator level except the usual collisional and radiative processes. We will construct a simple kinetic population model to describe the influence of field induced ionization on the relative population of the energy levels of hydrogen-like argon. We will present only the extreme case of no back tunneling in order to ascertain the maximum possible population depletion. As outlined above, the real situation could lead to results ranging somewhere from no tunneling depletion at all (the pure molecular orbital picture) to maximum depletion due to one way tunneling (the field ionization picture). We will not look for departures from absolute LTE populations but only relative departures from LTE among the upper states 81 involved in the spectral transitions; relative population differences are all we will be able to observe from experimental line spectra. We will examine the relative distribution of populations of the n = 3 and 4 levels of hydrogenic argon. The model includes connections between the hydrogenic excited state levels to the ground state and the fully stripped ion while omitting the n = 2 levels which are little affected by field ionization. At the high densities we are concerned with in this work, collisional processes will dominate the other population altering processes and relative LTE populations will be very probable if we do not include the field ionization. We consider only these few levels because we are mainly interested in the question of whether the field ionization effect can depopulate the levels enough to produce a significant change in the line shape. We will incorporate into the kinetic model collisional excitation, collisional de-excitation, collisional ionization, three body recombination, spontaneous emission, radiative recombination and field ionization. We can easily formulate a rate equation for the population Ni of a level denoted by the subscript "i". The number of excited quasibound levels is denoted by N, the fully stripped ion by the subscript "N+1" and the ground state level by "O". Our rate equation is dNi i-1 N+1 d~ = Nj neCi + Ej Nj neC d j=0 j=i+l N+1 i-1 E NineCj NineCf (3.145) j=i+l j=0 N+1 i-1 + E NjA,i E NiAi Niri j=i+1 j=0 82 for i = 1 to N + 1. Note that since the fully stripped ion has no electron to escape, FN+1 = 0. We will take the populations to be relative to the ground state population, No. This will give Nj -* Nj/No and No -- 1 for j = 1 to N +1. Since we are only interested in the steady state solution to these kinetic equations, we examine the case with dNj d = 0, (3.146) dt for all i. Next, we define the rate coefficients in Eq. (3.145): The collisional excitation rate is given by Cfj, where78 Cf = 3.75 x 10-5 T-1/21 (i Ij) 2 e-AEi,/kT cm3 sec-1 (3.147) Here, T is in degrees K, AEi,j is the energy difference between levels i and j, and (il F ij) is the atomic matrix element of r in atomic units. The collisional de-excitation rate is given by Cdj. This can be related to the collisional excitation rate by the principle of detail balance.79 This gives Cd. = Cf e-A j/kT (3.148) for states of equal statistical weight. The collisional ionization rate is given by COf e where80 N+1 = 3.46 x 105 1306) T-/2 e- {1 xi eIEl(xi)} cm3 sec-1 (3.149) Here, T is in degrees K, Ii is the ionization potential of level i in eV and xi = li/kT , (3.150) and loo e-xt Ei(x) = dt (3.151) is the first order exponential integral. The three body recombination rate is given by CfN+,i. This can be related to the collisional ionization rate by the principle of detail balance. The spontaneous emission rate is given by the Einstein coefficient Aij, where79 Aij = 2.142 x 1010(AEi,j)3 (i Ij) 12 sec-1 (3.152) Here, AEi, and (ij r' jj) are in atomic units. The radiative recombination rate is given by AN+1,i, where81 AN+l,i = 5.197 x 10-14 neZx3/2 eZEl(xi) sec-1 (3.153) Here, Z is the radiator charge, ne is the plasma electron number density given in cm-3 and zx is given by Eq. (3.150). For the steady state case and because we have taken No = 1, we can write all of the N + 1 rate equations in the form ao + alN1 + a2N2 + .' + aNNN + aN+1NN+1 = 0, (3.154) where the coefficients ai do not depend on the level populations. These equa- tions may be cast in matrix form to obtain C=RN . (3.155) 84 Here, C is a column vector of dimension N + 1 containing the a0 rate coeffi- cients, N is a column vector of dimension N+1 containing the level populations Ni and R is a N + 1 x N + 1 matrix containing the remaining rate constants. This equation is easily solved to give the N + 1 relative populations Ni. We have N = R-C (3.156) This effectively solves the relative populations problem for the N quasi- bound excited state levels plus the the fully stripped ion. For the total number of quasibound levels, we use the n = 3 and n = 4 levels of a hydrogen-like ion. This gives a total of N = 32 + 42 = 25 levels. The effect of the lack of inclusion of the n = 2 levels was examined by comparison with a more detailed kinetic population model.82 This showed no significant change in the relative populations upon inclusion of these levels. 85 -1.0.I -0.8 / - / / / _- )- / - S-0.6 / : / -0.2 - 0 -/ / 0.0 0 0.5 1 1.5 2 E/.o FIGURE (3.1) Field gradient term (Ezz)c as a function of the field for the APEX (- ), IP ( ...... ) and NN (- -) models at ne = 1 x 1024 cm-3 and kT = 800 eV. The field gradient is in units of e/r3,i. The three models are discussed at the end of section 3.1. -6 -5 -4 -3 -2 0) =f ct (d c, F: 0) a (d k ho -.** / 0 1 2 3 4 5 6 e/Eo FIGURE (3.2) Approach of the constrained average field gradient term (Ezz)e to the nearest neighbor limit for increasing density. (- -) refers to ne = 1 x 1024 cm-3, (- -) refers to ne = 5 x 1024 cm-3, (......) refers to ne = 1 x 1025 cm-3 and (- ) refers to the nearest neighbor model discussed in the text. All temperatures are kT = 800 eV. The constrained average is evaluated using the APEX model given by Eq. (3.50). -3.00 -3.25 -3.50 rzl -3.75 -4.00- "" -4.25 -4.50 3 4 5 6 7 F FIGURE (3.3) Example of the asymmetry of a resonate state discussed in section 3.3. This is the resonance with quantum numbers n,nl,n2,m = 4,0,2,1 or n, q, m = 4, -2, 1. The resonance center is given by (- ) and the width at half maximum is given by (- -). The field strength F is in units of a.u./1 x 10-4 and the energy E is in units of a.u./1 x 10-2. a.u. denotes atomic units (see Appendix A). 150 , 125 - 100 A o 75- C - S50 - 0 25 0 JI l I"I I I I "I II I I 0 0.0005 0.001 0.0015 0.002 0.0025 F (a.u.) FIGURE (3.4) Comparison of numerical and perturbation theory results for the calculation of a quadrupole atomic matrix element. The example given here is a diagonal matrix element with nI = 2, n2 = 0 and m = 0. The numerical solution as discussed in section 3.3 is given by (-) and the sixth order perturbation theory solution calculated from Eq. (3.129) is given by (......). All quantities are expressed in atomic units scaled for radiator charge, Z=1. CHAPTER IV RESULTS AND DISCUSSION In the previous chapters we have formulated a general theory of plasma spectral line broadening and discussed many of the approximations used to arrive at a calculable result. The theory that we use to make these calculations has been generalized beyond previous formulations in that it includes higher- order microfield effects on: atomic matrix elements, radiator energy levels, state lifetimes, and level populations. We will now present and discuss the results of these calculations. The presence of these higher order field effects can be expected to lead to several discernable changes in the spectral line shapes of radiators in dense plasmas. For the physical conditions we are examining the electron number density, ne, varies from 1 x 1024 to 1 x 1025 cm-3 and at a temperature, kT, of 800 eV. In this range, the lowest order corrections of the ion-quadrupole and the quadratic Stark effects will generally give rise to a blue asymmetry of the spectral line shape. This means that the intensity of the high energy (or blue) side of the spectral line will be enhanced over that of the low energy (or red) side of the line. This comes about by the preferential shifting to lower energy of components comprising the manifold of energy levels associated with a principal quantum number n. If all of the components were shifted by an equal amount, there would be no discernable change in the spectral line shape; only an overall line shift. The n = 1 ground state for these transitions in highly ionized hydrogenic ions is little affected by the field because of the much stronger binding potential for the n = 1 state. Additionally, the ground 89 90 state experiences no linear Stark effect. Energy levels that have been Stark shifted to the low energy side of the unperturbed upper state of the transition correspond to quasibound electrons that are found to have the maximum value of their probability amplitude on the upfield side of the radiator potential well. This corresponds to the side of the origin where the potential barrier has a maximum; the other side, the downfield side, corresponds to the potential barrier increasing without bound. The upfield electrons are more easily affected by the field and hence, their energy levels are shifted more. This preferential shifting produces a spreading out of intensity on the low energy side of the line, and thus, an increase in the peak height of the high energy side of the line. The addition of the ion-quadrupole effect enhances this trend but also generally produces a bunching, or lessening of the splitting amongst the energy levels on the blue side. This further adds to the blue asymmetry of the line. The consequences of the field dependence in the wave functions and their resulting matrix elements are harder to characterize due to less systematic results on the line shape. Therefore we study this effect only in combination with the other field effects. In this dissertation, we assume that the areas of all line shapes are normalized to one. The possible field ionization depletion of the radiator upper level popula- tions will lead to a lessening of intensity in the line wings. Again, the red wing should be more strongly affected due the lower potential barrier height on the upfield side of the potential well. For sufficiently strong fields, The Stark effect will cause the energy levels from states with adjacent principal quantum num- bers to overlap. Inclusion of this phenomenon is important for the accurate representation of line merging. 91 4.1 The Lyman a Line The La calculation contains the effect of the microfield on the atomic physics through field dependent matrix elements and energy levels. This gives rise to a field dependent fine structure correction to the radiator Hamiltonian (see Appendix E). The field dependence appears in the dipole matrix elements, and because we include the ion-quadrupole effect, also in the quadrupole matrix elements. We do not study the effect of possible field ionization on the levels of the La transition because, for this case, the resonance width r < 0.1 eV. Since for this transition the resonance nature of the states is not important, perturba- tion theory is adequate for the calculation of the atomic physics. At the lower end of the density-temperature range we are interested in (ne = 1 x 1024 cm-3, kT = 800eV), the Doppler effect is an important source of broadening so its inclusion is essential. For the densities and temperatures we are dealing with, the n = 2 and 3 levels are well separated in energy and do not overlap for any relevant field strengths. Consequently, we will not need to consider this overlap for the La line calculation. In Fig. (4.1) we have the La line at n, = 1 x 1024 cm-3 and kT = 800 eV. The higher order field effects produce only a slight blue asymmetry in the line which attenuates the red peak by less than 5% in magnitude. The fine structure splitting is readily apparent with Doppler broadening being responsible for more than half the width of the peaks. We estimate the Doppler width from34 hwD hwi,jkT/mrc2, where hwij is the transition energy and mr is the radiator mass. In this case hWD ~ 3 eV. As the density goes up the Stark broadening will increase and the Doppler broadening will become less important as its fraction of the total width decreases. In Fig. (4.2) the density has increased to ne = 5 x 1024 cm-3; the 92 Stark broadening has increased and begun to obscure the fine structure split- ting. However, the higher order field effects are still of only minor importance, accounting for changes in peak heights of less than 4%. For ne = 1 x 1025 cm-3 there is still no significant attenuation of the red peak due to the higher or- der effects. We conclude that, for the density range of this work, the plasma microfield is not strong enough for these higher order effects to become impor- tant. As the average plasma microfield strength continues to increase due to increasing density, it will eventually become of the same order as the central Coulomb field. At that point the higher order effects will be much more likely to be manifest. If we go much beyond 1 x 1025 cm-3, however, the electron degeneracy will have to be incorporated into the theory as we pointed out in section 2.1. 4.2 The Lyman 0 Line The L# line has its excited state electron in the n = 3 state. This makes it much more susceptible to the effect of the plasma microfield than in the La case. To investigate the higher order effects on this line, we will include the ion-quadrupole effect, field dependent atomic physics, the presence of a slight resonance width, F, for each energy level, and the overlap of the levels of principal quantum number n = 3 and 4. This last process allows for atomic matrix elements connecting the two principal quantum number manifolds. It also contributes more terms to the sum over intermediate states that is performed when evaluating the electron broadening operator M(Aw). In the field-free atomic physics calculation, this is also possible but the energy levels are widely separated at their unperturbed 93 energy values. Consequently, their contribution to the sum over intermedi- ate states will be minimal. The decreasing value for large Aw of the electron broadening many-body function G(Aw) will attenuate each added term if the energy separation between levels is greater than the plasma frequency Wp,e. In the field dependent atomic physics picture, the zero-order energy levels are perturbed by the Stark effect and can overlap. When this occurs, the G(Aw) function takes its maximum value for each intermediate state and the total electron broadening term will be larger. In general, the red and blue levels of each principal quantum number manifold interact most strongly with each other, while the red levels of one principal quantum number and the blue lev- els corresponding to another interact hardly at all.61 This quasi-selection rule causes the electron broadening of the n = 3 level to be most influenced by the red levels of the n = 4 level. This mechanism contributes a further blue asym- metry to the Lp line by way of a broader red wing caused by this additional electron broadening. These electron broadening effects are illustrated in Fig. (4.4) where we show the Lp line at ne = 1 x 1025 cm-3 and kT = 800 eV for the case of full field dependent atomic physics plus the ion-quadrupole effect. This is compared to the same line but with all connections to the n = 4 levels excluded from the electron broadening of the n = 3 to 1 transition. This is accomplished by limiting the sum over intermediate states in the evaluation of M(Aw) to dipole matrix elements connecting n = 3 levels only with other n = 3 levels. The sums for the n = 4 to 1 transition, however, contain all n = 3 and 4 levels. The n = 3 to 1 line with the normal sum over all n = 3 and 4 intermediate states is approximately 10% broader at half maximum intensity than the line with the restriction on the electron broadening sum. |

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