ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR THEORY OF HYDROGEN LINE BROADENING IN HIGH-TEMPERATURE PARTIALLY IONIZED GASES AFOSR-TN-57-8 ASTIA Document No. AD 115 040 Alan C. Kolb Otto Laporte Project Supervisor Project 2189 OFFICE OF SCIENTIFIC RESEARCH,. U*.S. AIR FORCE AIR RESEARCH AND DEVELOPMENT COMMAND PROJECT NO. R-357-40-6, CONTRACT NO. AF 18(600)-983 March 1957

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv ABSTRACT v OBJECTIVE v DISTRIBUTION LIST vi BIBLIOGRAPHICAL CONTROL SHEET xi CHAPTER I INTRODUCTION 1 CHAPTER IIo SURVEY OF THE EXPERIMENTAL SITUATION5.o Shock-Tube Experiments 5 2o High-Temperature Arc Experiments 10 CHAPTER IIIo THEORY 13 io The Classical Path Theory of Line Broadening 13 2o The Validity of the Classical Path Approximation 14 35 A Model for the Trajectories 17 4. The Stark Effect in a Homogeneous Electric Field 17 5o The First-Order Stark Effect in Hydrogen 18 6. Second-Order Stark Effect for Nondegenerate Systems 21 70 Quantum Mechanical Basis of the Classical Path Approximation 22 8. The Principle of Detailed Balance 24 90 Evaluation of m (t) by Perturbation Theory 28 10o Adiabatic Approximation 55 11o Average Over Collisions 38 12. Statistical Approximation 41 153 Phase-Shift Approximation 47 14o Nonadiabatic Effects in Degenerate Systems 51 15. Nonadiabatic Effects in Nearly Degenerate Systems 59 CHAPTER 7 o PREVIOUS THEORIES OF HYDROGEN LINE BROADENING 67 lo The Holtsmark Statistical Theory 67 2o Comparison of the Holtsmark Theory with Observation 73 35 The Krogdahl Theory 76 ii

TABLE OF CONTENTS (Concluded) Page 4. Spitzer's Theory of Lyman a 80 5. Inglis-Teller Theory of Electron Broadening 82 6o A Phase-Shift Theory for Electron Broadening 85 70 Griemns Theory for Hydrogen Line Broadening by Ions and Electrons 88 8 Summary 89 CHAPTER Vo HYDROGEN LINE BROADENING BY IONS AND ELECTRONS 91 1o I:ntroduction 91 2o Consequences of the Spitzer-Holstein Inequality 92 39 Electron Broadening in the Adiabatic Approximation 96 4. Errors in the Average Over Electron Velocities 102 5o The Debye Cutoff Procedure 104 6o The Effect of Close Collisions 105 7. The Broadening of Lyman a by Electrons 108 7ao Wavefunctions and Matrix Elements 108 b.o The Broadening of the Lyman a Line Using the Wavefunctions (5-7ao1) 111 7co The Broadening of the Lyman ca Line Using the Wavefunctions (5-7a.2) 113 80 Comparison of the Classical Path Theory with a Recent Quantum Mechanical Calculation 117 9. Comparison with the Inglis-Teller Theory 119 10o Hydrogen Line Broadening by Both Ions and Electrons 120 1Oa. Ion and Electron Broadening in the Adiabatic Approximation 120 10bo A Series Expansion for Small Frequencies 123 10c. A Series Expansion for Large Frequencies 125 lOdo Normalization of the Absorption Coefficient 127 o1. The Relative Importance of Ion and Electron Broadening 128 12. Broadening of the Balmer Lines by Nonadiabatic Electron Co lisions 137 135 Comparison of the Theory with Experiment 139 14o Critique- Some Unsolved Problems 143 APPENDIX Ao DERDIATION OF THE INTENSITY DISTRIBUTION I(w) 145 BIbBLIOGRAPHY 152 iii

LIST OF ILLUSTRATIONS Table Page Io Balmer Line f-Numbers and Stark Displacement Coefficients 94 II. AXcT 1 for the Balmer Line H~ 94 Figure 1. Shock-tube spectra of the Balmer lines Hp, HR H8. 7 2. Time-resolved spectrum of Hp with corresponding wave-speed photographs of the primary and reflected shock waves. The left-hand wave-speed photograph was exposed with white light and the center with only the light of Hp. 9 3. Microdensitometer trace of asymmetrically broadened and shifted argon lines. 11 4. Comparison of shock-tube and arc spectra of Hg with the Holtsmark distribution (log-log plot). 74 5. Correction factors Kn(pR) to the Holtsmark series for n = 2,5,4,5. 129 6. Comparison of the theories for electron broadening, ion broadening, and broadening by both ions and electrons. 132 7. Comparison of the Holtsmark theory with a typical HE profile obtained in the Michigan shock tube. 134 8. Comparison between the Holtsmark theory and the theory for broadening by both ions and electrons illustrating the density dependence. 141 9. Comparison between the present theory and the Michigan experiments. 142 iv

ABSTRACT The purpose of this investigation is to study theoretically the broadening of the hydrogen Balmer lines observed in the radiation of high-temperature partially ionized gases. The theory is based on the classical path approximation for the motion of the perturbers. The general problem of the broadening of a group of lines arising from transitions between "nearly degenerate" states is considered. The method employed is not restricted by the usual assumption of binary collision. The formalism is subsequently specialized to the case where the broadening due to the interactions between an ensemble of ions and a hydrogen atom can be treated as a static perturbation. The validity of this approximation is discussed in detail. The static ion field removes the normal degeneracy of the states of the hydrogen atom. The high-velocity electrons present in the electrically neutral plasma are then shown to cause phase changes and transitions between these nearly degenerate states. The phase shifts due to adiabatic effects and the collision-induced transitions due to nonadiabatic effects are of comparable importance as sources of broadening by electrons. The resultant profile caused by the electron-atom collisions is then averaged over the static ion field splitting with the Holtsmark distribution function. Series expansions for the line profile are obtained which reduce to the Holtsmark expansion for zero electron density and to the dispersion distribution for zero ion density. The principal result of these calculations is that both ions and electrons must be taken into account in the derivation of Balmer line absorption coefficients. This result is confirmed by experiments in the Shock Tube Laboratory at The University of Michigan (E. B. Turner, Dissertation, 1956; see also ASTIA Document No. AD 86309, Univ. of Mich., Eng. Res. Inst.)o The broadening of the Lyman alpha line by electron collisions is considered in detail for comparison with other theories. For this line, the nonadiabatic and the adiabatic effects are found to contribute in the ratio one to two to the broadening. OBJECTIVE The objective of research under ARDC Contract No. AF 18(600)-983 is the study of the hydrodynamics of and the spectra behind strong shock waves produced in a shock tube. v

DISTRIBUTION LIST FOR CONTRACT NO. AF 18(600)-983 No. of No of Agency Copies Agency Copies Commander 5 Director of Research and Air Force Office of Scientific Development 1 Research Headquarters, USAF Air Research and Development Washington 25, D. C. Command ATTN: SROPP Chief of Naval Research P.Oo Box 1395 Department of the Navy Baltimore 3, Maryland ATTN: Code 421 Washington 25, D. C. Commander 4 Wright Air Development Center Director, Naval Research ATTN2 WCOSI-3 Laboratory Wright-Patterson Air Force Base ATTN: Technical Information Ohio Officer Washington 25, D. C. Commander 1 Air Force Cambridge Research Director, Research and DeCenter velopment Division 1 ATTN: Technical Library General Staff L. G. Hanscom Field Department of the Army Bedford, Massachusetts Washington 25, D. Co Commander 1 Chief, Physics Branch Rome Air Development Center U. S. Atomic Energy Commission ATTN: RCSST-4 1901 Constitution Avenue, NW Griffiss Air Force Base Washington 25, D. C. Rome, New York U. S. Atomic Energy Commission Director, Office for Advanced Technical Information Service Studies 1 P.O. Box 62 Air Force Office of Scientific Oak Ridge, Tennessee Research Air Research and Development National Bureau of Standards Command Library PoO. Box 2035 Room 203, Northwest Building Pasadena 2, California Washington 25, D. C. Commander 1 National Science Foundation 1 European Office 2144 California Street Air Research and Development Washington 25, D. C. Command 60 rue Ravenstein Director, Office of Ordnance Brussels, Belgium (Air Mail) Research Box CM, Duke Station Chief 10 Durham, North Carolina Document Service Center Knott Building Chairman, Research and DevelDayton 2, Ohio opment Board 1 Pentagon Building Washington 25, D. C. vi

DISTRIBUTION LIST (Continued) No, of No. of Agency Copies Agency Copies Office of Technical Services 1 Dr. Eugene B. Turner 2 Department of Commerce Aeronautical Research Laboratory Washington 25, D. Co The Ramo-Wooldridge Corporation 8820 Bellanca Avenue Commander 1 Los Angeles 45, California Western Development Division Air Research and Development Dr. Roland Meyerott 1 Command Lockheed Aircraft Company ATTN~ WDSIT Palo Alto, California P.Oo Box 262 Inglewood, California Dr. A. Hertzberg 1 Cornell Aeronautical Laboratory, Commander 1 Inc Hq, Air Force Special Weapons 4455 Genesee Street Center Buffalo 21, New York Kirtland Air Force Base Albuquerque, New Mexico Dr. Hans Griem 1 University of Kiel Dr. W. A. Wildhack, Chief 1 Kiel, Germany Office of Basic Instrumentation National Bureau of Standards Dr. R. Go Fowler 1 Washington 25, D. C. Physics Department University of Oklahoma Professor Ho Margenau 1 Norman, Oklahoma Physics Department Yale University Dr. Zack Io Slawsky 1 New Haven, Connecticut Uo So Naval Ordnance Laboratory Silver Spring 19, Maryland Professor Eo Gross 1 Department of Physics Dr. Raymond J.o Erich 1 Syracuse University Department of Physics Syracuse 10, New York Lehigh University Bethlehem, Pennsylvania Dr Peter C. Bergmann 1 Department of Physics Dr. B Stromgren 1 Syracuse University Yerkes Observatory Syracuse 10, New York Williams Bay, Wisconsin Dr P Kusch 1 Dr. Otto Struve 1 Department of Physics Department of Astronomy Columbia University University of California New York 27, New York Berkeley, California Dr. Eo Montroll 1 Dr. Graydon Snyder 1 Institute for Fluid Mechanics Poulter Laboratories and Applied Mathematics Stanford Research Institute University of Maryland Menlo Park, California College Park, Maryland vii

DISTRIBUTION LIST (Continued) No. of No. of Agency Copies Agency Copies Dr. J. Evans 1 Dr. Alan Kolb 10 Upper Air Research Laboratory Navy Department Sacramento Peak, New Mexico Naval Research Laboratory Radiation Division, Code 7410 Dr. Donald H. Menzel 1 Washington 25, D. C. Harvard College Observatory Cambridge 38, Massachusetts Dr. Bennett Kivel 1 Research Laboratories Dr. Anne B. Underhill 1 Advanced Development Division Dominion Astrophysical Observatory AVCO Manufacturing Corporation Victoria 2385 Revere Beach Parkway British Columbia, Canada Everett 49, Massachusetts Dr. Lyman Spitzer 1 Dr. Shao-Chi Lin 1 Department of Astronomy Research Laboratories Princeton University Advanced Development Division Princeton, New Jersey AVCO Manufacturing Corporation 2385 Revere Beach Parkway Dr. J. L. Greenstein 1 Everett 49, Massachusetts Department of Astronomy California Institute of Technology Dr. B. Vodar 1 Pasadena, California Laboratoire des Hautes Pressions Place A. Briand Dr. Marshall Wrubel 1 Bellevue, France Kirkwood Observatory Indiana University Professor Irvine I. Glass 1 Bloomington, Indiana Institute of Aerophysics University of Toronto The Shock Tube Laboratory 1 Toronto 5, Canada Pennsylvania State College State College, Pennsylvania Cornell University 1 Graduate School of Aeronautical Dr. Henry Nagamatsu 1 Engineering General Electric Research Labora- Ithaca, New York tory ATTN: Dr. E. L. Resler Schenectady, New York Princeton University 1 Guggenheim Aeronautical Laboratory 1 Palmer Physical Laboratory California Institute of Technology Princeton, New Jersey Pasadena, California ATTN: Dr. Walter Bleakney Dr. S. S. Penner 1 New York University 1 California Institute of Technology Institute of Mathematics and Pasadena, California Mechanics 45 Fourth Street Dr. Robert N. Hollyer 2 University Heights General Motors Technical Center New York 53, New York 30800 Mound Road ATTN: Dr. R. W. Courant Warren, Michigan viii

DISTRIBUTION LIST (Continued) No. of No. of Agency Copies Agency Copies Special Defense Products De- Dr. C. W. Allen 1 partment 1 University of London Observatory General Electric Company Mill Hill Park 2900 Campbell Avenue London, England Schenectady 6, New York ATTN: Dr. Emmet Lubke Dr. R. 0. Redman 1 Cambridge University Observatory Professor Robert B. King 1 Cambridge, England Norman Bridge Laboratory of Physics California Institute of Technology Dr. H. H. Plaskett 1 Pasadena, California Oxford University Observatory Oxford, England Professor W. Lochte-Holtgreven 1 Director Dr. Z. Kopol 1 Institute fur Experimentalphysik University of Manchester 01shausenstrasse 40-60, Haus 20 Manchester, England Kiel, Germany Mr. R. Wilson 1 Mr. Jacob Pomerantz 1 Royal Observatory Hyperballistics Division Edinburgh, Scotland Room 4-172 U. S. Naval Ordnance Laboratory Dr. Walter Wada 1 Silver Spring Nucleonics Division White Oak, Maryland Naval Research Laboratory Washington 25, D. C. Professor C. F. von Weizsacker 1 Max Planck Institute fur Physik Mr. J. E. Milligen 1 Bottinger str. 4 Mrs. D. E. Buttrey 1 Gottingen, Germany Dr. H. Stewart 1 Optics Division Dr. Russel E. Duff 1 Naval Research Laboratory P.O. Box 1665 Washington 25, D. C. Los Alamos, New Mexico Dr. James Tuck 1 Dr. J. D. Craggs 1 Dr. Stirling Colgate 1 Electrical Engineering Department University of California Liverpool University Radiation Laboratory Liverpool, England Livermore, California Dr. Martin Johnson 1 Dr. Glen Seay 1 Physics Department Dr. R. Scott 1 Birmingham University Dr. James Tuck 1 Birmingham, England Los Alamos Scientific Laboratory Los Alamos, New Mexico Dr. H. S. W. Massey 1 Physics Department Dr. M. Gottlieb University of London Project Matterhorn London, England Forestal Research Center Princeton, New Jersey ix

DISTRIBUTION LIST (Concluded) No. of No. of Agency Copies Agency Copies Dr. M. Baranger 1 Dr. L. Spitzer Carnegie Institute Princeton University Observatory Pittsburgh, Pennsylvania Princeton, New Jersey Dr. H. Margenau 1 Professor A. Unsbld Physics Department Sternwarte Yale University Kiel University New Haven, Connecticut Kiel, Germany Dr. Arthur Kantrowitz 1 Dr. G. Elste 1 Research Laboratories Universitats Sternwarte Advanced Development Division Gbttingen, Germany AVCO Manufacturing Corporation 2385 Revere Beach Parkway Professor M. Minnaert 1 Everett 49, Massachusetts Utrecht University Observatory Utrecht, Holland Dr. L. G. Henyey 1 University of California Institut d'Astrophysique 1 Berkeley, California Universite de Liege Cointe-Schessin, Belgique Mt. Wilson Observatory 1 813 Santa Barbara Street Dr. J. C. Pecker 1 Pasadena, California Observatorie de Paris Mendon, Seine et Oise Lick Observatory 1 Paris, France Mt. Hamilton, California Institut d'Astrophysique 1 Astronomy Department 1 98 Boulevard Arago Vanderbilt University Paris, France Nashville, Tennessee Professor S. Miyamoto 1 Astronomy Department 1 Kyoto Astrophysical Institute Indiana University Kyoto University Bloomington, Indiana Kyoto, Japan Dr. A. E. Whitford 1 Professor Y. Hagihara 1 Washburn Observatory University of Tokyo University of Wisconsin Tokyo, Japan Madison, Wisconsin Professor S. Chandrasekhar 1 Perkins Observatory 1 Yerkes Observatory Delaware, Ohio Williams Bay, Wisconsin Dr. J. J. Nassau 1 Case Institute of Technology Cleveland, Ohio x

BIBLIOGRAPHICAL CONTROL SHEET 1. Originating agency and/or monitoring agency: O.A.: Engineering Research Institute, The University of Michigan M.A.: Office of Scientific Research 2. Originating agency and/or monitoring agency report number: O.A.: ERI Report No. 2189-3-T M.A.: OSR-TN-57-8 3. Title and classification of title: Theory of Hydrogen Line Broadening in High-Temperature Partially Ionized Gases (UNCLASSIFIED) 4. Personal author: Kolb, Alan C. (Otto Laporte, Project Supervisor) 5. Date of report: March 1957 6. Pages: 167 7. Illustrative material: 2 Tables, 9 Figures 8. Prepared for Contract No. AF 18(600)-983 9. Prepared for Project Code(s) and/or No.(s): R-357-40-6 10. Security classification: UNCLASSIFIED 11. Distribution limitations: See Distribution List 12. Abstract: The purpose of this investigation is to study theoretically the broadening of the hydrogen Balmer lines observed in the radiation of high-temperature partially ionized gases. The theory is based on the classical path approximation for the motion of the perturbers. The general problem of the broadening of a group of lines arising from transitions between "nearly degenerate" states is considered. The method employed is not restricted by the usual assumption of binary collision. The formalism is subsequently specialized to the case where the broadening due to the interactions between an ensemble of ions and a hydrogen atom can be treated as a static perturbation. The validity of this approximation is discussed in detail. The static ion field removes the normal degeneracy of the states of the hydrogen atom. The high-velocity electrons present in the electrically neutral plasma are then shown to cause phase changes and transitions between these nearly degenerate states. The phase shifts due to adiabatic effects and the collision-induced transitions due to nonadiabatic effects are of comparable importance as sources of broadening by electrons. The resultant profile caused by the electron-atom collisions is then averaged over the static ion field splitting with the Holtsxi

mark distribution function. Series expansions for the line profile are obtained which reduce to the Holtsmark expansion for zero electron density and to the dispersion distribution for zero ion density. The principal result of these calculations is that both ions and electrons must be taken into account in the derivation of Balmer line absorption coefficients. This result is confirmed by experiments in the Shock Tube Laboratory at The University of Michigan (E. B. Turner, Dissertation, 1956; see also ASTIA Document No. AD 86309, Univ. of Mich., Eng. Res. Inst.). The broadening of the Lyman alpha line by electron collisions is considered in detail for comparison with other theories. For this line, the nonadiabatic and the adiabatic effects are found to contribute in the ratio one to two to the broadening. xii

CHAPTER I INTRODUCTION This theoretical investigation of the broadening of hydrogen lines in a high-temperature, partially ionized gas was motivated by experiments performed in the Shock-Tube Laboratory at The University of Michigan.*1-4 These experiments showed that the broadening of the Balmer line HF was greater than that predicted by the familiar Holtsmark5Y7 statistical theory for broadening by static ion fields. In this dissertation the additional broadening due to high-velocity plasma electrons is calculated in the classical path approximation.** It will be shown that the electron broadening is comparable to the ion broadening at all densities and cannot be neglected in a theoretical description of hydrogen line profiles. An extensive comparison between the theory and the experiments has not yet been carried out. However, the main features of the theory are confirmed. by the existing data and it is probable that the additional broadening observed can be ascribed to the presence of electrons. The general conclusion that plasma electrons are an important source of broadening of hydrogen lines is of some importance to astrophysics. Until recently, hydrogen line profiles in stellar spectra have been analyzed on the basis of the Holtsmark theory.5-7 From these calculations one may derive the electron pressure and temperature structure of stellar *These experiments were carried out by E. B. Turner and L. Doherty under the supervision of Professor Otto Laporte. **The perturber trajectories are treated classically. 1

2 envelopes. Application of the theory developed in this idissertation to 8 problems of this type by Alier, Jugaku and, Elte modifies in a nontrivial way the calculated pressure-temrperature distribution in a typical stellar atmosphere. A survey of the existing literature on. line broaderning revealed that there was no theory available with which one could calculate a Balmer line broadened by both ions and electrons. An early theory by Spitzer9-11 was restricted to Lyman a since it did not take into account degenerate ground states. Furthermore, this theory was based on the binary collision assumption and did not take into account the simultaneous broadening by both ions and electrons, The results of Section IIIo15 constitute a generalization of the Spitzer theory that is applicable to Balmer line Ibroadening by electrons. moving in a static ion fiel]d. The resulting profile is then averaged. over the -static field with the Holvtsmark probability distributionO 10 -1 2 Other calculations 11 of electron broadenig in the classical path approximationl eithe r neglected nonadiabatic transitions — or else- took them into account by ro lgh order- of -magnitude es timat. a These var-oust theories will be di-cuscd in Chapte-r I1V:.nd co.mpared with the th-ory d;eve-loped in Chapte,. r il I arnd Chapter V, Chapter TI conta.i.ns a, bt)rief survey of hydrogen line broadening exper iments with shock t ub s andr high-tei mper'atre arcs,. I. Chapter:II the classical path tn.eory is disc;nlssed in detail. Much of this material is more or lesse well known. For example, the adiabatic. theory has received much attention by many autlhorso Our contr ibutionr to this top:ic is a new presentation in oiS ction IIl? and 1T1115 of the validity C- iteria for the phase-shift and statistitcali approximrlatioonrs Sections IIIo1-8 and IT ll1011 contain backgrou;nd material nlces saery for hsubsq uelt calcula

5 tions. This includes a discussion of the validity of the classical path approximation, the Stark effect in static homogeneous electric fields, the Fourier integral expression for the line shape,* the principle of detailed balance, the adiabatic approximation, and the method of averaging over collisions. In Section IIo9 an expression for the time-dependent dipole matrix element jCt(t) is obtained using first-order perturbation theory. This result is used in Section IIIo14 to derive the frequency distribution of spectral lines arising out of transitions between degenerate initial and final states. This formula was obtained earlier by Anderson, who considered the pressure broadening of microwave and infrared lines. Our derivation differs from Anderson's and does not contain the assumption that the average duration of each collision is small compared to the average time between collisions. For long-range forces the binary collision assumption fails, so this generalization is important for hydrogen line broadening by fast plasma electronso In Section IIIo15 a theory of the line wing is developed for the case,where a line is composed of a composite of lines arising out of transitions between nearly degenerate initial and final stateso. This investigation waas carried out because the statistical theory is valid for calculating the influence of the ion fields on the line wing. The static ion fields remove the normal hydrogen degeneracy so that the plasma electrons iinteract with nearly degenerate hydrogen atoms. Chapter V contains a detailed application of the formal results of Chapter III to the hydrogen lines. The electron broadening of Lyman a was worked out in detail in Section V 7 as an example It was found that the nonadiabatic contribution gives rise to two-thirds of the electron *This discussion follows that of Bloom and Margenau, Refo 15.

4 broadening of this line. These results are compared with a recent quantum mechanical theory'5 in Section V.8o In Section Vo2-6 the validity range of the theory is discussed further with numerical examples, errors in the velocity average are calculated, the Debye cutoff is introduced,16 the effect of close collisions is taken into account, the adiabatic theory of electron broadening is analyzed in detail, and new results are obtained for the adiabatic contribution to the line shapeO In Sections VolO and Vol2 the simultaneous effect of both ions and electrons is calculated Series expansions for the line profile are obtained which reduce to the Holtsmark distribution for zero electron density and to the dispersion distribution for zero ion densitye Sections V.11-15 contain a discussion of the relative importance of ion and electron broadening, nonadiabatic broadening by electrons, a preliminary comparison of the theory with experiment, and, finally Come remarks on unsolved problems connected with hydrogen line broadening in partially ionized gases,

CHAPTER II SURVEY OF THE EXPERIMENTAL SITUATION 1. Shock-Tube Experiments Before taking up the theoretical problem of calculating hydrogen line profiles, a short resume of the experiments which stimulated this research will be presented. Our attention will be focused mainly on experiments pertaining to hydrogen line broadening in partially ionized gases. High-temperature spectra can be obtained in the laboratory by a variety of techniques. However, of these the shock tube and certain arc sources provide convenient means of obtaining fundamental data on line broadening in high-temperature gases. In this chapter the results of shock-tube experiments will be briefly outlined. Recent arc data from Kiel University will also be discussed and compared qualitatively with the shock-tube results. The shock tube provides a homogeneous high-temperature light source in thermal equilibrium with which one can measure spectral line shapes under known conditions in a luminous gas. The shock tube constructed by E. B. Turner1 at The University of Michigan will be briefly described, This shock tube consists of a high-pressure chamber containing a gas (usually hydrogen and referred to as the driver gas) at pressures up to 155 atmospheres, separated by a diaphragm from a low-pressure chamber containing a gas (usually one of the rare gases) at a few mm Hg pressureo A very strong shock wave is generated when the diaphlragm is brokern The 5

gas behind this shock wave travels down the tube at a nigh velocityO When the incident shock wave reflects from the end of the tube, the gas behind it is brought to rest. The conversion of the kinetic energy of ordered motion into thermal energy heats the gas behind the reflected shock wave. By this means, temperatures up to 15,000~K can be easily produced in the gas which was initially in the low-pressure chamberO* The spectrum of the luminous gas is observed at the end of the shock tube in the region behind the reflected shock waveo Since te hluminosity is a transient phenomenon, it is desirable to measure the spectra as a function of time. This is achieved by moving a film placed in the focal plane of a spectrograph. The details of these experimernts have been presented in a dissertation by Turnero' In order to study the Balmer lines, a small aimount of' hydrogen was introduced into the low-pressure chamrber t.ogethen-r with the rare gas, neon.** The Balmer lines Hw^ H, HI, and 11. have been obserred in: this 0 way with widths ranging up to several hlundrcd3 An:g:2-trom<, diepe.nd.ing on the strength of the shock wave. The neon lines appear sharp in contrast to the hydrogen lines because hydrogen exhibits a first-order St tark effect, while neon does noto It is also observed that alternate members of the Balmer series exhibit a double maximnum., Fig.lre 1.-:hows a typical spectrum of the Balmer lines H, H,, aind HP, which illustrates this behavior, The double maximum in Hi is clearly in evidence i H is overex*Still higher temperatures can. be obtained when an explois've mixture of hydrogen and oxygen is ignited in the high-pressure chajr;bero **Temperatures up to 15,000~K can be reached in neon with very little continuum because of the high (21o5 ev) ionization potential of neon. ***Taken from page 107, Ref. o1

7 - - - - - ____._. __. __._. __. _ _ _ _i _ _ _I _ _ _ _ 14~~~~~~~~~~~~~~c i-~~~~~~~~~~~~~~~~~~~~c LI _ i L I t 4 ir 71 ~ ~ 1 1 T Ilrlr " 'i +~~~~~~~~~~~~~~~~~~~~~~~i i I I-1 IrI -Pi P4 -D-L-~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~L~~~~ Ii.-I. I ct5.I~ i-i __; -X III IIL llT~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l i.__ ~ ~ _ 0~~ dmi I i I ID " 17 I tpri Ii; 1, T 1-,, I~7i _Lli _ I 1!.ii~~~~~~ Lw -4 4;- Ff,~~f ~.~. - ~ f U i-I 1111- I~li~llI+ H IF 7~~~~~~~~ I: — 7- i - ~-~~~~~~~~~~~ ---, -— I!-I- ~ ~ ~ ~ ~ ~ Q 11!1 i ii:I_ e-l~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

8 posed, In Figure 2* a spectrum of H5 as a function of time at a fixed position in the tube behind the reflected shock wave is showno The constant width over 100 tsec indicates that the ion density is constant within this time interval, implying that thermal equilibrium has been established. The double maximum is clearly present in this spectrum. The presence of a double maximum is to be expected from the selection rules for hydrogen and the theory of the Stark effect since the odd members of the Balmer series, i.e., HE, Hb,etc., have no central Stark component in an electric fieldo The absence of a central Stark component in the shocktube spectra demonstrates the existence of a first-order Stark effect due to the presence of local electric fields of ions and electronso These general features have also been observed in the Oklahoma and Cornell shock tubes. -n Fowler's shock tube at Oklahoma, temperatures 17-22 up to 50000' -I have been reported. These high temperatures are produced behind shock waves generated by a high-voltage, low-inductance, electric discharge in hydrogeno Some recent experiments, initiated by the author 2 at the NTaval Research Laboratory, indicate that still higher temperatures can be attained in shock tubes of this general type. It is also possible to study the high-temperature spectra of pure low-molecular-weight gases (such as hydrogen or helium) by detonating high explosives in an atmosphere composed of the gas of interest. Seay and 24 Seely at Los Alamos studied helium line profiles and line shifts with a shock tube of this type at temperatures near 20,000K with ion densities up to 1018 cm 3 In addition to the great broadening of the hydrogen lines, the lines of the rare gases used in the low-pressure chamber of the shock tube are *Taken from page 115, Ref. 1.

0 0 -( 0 0 0 Figure 2. Time-resolved spectrum of Hi with corresponding wave-speed photographs of the primary and reflected shock waves. The left-hand wave-speed photograph was exposed with white light and the center with only the light of Hp.

10 seen to be broadened and generally shifted to the red by perturbations proportional to the square of the instantaneous local electric fields. For example, half-widths and shifts of the order of 3A are observed in 17 -3 the argon lines at ion densities in the neighborhood of 2 x 10 cm. Figure 3 shows a typical spectrum of argon lines shifted and broadened in the shock tube. The broadening in this case is attributed to the second-order Stark effect due to the local electric fields of neighboring ions and electrons. Further experiments are needed to establish conclusively the frequency distribution in spectral lines broadened by the second-order Stark effect. An investigation of the red asymmetry present in many of the rare-gas lines would be of especial interest and should have a direct relation to the ion density. 2. High-Temperature Arc Experiments It should be mentioned that hydrogen lines broadened by the Stark effect have also been observed in arc spectra. Special attention will be given here to the extensive experiments with a water-stabilized arc performed at Kiel University by Professor Lochte-Holtgreven and his coworkers. ~1 ' In this work temperatures of 12,000~K and ion densities of 5 x 1017 cm-3 can be easily maintained by circulating a stream of water about an arc column~ The Balmer line profiles obtained in this way are in general similar to profiles obtained in the shock tube; however, there seem to be certain slight differences. The arc profiles do not exhibit the H6 double maximum, while this feature is definitely present *Taken from page 100, Refo 1. See Chapter IV, Section 2, for detailed comparison with an H, profile.

ll i I I i II i',L,1,, i. I I! i i i i i _LL-J__IIi TM I I I i-! -i~ -:_ ~~ ij_~ '~!' _ _._i_ __'L L s~lI — t j i -.' 1. —............ *f_ i t 1 ~~~~~- V'_-ti{1 -'.. __.__!_____ i~ _-~:~...... _~-! ' —~ ZZZ'. i:-,!,,-_-, 1, _, -'-_ __-._, -_ 'ltl Ii~ I~ I - 1 _-!.-~ LL~~~~~~~~~~~~~~~~~~~~~~~~~c ~~ I i W Cd _ — tI XTH. _ L. 1wi I I_: 1 1 1 1 I _IV' I Illr~~~~~ ~~~~~~~~~~~~~ I_ _ i o4 -_. 5 - _; _ L t 1 I I I X ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~11!1 1 1 1 1, | 1 1 1 W 1 1 1 11 1 1 i I 1: 1 1 1XJ 1:l~ j l Illl4X-t-lllAL!-X ---1 ---l-lI -1 -- 11 1 tll111-0111l-< ~~~~~ LRiLo l iiii|t1l

12 in the shock-tube profileso It also appears that the dip between the central maxima in Hp is more pronounced in the shock-tube spectra compared to the arc spectra. One interpretation of these differences is that the shock tube provides a more homogeneous light source than does the water-stabilized arc. In any event, the shock tube may be used to verify and supplement conclusions reached at Kielo At the present time, the Balmer line profiles obtained by photographic photometry at Kiel are more accurate than those obtained in the shock tube because longer exposure times are possible with a continuously burning arco* However, the shock tube has the advantage that the ion density in the luminous gas may be calculated from hydrodynamic considerationso Therefore, various theoretical results can be compared directly with observations in which the ion density is a fixed parametero From such experiments it was conchluded that ion broadening alone does not account for the width of Hg in shock-tuibe spectra, and, in agreement with the conclusions of GriemlI (based on arc data), one must take into account the broadening due to fast electrons present in a neutral plasmao These observations are in general agreement with the theoretical results obtained in subsequent chapters 31 *The luminosity in the shock tube lasts a few hundred microsecondso

CFAPTER III TIJEORY 1o The Classical Path Theory of Line Broadening In this chapter those aspects of the classical path approximation to the theory of line broadening which are germane to the problem of hydrogen line broadening by ions and electrons will be discussed in detail. In the classical path approximation it is assumed that a quantum mechanical system (atom or molecule) is perturbed by neighboring particles whose trajectories may be described classicallyo This approximation has received much attention in the literature, particularly in. the adiabatic limit where the effect of collision-induced transitions in the radiating atom or molecule are neglected. However, this approximation has no justification for hydrogen atoms perturbed by high-velocity electrons, as will be shown latero Nonadiabatic effects, which involve collision-induced transitions, have been considered by Andersonl4 for radiation arising from transitions between, degenerate states with. application to the problem of pressure broadening in the microwave and infrared regions of the spectrumn Anderson's theory is restricted to the case where the duration of each collision is much less than the mean time between collisions so that only binary collisions are importanto The binary collision assumption should be viewed with caution. when long-range interactions cause the broadening, as is the case for hydrogen lines broadened, by ions and electronso A more detailed critique of other theories will be contained in Chapter IV in order to facilitate a comparison with the results of this chapter o 13

14 The theory of the broadening of hydrogen lines is complicated by the fact that both ions and electrons must be considered in different approximationso The ions move so slowly that they may be considered to be static (statistical approximation). The electrons, on the other hand, move so fast that one may neglect the duration of each collision (impact and phaseshift approximations),* A discussion of the validity of these approximations will be presented in this chapter and also in Chapter V, where numerical examples will be given. The theory is further complicated by the fact that the "static" ions remove the normal hydrogen degeneracy by the first-order Stark effect. In Section 15 of this chapter nonadiabatic effects in nearly degenerate systems will be considered. The results of this section constitute a generalization of an earlier theory due to Spitzer,9 who considered the transition region between the statistical and impact approximationso This phase of the theory is still incomplete, however, because of mathematical difficulties which arise near resonance (i e, near the line center). 2. The Vaiidity of the Classical Path Approximation In the classical path approximation to the theory of spectral line broadening9 the simplifying assumption is made that the Hamiltonian may be expressed as a perfectly definite function of the rperturbing particle coordinates which depend on timeo It is further assumed in practical calculations that the particles move along prescribed trajectories with constant velocity (see the next section)o Because of this latter assumption, one cannot treat by the classical path approximation inelastic collisions which involve a large fraction of the kinetic energy of the relative motion *This does not mean that one must necessarily restrict the theory to binary electron-atom collisions. The phase-shift approximation is disdiscussed in detail in III.1,5.

15 of the perturber and radiatoro In order to talk about an individual ion or electron at all, it is required that the gas be nondegenerateo The familiar Sommerfeld criterion for nondegenerate gases is (3-2.1) N1/3 << 1 where N is the number of particles per cubic centimeter and \ is the mean de Broglie wavelength (3-2o2) ( - (2mkT) -:-/3 Since the mean distance between particles, r, is of the order N, the inequality (3-2.1) may be written (5-2~5) X A r ^N -/3 The Sommerfeld criterion for nondegenerate gases simply states that the mean de Broglie wavelength be much less than the mean distance between particleso Substituting (3,-2,2) into this inequality gives 2 2/3 2 — 2 (3-204) m >> 2- / * T (k. in 2kT In a typical case of astrophysical interest the temperature is of the order 104 K. For ion and electron densities in the range N = 10~2 - 108 cm-3, the inequality (3-2.4) becomes (3-2.5) m >> (1C0-7 - 1033) gm. This inequality is clearly satisfied by both electrons and ions. Now since the de Broglie wavelength X is approximately Tf/mv, (3-253) may also be written (3-2,6) m-r- >> o This inequality expresses the fact that the angular momentum corresponding to collisions with impact parameters near the mean distance between particles must be much greater than the unit of angular momentum Al. For long

16 range Coulomb forces it is just these collisions that are mainly responsible for hydrogen line broadening by electronso For large quantum numbers one may replace the angular momentum t* i by the classical angular momentum mvro The quantum number 7 corresponding to the conditions of temperature and density mentioned earlier is - 1/3 (3-2.7) vr m = 5000 - 50 With. sucf, large quantum numbers it is not -unreasonable to treat such collisions classicallyo These inequalities also satisfy the validity requirement of the WKB approximation to the wavefunction describing a passing perturhbere If the perturber interacts with its neighbors by an interaction of the type (5-~28) V (r) - con.start/r1 then it is required that the change in (r) -be small over a de Broglie wavelength a ( (3-2o9) L ' * VlTAvg < r (53-210) x <c< < Foley3" has shown, in some deta.l t-lhat a wave mecthanical treatment of the perturb ers in the WKB approximation..eads t,o the classical path theory of line broadening if the classical angular momelntum mvr is sub'stituted for From these considerationls it appears that the classical path approximation may be employed for the calculation of hydrogen line profiles if the broadening is assumed to be main:ly caused by long-range interactions with ions and electronso

17 35 A Model for the Trajectories It is usually assumed that during the time of interaction the perturbers move in a straight line relative to the radiating atom, with constant velocityo If v and p are the velocity and distance of closest approach of the perturber, then the interaction distance is given by (3-3-1) r2(t) = v2t2 + p2 where time is measured from the time of closest approacho Consider interactions HI(t) of the type An An (3-3o2) Hi(t)- = rn(t) - (v2t2 + p2)/ ~ ' The time at which Hi(t) is half its maximum value is found from Hi(0) -An _ An 2, ) 2pn (v2t2 + 2)/2 so that 2/n -P / P (5-354) t~2 = ~ (2 '/ - V - V The duration Td of the collision is of the order of the width of Hi(t) at half its maximum value 2p (3-3o5) Td ~ 2t., p o The characteristic time Td will appear frequently in later arguments and is simply the time during which the principal contribution to the perturbation by a passing particle (eogo, ion or electron) takes place. 4, The Stark Effect in a Homogeneous Electric Field For sufficiently distant collisions the electric field due to an ion or electron is nearly homogeneous over the radiating atom. If the Bohr radius n2ao is taken to be a measure of the linear size of the radiating atom, then the field is essentially homogeneous if the interaction distance,

18 2 r, is much greater than n ao r rn2ao = 4.8 A HQ (n=3) (3-4.1) = 8,4 A Hp (n=4) = 13 A H (n=5) 0 Since most of the interactions take place with r > 100 A for the electron densities with which we are concerned here, the effect of inhomogeneous fields is probably not too important, This is confirmed by more quantitative considerations due to Margenau and Meyerott33 and Spitzero9 Before embarking upon a discussion of the line broadening theory we will first review briefly the theory of the Stark effect in. a homogeneous electric fieldo The theory is, of course, 'basically different for degenerate systems (hydrogen) and for nondegenerate systems It is included here because it serves as the foundation for later calculations and discussiono 5~ The First-Order Stark Effect in Hydrogern iConsider now a hydrogen atom in an. external electric field of strength F in thl-e z directionrO The potential energy due to thi-s field is -eFzo The total Hamiltonian is therefore, (3-5o1) Hi = - -V2.. eFz o Schr'dinrger34 and Epstein35 have shown that the Schrodinger equation with this Hamiltonian is separable in. parabolic coordinates and that <eFz> is a diagonal matrix in this representation. The calculation of the Stark shift in the energy levels is then reduced to a simple calculation of the z matrixo *lThe following presentation follows that of Condon and Shortlevy pages 398-5399 Refo 36,

19 The Schrodinger equation, (3-502) H = with the Hamiltonian (3-5.1) and the following well-known change variables x = 1i2 Cos y = 't152 Sin 0 (3-553) z = 1 (1-t2) = F.(S1)F2(t2) t (0) g reduces to the following: fimo (3-5.4) eim /2x and.m6 -2 2_2 L Hi,-'. ) 5^ (S...e^)re~~E' —.- -- a ' i aSi ki + ml If we let qi = ti/rnaos where ao is the first Folr radius and n is the principal uantlum nulmber then for the electric field streajgth F = 0r qFiatloly with th canre solved i. terms of Lagierre polyntiomials ii- the suatsl wayt Thae so kt mon is (5-5 o86) eikm = k k L ) k(i +1 | I where ki and k2 are thte parabolic q,_-antuim nufmbers which satisfy the relations kiF + |ml + (3-507) n = k + k2 + Iml + 1 g i = e Finally, with thI.e correct normalization. the wavefunction which characterizes the state n, kj^ m is 1/2 tnklm = /a3 n4 [(ki + \}l)2, (k + 1m/)J3) im0 (3-5c8) ~ Fi(Ti) F2(rI) -- 0

20 The first-order change in the energy levels of a hydrogen atom due to the electric field can now be calculated using the properties of the Laguerre polynomials (3-509) AE = - eF < nkmlzlnklm > = (ki - k2) eaF In the classical path approximation the field strength F at a particular time depends on the instantaneous configuration of the assembly of ions and electrons, namely, (5e-10o) F' k (Fe)k (Fk. where (i)j9 and (Fe)k are the Coulomb fields of the jt ior and kth electron at the radiating atom. '* ' (i)/[r.(i):3 r ke ) (e) 3 (3-5.11) (Fi) = e (i)Lrj j (e)k -e /Irk ] Because of the time-dependenrt nature of the interaction distance, these fields are a fnniction of time trajector y (e)(tk-t) — '-', k I perturber ( )t ) /(e)) P krk radiating at or " so that(e) t e (t t (ke (t) = P 'k (t - tk) (3-Por2r PPk kk (5-5o l2) 3~ vk(e (e) = 0 -With (3-5 12) we have ~ ^() ^Pk + (e) (t tk) + <e) (/-5~,~3 (1)k =el — b- {~ (e(3-50k) {Pk + [Vk (t - tk)1}/2 and a similar expression for the individual ion fieldso Substitution of (3-5il3) into the expression (5-5l0) for the the total instantaneous field F is seen to have a very complicated time dependence. However, because

21 of the relatively high and low velocities of the electrons and ions, respectively, it is possible to approximate the interaction Hamiltonian considerably for certain ranges of temperature and densityo These approximations will be discussed at length in connection with our discussion of various approximations to the general theory of spectral line broadening in the classical path approximationo 6. Second-Order Stark Effect for Nondegenerate Systems If the electric field is in the z direction and JLz is the z component of the atomic electric moment, then the appropriate Hamxiltonian is (53-6l) H =- Ho tz F where Ho is the Hamiltonian for the unperturbed atomo Now for nonde= generate (nonhydrogenic) states having definite parity, the diagonal matrix elements of. z are zero since pz has odd parity0* The shift of the unperturbed energy levels must then be calculated by second-order perturbation theory and is given by the well-known formula Enim+~En ' 'm (356.2) E =- FE2 E < n~m Ilzl n''m' > < n'Lt ' flzi nlm > n' i'm' EnIm En', 'i where the Enim are the unperturbed eigenvalues of the unperturbed aamiiltonian9 Hoo The above expression defines the polarizability aC of the atom, which may be calculated if one has sufficient knowledge of the energy levels and wavefunctions of the unperturbed atomo In the classical path approximation a is left as a phenomenological parameter in the theoryo Fortunately, however, a great number of experimental Stark effect data are available for many spectral lines from which Cx may be determined empiricallyo These remarks apply only for weak electric fields that do not cause the normally *See, for example, Condon and Shortley, page 410, Ref0 356

22 unperturbed, nondegenerate levels of interest to overlap, 7o Quantum Mechanical Basis of the Classical Path Approximation The intensity distribution I(w)) (energy radiated per second) of dipole radiation from a classical charge distribution with a time-dependent dipole moment p.(t) is proportional to the absolute value squared of the Fourier transform of the dipole moment.* lim 2 14 iTt 2 (5-7el) I() = T->oo 5tc3 T dti(t)elt 0 The formal quantum theory can be developed in a variety of ways. According to the correspondence principle, the dipole moment hi(t) is taken to be a quantum mechanical operator0 The intensity distribution is found by taking matrix elements of this operator between the initial and final states involved in the radiative transitions. The initial states must, of course, be weighted with appropriate weighting factors (ioe., the usual 14 Boltzmann factors for a system in thermal equilibrium)> Anderson has shown that either correspondence-principle arguments** or a quantum theory for spontaneous radiation leads to the same quantum mechanical generalization of the classical formula (53-7l)o Bloom and Margenau13 also obtained a similar result by considering absorption processes according to semiclassical radiation theory. Since these results form the starting point for the theory presented in this dissertation, the derivation of Bloom and Margenau is given in Appendix A0 The results of Appendix A are summarized in the remainder of this section. The intensity distribution is found by calculating the proba*See, for example, Ref 057, Ppo 125, 195o **Following the ideas of Klein (Ref0 58) and Pauli (Refo 09)o

23 bility of a radiative transition between states of a radiating molecule or atom perturbed by interactions with neighboring particles (molecules, atoms, ions, or electrons), as well as by a thermal radiation field, Standard time-dependent perturbation theory is used throughout, If Ho is the Hamiltonian of the unperturbed atom or molecule and H1(t) is the Hamiltonian which describes the interaction of the radiator with the assembly of perturbers, then there is a set of wavefunctions %n(t) which describe the time development of the perturbed molecule as it collides with the perturbers. The 'tn(t) satisfy the Schrodinger equation (3-7.2) ii(t) [Ho + ni(t)]fn(t) Hi(t) contains explicitly the coordinates of all the particles in the gaso In terms of the wavefunctions ^.n(t ) it is shown in Appendix A that the intensity distribution, for dipole radiation is given by the expression 2a, P~ c - i~' '37~3() -n()l / citH^dt4nlm(t )e t(3-7~3) o Avg a.verage over collisials where rlm (t) is a dipole matrix element defined by [ r (t )w*L m(t) ] n and Pm(0) and Prn(O) are Boltzmann factors for states of the radiator having unrlpertured energies E4 and E~, respectivelyo Tlhe term containing Pm(0) in 'the above su.m involves transitions frorn the states m —n and serves to populate the state.i.o The second term involves transitions n-m and serves to depopulate the state n.o If Em is greater than En, then the nr*n transitions correspond to induced emission while the nhem transitions correspond to induced absorptiono If En > Em then the opposite is true, The sum over alL states m and n therefore includes all processes which involve

induced absorption and emission0 Spontaneous emission is, of course, not included since we have not considered the interaction of the radiating system with the electromagnetic vacuumo It can be shown easily that if E~ > Eg, then the term in (3-7o3) containing (-ict) produces resonance while the term with (+icnt) is nonresonant and can be dropped in comparison to the resonant terme In. the microwave region. the nonresonant term is sometimes important but will not concern us here for our considerations of the optical region of the spectrum0 The expression (5-703) is to be compared with the classical formula (3-7ol), which has the same formo The averaging process denoted by the subscript Avg in (3-735) means that one must average over the positions of the perturbilJ.ng particles since this expression was derived with. a def:irnite configuration of pert urbers in mind [a definite time-dependent perturbation i,,(t) ], The configlration average will be dealt withl in detail in subsequent disc-ussions. 8~ The Principle of Letailed Balance For an atomic or molecular system In. itrmal equilibriu with a. radiation fiel'dl the energy emi.tted and absorbed L: the matter is related by Kirchoff s lawo This Law is employed directly in,. t}'e th1eo:ry of radiative transfer in stellar atmospheres,, for example, to reiate the coefficients of emission and a-brsorpticn w:hen o:n.e or the othl`Fer i,s caLculat-ed theoreticallyo Fowever it is known:.i' that th'e line broadeninrg theory under discussion here apparently does not satisfy the principle ofi dietailed balarncin-g at individual frequenciese Zt is of some importance to reconcile this difficulty because the application of the semiclassical radiation trneory for induced absorption and emission or the spontaneous emission theory of Anderson14 lead to nontrivial numerical discrepancies in the calculation of the shape of *See, for example, Bloom and Margenau, RefO 13o

25 highly broadened hydrogen lines. The difficulty seems to be connected with the choice of the Boltzmann factor. Pm(O) and Pn(C)0) in the intensity distribution (3-7~3)o In this section it will. be demonstrated that the Boltzmann distribution involves (E~ + AonA) instead of the unperturbed energies E~ and that this leads to detailed balanceo The quantity iAcon is the perturbation energy of the state n at the initial time t = O0 The proof is restricted to the "static"t limit'*' where the perturbers move so slowly that the perturbber configuration does not change appreciably during times of the order (Ao )- o Van Vleck and Margenau showed. that the frequency distribution in the spectral lines of a classical harmonoic osci.llator a:nd a LDebye rotator ar-e *the same in absorption. and spo-ntaneo&us ernission, if the radiation density obeys the Rayleigh-Jeans lawo These caLcul-ations were restricted to the strong collision theory where each collision is assumed to interrupt the radiation process completely (ioe, destroy all phase relationsips)e it was also assumed that the time durationr of each coll.ision is -inf,'nitely short o Let us corinsider ere an, assemr ly of lhydrogen atoms in thermal equilibrium with a Planck radiation fieldo S-uppose t;hat the -bOroadeninrg of the hydrogen Falmer lines is due to the presence of ions only- (the electron broadening will require another arg-ament j) it is now assumed that the ions can be treated as if they were stationar-y so that the insttanteous perturbation of the hydrogen atoms can 'be taken to be time independento Let the time-independent perturbation of the state n and m be A3n and Ai measured in -units of circular frequency. The perturbation energy, of course9 depends on the instantaneous configuration of the stationary *See Section 12 of this chapter f a more detailed discussion of the static or statistical limit of the theory0

26 ionso Using the results of Section 12, the statistical (static) limit to the theory reduces to the average of a delta fuinction T 2 (te =T( [b(,m2O m_A%.A%)]Avg, (3-8l1) lim 1 dt>cn (t)eit = | () T+o0 T r t o= Avg where cnm is the unperturbed frequency corresponding to the transition n-m and ~1ATh is the perturbation energy of the state n, Since in the static limit the perturbers do not move during the times of interest, the instantaneolus energy at the initial time T0o — -Ois taken to be constanto The density matrix* at i,:-0) therefore involves the total energy, including the perturbation energy, and is given by E~_ -(o,A' )/kT (3-8. ) A~;(0)An (0) = p (O) 5m' 7 - rn,: where Z(t) is t'he sum over states. It wiIll now,be shown that wit'?) these B.octzmann fIactorYs one obtains deta-iled Ltalanrce at all freqluencies, With, (3-8o.) the irtensity distribution (3-7,3 ) becomes ir the static lXnat (Em+fiSXtnr (E.Ao a. Arnderson's theory for spontaneous emission yields the following result for the intensity Tg(s) in the static imit:. 3 frwc3 mZ [ + *^See Appendix Ao

27 [E- E - (L(Ac - Awn)] 3 2c 3Inm(O)L [I()- Is(] { (e'D/kT _) I 1 (53-8.5) e-(E+i5AT )J/kT 0e Z-E(TR~)/kT [ 5 [) - (c.m + Acn Am)]Avg Now since the delta function is nonzero only for 'D = o~ + AO n- A(n' where i um = E - Em~ one observes that the right-hand side of the above expression is identically zeroo Therefore, the spontaneous emission just balances the induced absorption and emissiono For very -wide hydrogen lines the correction to the Boltzmann distribution due to the static ion field can lead to a measurable asymmetry in the line profileo For an emission line, the ratio of the Boltzmann factor corresponding to the red and blue wing is exp - (2,Aoa/kT)o For T 10, OC-"K and LAn1 5xl0'13 (50A from the line center), the ratio of the red to tble intensity due to the corrected Boltzmann factor is e(2flACj^)/kT ^ 1,1, so that the asymmetry is about ten percent, For wider lines asymmetry is correspondingly greater, i oe, for lines 200A wide, such as are observed in, white dwarf stars and in. the shock tubev the asymmetry is about 17 percent due to this cause. The existing experimental data are not accurate enough to verify these conclusions, although the shock tube is a promising instrument for the investigation of asymmetries in the Balmer lines. Further work is needed to generalize this discussion to the case of time-dependent perturbationso It is not clear at the present time what one should take for the density matrix p(O0) at the initial time t = 0. For narrow lines, however, one may safely assume that pn(O) is very nearly the Boltzmann factor corresponding to the unperturbed energies, although the theoretical situation is not completely satisfactoryo The difficulties

28 are closely related to the problem encountered in deriving a statistical transport equation from quantum mechanical perturbation theory, where it is usually assumed that at some initial time the density matrix is diagonal in the unperturbed energyo Although this problem has been dealt with extensively in connection with transport theory, it has been largely ignored in the theory of spectral line broadeningo 9. Evaluation of Pnm(t) by Perturbation Theory Let us consider the broadening of a spectral line originating from a transition between, an initial degenerate state i and a final degenerate state fo In hydrogen the degeneracy is 2n2 ~ where n is the principal quantum number. For nonhydrogenic levels of other atoms there is also a (2J+l)-fold spatial degeneracy in general. The exact calculation of the line broadening would involve the sol'ution of a Schrodinger equation wit h a time-dependent perturbation due to the entire assembly of fastmoving electrons and slow-moving ionso (We will amplify later the exact meaning of fast and slow ) It would then be necessary to calculate the time-dependent dipole matrix c S(t) with the wavefunctions obtained from the appropriate SchrSdinger equation, Finally, one would have to substitute this matrix into the formula for the intensity (3-753)7 perform a statistical average over all configurations of the ions and electrons, and finally do the resulting integralso The approach to the problem as outlined above is prohibitively difficult, so further approximations are required to obtain quantitative resultso In the following discussion, for convenience we drop a factor 4C4[3c (exp o0/kT-l)] in the expression (3-7.3) for the intensity distribution and focus our attention on the term

29 (~ ) ( ( X i dt T0 o Tm d (t) e | > E > (3-9.1) ^ (~1W") -) I (W) 2-A 0 nm ' m m0 Avg Induced absorption from all states in which serve to populate the state n are taken into account in the above sum since it refers to transitions for which E~ > Em and since initially the state m is occupied. The term n i pm(0) is the appropriate Boltzmann factor. The above expression defines the quantity Imn(o), which is not to be confused with the total intensity!(O) in (3-735). For simplicity we will refer to _n(zz) as the absorption coefficient in the following sections. Let -us now consider a group of spectral lines arising from transitions between an initial state i with substates aO and a final state f with substates ao In the remainder of this work Greek letters refer to substates of the initial state i and Roman letters to substates of the final state fo In (3-9ol) the index n + (a,f) and m + (ai)0 Summing over all transitions, the absorption coefficient can now be written as a sum over the degenerate substates of i and f (3-9~2) if((~) = X a(-) lin 1 Tdt e t (f Pa )| = 7,() ia(i) o 2rI T T Qoo T aLJ a acz,a Avg (a The expression a(t) describes the time development of a state tfuntio:n which at t = 0 reduces to Oa, the wavefunction correspondinrg to the energy Eaf of thre a su'ostate of the unperturbed state fo These time-dependent wavefunctions satisfy the time-dependent Schr'dinger equation (3-7s.)e In the following analysis it is assumed that the time-dependent perturbation does not cause transitions between the states i and fo In the case of hydrogen line broadening this assumption means that collision-induced transitions between states of different principal quantum number are

50 neglectedo This is a fundamental limitation of the classical path theory. The energies involved in the collision-induced transitions must be small compared to the kinetic energy of" the perturbing particles so that the perturber velocity is not changed appreciably during a collision, A more exact quantum mechanical theory would be required to treat inelastic collisions where the energy changes are largee f 1 It is now convenient* to expand the wavefunctions (t) and (t) in terms of the finite orthonormal set of functions 0f(t) and C(t) which satisfy the instantaneous Schrodinger equation %(t ) E^(t) W(tn (5-9o5) [Io + -, (t) a) I f. 0f (t)J LE(t) a(t) where th.e time is taken to h'be a pariam'eter iLr the stationary Schr'odinger eQuatioT0 The set of funlctiLonrs 1(t) and. (t) is chosen to be the unperturbed set and - a at t - 0o This implies that at some initial time the t;ime-depenydent perturbation i(t) is effectively zero. We have seer already in oar discussion of the principle of detailed balance (Section I1.*, 8) that taking i' (t) to be zero at the initial time is incorrect in the Tstatistical limit" whiere Hi(t) is a constant for the times of interest in the Fou:rier integral expression for the intensity, However, it was also shown that the error introduced was small for lines that are narrow compared to the unperturbed energy difference EU(O)-Ea(O)o For perturbations that vary rapidly in time (e<g,e fast electron collisions) one must average somehow over the different possible initial conditions in a more precise theory. Taking Hj(t) to be zero at t = 0 avoids these difficulties which are inherent in time-dependent perturbation theories *See, for example, Lo:IL Schiff, Quantum Mechanics, po 208, McGraw-Hill (1949).

51 where the perturbation is never zero; eeg., in a gas where there are longrange interactions so that a radiating atom is never isolated. For narrow lines, however, the assumption that Hi(O)=O probably does not lead to serious errorso Dropping the i and f superscripts on the wavefunctions and remembering that the Greek and Roman letters are associated with the substates i and f, respectively, we can now write down the expansions for %c and _ I / E (t )dt" (3-9o4) pt ikd t a(t) C=Cba(t) ' b(t) e -o Substitutioni of (3-90 4) into the Schrodinger equation (3-7T2) and using (3-903) yields an expression for the expansion coefficients: i t Edb (t' )d-t ECatt) - ' - da( bcd) e d (53-9o5) ~ i -. Ol E),~(tl )dU, C (I e 7 where Edb(t) = Ed(t) - Eb(t) and E (t) = E7(t) - EP(t)o Now in the usual fashion one can find an alternative form for the matrix (-5lg^) by differentiating (3-9o3) and multiplying 'by 0: (3-9.6) (Oa9 P0;,) + (^ Ei) = (0p, Lae) + (0p E4a); H - Ho + Hi(t). F'or -=a, (~=c F- 5 ) = 0 because of the orthogonality of the wavefunctions; also, because of tne Hermitian character of H,

32 (3-9 7) (B ^;, ) = (EHp, ) = Eg3(B3) 9 so that (3-9~6) reduces to (5-9.8) -(0p) - -H = |a B a E EP -Ep ECp Suhbstitution of this identity into (3-9 5) gives.g - E i F Et )dt) (3-9 9) CP = C C j - )dt C( ) (with tnhe usual notationL l - Z ) and sirmilarly for (t) 7?l 7 ' ba The set of equations (3-9o9) form a finite set of coupled linear differential eqjuat:ions. T7hese equations have a simple solution in two extreme cases~ (1) if the petrtunroing field is essentially static9 then,(t) O 0 and the:noisdiagoral e;lements of Cp, tt) vanish.; (2) for long-range interaction s (Large inpact pa.rameters) it is sometimes the case that the ('C(t) <K< I; t1her; only tie term >-j,Ixt) h: '7(0i) - 1 con:tritbuites to the rigLt-hand silde of equation (3-9') for the e xpansion coefficients et (t)o If th:e;ondilagoal e:le.ments, '.';, (t)t 6 are neglected completely, then onle is dealing with the rsua.L adiabatiec theory of spectral line broadeningo It is shown in- Sections, 1i. and 12 of this chapter that this approximation is valid for,hydrogen lines oroadened yya slow-moving ions In. Chapter V, nowever it will bee demonstrated` t,.at the neglect of collision-induced transitions by plasma electrons "Leads to serious errors, Xn order to take the electron broadening into account properly, the time-dependent dipole matrix.a (t) is calculated in this section ip to first order in C (t) [or equivalenrtly to first order in HEi(t)]o It turns out that this "weak collision?' theory is adequate for our problem because of the long-range nrature of the Coulomb potential0

33 With the weak collision approximation [Ca(t) % Ca1(O) = 1;ICoI(t)l<< 1] the amplitude equation (3-9 9) becomes i Ea(t')dt' (3:-9lo) rC(t) = - e 0 (t')dt' since Hsa(t) = [Fo0 + Hi(t)] = [Ii (t)]5 Integrating (3-9,10) directly yields t / ' H ) E (tl')dt' (3-9. 11) Ca(t) = - / (t l' e dt 0 If the perturbation is due to a finite number, k, of passing electrons whose time of closest approach tk is large compared to Td (where Td again is the collision duriation discussed in Section 5), then HI(0) Mng Oe Making use of tnis, together wxith c e relation, HF (W)=-0O a partial integration yields t Lt-J ^E J^ v(t )dt" 3(-9 L12 ) limJ LC (t )] e.. o i" i -~ - U /Et -- &ti J L 1; E dt" _ /S EPdt Ea " Ea(t) e U C (t)(a(t) e where to first order Ca(t) '- Caa(t) - 1 With these wavefunctions the dipole matrix a4(t4) becomes

34 t - i i( Ea - Ea)dt' (+a~a) - - ac(t) = (a C0a) e o t _5-.14 i/ - Bo~ ) idt' +Z' (Sb a)Cba e o b (3-9.14)- 1 ( - E )dt' + E(Oa, 4O) C(,z e 0 _ i (E5 -? ) dt + I (O5 L~bk)C)cz Cta e o Colsistent with our perturbation treatmnent it is ass-umed that the line strength is given to first order by the unperturbed strengths: (3-9015) [~a(t) W^(t)] - r[a(o)~Fe( o)] - no( o We also drop the last term in. (3-9014) as small compared to terms of order.C (t) and C-ba(t)o The dipole matrix is:now.- 1',(E; - Ea) d.t,.C C0 aa ~, t 4 -(...16 ' ~ ~ ^( t ) (e. -^ (Ep Ea)dt' ^.'he time-depeenlden.t en.ergy i.s given. by (39o:-) (t) = [t0)(t):o,Ea(t)] + ~i(t)z )t(t) ()] To first or1der this 'b.ecomes (-.9:ol6) r(t) =:lZ(0) (: 0Xo, (o)] + [i(0 )),(t) (ou)] = E0 + ['=(t)]a oF (5-9i19) (t) = at, + A())t)( 0 o wh.ere t](t) ~- E(t)j and,caq(t) = [.((t) ]aod ( The expression [Kl (t) ]~ is simply the time-dependent shift in the position of the state a as a co:r.,seqnence of the perturbation~

35 Then t rt (3-9.20) J Eo (t' )dt' = ua t + J Aw(tt )dt For further simplification let us abbreviate the integral on the right side of this last identity, t (3-9.21) Pc(t) =-t o Awc(t')dt. Employing these definitions and factoring out pa~ and exp i (u - ~c)t from (3-9.16), we obtain the following expression for ac,(t)~ () = e i(o~a - )tWo o -i[PPo(t)-Paa(t) (53-9.22) + i C (t)- O) t + i [Pbb(t)-Poa(t)] -(3-9 (t) ei(a b + 0 C(t) ei(Wo - o() t + i [Paa(t)-P (t)1 ap Cg,(t) ePa a The spectral distribution in the line i -- f is related to the absolute value squared of the Fourier transform of >2,(t) according to (3-7-3). The so-called adiabatic and impact theories of line broadening are contained in this result as special cases. 10. Adiabatic Approximation In the adiabatic approximation to the theory of line broadening the expansion coefficients Cba(t) and CO(t) are taken to be zero so that the nondiagonal matrix elements of the perturbation Hamiltonian do not enter into the problem. As is well known, this approximation neglects collisioninduced transitions and thereby allows a great simplification in the theory. Since the adiabatic theory has been extensively applied by other authors to various line broadening problems and because it contains much of the

36 essential features of a more accurate classical path theory which takes into account collision-induced transitions, we will discuss this approximation in some detail. In the expression (3-9.22) for a0((t) the expansion coefficients Cba(t) and Cpd(t) are taken to be zero in the adiabatic limit, so that the dipole matrix becomes (3-101 ) [ia (t)]Ad = [aa e-i cDt-i[PM(tt)-Paa(t)] where we have put a - -- a o > 0. Substitution of this time-dependent matrix into the general expression for the absorption coefficient (3-9.2) yields 'T -iA0~t-i[PO(t)-Paa(t) ] (5-10o2) I (w) = lim |a-2 1 dt e ivg QtP t)- aat Av a- To 2it T o where AG o-. The usual correlation function form of this formula is found by writing Iacx() in the following form: T T I ()= -lia | |aa /dt2 dtl e (3-10o.3) -i[ P (t ) -Pce (t2)+Paa(tl)-Paa(tl)] ~ e +tjl~ ~Avg where Pc(t1)-Pa(t2) = / Au(t')dt';Paa(t2)-Paa(tl) = - Aaa(t')dt'. t2 t2 Putting t1 - t2 = T, and integrating over T, holding t2 constant, yields T+pt2 -i (Aot-Aoa)dt' T T-t2 _iA (5-10.4) I a( I) = lirr t l|a2 dt2 dT e iAa T e T^0 2 iT vg Now in a time interval T, the initial and final times do not affect the statistical average for statistically stationary distribution of perturbing

57 particles. Therefore, in the adiabatic approximation the absorption coefficient is simply o 2 00 0 (3-105) IaU () =dT e iA T e a-A t -00 L- -I Avg In the static approximation, where the perturbation Hamiltonian is time independent, the phase integral becomes t p X (ACO -AaC)dt' = (A(Da-AcAL)t 0 In this approximation the absorption coefficient (3-10.5) reduces to the average of a delta function over the perturber coordinates (see Section 12 of this chapter for a more detailed discussion): a 0 3-10 6) Ia (1) = l 2 { ~[ -(A aAc a)]} Avg It can also be shown in general that if ADa-ALa- is an even function of time, then F - -iAaoT [ij (Aa-A%)dt dT e e - L ' Avg 00 e -i C (Awa-A%)dt ' - / dT e+AWa T e o Avg so that (3-10.5) can also be written O12 oo o i (Ama-AaU)dt' (3-10.7) Iaa(c) = C Re dT e a Le 01J~~ o ~~~Avg The factor iT is a normalization constant such that 00 Ia,"(a~) dao I o 2 (3-10.8). Iaa(fJ) du = au i ~ This will be verified in Chapter00 This will be verified in Chapter V.

38 The integral (3-10,7) can also be simplified considerably if the perturbers move sufficiently fast or slowly, These approximations are commonly referred to as the phase-shift and statistical approximations, respectively. In Sections 12 and 13 of this chapter these two limiting cases will be discussed in some detail and rather simple derivations of their range of validity will be presented, In addition it will be shown that the usual assumption of binary collisions can be removed from the phase-shift theory for scalarly additive perturbations, 11, Average Over Collisions The phase factor which appears in the adiabatic theory (equation 3-108) must now be averaged over all types of collisions, Let us focus our attention on a particular radiating atom perturbed by an assembly of particles (ions, electrons, neutral atoms or molecules) assumed to move on linear trajectories. In particular, take the case of scalarly additive perturbations of the type (3-11.1) AtDa^A = Nk Aa- - = z 2jk )( j) jk j r (t-tk) jk [vk(t-tk)2 + P]/2 where tk is the time of closest approach, vk is the velocity, pj is the impact parameter, and Njk is the number of perturbing particles described by the variables (oj,pj,vktk). The proportionality constant Aaa(oj) is taken to be a known parameter for the interaction of interest. It can depend, in general, on other collision parameters denoted by aj, For example, when a fast electron interacts with a hydrogen atom, Aaa(a) has a cos Gj angular dependence, where ej is the angle determined by the position vector r?(t-tk) of the electron and the dipole moment vector of the atom (see Chapter V)*

39 For scalarly additive perturbations of the type (3-llol), the phase factors to be averaged are T L J Avg r~(t tk) Avg where the quantity PnaCjk(T) has been introduced for simplicity. The index n denotes the power in the rj(t-tk) dependence of the interaction, a and a refer to the atomic states in question, and (j,k) refer to collisions described by the variables (vk, tk, pj, Gj)o For sufficiently dilute gases it can be assumed that the individual perturbers move essentially independently of one another so that the average phase factor (3-11o2) can be written as a product, (5-115s) Leik jkPnajk (T iNjkPnaajk (T.L JAvg L -— 'Avg The probability that Nk(vk) of the particles with velocity vk have a time of closest approach in (tk, tk + Atk) is given by the familiar Poisson distribution for purely random tk: Nk - At (53-11.4) WNk(tk) = (T) e Tk WNk(tk) =Nk' ~-k k where Tk is the mean time between collisions with velocities in the range (vk, vk+dvk)~ If the assembly of perturbers is contained in a spherical box of radius R, then Tk is given by the kinetic-theory result (3-ll5.) 1 =- CR2 Vk Nk (vk)dvk o Nk(vk)dvk is the average number of perturbers per unit volume with velocities in the range (vk, vk+dvk) and is given by the Boltzmann distribution 2 m V_ (5-116) Nk(vk) = ) Vk ee 2 kT dvk - W(vk)dvk o N is the number of perturbers per cubic centimeter N is the number of perturbers per cubic centimeter~

40 The probability that of these Nk(vk) particles, Njk have p and 9j in the range (pj, pj+Apj), (oa, aj+Acr) is given by Bernoulli distribution for randomly distributed variables: Ni ^2ipApj Ap.ijk (5-11.7) WNjk(%,Pj ) N (2C? AR2 p$ where e Njk = k = N k j k (5-11L8) 2iTp.Ap. = tR2 \ Ac. = Z. L j The element of phase space (ACT APjAtk) can now be taken small enough in the limit so that it can contain but one particle at a time; then Njk = 0,lo Weighting each configuration with the probabilities WNk(tk) and WNjk(Pjpj the average phase factor (5-11,5) can be written (k iN P (T} jk1 iN P (T) j k na.t k j k naa k, ^ Tie j7nj n Cg~ WNk(tGk)WNjk (pj Crj) e N LAvg jk=O (5-11.9) jk Atk Njk N - Z e Tk 1 (2TPiApJ Aj Atk eiNjkpnaujk(T) IlrC eT I ('pe e Jtl\ j 1 0 N^' tR2 Z k jk=O jkZ Substitution of (5-11.5) and (5-11.6) into (5-11L9) and summing over Njk=O,l yields* jk Atkn-k ) { T k AGx (5-11.10) |ie k 1 + 2irNvkPjAPj n-r Atk W(vk)dvk e Now since lim e = +, i 00 *Summing Njk = 0 -, oc yields identically the same result.

41 the average phase factor (3-11,10) is aa~ ~ ~~i naCaj k F Ak zjk{2iNvkPAp. Atk W(vk)dvk e (5-11m11) lim e jL k tJenc Zi ( )Ak ApjA ajAtk->K Replacing the sums by integrals over the continuous variables tk, pj, Pi vk, and letting R-oo, yields T ~ co 0,00 j p (T i(Acua-Aidj)dt 2:tN 1W(vk)vkdvkjpjdpj f i dt[e naaj -] J J J J L~~~~~~ J (5-11l12) =e 0 0 0 Avg 12. Statistical Approximation The statistical approximation to the adiabatic theory of line broadening has been extensively discussed by many authors.,5-7'9-1214y4l 47 In this 47 section the validity criteria derived by Holstein for the statistical approximation to the adiabatic theory will be obtained in a simple fashion, It will then be demonstrated that in the validity range of the statistical theory collision-induced transitions can also be neglected for degenerate or nearly degenerate systems under certain conditions. The present investigation was carried out because it was not clear to the writer that the Holstein validity criteria should necessarily be the same with degeneracy. It was also of importance to determine how to incorporate the statistical theory for ion broadening into a quantum mechanical theory which also took electron broadening into account, The phase Pnaajk(T) due to the passage of the nearest perturber is written P (T) au ^ W na~jk ) 2 + 2]n/2 na~jk~ " [(t - tk)vk + 2n 0 3 (5-12,1) Aa5( j) T'T dt P11 J vkit- tk ]2 n/2 j ~ U pj J L

42 Expanding the integrand by the binomial theorem yields n 2 Aaa(p ) T E v k 12 (an. dt 1 - Vk - Pnace j jk f 6 L 2 P' PJ (3-12.2) A 2 n ( T - -6 r tk)3 = -^f^ '-^-^-^ +tL +.4 Now let T-tk=AT, and for convenience shift the time origin to to so that (5-12.5) Pnajk(T) ( 1 _ k (AT)2 + L n 6p Pj 6 In the statistical limit it is required that the higher-order terms in (3-12.3) be small compared to unity: (3-12.4) n aj - 2n1) where Td = 2Pj/vk where Td is again the duration of the collision. For frequencies determined by AWAT ", 1 (3-12.4) yields (3-12.5) n << When this inequality is satisfied the time-dependent perturbation can be considered to be static. The shift in frequency corresponding to the static field is (3-12.6) Ac = - ( so that (5-12.7) pj = 1j (a,), j n J AO)1

43 Substituting this expression into the inequality (5-12,5) yields the 47 Holstein inequality4 v 1 n/n-l (5-12.8) CWA ~ Lac j nn- l n() Aaa (a 6/ These inequalities also imply that the total phase shift is large compared to unity in the statistical limit. This fact will be important in our discussion of nonadiabatic effects for degenerate systems. To see this, substitute (5-12.6) into the inequality (5-12.5); then Aaacr- -61/2 (3-12.9) A j) () ~. 1 (3-~~~~2.9) ~~n-l n P. Vk The total phase shift per collision is given by 00 Aqa(#c) dx Aaa(qj) P 0 (5-12.10) PaQkG) 2 n-i na1.j k (0 - (V2 x2 + p2) YT2 pn- vk o E2 1 -00 k j Substitution of Pnaajk(o) into the inequality (5-12.9) gives - 00o (3-12.1) Pnaajk(Co) ~ J d, - 00 (2+1 /2 The line profile in the statistical or static limit can now be found from the adiabatic theory (equation 3-10.7) as a special case. According to (5-11.15), the average phase is given by 7. 00 ip PT poo prj 0 00naaj k i/ (Acw-)acD Ai)dt 2jt f/ W(vk)dvk P.dp. d /vkdtk e (T)-l, (5-12.12) e = e o o 1- -+~~Avg where, according to (5-11.2), Pnaj() j T Aaq(aj)dt [(v tk vkt) +2 2]n1 Consider now the integral

44 ir in Aa (Oj )dt 1 p J [(vktk - vkt)2 + p2]n2 (3-12.13) p pjdpj vkdtk e 0 0 The quantity vktk - k is the distance traveled by the kt perturber during the time interval (t=O+tk), where tk is the time of closest approach. Now in the static limit to the adiabatic theory one is concerned with times in the Fourier integral that are much shorter than the duration of a collision. For such short time intervals it has been shown that the perturbation is essentially static. Therefore, the distance r(t-tk) to the kth perturber given by r(t-tk)=[r(t-tk)] t, the position at the initial time t=Oo The t =0 integral (3-12.13) can therefore be written in the static limit Tf irAa(caj )dt (3-12o14) o [(vkt + 2]n/2 0 [ (vktk) a + PJ lim / pjdpj d - vkO 0 0 Consider now the diagram k t=o -- -i t=tk r\\, Pj For fixed r (since the perturbers do not move in this approximation) and vk we have ~k - = sin, cdk = r cos d dO r (3512015) lim djk = lim r cos 9 d = lim 2r = o 3115dk 0cos 9 dG lim =r 2pj vk+O o vk vk+o 2 The integral (3-12.14) is now iAa j)/pnJ (5-1s216) 2 r p. dp. [e J - 1] (3 —12.16 ) 2 P X 2 J J J 0o

45 Substituting this expression back into (3-11o13) yields the statistical approximation to the phase factor AaU ) (3-12.17) i (Ama-Am)dt 4 ItN p jdpj j iA Pa -) i 1 Avg = e The absorption coefficient with (3-12.17) is now A 1 Ia 12 poo -iacjT + 4)-N pd p j -e pni A1 (3-12.18) Iaa() = Re / d e n J o and agrees with the usual form of the statistical theory of Margenau,42 5 46 Holtsmark, Chandrasekhar, et al. The situation when the perturbations add vectorially can be treated by the same methods and will be discussed in detail in subsequent chapters in connection with the theory of hydrogen line broadening by ion fields. The influence of nonadiabatic transitions can now be investigated in the statistical limit. According to (3-9.11), the expansion coefficients Co(t) are given by rt iJ Cg(t) = jt dt" [Hl(t" )] e h J Ep(t')dt? E:a (t ") where |C a(t) 2 is the probability of a transition to the state D if at the beginning of the perturbation the radiating atom is in the state a. Consider again perturbations of the type d Ay(aj) (3-12.19) [Hi(t)lo - HL(t) = 2(t tk)2+ p2]n/ = Eap(t) Then With (3-12.19) and (3-12.20), C\n(t) becomes

46 c 3a~(t ) -= n2 _ r ( t " -2 t k) e p(ro Pv2 (t - tk)2 ()L p(j) t,2 + In the statistical approximation one is concerned with times that are small compared to the duration of a collision (AT~Td) so that one may neglect terms of the type V2(t - tk)2 P2 (5-12.22) C^(AT) =,_1 n2 2J dt-t.e pn Integrating yields obtain theo Bxp n f AT A)-(j) AT jCa(A\T)- (ajcnj L2v, cos A2 pn (3-12.25) 2 ( l) -/ 2 p ( | P vap ^a-j) A2(g) A-A -Ig- Sii 2 PnJ Consider the first term in the bracket ) n-2 2S - ^-~! n (3-12.24) A cos A- ~(j) AT( n-e 2 Aao s j< ) 2P Also, since AT K< rd = 2 p/v, we have

47 mp-2 vT ( T 2p v1 (3-12.25) -2 cos T 1 << Acep (j) i 2 Ap A2(cj) In addition, the second term satisfies the inequality ~n-1 2 -2 (3-12.26) 2 vp sin < 2 V P A (aj 2 nA(a) In statistical limit it has been shown that [vpn-1/Ac5(aj)] << so that Cpc(AT) satisfies the following inequality: (3-12.27) ICp(AT)| << (j n It will be shown in Chapter V that B<<x(aj)and A) (aj) are comparable numbers for hydrogen when the perturbers move very fast (e.g., electron collisions). For static perturbations the nondiagonal matrix elements of the perturbation vanish identically according to the usual theory of the first-order Stark effect. Therefore, the ratio of BaB,(aj) and Ad(a0j) ranges from zero to about unity for perturber velocities ranging from zero to infinity. This means that the elements |Cac(t)l are small compared to unity and approach zero in the static limit. 15. Phase -Shift Approximation In the phase-shift approximation to the adiabatic theory of line broadening the phase integral Pnaacjk(T) is replaced by the total phase change Pnaajk (o) ( 3 Alac(aj ) dt Aa + r(2) (3-13.1) Pnaajk(T)+ Pnaajk(o) [(t-tk)2v + p ]ln/2 n- k k.3 vkPJ

48 This approximation is valid for the core of a spectral line for sufficiently weak interactions [small Aajc(aj) and/or high temperatures]. A validity criterion for the phase-shift limit will be derived in this sections These results will be applied in Chapter V to the problem of hydrogen line broadening by high-velocity electrons. It is of interest to compare the mathematical basis of the (1) statistical and (2) phase-shift approximations to the adiabatic theory: (1) in the statistical limit one is concerned with frequencies (measured from the line center) that are high enough so that the Fourier integral expression for the line profile receives contributions from times AT that are much less than the time duration of the collision Td, i.e., the perturbing particles do not change their positions appreciably during the time AT; (2) the phase-shift limit is valid for frequencies that are low enough so that the Fourier integral receives contributions from times AT that are much greater than the time duration of the collisions Td' The phase integral Pnaajk(T) (3-12.1) can be written ACa(aj) k dx (3-13.2) Pnacjk() k= a+ d vk k v For x > 0 (r < tk tk > 0), the integrand may be expanded in inverse powers of x: Aaa Wgj) /"'dx- n r Pnaajk(T) = Vk- k - r (1 + - k k k (5-155) = vk (n-l (tk) - (nl)(tk)n + - ] with T < tk, tk > O~

49 For x > 0, (T > tk, tk > 0) the phase integral is written ( ' -tk P -~., (r) = P., (CO) -. A^ j)n / ______ dx __I Pnaajk(T) = naajk(0) J Aa {( j )x k -00 Xn1x + ( Vi(3-1354) + / dx J P.;/' n 2 7-tk x + i 2/ Again expanding the integrands by the binomial theorem and integrating term by term yields Pna (T) P n k() A j) 1 1 naajk) Pnaajk - A j) (n-l)(T-tk)n1 (n-l)(-tk)n-1 (5-1355) with; > tk > 0 The main contribution to the integral arises from times in the neighborhood of T=tk so that with T-tk=AT, (3-1355) and (3-13,3) become (3-1536) Pnajk(T>tk) Pnaajk() Pnaajk(<tk) + - (3-13.7) P (. Pnaajk(T<tk) (n- ) ( AT)n-1 + In the phase-shift approximation [PnaCjk(T) - Pnaajk()] it is required that (5-.8) ~ IPPnaajk(T<tk)l << 1 (3-13.8) IPnaajk(T<tk)l << IPnaajk(o ) With (3-1351), these two conditions lead to the following inequalities: (3-13 9) 1 ( n- )v n A) C Aaa (a < F A-T and A__ 1 Ir F2) n-l (5-13l10a) Td 2 L (n-l) I )

50 or 1 r (n-)p n-o (3-13.10b) T A <K< Vk These inequalities restrict the phase-shift approximation to weak interactions [large impact parameters and small Aaa(oj)] and high velocities or temperatures, In contrast to the statistical approximation, the condition (3-13510a) demonstrates that the duration of a collision must be small compared to times of the order (Ac). The validity criterion (3-13o9) is essentially the same as that derived by Spitzer9 by a different method, The average phase factor (3-11,12) reduces to the following expression in the phase-shift approximation w PT re (AwDa - ACX)dt (5-15.11) ~ | = e-(Yacr-iau) T J The expressions 7aa and Z are defined by (/2 Y (7a\ Re 00 pZ do. FiPraaj k ( 7 (3-1312) -- 2fN W(vk)vkdvk pjdpj -: aCZ/ o o o where, subject to the validity criteria (3-1308), 00 PnaDjk(T) + Pnaajk(0) for tk<T; dtk X T 0 The equation (3-13o12) for 7aa and yao, contains a further approximation in that the integral over the velocity and impact parameter is extended from zero to infinity. This is inconsistent with the Spitzer inequality (3-1359) since the phase-shift limit does not apply to collisions in which the velocity is near zero0 However, it will be shown in Chapter V that the error introduced by this approximation is not significant for electron

51 broadening at high temperatures since the main contribution to the integral arises from velocities in the neighborhood of the average thermal velocity and from large impact parameters. 44 Our expression for the parameters 7aa and ya1 reduces to the Lindholm44 Foley32 half-width and shift parameters if W(v) is taken to a delta function 6(v-v) where v is the average velocityo The errors introduced by this approximation are not entirely negligible (about 27 percent in the case of the first-order Stark broadening of hydrogen by electrons) and will also be discussed in detail later. It is also of some interest to point out that we have not restricted our derivation of the phase-shift theory to the consideration of a succession of single encounterso In deriving the expression (3-13512) for 7aa and 7aa, the simultaneous interaction of many perturbers with the radiating atom was taken into account and is therefore somewhat more general than the derivations of other authors10'11'14,32 who required the assumption of binary collisions and low densities in evaluating the phase-shift integralso The line profile corresponding to the phase-shift approximation is found by substituting the phase factor (3-11o12) into the Fourier integral (3-10.7), giving the well-known result Iacy(w) I ReJ i T eiac) (Yax - i7 aC)T Iaa(w) i! Re - dT^ - a7 (3-1l513) 1-0Ik |2 y7a,, (Ag(a O- 7 )2 + (aa)2 14, Nonadiabatic Effects in Degenerate Systems Let us now consider the situation in which a group of spectral lines are not resolved with respect to their breadth0 These lines are thought

52 to originate from transitions between the degenerate substates a and a of the initial and final states i and f. This presentation constitutes an extension to optical problems of the nonadiabatic impact theory worked out first by Anderson4 and later by van Kranendonk,48 using somewhat different methods. These authors were mainly concerned with the pressure broadening of microwave and infrared lines arising from dipole transitions between states i and f having angular momenta Ji and Jf. The theory was developed with the following basic assumptions: (1) the socalled impact assumption that the durations of the collisions are short compared to the time between them and (2) that only binary collisions are important. Both of these assumptions are highly restrictive for the problem at hand; namely, the broadening of spectral lines by ion and electron electric fields in a high-temperature plasma. The basic difficulties can be illustrated in the case of hydrogen line broadening by high-velocity electrons. The average duration of each collision Td is of the order 2p/7 (5-355), while the average time between collisions Tc is of the order (3-14.1) 0c - 2 Nev The ratio Td/Tc is therefore of the order (3-14.2) Td 2p3r TC t A typical interaction distance p can be taken to be of the order of the -i/3 mean distance between electrons p gv Ne so that there is a significant number of collisions for which Td/Tc _ 1. This argument is, of course, valid only for long-range interactions where distant collisions are important. Since the average time between collisions and typical collision times are of the same order of magnitude, the binary collision assumption

55 must also be viewed with caution. The theoretical development of this section will be based on the expression (3-9.22) for the dipole matrix aC (t). It will be remembered that this matrix element was derived under the assumption that the expansion coefficients Cba(t) and CgC(t) are small compared to unity. The case of "strong" collisions, where this approximation is no longer valid, 49 must be considered separately and leads to the Lorentz type of theory. However, it will turn out that the distant weak collisions are mainly responsible for the electron broadening If the initial and final states i and f are degenerate, then - - Ea - = (3-14.53) 4X~ -- EC - E = 0: This condition allows great simplification of the formal theory. Substitution of our expression (53-922) for tac(t) into the Fourier integral 3(-9.2) and using (3-14,3) yields the following intensity distribution: T 0 if () Pa(O) li m 1Ti dt e if P if() aa 2,, T- T aY 0 (3-14,4) + - - -a C (t) e -i (t) + C (t) eiP b 1a0 C bta Cba 4] Avg where again a = if = CD - cf the unperturbed optical frequency Awif = c - cif frequencies measured from?if Pa(t) Paa(t) - Pca(t) - (A)a - Ac)dt' 6 — = unperturbed dipole matrix Ea(t) - Eo(t) = O (cif + ADa - AaTc)

54 c,(t) EKH(t)cf(t - t Ao - A_1 ) exp - i - (Apt - AC )dt dt" o /t Assuming that the collisions are weak so that ICsa(t)l << 1 and |Pa (t) << 1, and keeping terms in (3-14.4) up to first order in these quantities, yields If( ) E p (O) 12 lim 1 d iAtf Consistent with the "weak collision" approximation this may also be aC6 2xc aC6 T-too T a Avg 2 written where aa,0 o0 Xac(t)) t- +2 a a 0(t)a + Z' pa:b i a(t) * + ' a This formula is a generalization of the adiabatic theory in which the nondiagonal matrix elements of H1(t) were completely neglected, By the same arguments used to derive the adiabatic formula (3-10o7), one can write (3-14.6) in the usual correlation function form, 0i2 C -iofa (3-14.7) Iif(o) = ac P(0) c Re dT e-iATift-ia( 0 Avg According to Section 13, in the limit of high velocities where times (AT > Td) contribute to the Fourier integral, one may replace t 00~~ 1 / EH(t' dt' by / Hl(t ) dt' 11aU H(t) - i dt) 0 ~~~

55 In this approximation we have Pa (H) i [H(t)aa t dt -[ J s(t dt (5-1^8) C(l-). "Hk)le^ t 1 -00 i co i PBr(t (3 14[H1(t)] e d -00 00 00 I / [i1 (t) ] e at ~ = at -00 -00 where we have simplified the notation by introduicing the matrix ax defined above0 In the high-velocity approximation ^aO(r) is now (3-l419) 0aa() a - (h) (O/ * ' a^ aa) 1L~aai P.?L) The final expression for Tif (w) canll Iow be found from531-4, -7) with (5-14.9) by putting exp -(iaQ(oo) l-i0(0oo) [since:aQ0(0) << 1 in the weak collision theory]: Iif(Qf) = R, ai L ~ at e LIDf [ - ai()] Avg Tr a J (,.14.10). ^ e' — f[g, s iB.()] 0 Pa~~o~~rF~~o,^ rI, -iA T,rl00) 00 ~ ~ ~0 _0 f if 0~ (5-14.ll) R e t e K - ~ ^ ~2 -0a a a a't aC4 Avg (-43_so that laO PU 0)i a 0 aa J Avg

56 ipa(o)c~12 ma(c) u(0) 02 00 -_if T Z 0I12 (5-14.12) Iif(L) -= ' c leacJ Re d e a 0 ~~L iJAvg with (3-14.6) it follows that, a l[au12 aa (() ao o ( (aa/aboba) (3-14 13) ) - _ E 1~l E o |2 aa I~ ^ l au ~au II aaL 14 This is the Anderson impact formula. It can be used to describe optical transitions between degenerate initial and final states as well as transitions between rotational states giving rise to microwave and infrared spectral lines. However, in addition, this derivation does not contain the usual assumption that the mean time between collisions is small compared to the time duration of the collision. This generalization is important for Coulomb interactions. Now, to first order in Hl(t), we have with (3-14o8) and (3-14.9) ~ 1 2 a( ) o aa au aaPac~bB ap ~a a (3-14,14) 0 o - (aa [Hi(t)]ab If Hi(t) is a scalar sum over all the perturbers, then the statistical considerations of Sections 11 and. 13 of this chapter apply directly to the general nonadiabatic case since the phase factors i E 1\[i 12 (0)/ L. 1'1;~ a1 e- ~a aa au, aa 1'a1 _ ab p.-[Avg can again be written as a product of individual collision phase factorso The method of evaluating (53-1411) is then identical to the adiabatic case and the absorption coefficient is given by

57 (3-14.15) Iif(a) = I oj 1 (o0 -72)2 + 72 + 2 2 i t Caa (-(_o -Y_)2 ) ( - + 7 if where 71 and 72 are the real and imaginary parts of -7, Re p~~ p00 d^ ^ -^naailj k00) ak (3-14, 16) lh = e 2jrNvW(v)dv pdp ' exp 13 Thm e%0 L ab Ja! The above expression* is valid for scalarly additive multiple interactions. For completeness we have included the nonresonant term in (3-14K15), which follows from the general intensity distribution (3-7-3). The result (5-1415), which has been derived on the basis of the weak collision hypothesis [small phase shifts and amplitudes C^a(t) and C d(t)]G is of the same form as given by the "strong binary coliision"50 theory of line broadening in which 1/7y is replaced by rc; the mean time between collisionso There is a slight differences however in that the more distant weak interactions can cause a shift in the frequency maximum corresponding to 72y The Van Jleck-Weisskopf theory50 does not contain a shift (72=0)' Because of the fact that the strong and weak coll.sion theories yield such similar results near the line center, it is extremely difficult to determine intermolecular forces from line broadening data aloneo Both strong and weak collisions can be taken into account approximately by consiAering two types of collisions with impact parameters ranging from O0Pc and pc+~o~ The impact parameter pc separating the two regions is chosen such that the matrix element of the phase integral is unity. The weak collision theory fails for close collisions with such large matrix elements since each collision can be considered to induce a transition = = aik(0 where (najk) is the contribution of a single collision to ~a(~~)

58 with nearly unit probability. That is, for p < pc the contribution to the half-width according to the strong collision theory is just the mean time between collisions. For collisions with p > Pc the weak collision theory developed here is assumed to be applicable. (3-14.17) (Y7)p < = p2 N 0p 0aE naajk (oo) 1a 12 (3-14.18) () = - Re 2NvW(v) dv pdp / exp I aa j i o o aU aa2 Dc (5-14.19) = (Y)p p+ (7)p >P where pc is to be determined from (3-14.20) P (oo) = 1 (unit phase shift). Collisions with p " pc are clearly treated incorrectly because our interpolation procedure is of doubtful validity in that region. A more exact theory would require the solution of the set of coupled differential equations (5-9-5). This is prohibitively difficult if many eigenstates are involved, as there are in the degenerate levels of hydrogeni However, if either strong or weak collisions are mainly responsible for the broadening, the above method of approximation is probably fairly accurate. This interpolation scheme is somewhat simpler than that proposed by Andersono In the above derivation of the line broadening to be expected in the optical region for transitions between degenerate states, it was assumed that jC~ is zero (the condition of degeneracy) in the expression (5-9.22) for the dipole matrix Pa(t)e If the degenerate

59 splitting is not zero, but small in comparison to the width of the line, then the theory is still applicableo To show this, consider the frequency ACD? defined by 10 0 o o o o o0 (3-14,21) )(B a + - a) = a a~o + ac = WW aC +Wo a For frequencies much greater than the degenerate splitting (Aa ~ oD ) we have (3-14. 22) Ao o A 9 so that ACDa can be taken to be independent of the degenerate indices when the line width is much greater than the degenerate splitting0 This approximation leads directly to the nonadiabatic formalism of this section, These results can be used to estimate the nonadiabatic effects due to electron collisions in calculating the shape of the Balmer lines in the far wings (Aco large compared to the average static splitting of the Balmer levels due to ion fields)0 The situation when the static field splitting and Agp are comparable (the core of the Balmer lines) leads to a much more complicated theory and will be discussed in the next sectiono The iformal results which will be obtained are similar to those of Spitzer,9 who considered the broadening of Lyman a in an ionized gaso 150 Nonadiabatic Effects in Nearly Degenerate Systems The results of the previous section will now be generalized to take into account transitions between nearly degenerate substates of an initial state* i and a final state fo By nearly degenerate is meant that 6 Coab and 6 oip are not necessarily zero but small compared to the kinetic energy of the perturbing particleso Considerations of this type are relevant to *The substates of i are denoted by the subscript a and those of f by a, as in previous sectionso

6o the problem of hydrogen line broadening by both ions and electrons, The slow-moving ions produce a static electric field which removes the normal o o hydrogen degeneracy., The static splitting cab, c0 can be calculated by the ordinary theory of the Stark effect and averages over this splitting can be performed with the usual Holtsmark probability function (see Chapter V). According to the weak collision theory, the time-dependent dipole matrix is found from (3-9422): a( t) = 1 e [1 + iPaa(t)] + E. bt CZ(t) e (5-15.1) + L.ba Cta(t) e Exponentials of the type exp iP a(t) were approximated by [l+iPFa(t)] in the expression (3-9.22) for [ac(t) and terms of the order Pa,(t) Ca(t) were neglected in first order. This is consistent with the weak collision theory in which |Pa(t)| and ICPa(t)l are assumed to be small compared to unity, This approximation was also employed in the theory (previous section) for transitions between degenerate initial and final stateso The absorption coefficient is found by substituting the dipole matrix (3-151) into the Fourier integral expression. (3-9~2) for the line shapeo -1 im 1 iAct e ) IaO(CD) - 21 Ti oT T a [1 + i Pat) (3-15o2) 2 + E Elba Cba(t) e b + T1o (t) e at Avg To proceed further, consider a typical term in this equation:

61 -iAt * t Cs =~a dtC~t0e 0 T t (3153) C dt C (t) e b dt Cba (t) e 0 0 o o The expansion coefficient Ca (t) is given to first order in H1(t) by (see 3-9.12) t -i 0t' (315 4) C (t) - dt' [Hl(t')]a e (3-15.4) C c(t) =a -~o U-la ^o - a 0t >> Td This expression for Cs,(t) is valid for times which are large compared to the duration of a collisiono For scalarly additive perturbations, H1(t) is a sum over all collisions with times of closest approach tkI velocities vk, impact parameters pj etc. (5-15.5) Hil(t) = Z Hj(t-tk) k, J The typical term (3-15.3) can now be written with (3-15.4) and (3-15.5): O 0 - i~o = dt e (3-15.6) j" ^ ao — dt e ap tdt' [Hj(t-tk)] e o 0 Letting t'-tk=T' and t-tk=T yields 0 0 i o7'in ey-i -"iA" T Coi 0 e- iactk T-t C 'a t aP. e a- kdT e (5-15.7) -tk T -i 0) T' dT' [H (T)]] e -tk A partial integration gives

62 -iAutk -iA ~ Tu C = - Z e k dT[H. (T)]a e ca 0g kj J Pa -tk (-15-.8) _~.wo ' T_-i1 0. T au k j J -ie AaT T-tk d [Hj (T) ] e -tk In the phase-shift approximation, where one neglects the duration of each collision, Cogo becomes 1T -iAJ?%t p0 -iAW T ap o -i Aw~~ jvko -~ (3-15.9) -iAcAaCIT P.oT~~ ez. / dT [Hj(T)].e e kj where the sum over tk has been replaced by an integral. Substituting terms of this form into the Fourier integral (3-15.2), and taking the square of the absolute value, yields a delta function from the first term in (3-15.9): 2 (3-1~.1o) o( (3-15-.10) 1 lim 1 dt e (A=O) 2i T+oo Ta 0 This function is zero for frequencies Axaz ~ 0. In addition, cross terms involving exp -iAwaoT do not contribute to any time averages over finite time intervals. For these reasons C o can be written* *Similarly, Ca is given by ba kj C ab

63.O (3-15.11) C = O E dT [H.(T)] e for Awaa Ca -oo One can now give a physical interpretation of this result. The atomic system is initially in the state a. A time-dependent perturbation causes a transition ca-+ with the subsequent emission of a photon resulting from a transition to a final state a. The term C> takes into account these processes. The divergence at Aci=O reflects a lack of refinement in our analysis, which can be traced to our weak collision approximation. This approximation requires that matrix elements of the phase integral, summed over all collisions, be small compared to unity. For large times this approximation fails due to the cumulative effect of many collisions. The large-time behavior of these matrix elements determines the behavior of the absorption coefficient near resonance. This follows from general considerations of the properties of Fourier integrals. The absorption coefficient (3-15.2) can now be expressed in terms of the C Ia(W ) = 2 T>oo tk jvk T a + a (3-15.12) 2 1 Ii It 2 C +b Cba j,vk Z b Avg Inthe adiabatic case, the abnondiagonal elements are taken to be zero and the above expression reduces to

64 - 2TJ~j\C + Ca [Ia(c) ]Ad = 2 vk O aa (3-15. 13) 2 iad i2 1 a )JVk 2(T)) ] [H (T) ]aadT 2 j lajvk Avg This result is to be compared with the results of Section III.13 where, according to (3-13513), [I-WI - aC Aaal A Z >> Za d Is (3-15.14) [Ia( )]Ad (AWO )2 aI o, 7 The half-width parameter yac can be written (see 3-13012 and 3-1351): p00,00 Zdaj iPnaajkF() aa = 2 -NRBe W(vk)vkdvk Pjdpj iPnC J0) o o o | y TJ ([Hi(T)]aa- [H (T)]) dT =2i2 J n ( j(T)aa- d [r1o]a) + *.=Re Ze -00 -- [(Haa- [(T)] d) + Avg The two results (3-15.15), (3-15.14) and (3-15.13) agree with one another to first order in the phase integralso Since our expansion procedure for the weak collision approximation requires this approximation, the two results are consistento It is therefore evident that the formula (3-15.12) is valid for frequencies which are large compared to the half-width. For example, consider a three-state atom with an energy-level diagram of the type

65 a and spectral lines with the general appearance I(n ) _ o 043 a a The nonadiabatic theory developed here is not valid for the shaded portion of the lines, i.eo, near resonance. Furthermore, the splitting -mad must be small compared to the kinetic energy of the perturbers in the classical path approximation. The results of this section constitute a generalization of a theory due to Spitzer for transitions between a doubly degenerate state and a nondegenerate S-state if we put [Hj(T) ab = 0 (nondegenerate final state) (5-15.16) |iaJ21 = 2 2! (lines of equal intensity) % = 0 (no degenerate fine structure) and furthermore neglectr all cross products in (3-15.12). The assumption that the final state be nondegenerate restricts the applicability of the Spitzer impact formula to the Lyman series in the case of hydrogen. The assumption that all the dipole matrix elements i1I I are equal restricts the theory to a composite of lines for which the line strengths are all equal. This condition is not satisfied for hydrogen since the various Stark components have, in general, different strengthsa Finally, the

66 condition that there be no degenerate splitting restricts the theory to the case in which there is no static perturbation field due to ions. To include both ions and electrons in the theory of hydrogen line broadening, the static ion field is important and cannot be neglected. These results will be applied to Lyman a line in Chapter V and a procedure for calculating Balmer line profiles will be outlined. This completes our discussion of the formal aspects of the classical path theory of line broadening for weak interactions. We will now proceed to a discussion of some explicit calculations by various authors in connecton with the theory of hydrogen line broadening by ions and electrons. This historical resume was postponed so that we could compare the different theories with the formulation given in the present chapter.

CHAPTER IV PREVIOUS THEORIES OF HYDROGEN LINE BROADENING 1. The Holtsmark Statistical Theory In this chapter various classical path theories of line broadening will be reviewed. It is not our purpose to give a comprehensive summary of the entire subject but to focus our main attention on those aspects of the problem which pertain to hydrogen line broadening in a partially ionized gas at high temperatureo The relationship between the theories and experimental observations will also be discussedo HIoltsmark5 recognized the need for a theory of spectral line broadening which took into account explicitly the interaction between a radiating atom and the perturbing particles. He was dissatisfied with the 49 earlier classical theory of Lorentz,49 who conceived of collision broadening as arising from the interruption of a continuous wave train at each collision. Prompted by the then recent experiments of Stark on the shift and splitting of spectral lines in an external static electric field, Holtsmark calculated the probability distribution of the strength of the field produced by a static random distribution of ions, dipoles, or quadrupoles. We are interested only in ion broadening here and will therefore omit a discussion of dipole and quadrupole interactions. The justification of the static approximation for ions follows from the Hol47 stein inequality given in Section III.12. Further discussion of this point will be given in the next chapter. The Holtsmark theory for broadening by static ion fields, however, is not valid for electron broadening 67

68 because of the large average velocity. Holtsmark based his calculations on the Markoff 45 method* of calculating statistical distribution functions. Because of the importance of the Holtsmark theory in practical applications, the basic theory will be given here also, but from a slightly different point of view. The statistical theory follows as a special case of the classical path theory of line broadening developed in the previous chapter, so it is convenient to adopt the notation and ideas embodied in the Fourier integral forma lsm. l.:e results to be obtained are, of course, identical to those of the Markoff t;ype theory. It was shown in Section tIh.10, that the statistical approximation follows from the adiabatic distribution function |i /9 (Acx-A^.)dt | (-10.1) I o v2 d1 e e a a9= aa1 2L J^ dT -ie Avg g when the p.ase integral is evaluated in the statistica l iitc by taking the velocity of the perturbers to be zero: (4-1.1) iim Jo (A/a-Aa~)dt = (Aaa-Am~)r * The adiabatic formula for the absorptiLon coefficient i1s now reduced fto ~(4-1.2) Ia(w)I 1 2d - d e 00he case of scaary a T1he case of scalarly additive perturbations was discussed in Ch~aptjer III and the perturbation was written in the form @Aa-^A =, ( A t-A%)kj - k t xce nt reve artice of *cSee the excellent review article of Chnandrasekhar, Ref. 46.

69 The averaging procedure was also given for this situation. Suppose now that a spectral line arising from a transition between the states (i,a) and (f,a) is shifted due to the presence of a static electric field and that the shift is proportional to the absolute magnitude of the instantaneous field strength IFI - F. For hydrogen the shift can be calculated from the relation given in Section IIIT5: (3-5'9) AE = 3 n(kl-k2) e ao F 2 The field strength F is given by the absolute value of the vector sum of the individual ion fields: (4-1.3) F kF lrj(t-tk)l (t-t) + jk [rj(t-tk)]3; Ir-tk ~ vk(t-tk)2 + k. For time-independent field strength F is given by (4-1.4) lim FI. L Vk+O J p3 and the phase integral becomes (4-1.5) = AXaUy Z iAT where A and Xaa are defined by 3=T 3 e2a a (4-16) A - e ad Xa - [n(kl-k2)]a - [n(kl-k2)] The absorption coefficient corresponding to this type of interaction is then Iac(O))do~ = [aX d e0, e \Aa-ka~ iZ (4-107) -2o L a ) A Tvg z - AXao

70 For convenience let us define a vector A in the direction of F whose magnitude is (4-1.8) IAl -waa AXa_ AXa - then (4-1.9) d I| -| = Aa AXa~ The absorption coefficient may be expressed in terms of I |: IaC~(!~[)d[j = [: [ j p d] = ~a~l 2x ~ x Avg (4-1.10) -= 2a di|l W(i|t) Consider now the probability distribution of the x component of I: (4-1.11) W(Ax)dAx - dx /~ ds e- Ax - j s W(IVx)dAx d=J dA ~ eIrvg and similarly for the y and z components. The distribution W(A) dA of A is given by the product (4-o12) W(l)dA = W(Ax)W(Ay)W(Az)dAXdAydAz. Since the distribution W(A) is independent of angles for an isotropic ion distribution, the scalar distribution W( AI) is found by transforming to polar coordinates (4-1135) dAidAAdAz = dIAA| dil and integrating over the solid angle dQ (4-114) W(IA|) dilA = J d | d W() = 4 (A) | dA

71 The distribution function W(|A|) is now given by substituting (4-1.12) and (4-lo11) into (4-1.14):' w(AlI)dlA = 41 t|2w(A)w(Ay)W(Az)d1Al 2 0000 =(4-1. 15) A 2 d,J lA [ L j (2t) j J JAvg -0 -00 -00 where dS - didid( + Z+ -+ z Z Z (A-j pj/pJ) (-Pjx/j) + (Ay jpjy/P) + (A J Pz This is precisely the integral encountered in the usual Markoff theory. Transforming to polar coordinates yields + (4-1o16) W(|AJ)d]Al = i / | d| d 2d i d Avg -1 O where d do dai |12 d|1I and integrating over the angles gives - 0+ oo +00 i j- /P\ + (4-117) W(JA|)dAl = IAI. I d sin |Al F ) dIAl JI~ \ j3.o/vAvg 0 The static phase (4-1o18) iZ = i 1>j =. 3 Z = i J P3j j is now a scalar sum and can be averaged according to our general formula (3-12017) for the average phase arising from scalarly additive time-independent perturbations~

72 iS fl/ph 4N Pj dpj ' (- 1 (4-1.19) \- Avg = e dpj 1 where the variable cj in (3-12.17) is the direction cosine,ij in this application. The angular integration gives for the argument of the exponential, c002 I- sin (/P? (4-1020a) -4NN p2 dpo (J s — 7p 0 0 3 0 with x - S/p.; (4-lo2Oa) becomes (4-i.-lo) 3/2 dx (1 s (4-'~2o~)~ cN 512 K x Integration by parts gives the final result: (4-1.20c) 16 43/2 421 N /2 15 2 / The distribution function is now 3/2 (4-121.) W(2jA) dWAJ = ' jA- dAASJ d| sin JA e O 0 Another convenient form for this integral is found by putting E AXax Y, then (,/ 3/2 (4-lo22) 421 I3, /4 N = 4o2l (AXa'dy)3 N, and defining a new constant Xka a 3/2 4o21 (AX )3/ N (4-l023) a(4- = 2o61 AXa N2/3 = 452 Xa N2/3 the absorption coefficient is now 3/2 (4-1.24) Ia(iao )dLczD = 1a ' AO dy-y sin Aoaaye e (dAc) o 0

73 Integrals of this type will also be obtained as a special case of formulas in Chapter V, so only the results of the integration will be given in this section. Defining a new dimensionless variable 0 (4-1.25) p = — a a0aa the Holtsmark statistical theory yields the following series expansions: W(p)df = Ia(P))dP 4 7 (-_)n (1) sin (n-1) i 40C 2 3t n=l n1 (n ] 2 a o a (4-1.26) P << 1 vn-l 2/ P ~ 1 2: EC n) ( 2+ sin (3n+1)t/4 The numerical consequences of these equations have received adequate discussion elsewhere, so we will close our discussion of the mathematics connected with the Holtsmark theory at this point and compare the experimental observations with the theory. 2. Comparison of the Holtsmark Theory with Observation 12 2 Turner 2 and Doherty2 compared experimental profiles of Hp obtained with the Michigan shock tube with the Holtsmark theory and also with a pro11 file obtained by Griem with the water-stabilized arc. The temperature in these experiments was about 12,000~K. Figure 4 shows the results. The ion densities were determined in the shock-tube experiments from measurements of the shock velocity with a rotating-drum camera. The theory predicts that on the wing the intensity is inversely proportional to J5/2 o The observed intensity on the wings, however, is (Au), where p is definitely less than 2.5. The measurements indicated that the intensity on the *Private communication with E. B. Turner and L. Dohertyo

74 100 80 60 - 0'6 HOLTSMARK 5.5 x 1016 GRIEM / 40 3 x1016 20 \\ 1 2 4 6 8 10 20 40 60 80 100 8LL O3 6 z z -4 -1016 I - 2 4 6 8 10 20 40 60 80100 AX (ARBITRARY SCALE) Figure 4. Comparison of shock-tube and arc spectra of Hp with the Holtsmark distribution (log-log plot).

75 wings did not drop off as rapidly as the Holtsmark theory predicted. It should also be noted that the line shapes determined in the water-stabilized arc and in the shock tube agree closely with one another on the wing. Griemts photometry was more accurate than the time-resolved shock-tube spectra because longer exposures could be made with the arc. The shock tube has the advantage that the ion density can be determined independently from hydrodynamic calculations and that the radiation is produced in a homogeneous region so that no corrections for temperature gradients are necessaryo The qualitative conclusions drawn from these data are that there is about a 20-percent discrepancy between the observed half-width and the Holtsmark half-width and that there is approximately a factor of two between the measured and Holtsmark wing intensities over the frequency range observed. The laboratory data are also consistent with astrophysical observa52 55 tionso Since Struve and Elvey recognized in 1929 that the Stark effect plays an important role in the broadening of hydrogen lines in stellar atmospheres, a great number of data have been accumulated and interpreted according to the Holtsmark theroyo The general impression that one gets from this work is that the densities, temperatures, and surface gravities derived from the observed profiles using the Holtsmark theory and the general theory of radiative transfer are inconsistent with those derived from the continuous-spectra, metallic line intensities and also masses and radii of components of eclipsing binaries. It appears that the observed widths of the Balmer lines are greater than predicted by the statistical theory, which can treat only the broadening by ions. It has been suggested by Odgers54 that electron broadening as well as ion broadening

76 might account for the extremely wide hydrogen lines (2004) observed in A-type stars. Miss Underhill55 also finds that the Holtsmark theory seems to yield too narrow a profile for H7 in the C-type stars a Orionis and 10 Lacertae. Similar conclusions were reached by Aller5 and Miss McDonald57 in connection with their studies of type-B stars. The hydrogen lines observed in the solar spectrum are also wider than is predicted by the statistical theory. This list is incomplete and serves only to illustrate that the Holtsmark theory seems to be inadequate for a description of hydrogen line profiles in high-temperature atmospheres. The as4rophysical observations must, of course, be viewed with caution since some mechanism other than collision broadening (in the general sense) might be responsible for the wide hydrogen lines. Thus, de Jager has suggested that the additional broadening of the solar Balmer lines might be due to microturbulence in the solar photosphere. He found that the temperature fluctuations necessary to reproduce the observed profiles must. be of the order - 100G~,K. However, the astrophysical evidence coupled wi-h data obtained from shock tubes and arcs strongly suggests that certain modifications to the Holtsmlrrk theory are necessary. 3. ThLe Krogdahl T'heory rT]he HoIltsmark statistical theory follows as a special case of the adiabatic approximation to the classical path theory of line broadening. The perturbation is taken to be time independent in the statistical limit so that the motion of the ions is completely neglected. Therefore, according to the Holstein inequality, the theory is not applicable near the line center. Mrs. Krogdahl59 sought to modify the Holtsmark distribution by taking into account the motion of the ions. For scalarly additive timedependent perturbations, the formula (3-10.5) for the absorption coeffi

77 cient with the average phase (3-12.12) may be employed to calculate the line profile. The integrals involved are perfectly definite and can be done numerically in the intermediate range of frequencies where neither the statistical nor the phase-shift approximations are strictly valid. A method for carrying out these integrations was described by Lindholm44 and improved recently by AndersonO However, for vectorially additive time-dependent perturbations of the type encountered in the Stark broadening of the Balmer lines by electric fields, the average phase (311o12) derived for scalarly additive perturbations is not applicableo Mrs. Krogdahl argued as follows:59 o o since the perturbations are to be added vectorially we must regard the frequency cX as a vector as well as the quantity t [time], as used in evaluating the phase shift. While at first sight it may seem somewhat artificial to regard time and frequency is vectors, it must be remembered that the quantity t, as it occurs here, is really derived from the trajectory, which is a directed quantity dependent on the time. Likewise frequency differences are always proportional to differences of momenta o.. The absorption coefficient (5-:0o.5) was then wrritten +o o 2 1aa (453-1) Ian(Aac) = kia 2 (2-1 3 c t d. e vg and the distribution Ia( ac)o was found from (4-3.5 2) aW(at )1 d a =u [j(O)2 dau Ia Wa( ) C where dAw 4 4tA( a )2 dNou^ \= Aoc o_ The numerical calculations of Mrs. Krogdahl were based on these formulas

78 44 and followed the integrating scheme of Lindholm. She has found that the corrections to the Holtsmark theory obtained in this way were generally quite small and certainly could not account for the observed discrepancy between the experimental data and the Holtsmark theory. The argument that the frequency and time must be taken as vectors for vectorially additive perturbations because trajectories and momenta differences are involved is not self-evident. In this section it will be shown that the Krogdahl formula follows from statistical assumptions and is not connected with momenta changes or pertuber trajectories. However, we have not yet been able to justify these assumptions from first principles,* The basic expression for the absorption coefficient in the adiabatic limit is again PT p~ -iAcDa - i (6ma ZJdt, o a dji) /a -A~ d d (310.o5) Ia(W)dJ = I a2 F o2 Ja dT e I Avg where the time-dependent phase for the first-order Stark effect is (4-3-5) 0 (Ma-ALuc,)dt = AXaa.j-t.dt tk.) i O,Jo jk 33 J 3 rj (t-tk) The notation here is the same as in the previous section pertaining to the Holtsmark theory for time-independent perturbations Letting z AXagT and using (4-353), Iaac()) becomes /AXaa \ 2W0 i (to / a/ djk rjQ t (4-5.4) Ia(co)dm = J do dJ e dg Let us now define a vector ~ ( ) whose absolute magnitude is a constant (but whose direction is time dependent): *A preliminary report on this phase of the work was given at the Stellar Atmospheres Conference, Indiana Univo, Oct, 1954, and before the Astronomical Society, April, 1955, Refo 351

79 (4-3.5) |A(T)| _ h. AXau The direction of A(T) is taken to be in the direction of the instantaneous electric field, + (4Z 6)3 +6oz Z a /Zr31 (4-3.6) A jk rj/rj AX j, with this substitution in (3-105), the following result is obtained: (43~-7) Ia a( =)ddm = Ia Aa ( 1i) d o / aa AXa d|/| JdA( -i jA J k rj/rj dt z W iA|d|A, = I~acdJAI jk ': J g - -Ia! W i di j In obtaining (4-357) no assumptions were necessaryo The only mathematical operation involved was a simple transformation of variables by -introducing the auxiliary vector A(T). Since we have chosen A to be in the same direction as the instantaneous electric field, the x, y, and z components of A are directly proportional to the x, y, and z components of the instantaneous electric field strength, respectivelyo To obtain the Krogdahl formula, it is necessary to assume that the probability distribution of a component of A, say Ax, is given by dxi Ax rrj/r d (4-3.8) W(Ax)d.Ax dAx d and that the distribution of A is given by the product (4-3.9) W()inA = W(Ax)WAy)W(Az )dAxdAydAz - 4j| |2 dli\ W(Ax)W(Ay)w(Az) o

80 Then by exactly the same methods used in the previous section one obtains the Krogdahl formula (4-31i)o In the static limit, where A (T) has a fixed direction, this formula reduces to the exact Holtsmark distribution. For time-dependent fields it is not clear that the probability distribution of the x, y, and z components of A is statistically independent so that the product distribution (4-359) can be justified. Because of this uncertainty, the modifications of the Holtsmark theory derived with (4-3.1) and (4-352) may not be correct. It appears, therefore, that the refinement of the Holtsmark theory to take into account simultaneously the motion of the ions and multiple interactions remains to be done. 4. Spitzerts Theory of Lyman a In Section III.14 it was demonstrated that the weak collision theory 9 leads as a special case to a formula given by Spitzer9 for describing the Lyman a line. The averaging procedure employed by Spitzer was based on the assumption of binary collisions. The difficulties inherent in the Krogdahl formula are thereby avoided sinces the vector addition of the electric fields of several ions does not enter into the problem if only binary collisions are considered. The numerical analysis contained. in Spitzer2s work showed a gradual transition between the Holtsmark statistical distribution valid on the 61 wing of the line to a Weisskopf type impact distribution valid for the line center. It was also stated by Spitzer that the broadening due to fast electrons can be neglected, in general, in comparison with the broadening due to an equal density of the slow-moving ions. This assertion will be ii re-examined in connection with our work and also that of Griemo The Spitzer theory included an effect due to thte rotation of the coordinate system whose z-axis points in the direction of the passing ion. In the

81 usual theory of the Stark effect, the z-axis is chosen to be in the direction of the external static electric fieldo The radiation field and transition matrix elements are then described with respect to the same spacefixed coordinate system. For the time-dependent electric field due to a moving ion, the matrix elements calculated with respect to the space-fixed coordinate system used to describe the radiation have an additional time dependence due to the rotation of the z-axis. For the long-range, weak interactions which are important near the line core, Spitzer showed that the electronic quantum states doe not adiabatically "follow" the rapidly rotating field and that each distant collision produces a phase change of -t in the radiation. Spitzer showed that this rotation effect can be taken into account in a careful analysis which included a detailed account of all nonadiabatic processeso Furthermore, it was demonstrated that the statistical distribution applies to the line wing and that the rotation effect is not important. In our work only the line wing is considered so that the ions can be treated as if they were stat ionaryo Consequently, the usual theory of the Stark effect can be employed if a coordinate ssystem is chosen to be in the direction of the inntantaneous ion fieldo In treating the fast electrons, the rotation effect is avoided by this choice of a stationary coordinate system determined by the instaLntaneous ion configurationo This procedure is valid because, according to the Holtsmark theory, the probability of zero ion field is zero. Consequently, the radiating atom always finds itself perturbed by ions. This perturbation removes the normal hydrogen degeneracy and the resulting wavefunctions are given by the usual Stark effect theoryo Each electron collision is then considered to produce phase shifts and induce transitions among these nondegenerate states, When the electron broadening is calculated in this way

82 it is found that it is not negligible compared to the ion broadeningo In addition, the binary collision assumption of Spitzer can be completely removed from the theory. In this connection it should be noted that the Spitzer binary collision theory and the Krogdahl theory lead to completely different distributions in the line center. There could be several reasons for this (1) the corrections to the Holtsmark theory due to the motion of the ions are important in the line center where the nearest neighbor (binary collision theory employed by Spitzer) approximation also breaks down, and (2) the Krogdahl formula may not be correct for the reasons discussed in the previous section. For these reasons the theory of Balmer line broadening near the line center is very unsatisfactory and we have been unable to improve the situation. 5~ Inglis-Teller Theory of Electron Broadening,. -...........12 Inglis and Tellerl2 attempted to calculate the manner in which the spectral lines in hydrogen-like spectra merge into a continuum due to Stark broadening. The broadening by both ions and electrons was considered by these authorsO However, they indicated that the electron broadening is not as important as the ion broadening. The Holtsmark theory was used in treating the ions, while the electron broadening was found from order-ofmagnitude considerations. In our notation, the Inglis-Teller estimate follows from the following argument for distant collisionso According to (3-9.13) the wavefunctions describing the radiating atom in the weak collision approximation are of the form i ot i t (9ltJoi Ej (t )dt' I ( EJ (tB)dt+ (3-9'13) 3C,(t) =0 e+ C (t) 05(t)e

83 The coefficients Cpa(t) describe the nonadiabatic transitions and the probability that such a transition has occurred is (4-5.1) ' ic^a(t)12 In the high-velocity limit (phase-shift approximation or "col.lision" theory in the terminology of Inglis-Teller), using (5-13=54), with c =, and an inverse-square perturbation, the transition amplitudes are 2(o K_ _ Acdt 2 iC r(t) 2 C (C) 2 - 1 r [H(t)l'a dt' | A0 _ (4 /5'2 IJ - " J (v2t2+p2) (4-5 2) 1i ApcRev I2 but I[H(o)] - = 3 Pa:2 thus 2 2 (4-553) Ic2 1(0) o The total transition probability for the nonadiabatic contribultion to the broadening is now approximately (4-5 4) f iC a() f-MA E i. t (o):1 This expression differs from that of Inglis-Teller in two reepectsO The factor t2 does not appear in their argument and their sum over f includes a term [BH1i(0)jI|2 which does not enter into the nonadiabatic transition probabilityo It has been show in in Chapter III that the diagonal andy non diagonal matrix elements of the perturbation enter formally into the weak collision theory in an identical fashiono One might say, therefore, that

84 the phase shifts act like transition probabilities since 'they cause a transition from an unperturbed wave to a new wave that is out of phase with the original wave. The total transition probability, including transitions to a wave of different phase (that is, the adiabatic as well as the nonadiabatic transitions), is then given by the ruprimed sum (4-5.5) (v)2 Ji[HI(O )! ( L I f(O)] O Now except for the factor r2, this i'the expression obtained in the InglisTeller paper. However, their argument that for distant collisions there are two regions in which the broadening is purely adiabatic or purely non= adiabatic is not correct. These two effects cannot be separated in such a simple fashion, The broadening can now be estimalted. with 4-55) nglis and. Teller take [H2(O)]. to be of the order (4-506) c (o)H) ' where a is the Bohr radiuso The number of colli,.-r:ions pe-r s::econd with im pact parameter between p a:nd p+dp is- given by vN Pnopdp so that there are (4-5.7) (1Nv.2pdp).2 H..) 3,e ' ip transitions per second, The total number of no t:rantion5 per second is found by integrating from Pc to pr, w here p. is the critical, impact paramr eter for which the phase shift is unity', For larger phas.e,shif'ts the weak collision theory fails. The impa.ct parameter P:i. a maxn-imm imrpact parameter whic.h takes screening ianto accoontlm this will be dicussed i:n the next chapter in some detail The line widthn is now fo.und from the uncertainty principle

85 AE,vi e transition probability (4-5.8) 213Na2e4 - ---- tn p /P which agrees again with Inglis-Teller exceptr folor the faor r the factor it in [Hl(0)]O'c These arguments clearly do not follow in a rigorous fashion from the general weak collision theory szice the above quantitative results are based on an order-of-magnitude estimate of the matrix elements of the perturbation. With the availability of relatively good experimental data it is of interest to re-examine the influence of nonadiabatic effects more precisely and to investigate how one should include both ion and electrons simultaneously in a theory that includes the possibility of multiple collisionso It should be remarked here, howeverr, that these improvements are in essential agreement with the order —of-magnitude estimates of the Inglis-Teller theory. 60 A Phase-Shift Theory for Electron Broadening The astrophysical importance of the theory ofI hydroger, l3ne broadens1 10 ing for high-temperature, partially ionized gases led UnSrold to consider perturbations due to electrons as well as ions. He applied the adiabatic 44 AQ phase-shift approximation of the Lindholm -Foley' typpe to electron broadening. These results were later used by Griem to interpret the Balmer line profiles obtained with a water-stabilized arc A brief discussion of Griemts calculations and experiments will follow in the next sectiono The use of the adiabatic approximation to the classical path theory for electron broadening as applied by Unsold siffers from three major defects: (1) the collision-induced (or nonadiabatic) transitions are not considered; (2) the relative direction of the electron-impact parameter

86 with a space-fixed axis of quantization is neglected; and (3) the average over electron velocities was done by replacing the Maxwell-Boltzmann velocity distribution by a delta function 6(v-a), where v is the average velocity. It will be shown in Chapter V that all three of those approximations lead to appreciable errors. In order to compare our results with this earlier theory, the Unsold half-width formula will be derived here as an approximation to the formal theory given in Chapter III. The interaction in the classical path theory between an electron and a hydrogen atom is given by the diagonal matrix elements of -(er ~ e) where er is the dipole moment operator for the radiating hydrogen atom and Fe is the instantaneous total electric field of all the individual electron field strengths: (4-6.1). e -Fe (Fe)i In the Unsold calculation the interaction was written Hi(t) = lerl lFeI i.e,, r was taken to be in the direction of Fe. This approximation is not justified and neglects an important angular dependence in the interactiono The matrix elements of ler were taken by Unsold to be those given by the usual Stark effect theory. This approximation neglects completely all nonadiabatic and "rotation" effects. If it is also assumed that only binary collisions are important, then one can write the ilami_ tonl.an I.(t) as (4-6.2) Hi(t) r Hi(t) = - er () With these assumptions it is now a simple matter to obtain the halfwidth in the phase-shift approximation. With the notation of Chapter III the phase integral corresponding to the perturbation (4-6 2) is

87 Jfo (Awa-A c)dt - E J tH(t) aa L H (t) dt 00 dt (4-6o3) AX v2t2+p2 a= Aac 00 v ta+pAXaa Piv The half-width parameter y7a is found by substituting this phase integral into the expression (3-li14) with W(v) 6(v- ) (4-6.4) 7a = 2cNVj pdp a \ p vcos...._ where the integration extends from zero to a maximum impact parameter Pm determined by the screening of other ions and electrons (see Chapter V). Letting (4-6.5) x - -X p vT and. (4-6o6) aa == b a v Pm the half-width parameter Yacz becomes 00 3N 2 d.X (4-6.7) a = 23-t (AXac) 6a 3 (i - cos x) m adC A partial integration with 5m < 1 yields (4-6,8) 7Ya = - -(AXaa) 0, 923 - n 6m + ~ + ~ It is easily verified that the phase change 6m is small compared to unity

88 for the inverse-square interaction so that the firsft, two terms in the series expansion give a high degree of accuracy for the integral. The result (4-6.8)is that obtained by ULnsold. According to the theory given in, Chapter III, there is also a nonzero shift, 1 in the position of the? aa line center, Unsold argued that the shift could be neglected since the Stark components are symmetric about the -urnpertuLrbe'-d line center, This result should follow from the formal analysis. It will be shown (Chapter V) that the shift is in fact zero in the adiabatic approximatior if tlhe angular dependence in the interaction is taken into accouLnt, It is therefore unnecessary to take y' to be zero in an ad hoc fashion. It will also be shown that the binary colliion a ption ois not necessary in treating electron broadening. 7' Griem s Theory for Hydrogen Line Broadening by ions and Electrons 'Discrepancies between Balmer line p rofile s determined expe rimen-tally (with a water-stabilized arc) and those predicted by the Holits-mark theory 1 i for ions l.ed Griem to investigate furt:her the effect of electron col lisions on these profiles His calc-iuations,were 'based on the half-w-idth parameter obtained by IUnsold (previous section) wi the adiabatic pha.se shift approximation. Griem recogn izLed that i'the nornaiabatic. trans it io and the Spitzer rotation effect* (see tnhe d.3iscussion iln Section IVV,4 are 10 44 not taken into account in the Unsol_'d -iYn:dhoolm4 tneoryo An approximate (ad hoc) procedure for taking nonadiabatie effects into account for' degenerate systems is to apply the adiabatic formula for -the nondegenerate case with an intera:ction derived from an unweighted a verage of the Stark di G *Neglecting the '"rotation effect" of Spitzer is equivalent to neglecting the angular dependence of the perturbation Hamiiltonian discussed in bection 6 of this chaptero

89 placements in a homogeneous static electric field, This method tended to overestimate the electron broadening* but reduced considerably the discrepancy between the theory and experiment. Our theoretical development is essentially an attempt to calculate the phase-shift and nonadiabatic contributions to the line shape in a more rigorous fashion. The results will also be compared with the estimates of Inglis and Teller12 in the next chapter. It should be remarked also that Griem treated the ions in the statistical or static field approximation. It follows from the Holstein inequality47 that this approximation should be quite accurate for the Balmer line wings and is also used in our analysis. Griem was also concerned with the justification of the binary collision approximationo He indicated that since the close collisions are the most important in determining the perturbation field, the more distant collisions may be neglectedo This conclusion is not supported by our work, where it will be shown that the main contribution to the collision cross section arises from distant collisions for inverse-square interactions. When one considers long-range interactions, multiple collisions might be important. It was found (Chapter V) that it is possible to include long-range multiple electron collisions in the theory without difficulty. The results are exactly the same as for the binary collision approximation. Because of this fact, the limiting density derived by Griem at which multiple electron collisions become important is not a fundamental limitation of the theoryo 8. Summary The Holtsmark theory for hydrogen line broadening due to ion perturbations is valid on the Balmer line wings where the Holstein inequality *Private communication with H. Griem.

90 is satisfied. This conclusion is also supported by the work of Spitzer, who considered deviations from the statistical theoryo The correction terms obtained by Mrs. Krogdahl are questionable because the Fourier integral used in her work cannot be shown to be rigorously justified. A theory that takes into account multiple collisions in the line center, where neither the statistical nor the high-velocity approximations hold, is still to be desired. The theory of electron broadening can also be improved in several ways. For long-range interactions where the binary collisions approximation might fail, it is of some interest to investigate the effect of multiple collisions. This calculation will involve the consideration of the relative direction of the electrons and the direction of the instantaneous static electric field due to the ions. One can also take into account nonadiabatic effects in these calculationso There is also the question of what line shift is to be expected due to electron collisions. This point was not considered by Inglis and Teller and was not discussed in a quantitative fashion by Unsoldo Furthermore, none of these theories averages properly the electron velocities over a Maxwellian distributiono In addition, the existing theories in the classical path approximation contain only order-of-magnitude estimates of the various matrix elements involvedo Consideration of these factors will be the main concern of the next chapter.

CHAPTER V HYDROGEN LINE BROADENING BY IONS AND ELECTRONS 1o Introduction In Chapters III and IV the formal theory of line broadening in the classical path approximation was discussed and compared qualitatively with the results of otherso In this chapter the theory will be applied quantitatively to the problem of hydrogen line broadening in a partially ionized gaso Particular attention will be given to the Lyman a line (n = 1 -+ 2) 1216A since it has a relatively simple structure and also because it has already received much attention in the literature. This detailed study of Lyman a will serve to illustrate the main features of the theory as a guide for the calculation of the profiles of the Balmer lines which have a more complicated structure but for which experimental data are available. Before discussing Lyman a in detail, however, the validity of the approximations used for both ion and electron broadening will be first examined on the basis of the Spitzer9-Eolstein47 inequality. The adiabatic contribution (involving the diagonal matrix elements of the perturbation) to the electron broadening will then be presented together with a discussion of the Debye screening length. These results will next be compared with the adiabatic theory of Lindholm44 and Foley2 as applied by Unsoldl! and Grieml1 to the first-order Stark broadening of hydrogen due to electron collisions. The section concerned with Lyman Ca will include both adiabatic and nonadiabatic effects and a comparison of the numerical results with the 91

92 9 12 work of Spitzer,9 inglis-Teller,2 and the recent quantum theory of electron broadening due to Kivel, Bloom, and Margenau. 5 The classical path and quantum theories will be shown to be equivalent. This lends support to the arguments in Chapter III which were intended to justify the use of the classical path approximation in the theory of hydrogen line broadening by ions and electrons. The remainder of the chapter will be devoted to some approximate procedures for calculating Balmer line shapes when both "static" ions and "fast" electrons are responsible for the broadening. This phase of the investigation has not yet been completed due to the large amount of numerical analysis involved. The results obtained so far will be compared with experimental Balmer line profiles obtained in the U-niversity of Michigan shock tube by Turner1'2 and Doherty.2 Suggestions for future calculations will also be outlined briefly. 2. Consequences of the Spitser-Holstein Inequality In Sections III.12 and III.13 the Spitzer-Holstein inequality, which expresses the validity range of the phase-shift and statistical theories, was obtained by expanding the phase integral in powers of the velocities 1/v and v, respectively. It was found that the statistical theory is valid if the perturbers moved s'lowly enough so that the duration of a collision is long compared to the times (Atrl/ALW) that contribute to the Fourier integral expression for the line shape. Ile phase-shift approximation, on the other hand, may be used if the perturbers move so fast that the duration of each collision is small compared to AT. In that case each interaction behaves like a delta function in time and the phase integral can be replaced by the total phase change per collision. Both the statistical and phase-shift approximations allow great simplifications to be made in

95 the theory, as was indicated in Chapters III and IV Fortunately, these approximations turn out to be excellent for many applications to the study of high-temperature phenomenao According to the Spitzer-Holistein inequality, the statistical theory is valid for frequencies satisfying the relation (see 3-12,8 and 3-1359) (5-2j1) C > >. AC>, AXa where the perturbation behaves like AX aj/r2 for the first-order Stark broadening of hydrogen. The strength of the interaction, AXacz, is given by (4-1L6). The phase-shift approximation can be used when the above inequality is reversed according to (3-1359)o In Table I thle X, a:re tabul ated for the Stark components of the Balmer lines Hc, H5, 1i, and HE, together with the relative f-numbers, fa, which are proportional to the frequency MCa and the dipole matrix elements o 62 BaOa ^~These f-numbe-rsn were taken from. the work of Schrodinger and include a factor of t-o, hich takes inJ1to account both the x and y matrix el-ements for the perpelndic Aar radi atio-. (a corrponents ) This factor of 6 two was not inctleludld in the ea-lly caiationi of Verweij, who applied the Hol, saemark theory to tlhe Ba-lmeL.' lines. of Kydrogeno With these values of Xac one can now readtily.etermnr them f:reqauencies and temperatures for which the statistical and. phase -hiaft -type theories are applicable. In Table II we have tabulated AXk for Hpo AXe is defined by the relation (5-2.2) AX Ac v22 2 2c 2c AXa 2- c The broadening is assrumed to be due to electrons and ions so that v is taken to be the average electron and. proton velocity in a center-of-mass system with respect to a radiating hydrogen atom.

94 TABLE I BALMER LINE f-NUMBERS AND STARK DISPLACEMENT COEFFICIENTS Xau fag Xaau fag XaaH faf Xaua fag 8 1 12 16 22 7 32 2 6 36 10 373 20 80 30 18 5 32 8 384 18 1318 28 242 4 1618 6 669 17 52 26 10 3 2304 4 912 15 1152 24 180 2 729 2 153 13 1664 22 250 1 3872 12 166 20 18 0 5490 10 1760 18 198 8 15 16 18 7 116 14 4 5 192 12 72 3 932 10 196 2 156 8 32 0 1416 6 396 4 8 2 36 TABLE II AXc/T FOR THE BALMER LINE Hg Xa AXAc (electrons) AXc (ions) Q./ -m -, T O T 2,21 A (OK)l'.23 x 103 A (~K)1 4 ol o12 x 10-3 6 o070 o076 x 103 8 0553 o058 x 103 10 o042 o046 x 10o3 12,055 0o38 x 10

95 In Table II it can be seen immediately that for temperatures in the range 10,000~K An, ranges from 2100 A to 3550 A for electrons and from 0 0 0 2.3A to.38 A for ions. For Balmer lines of the order 10-200 A wide (which are observed in spectra from the shock tube and from stellar envelopes) one may apply the statistical theory for the ions and the phaseshift approximation for the electrons, except in the line center and far wings of the lineso For temperatures much greater than 104~K the statistical theory cannot be used to describe the ion broadening near the line coreo For Lyman a (1216 A), a line with which we will be concerned later, there are two displaced components (for which Xa = ~ 2) and two undis0 0 placed components. For the displaced components A\c = 120 A and 0.14 A for the electrons and protons, respectivelyo If the ions which are responsible for the line broadening are not protons but heavier ions, then (Ac) ion = MP (Aec)protons where Mp and Mi Mi are the reduced masses of the protons and ions interacting with a hydrogen atom of mass mh:. mp mmh mp (mhr mp) ip+mh, (5-2.4) mi mh, 1Mi +i mi+op where mi = Amp is the ion mass and mp is the proton mass. If A, the mass number, is large compared. to unity, Amp (5-2.5) M = ) mp (A+i)mp Then (5-2o6) (A) ions 2 - rotons

96 The statistical theory is therefore valid for somewhat smaller frequencies if the broadening is due to electrons and heavy ions instead of electrons and protons. This is of some interest since shock-tube experiments on hydrogen line broadening can be performed in which the ions responsible for the broadening are mainly heavy rare-gas ions.* From the considerations of this section one may conclude that since the Balmer line wings receive their greatest contribution from the displaced Stark components with large Xa, the statistical theory may be used for the ions to describe the line wings, In the remainder of this chapter we will be mainly concerned with how the electron collisions tend to modify the theoretical Holtsmark distribution for static ion fields It will be seen that the electron broadening is in general not negligible compared to the usual Holtsmark ion broadeningo 5. Electron Broadening in the Adiabatic Approximation In the theory under discussion here the broadening of hydrogen lines due to collisions between fast electrons and radiating hydrogen atoms in a high-temperature plasma is considered to arise from two effectsO The passing electron can (1) cause a phase change in the atomic wavefunctions or (2) induce transitions among the normally degenerate substates of the unperturbed hydrogen atom. The first effect is characteristic of the usual phase-shift theory for a two-state (nondegenerate) atom and will be referred to as the adiabatic broadeningo The broadening which arises from collision-induced transitions will be referred to as nonadiabatic broadening, It will be seen later that these two broadening mechanisms are es*Hydrogen can be introduced into the shock tube as an impurity with a rare gas so that there is a relatively low proton densityo Also, in the photosphere of the sun the ions come mainly from metals with a lower ionization potential than hydrogeno

97 sentially incoherent processes and do not interfere with one another, It is therefore convenient to discuss first the adiabatic broadening (which involves only the diagonal matrix elements of the perturbation Hamiltonian)o The analysis of the adiabatic broadening is quite similar to the nonadiabatic analysis, so it is instructive to consider first the simple case in which collision-induced transitions are neglected. The analysis of Lyman a in following sections indicates the method of including both adiabatic and nonadiabatic electron broadening in the theory. The electric field of the electrons in a plasma can interact with a radiating atom causing phase shifts and transitions among the degenerate substateso The dipole interaction is given by (5 -3 1) Hilt) ' F,(t) Crj (t-tk) (5-5.1l) Hi(t) = er Fee(t) e= t e k) jk [rj(t-tk)]3 where er is the dipole moment operator of the radiating atom defined with respect to a space-fixed coordinate system and rj(t-tk) is a vector describing the position of a passing electron with time of closest approach tk and impact parameter pj. GEOMETRY OF A COLLISION tk vk t \ gk -i r Pi / rj (t-tk) The vector rj(t-tk) can be expressed in terms of vk and pj~ (5-5.2) rj(t-tk) = + (t-tk) vk

98 Let pj- cos G. and k E cos Gk be the direction cosines defined by the vectors (,pj) and (r, k). The interaction (5-3.1) can be written with (5-3.2) in terms of kj and pk: E e2r Hi1kj [v(t) tk)2 + p2]3/2 [Pjj + VkLk(t-tk)] (5-3.3) X, e2r jk (vx2 + p ')3/2 (Pjj + Vk)k X) where t-tk - x. In the phase-shift approximation it was shown in Section III.13 that the integrals pt Hi(t) dt tJ O can be replaced by integrals from -oo -+ co: t0d Hl(t)dt +f Hl(x)dx = e2rpj j (x2 + p)3/2 Jo -~ -kj J J-m (VkX + Pj) (5-3.4) (J e2rp. j 2 kj J Vkp2 The term involving J (vk x + pj vanishes. The phase change per collision is found from the diagonal matrix elements of (5-3.4) co 00 dt (Aa-)Aa-)dt =J [(Hl)aa-(HI )C1] o0 — 00 (5-3.6) Z 2ekj Pjvkd

99 Now j = cos Qj is the direction cosine between the directions defined by r and pjo This direction cosine can be expressed in terms of the polar angles (@,0) and (Gj,0j), which describe the orientation of? and pj with respect to some space-fixed coordinate systemo The atomic coordinates are r and, while G and 0j are coordinates which describe the position of the moving electron relative to the radiator. It will be necessary to perform a statistical average over these collision angles G. and j.o According to the rules of spherical trigonometry, [Ij is related to G, (, pj, and 0j by the relation j Co= c os ~ cos j + sin sin cj cos (0-0j) (5-3s7) = cos ~ cos Gj + sin Osin [e i(G + eij)] - 2 The diagonal matrix elements involve only the first term since integrals involving ei1' vanisho In parabolic coordinates the diagonal matrix element (n klm Ir cos ~j n klm) is given by (3-559): (3-5 9) (nkim Ir cos ~ i nkm) = 3 (ki-k) ao The phase change per collision is therefore CO t pe2 3 ~ (JDOa'w)dt Z kj 2e. cos j n(klk2)] -[ (5-35.8) P 2AXaQ kj _ coa 0j, Pjvk where A and XaU have their usual definitions (equation 4-106)o The halfwidth 7a. and frequency shift 7a corresponding to the phase shift (5-3~8) can now be found from (3513 l 2): (545.12, _ (e\ ^ i Aa (oj) dx (3-13.12) = i( 2 NvW(v)dv pdp d o e (v 2+p2)11/2 1 1 \ac \Im o o Z

100 For the interaction (5-503) the above expression is specialized to the case where n =3 Aaa(aj) = AXaQ cos Qj (5- 39) v v daj 2 dj in jdj Jo Z Jo d2 k i o 2 G d E00 PM. 2aAXag (5-3.10) ac = 2nN vW(v)dv pdp i _ pv~ ) 2AXa9_ 7pv-/ Following the discussion in Section IV 6 of the phase-shift theory as ap10 plied by Unsold, let (5-3.11) x = 2 Aa pv and 5 12 2^a) _C AXacz 2(1 737) Xaa (5-312) m = 2 = v, v 2 Pm v P where Pm is a cutoff parameter to be discussed in Section 5 of this chapter. With the above change of variables, ya is given by Yaa = 8 tN(AXaT ) 2 W(v) dv aa dx sin x (5-3513) C^aamx ' ~ ) The integral can be done by parts: '( l)J d (1 - x) =2 - in cos + 6 1- (5-5~~4)~x = 25a 353 65a 65 6

101 where Ci(6) is the cosine integral, cos X 62 (5-315) Ci(6) d= -dx = 0o5772 + in 6 + T + o. 6 up to the power (8m )2 Ya7 becomes __o + — a ( a)2 (5-3518) YaC = 8tN(AXac)2 w(v) dv [0 2094 - n m + (4 + 0 v 6 24o Other authors, ' 115 in their consideration of electron broadening, took W(v) to be a delta function 6(-v), where v is the average velocity. This leads to a slight error since (55-3o7) [W(v) dv] _ 4 1 1.273 JO v vV. q V where W(v) was taken to be the Boltzmann distribution for a gas in thermal equilibriumo Replacing W(v) by a delta frunction is seen to lead to a 27 -percent discrepancy in the average half-width parameterO Since the depenIdence of Yad on il is essentially logarithmic and therefore is a slowly varying function of v, the approximation is made that (5-3.18) J V) dv ir _ 3 47Xn.3347 vO v PPmv v Prm In the following formulars 6m is taken to be -vaaC 3.474 Xa, (5-3019) b(, Pmv so Y a is now 32 N (AX) (5^*320) / ) ( O~a( t ) G(a) ( (2094 - ',aE (m6m )2 \ E ~ ~ nII m + 240

102 It is of interest now to compare these results with the Unsold1 phase-shift calculation in which all angular dependences were neglected. Taking the ratio of the Unsold result (4-6.8) and our expression (5-3018) for Yaca yields (55-321) Yacu = l (1o708 -,n 5 + o 2) (a Unsld 33 (o 923 -n 6Srr) ~aa auz (note that 6m 2 m / ) The theory discussed by Unsold and later applied by Griem to the electron broadening of the hydrogen Balmer lines yields a result that overest:imates the adiabatic contribution to the broade:ing by perhaps a factor of five This agrees qualitatively with the statement by Griem* that his calculat+ions tended to overestimate the electron broadening. Furthermore, it is seen that the present theory gives zero shift in a natural way without recourse to additional assumptionso 40 Errors in thel, Average Over Electron Velocities According to the Spitzer inequality,9 the phase-shift approximation is valid for frequencies satixsfying Am << v/AXa The averaging process discussed in Chapter III included an average over tIe velocit+ y v from 0 + oa, Thi. is ciearly incorrect for velocities less than!nJXaAw since the phase-shift approximation fails for velocities not satisfying the Spitzer inequality. Let us now compute the contribution to 17v from velocities less than wAXau Aw in order to estimate the errors introduced into YaC by our averaging proce dure o The Boltzmann velocity distribution is given by *H, Griem, private communication o

105 W(v) = 4- v2 e-a2 a _ m a 2kT " so (54.2)~1 j W(v) v= 4 t a. (5-4.2) V = V ^ "^ ~v Jo ~v \~2a The contribution to T- from velocities less than s AXadA is given 'by Jo v \2f = With (5-4o2) and, (5-4~3) we find the percentage contribution to 1/v from velocities greater than fAomAX a: i(_ - auA AXa d (5-4o4) 100 Z v < 4i x.DA7. 0 '100 2 +.... 1. (486 For temperature, of 10,000~K, a equals 5o5 r o106 Consider Ht (486ll) 0 at 30A from the line center An ist of the orer of 5 x 1012 and Xa isf of the order 6. For these conditions, aAwAXaa, 0.1o so that 90% of the contribution to 1/vcomes from velocities which satisfy the Spitzer inequality. For smaller frequencies the errors introduced by treating the low-velocity end of the velocity distribution in the phase-jshift approximation are correspondingly smaller. The errors increase slightly for larger Xao However, this is not serious for H., H,, and H since the Xaz are of the order 1-10o A more exact -theory would involve extensive numerical integrations and might be of interest with more accurateexperimental data on hydrogen line broadening at high ion densities where the lines are very wideo

1.04 5. The Debye Cutoff Procedure It was necese sary to cut off our i:ntegrals over the impact parameter at some value Pm because of a logarithmic divergence o This difficulty is not uncommon in theories dealing with Couloimb interactionso Following 6 6(5 Bohm and Aller and Spitzer et alo, the cutoff pm is taken to be the collision parameter 'beyond which an electron is effectively screened 'by the neutral plasma; of ions and electronr.;o This di.stance is commonly referred to as the Dlebye length and may be derived as fol:lows according to the discussioln by Bonh and Aller, The electric potent:ial 0 sat i.sfi:e: the Poisson equation (5-51o) v2 -= 4r.p = - 4.e ( 7.i 4 e kT-:e e- t where Ni and NF are the partllILe dens:lities of ions and electrons and Zie is the ionic charge. On expand ing the exponentiala s and making use of the fact that the electron density is related to the ion density by the expression N = ZiNi, and that for high temperature e0/kT i small compared to unity, the Pois-so:n e quation reduce-s to (5-o?5.2) V2^ e ( ) 7... kT Prn The solution corre.s>!ponding to a poiznt ch'arge having a finite poten tial at infinity is (5-5.5) 0 e r /p where Pm is given by (5-5.o4) p,,l (1+7i )e1

105 If the ion and electron densities are equal, then Zi = 1 and Pm is ( kT /2 (5-505) Pm - (e l /2 "aa and the maximum phase shift m is given by ~aO 2AXo~ -6e (5-5 6) & 2AXa 1.80 x 106 Xa m paa 0am For temperatures of the order 10,000 K and ion densities of the order Ni ^ 105 cm 3, the Debye length Pm and the mean distance between ions ( Ni 7/) are very nearly equal. Therefore, on the average, an electron feels only a single ion at a time, ioeo, only a single ion is contained on the average in each Debye sphere. This is a physical justification for the validity of a binary collision theory of the Foley2 typeo However, at higher temperatures, where the Debye length exceeds the mean distance between particles, the binary collision assumption necessarily failso For this reason earlier phase-shift theories 'were expected to fail at high temperatures Our theory, however, is not subject to this restriction since we have included multiple interactions in the formal theoryo 6o The Effect of Close Collisions In Section V 3 the half-width 7a, 'was fowud to be 32N p aGz 7aa = -3 (AXaa) G(m ) (5-3o20) G( ) = 0.2094- 1/6 En bm + + o /6 (1,2564 1 n Sm + o) o The integration over the Impact parameter included the range p = 0 - p

lo6 where Pm was taken to be the Debye length. The above expression for the half-width parameter 7ac must be modified to take into account the close collisions. The weak collision theory developed in Chapter III fails when the matrix elements of the phase integral co 1 / H1(t)dt co,00 are not small compared to unity. An approximate scheme for taking into account the close collisions was discussed in Section III.14 (see equations 3-14.17-~3-14.20)o The contribution to yaCx for collisions with impact parameters small enough to give phase shifts greater than unity is given by the kinetic-theory result (5-6.1) (7a) i= L = pc Nv = mean time between collisions, P< Pc Tc where Pc is determined by the condition that the phase shift per collision is unity (5-6.2) = 1 = c = 1 - V so (5-63.) Pc = 2Aaa v Hence (5-6.4) _.a.)p<p = (AXaG) N (TaCZ)P <Pc v The contribution to 7an from the weak collisions is found by integrating over the impact parameter from Pc + pm instead of from 0 + pmo (5-62 (Ax = [G(bm G(1)] P> Pc V

107 According to the expression (5-3520), G(l) is given by 1 (5-6.6) G(l) =.2094 + = 2136 so (a)p > = )32N (AX )2 (-1/6 In -.0042) auP > P caQ m (5-607) 16N aa 3- - (AXaC) n vm ~ The total width, taking into account both strong and weak collisions, is found by adding (5-6o5) and (5-6o7)~ (5-6o8) 7 = 3-6N ( (AacX) (2o36 ~- n am +,) In the pure weak collision theory (where the integration over the impact parameter is carried out from p = 0 -+ m) the half-width parameter was found to be (5-5320) (7a ) = N (26 - n m' + oo) auX weak l - 6m n... The difference between the two results is negligible if (5-609) (2036 - lo26) = lolO << n iC 6m When this criterion is satisfied, the distant collisions are mainly responsible for the broadeningo For example, when Ne = 10o, T = 20,000'K, and Xaa = 1 l0, then In 7Ca, ranges from 5~86 to 35560 The rather ad hoc procedure for treating the close collisions introduces an uncertainty of perhaps 10-20 percent into

108 the expression 5-3520 for the half-width parameter 7aWo~ 7. The Broadening of Lyman a by Electrons The broadening of the Lyman a line due to electron collisions will now be considered in detail. The line originates from a transition between a fourfold degenerate state (n=2) and the ground state (nr-1) of hydrogen. The passing electrons cause both transitions among the degenerate substate- (nonadiabatic effect) and phase shifts (adiabatic effect). It will be shown that the adiabatic and nonadiabatic contributions to the broadening are comparable. The quantitative results will be compared with other theories. 7a. Wavefunctions and Matrix Elements It will be convenient first to calculate the matrix elements of interest in the Lyman aC problem. The wavefun.ctions n,m corresponding to the states with principal quantum numbers n-2 and 1 are r -r/2ao ois 0 0: = -'-7 —7~.e e?sln O 8 \/ x ao/':-r-_ -'r/2ao (578^0. n *= /2 e CoB O (5-7a.i) ^ c~ (2 r) e-r/?2.ao IT o 1. r/a,oo, ao- The appropriate stabilized eigenfunctions for a. hydroge om in an external static electric field are given by

109 1 0+ - X (02,0,0 + 02,1,0) (5-7a.2) - = (02,o,o - 02,i,o) 02,1,~1 l If the ion broadening is neglected so that there is no associated static electric field, then the electron broadening can be calculated with either set of wavefunctionso If, however, in addition to the rapidly fluctuating electronic electric field there is a static ion field, then one must employ the set of functions (5-7ao2) since these wavefunctions describe the perturbed hydrogen atom between electron collisions. This point will be discussed at length later when both ion and electron perturbations will be considered simultaneously. For the electron broadening one is interested in matrix elements of the phase integral 00 p / Hj (r )dt which, according to (5-3.6) and (5-357), involve the matrix elements of CO CO sin G sin - r -j -i(-0) (5-7ao3) r J. = r cos 9 cos 9 j + s i e + e J U L J 2 L 13 The radial integrals involved in the computation of the various matrix elements are trivial and can all be done easily by a series of partial integrationso The result of the calculations is as follows: <2,1,~l1 rj i2,l,~l>=0 <2,1,0 |rpj|2,1,0>=0 J

1.1.0 <2,0,0 rpj 12,0,0>=0 <2,1,0 Irijj2,0,0>=5ao cos Q. 3aO e+iodj <2,0,0 |r~j|2 L,+l>J~sin Qj e JF <2,0,0 |r[j I2,l,+>1. sin 0j e. <2,0,0 Irpjijj2,l,~1>iao < + |rij| + >=53ao cos G. < - |rij| - >=-5ao cos j < + |rI |2,l,~l>-4 a sin ~j eG,,, i 5 ~ ~~~~~ ~i(^ * (5-7a.4) < I r~jij 12, 1 ~1>=7 ao sinG e < + Ir^| - >=O, where ao is again the first Bohr radius. The intensity of the observed radiation also depends on the dipole matrix elements which are tabulated below: <2 1+l~ix~l-, 0~>i 6a, 0 <2,ll Ixjl,0,0>=+i R 6ao P00 r "5r/2ao R^21 R? r4 e, dr <2,l,~l y.-, 0,0>= a4 <2,1,0 |z11,0,0>= -R 6ao (5-7a.5) <2,1,0jx,yjl,0,0>=0 <2,l,~lJz l,0,0>=0

11l R2: < + 11,0,0> < + xy l,0,0>-= < - ix,yll, 0,0>-0 7b. The Broadening of the Lyman a Line Using the Wavefunctions (5-7aol) Let us now compute the broadening of the Lymnan a line using the hydrogen wavefunctions (5-7a-ol) Since we are neglecting the ion broadening here, Do= 0 and the expression for Ia5(w) reduces to (5-7bo 1.) Ia() =- a where raa is defined by ra - L = ro o)a 12. j dT t 4 (,o,o)p L j()a Avg (5-rib.2) a = ( )(2,,0), (2,1,0), (2,.1.,),,l - ). With the table of matrix elements (5C:7ao) andl,. (5p —ao5) it i:s ireadily verified that only r(Ioo) (aoo) is nonzero I ),n;'.0 ) Or) ^i,o,o)(,o,o) d J J T0ilono) = o oHj~rA,( (5-7b 3) + t1,0,0) (2,1,0) [Hj(T)] (2,l,o)(2,0,o + (1,o0,0)(2,I,) ~ H:(.2,l,1), +.,oo)(2,.,0,0 ) (T)i(.,),,)(2,o,o0 ) According to (535o4), the phase integral is given by [HJ('']2,.,)2.o + '(:.O.,o)- 1(a.l.=l[E(r)]2l. z.(2,oo

112 (5-354) Hj(T)dT = 2e rp, where rij is given by (5-7ao3). Since <2,0,0 r j!,,00>=0, (1,0,o)(2,oo) is now + I(1,0,0o)(2,1,1)[rji (2,l,)(2,o,o) where we have abbreviated the matrix elements < a rijl > by [rizj ag, Now u, receives contributions from the x and y matrix elements corresponding to radiation with different polarizations and can be written '(l,0,o)(2,l,~l) " = '[ 0 7 ~ ) 22 L ) e r l.: t) (5=T7bo.5) [xJ(LOO)(2,LL) (.i,0)(, ) + + [Y]( _,0o,o)(2),.,~) = + ) ( 67b o ) x O (2,, )(,oo) l ( Y. (0L,1)( LN ) On averaging over the random phase ~r between the y and x radiation one finds 0 [x 2 ] (5~-7b.7). (... () - 2 y (l)2 ~ On substituting for the various matrix elements into (5-7b 4), using the

11.3 above relations with (5'7ao4) and (5-7ao5), the following re-;ult is obtained: ~f - ~ f 2e4 A 5 L (1i,0,0)(2,0,0) j 6a 3ao cs (eia +l)e, 3~sin j +(e =1.) s in j e (5r7b.8) t) R~6aor Ljo On squaring and averaging, all terms involving exp iW and (sin Gj cos Gj) vanish. Hence Z Fr/6e 2ao\ B21 (o2 (1,0,o0)(20,0) = j ' 6a (cos~ j (5J-7b9) + sin2 o j cos oj eli + sin 9j vj) e s-a the 0e 1 (5-'7b.O),0 T e ao ': 1+(,oo) j \/ j v^Pj.Avg This broadening arises solely from nornadiabatic transitions s:ince the di - agonal matrix elements all vanish with the, 0 o-,o,,, et of i2,n' (0 0 2,.., o wavefunctions Let -us now compare these results with a calcuiatio2r using the Stark wavefunctions (5-7a.2). 7c. The Broadening of the Lyman a Line Ugsig the, Wavefunctions (5J-7a2) The broadening of the Lyman a line of hydrogen will be computed again with the set of wavefunctions (5-7ao2)(i.e., the set.,,). The final result is identical to the result obtained with the set of wavefunctions (5-7aol) and is included in order to illustrate the importance of

114 nonadiabatic transitions in the theory and to facilitate the comparison with other theories of hydrogen line broadening. It is also to be remembered that one must use the Stark wavefunctions when a static ion field is presento It is readily verified that l(1 0o) = f(1 0,o) for the displaced Stark componentso Since the calculations are similar to those of the previous section, the details will not be given. In addition, the term r(1,oo)(2,1,+~) is again zero (previous section)o Let us consider now the terms F(1,o,o)+ = (1,oo)_: Z(1~0~0)+ e2 \2ve~~1 0 o o0 (1,0)+ = j 2 (1, 0,)+ [rLj]++ + P(1,0,0)(2,1 1)[rj](2 1 1)+ (5-7c.1) + (l(,0o)(2, ll)[rtj](2,,1-l)+ l A Now according to (5-7a.4) and (5-7ao5), 0, 4(1,0,0)(2,120) - (1,0,0)+ ^ (5-7c.2) [r](211 = ri (2,1,1)(2,o,o) [rj]++ = [rij](, 1,o) (2,o,o) With these substitutions, Fr(1o,o)+ becomes "i Z 2e4 o r(l,o,o)+ = j { 2pef2E2 (, (l,o)(2,l,o)[rPj](2,l,1)(2,0,o) (5-7c.3) + 0 [rni] + (1ioo)(2,l,1) [ri] (2a,,11)(2,0oo) + I(1,0,0)(2,1,-1) [rtj](2,1,-1)(2,0,o) ~gv

115 Comparing this expression with (5-7bo4), one finds that (5-7c4) F( - r O)(o ) r(2, oo)+ = The broadening arises from both r )and (1 0 so that (~o ad.o) (~,oo). so that (5.7c.5) I(cL)= (io"O)+ + 'oo = ~o)(2 Lol ~ I(&_0)2 T (&D)2 Therefore, it is verified that either set of wavefunctions leads to the same result. There is, however, an interesting comparison between the two calculations. With the set of wavefunctions (5-7ao1), all the broadening arises from nonadiabatic effects. However, the broadening is partly due to nonadiabatic effects and partly due to adiabatic effects with the Stark wavefunctions (5-7ao2)o The ratio of the adiabatic to nonadiabatic broadening in the second case is found from the ratio of the two terms in (5-7bol0): c adiabatic <C2 j>Avg (5=7c06) 1nonadiabatic <sin2 jAvg so that (5-7c.7) r(1o,0,)(o2,0,) = F(,oo)+ + r(1,o.o). adiabatic + 2radiabatic and the absorption coefficient is accordingly (5-7co8) I(X) = A From this example it can be seen that it is necessary to include both nonadiabatic and adiabatic effects in the theory of hydrogen line broadening by electron collisionso

1.1.6 If one requires that the absorption coefficient (5-7c.8) be normalized to 2[j(1 0) I' (iLe., the line strength of the displaced Stark components), then one can determine the appropriate damping constant. This procedure leads to the following result: (5-7c.9) I(Xc) 2 \ loo) __ Ex tAm2 + y2 where (5=7colO) 7 = - 2 1,8 (a —.Z 2 14=(1, 0,.0)1 J \ PAvg Now there are 27pjd.pjNv collisions per second with impact parameter between (pj,pj+dpj) so that y is given by 18a2 e4 P M (5=7c.lla) 7 = _0 E 2N dp2 p. c J J/Avg over v 2 4 (5-7c llb) 7 _ N In Pm ha- Pa But L 4 L v 7: v so that y is finally given by 144 a2 e4 0 N PM (5-7c.12) 7 2 an P and 2 4 y 48 ao e N Pm (5-7T:c l) 7^'Ad 3 2d 5v Pc This result is to be compared with the result of the adiabatic theory (5-607)e

117 16N Pm (5-6~7) 7ac = (1,o,o)+ = (10,0)+] PC For Lyman a, Xa = 2 so that with the definition of A (3-~16) (5-7c.14) AX ( e2ao 2 and 48 a 2 e4 (5-7c 15) Y (,o,o- 4n (l,0^0j+ ^2 v PC which agrees with (5T7co13) as it should, 8 Comparison of the Classical Path Theory with a Recent Quantum Mechanical Calculation 15 In a recent paper by Kivel, Bloom, and, Margenau a theory of the electron broadening of the Lyman a line was presented' This theory was completely quantum mechanical. The electrons were described by plane waves (Born approximation) and the states of the hydrogen atom were ta= ken to be the set of Stark wavefunctions (5=7ao2)o It has been verified that the perturbation matrix elements computed according to the classical path theory and those computed quaWntJum mechanically are identicalo The reason for this is that the distant collisions are mainly responsible for the broadening so that the plane-wave approximation for the electron waves function is quite goodo The classical path approximation is also valid because of the large orbital angular momentum associated with the distant collisions, In order to show the equivalence of the classical path and quantum theories, it was necessary to compute the diagonal perturbation matrix element with the wavefunction ~+ in the quantum mechanical theoryo This matrix element (which was not computed by Kivel, Bloom, and Margenau) is

118 large and contributes one-third of the broadening using the Stark wavefunctions. This is equivalent to the adiabatic part of the broadening in the classical path theory. In the Kivel, Bloom, and Margenau (KBM) theory this is referred to as the "universal" broadening. The broadening computed in KBM was due mainly to the (+) -+ (2,l,l1); (-) - (2,l,+l) transitions induced by the passing electrons. This was called the "polarization by induction" broadening and is the "nonadiabatic" effect referred to in our classical path theory. A third source of broadening due to (+) - (-) transitions was found to be small in the quantum theory and zero in the classical path theoryo This broadening was called "polarization by reorientation" by KBMo The only other differences between KBM and our treatment are: (1) in the choice of the critical impact parameters with which one cuts off the divergent integrals that appear in the theory (because of the logarithmic nature of the divergence this difference is small, about 20 percent); (2) our treatment includes a velocity averaging; and (3) the KBM theory did not include a discussion of how to weight the various perturbation matrix elements with the dipole matrix elements t~O Including the diagonal matrix element [H1]+,+ together with [H1]+,(2 1 +) in the quantum theory and weighting these elements with +, (1,o,o) and l(2o,1 i ) - - to yields a half-width y that is three times that given in KBM. In comparing the electron broadening with the usual Holtsmark ion broadening it is also to be remembered that there are two displaced Stark components. The wing intensity due to electron broadening is given by I(@) = 2 14o ) = lB 2 _ 2 0o I7 - 1 +( o oKBM - - +(io,) aW+y2 goo) A+57BM)2 0~ 'IS~~~~~~~~~~~YB

119 Accordingly, the ratio of the electron broadening to the ion broadening is some six times that given in KBM. A mpre detailed comparison between the electron broadening and ion broadening will be made in subsequent sections of this chapter. 9. Comparison with the Inglis-Teller Theory In the Inglis-Teller1 theory of electron broadening discussed in Section IV.5 it was found that the broadening of the 0+ state was proportional to the following matrix element: L H (T) dT If one employs the phase integral given by (5-5.4) and computes the above matrix element exactly with the wavefunction 1 O+ O -+,,) 22 02^0 2Y1Y0 instead of using the Inglis-Teller estimate, then the result for Lyman a is in complete agreement with our calculations and also with the quantum theory of Kivel, Bloom, and Margenau.15 The agreement between our classical path theory and considerations of the Inglis-Teller type is not expected necessarily to give identical results for the other Balmer lines. The reason for this is that the matrix element f [H(t)] dt contains all the nondiagonal matrix elements [Hj(t)]a, including those between states of different principal quantum numbers. Our theory considers transitions only among the degenerate substates of hydrogen and neglects transitions between states of different principal quantum number.

120 Thie classical path theory cannot be used to treat transitions between states whose energy difference is not small compared to the mean kinetic energy of the perturbers. 10. Hydrogen Line Broadening by Both Ions and Electrons In a partially ionized gas the spectral lines of hydrogen are broadened by collisions with both ions and electrons. In our "weak collision" theory for the broadening due to high-velocity electrons, a static field splitting of the normally degenerate hydrogen states was introduced. This splitting is presumned to be due to the Stark effect of the static ion field. In this section bot-. ions and electrons will first be taken into account in the adiabatic approximation to the theory. This can be done exactly for slow ions and fast electrons where the statistical and phaseshift approximations are valid. Th.e results can then be generalized to include nonadiabatic effects for frequencies large compared to the half-width for electron broadening. For frequencies smaller than the half-width due to electron broadening the divergence of our basic equations does not allow us to obtain explicit formulas for the line core (Aw < 7a) in the nonadiabatic case. lOa. Ion and Electron Broadening in the Adiabatic Approximation According to the adiabatic approximation to the classical path theory, the absorption coefficient is given by (107) () = Re dT e i L (A-A )dj (3-10.7) lat(CD) =l l Re dr e1 eT e o ' " Jt ^~~o - ~J Avg If the perturbing particles are ions and electrons, the phase integral is given by

121 T0T / T (5-1Oa.1) J (A/a-c)dt =Jo (a a-L)ions (t + Jo ( a-c)electrons dt In the statistical approximation the perturbation due to the ions is time independent: (4-1.5) (Ama-Wac)ions dt = FT J~~o~ e where e is the charge of the ion and F is the instantaneous ion field strength at the radiating atom. The absorption coefficient is now.T 1 2 A [ (iaCi AXa T+i o (Da-Aa)electrons dt (5-10a.2) Ia5(o) = aa' d e oAvg idut o If all correlations between the electron and ion positions are neglected in the statistical averaging, then the results of Section V,3 can be used directly for the average over the electron coordinates (5-10a 3) rAvg (5-lOa.~) e i o( a- )electrons dt = ea and the absorption coefficient becomes 5-10a.4) iac( ) laC Re ei( a - a T7aC T dT it,-,o ~-1Avg(ionss) The integration over T yields I o 2 ^eC Ya12Y Ia-(Oo) = e_ -2 2 aI - I aa - AXa ) + Ya Avg(ions) (5-lOa.5) o 2 1 - 7aG Jo w (A)dA (, AXA). Yau~a1 I:,,~,,,,,(~ ~(,- AXaz)2 + 7ac

122 where W(A )dA is the probability distribution of the quantity F/eo According to (4-1,10) and (4-1d21), W(A)dA. is given by the Holtsmark distribution *-> ''2 r,]-4. Ni N 3/2 (4-1o21) W( A I) dA. A W(A)dA - AdA / S d: in A. e i /0~~ where Ni is the ion density With the above expression for W(A), the absorption coefficient is now~. i 3/2 oo (54Q-10a.6) o = 4 'a d. e 4 2112 d ( sin A )) AdA it o '( - Jo) i Jo0 0 (aloa-AXaUoA) +-7a.a Since the broadening is symmetrical about the line center, the integral over A must include both the high- and low-frequency Stark components. The absorption coefficient for all frequencies:is then 2 /~ _ IP r 4.I21Ni /3/2 (510a,7) o,,AX' ~X - __..-+ ( XaA )2 +7 ', Jo l A (.+AX^V + (7ao-"AX"c )+yj ] *b at-Xa but A 8in A, dAJ 0 A in A i dA (5=10ai 8) 0o ' (S(a + AX a) J4 (AA Q AX )+ so that (5~10ao7) becomes o 2 _,4 21N" h pi AT-?3/2 r^ A sin A dA (5-10a.9) Iad(a~) C " — o 1^1-a06 oAa aceA aa The integral over A can be done.imply 'by a coutour integration (5)=iOa~Ol)/ A:,n A (,d. A.=_: A AXa, m~Cz + cos __ac_ (50a~J00 = (acoaa-A XacL) +?oc (AXacx)27\arx hCS s-5U~~~ ^S^51" *

123 The absorption coefficient is therefore o2 2 00 3/2 1 Ia (X) = 2 l(aal 2 f oa e(- 421 AXacNi 3/ -7aa ) a (AXaa)2 o (5-10all). sin + 7_a cos Xa (Auac sin CaQ Yac 005 AXa AXaU It is immediately verified that this reduces to the Holtsmark distribution (4-1.24) in the limit of zero electron density (Ya7C + 0) and to the dispersion line shape for zero ion density. 10b. A Series Expansion for Small Frequencies For a given spectral line one may calculate the electron half-width parameter 7ace according to the weak collision theory and then evaluate numerically the expression (5-10a.ll) for the absorption coefficient Iag(w)o Although the integral in (5-lOaoll) is not expressible in closed form, it is convenient to derive asymptotic limits of the integral for large and small frequencies nmaa to facilitate the numerical calculations. Consider now the integral sin o0, 41(-4 21 NiAXa 3/2 (5-10b.ll) o e AXa cos AXaz Now let A = AXax,, then this integral becomes (5-10b.2) (AX )2 o xdx sin A, x e(a 2 a (Aade xd i o _aQ/ -aOSXa, cos where %ac is the ion half-width parameter defined earlier in our discussion of the Holtsmark theory, (4-1.25) kaG - 4o 52 XaNi23 / The expression XaC is simply the Stark shift in units of circular frequency

124 that corresponds to the familiar Holtsmark mean field strength. With the above change of variables, the absorption coefficient is given by Iaa(aa Ldaa U= -- laal2 daa o xdx (5-10b.3) 0. (o o -(%aux)3/2 _ YaCaa *(aa sin wao&x + YaG cos Waax) e a a It is now convenient to measure frequencies in units of kaX and to introduce the ratio, R, of kao and YaG. Therefore, R is a measure of the relative importance of ion and electron broadening: au %aa (5-10b.4) R LaaG 7aa If we now make another change of variables in (5-10b,3) by putting kaax y2/3, then the distribution becomes Ia(f5f)d5 = ~ oI2 4 d 0 yl/3dy(p sin +yc2/3 + cos (5-10b.5) 0 2/3 1 - 1aGl 3n dP Re (-ip + R o y/3 dy e - Expanding exp [_y2/3 (l/R-i3)] in a power series and integrating term by term yields I 1 / 1 00 n2n-n (5-10b.6) Ia,(P)d = |ao2 4 d E (R-1) JRe dy e-Y y 3 3~nn=l -l). The integral is given by a gamma function, so for small P we have finally (5-10b.7) Ia(B)d 4 d l (-1) n-l) -1 r)c os (ntg- ) I a~l2 5 " n=l (n-l) +

125 For the case of zero electron density, R + o, so the above series reduces to the well-known Holtsmark expansion _(_)_4d 1 (4)n n [ (n+l) cos 2 au 1i'Wa2 5dt n=liT5^7T (n J 2 (5l0obo8) 4 dp Ir(2) P2 - P4 + o lOc. A Series Expansion for Large Frequencies If we let z = y2/3 in the integral (5-10bo.5), then the absorption coefficient can be expressed in another convenient form: (5-QlOcl) Iac(P)dO l=4-2 do zdz sin z + cos e Expanding exp - (z/p)3/2 in a power series and integrating term by term gives I 22dB) n- 10 dz [(n+1 1) _ z oP)d 2 A -^ = f(e P (a T P n=l (n- G3(n-l)/ 2 (5-10Co2) ( (2 sin z + 1/R cos z )J The integral. in this expansion can also be expressed in terms of gamma fmuctions: z kl e R i 1 k (sin i (5-10c3.) J dz z e B in z k i t PR With (-1 ) te asorion st Z) comes s g With (5d10c03) the absorption coefficient (510cc.2) becomes

la(P)d = ikc~l d (1 ), (4-l0c 4) ( t R) + os ( tg 3n~l + ( 2> Using the following identity, (4-10C~5) PR I _.. cos (tg' 31. ) one finds 3n+ l I ] 1 n+1. + si/ R0 tg R (5-10c.o6) With this identity, the absorptiorn coefficient, becomes finally density or the ion density is set equal to zeroo For zero electron den~ 9$Pto-~e 1 )6,7 (5~10-co) /o? \G dt3il r,-. + LI n 3. rfI cos Itg J ) Let us now examine the two limitfng cases where either the electron density or -the ion dens-lity is set(- equal to ze-rYo For zero electron density R -+ oo, and. one obta;in:. tlhe us,,ual., Holtsmarrak expansion \ 7. -

127 For zero ion density + o, so only the first term in (510c,7) is nonzero With the definitions of B and R the first term is given by p2. Y7aa (5-10Co9) Iac(2uoaao) = Ii~ 1 _ _o_ )_ t_ a(~a) = x ac (Aa~ C)2 + 72 au aO This is the familiar dispersion line shape to be expected for electron broadening onlyo lOd. Normalization of the Absorption Coefficient It is now necessary to discuss the normalization of our distribution function. From (5-10c.1) with z pp, 00 2 2 r00000 / /1.p/R-p3/2 (510dol) J a(p)d - lal Jo Jo ^pdpdp (sin P + - cos PP e With a partial integration of the first term tw ice and the second term once, this becomes >l )d voJs 0 2 2 / in p d2 ( -pl/R-p (5-10od.2) s in jpddpL R-pe + dp ~ Silncee do p > o in d p> p>o, the integrated absorption coefficient is now '00 o, 00 2 1 p/Bp 3/2 T ([i)dOU1Iac dp II + Ld pe o 0o \dp Rd dp (510d.53) ic02(> J ) -p/R-p3/2 0o \ / ~~~~~0

128 so that finally r?- CO (5-lOd.4) Ia )d 1 2 0 and, since the distribution function is an even function of P, oo Iao P)(5d 2lal (5-10d.5) J Ia ) = 21 Co It is to be remembered that |p~a is proportional to the line strength of a single displaced Stark component so that 2aI|O12 is proportional to the strength of a pair of displaced components, In calculating the frequency distribution in a hydrogen line which is composed of several Stark components, one must, of course, sum over the components, using the distribution function (5-10cl1) or the various series expansions derived earlier: (5-10d.6) Iif(C) = E P(o) Ia(AO) where po (5-10d,7) Iif(c)d = Z 2 p(O) I 12 I 0 This is the usual expression for the total absorption coefficient of an unperturbed hydrogen line, 11. The Relative Importance of Ion and Electron Broadening In this section the theory which takes into account both ion and electron broadening will be compared with the usual Holtsmark theory for ion broadening, For convenience, a set of functions Kvn(R) will be introduced0 These functions are correction factors to each term in the Holtsmark series (5-10c,8) and have been plotted in Figure 5 for n =

L29 1.5 t rrl t i i 1.4 I.I= Ji 2 I _ A4t tifiitt4 illdi 1.3 O 10ti -fi -WS 8 'X1-$ 1.2 d I f i X 2 t ~ i: ' __ - -- t * t~~~~~~~~~~~~~~~~~~.... +tT. -.. +. +t'. ' — ' 1.2 '' ' '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i 1 ~14 2.T T -:..:. | |- |i. 7 > r — ~ ~~ ~ ~~ > XT.......+ -/; 1lt t | | I |i |/ | | | |~~~~~~~~~~~~........I ' + + f I I,~' iiiji 1 _t -- e',;_t- H --- —q| - l,, ~~~t i.. -Xfl- -1il - g -- '1 1 —=:::- -i --- -ol V II -~~~~.7.,....,......,.,. 1 l. ~.8t........ i T-, _,____: -1.0 -~.......: — = —: —' — '; — ':: — ~-~ 100: X-'-/1i:..... -I.3 1-. ~-1-1-1141/:... t......:-...l -- ~~~-.6 i= --- ~~ -~ I t - 1.- -.; I '-t — —...I.' f.J; — 1.121 W I; 7 -, t — 10- 10 i~01,B R = Yaar Figue 5 o Correctioni facto~r: n(5-R) to the HoltsmarkZ series for n 9= 23,,4,,,,5o 7..,.,., 7:r c -... Ttf ' ~~ —f~~f~ r-.h..-~~-f- '1-~~I -I-T 'f

150 2, 3, 4, 5. They are defined by 2l ga 11n(5-11.1a) Kn(PR) F= n | for co (ci - (5-11. ob) K^(PR) - cos (3 tg 1 +) n rd -3^ cK,, (R) os 3n ~3~ ~ (R:)O for cos, = 0o The absorption coefficient written in terms of Kn(PR) is a+ g ~'la)" (p2R +1+ I n2 i ( 1 > f rm 1 i \ R +r -2 C 0 co ( XT' Kr,(PR) ~(5~~11.) 2! \ o n- ( \1 0 + Il -5 n~:i - o~ o (L,)) CS l ( ) n ) 0 Ia5()d ~ 2 R- 2 o._ K( + [2 K ( ItR ) ( ~ 5:+L e/ I.... 3309 + 2:10 ( ) ] on1 i, a 2jB RJ. + 2 K pR)35 KPR) 12K (PR) pii/5 /2 PI

131 In the limit of zero electron density these functions are constants and independent of the frequency: Kn(PR) - 1 cos (-n 9) 0 (5-114) \( Kn(PR) O 0 cos (- = 0 From the graphs of the functions Kn(PR) it is seen that they deviate considerably from their Holtsmark values for PR < 100 (or for frequencies measured from the line center that are less than 100 times the electron half-width parameter 7aa). In addition to the correction factors, Kn(BR), to each term in the usual Holtsmark series there is an additional term in the distribution (5-11.2) that depends only on the electron density and has the well-known Lorentzian frequency dependence. This term becomes large on the line wing. In Figure 6 the theory for electron broadening alone, ion broadening alone, and for both electron and ion broadening are compared for the following conditions, Ne = Ni = 1016 cm-"3 T = 15,000~K, and Xau = 10. It was found that the series expansion (5-11.3) for 3 2 5 was accurate to three significant figures. For P < 5 the absorption coefficient was calculated by numerically integrating (5-10a.ll). In these calculations the nonadiabatic contribution was estimated by taking the contribution to /Ya by the weak collisions to be twice the adiabatic contribution (see 5-6.7): (Ya)weak = -2 - e (AXac)2.gn m (5-11.5) (7aa)strong = 2536 [13 N (AXa)2 7aa = (Yaa)weak + (7aa)strong *For Lyman o, the half-width was three times that predicted by the adiabatic theory for electron broadening and it is to be expected that the adiabatic and nonadiabatic contribution to the electron broadening for other hydrogen. lines will also be comparable.

152 ~~~~~m0 d-0 0 l -o N rOU )) - *1 a c) '-'" a) ~0 H CH L. m - c,c 44~~~~ -. E ~ ) ~rd 0o~~~ 0 0 c o o.~. 3 -P 7E C.)\ /,/ 0 0 0 0 0 0 0 0 0 0 0) 00 o OP (60.1

1553 The detailed computation of the nonadiabatic contribution to gaa for hydrogen lines other than Lyman a has not yet been completed. It is evident from Figure 6 that the half-width predicted -by our theory differs only slightly (21 percent in this case) from that predicted by Holtsmark, even though the frequency distribution is greatly modified by including electrons in the theoryo For this reason the experimental determination of the Balmer line half-width is not a sensitive measure of the validity of either theory. Note also from Figure 6 that the position of the Holtsmark maximum does not depend on the electron density. This behavior is borne out experimentally. In Figure 7 an experimentally determined profile* of Ho is compared with the Holtsmark theory. The qualitative features (Figure 6) of our theory agree well with the qualitative features of the experimental profile. In experimental, applications, therefore, the measurement of the frequency splitting of the double maxima in HE or Ho affords an easy method for the estimation of ion densities in a high-temperature plasma. The importance of electron broadening can also be demonstrated by examining the limit of (5-10c.7) for large frequencies: With P `)aa and R - a/yaa/, this equation can also be written baa 2 ~12.707 r au / 2 (5-11.6b) Ia(NUa)daDaU )da) ( W 0 )dAa - iakn ) g 17Yac Rc aa *Taken from page 157, Ref. 1.

134 0 ro U Ilr I (D:4 r dI IN ~w3;tOS 00 //' 0 I.- o0 rE / 0) )~// u -.I — ~ < fi a) ~~~~ 0 i~04 /o f 00 — T —~ —~ —~ —t. - ~ ^- ro N c - C)0(T a N- i - ro N - (39TVDS A1VyLlSyV) A1ISN31NI (3TtVOS A8V~tile8V) k lSN3iNI

135 In this formula the expression for a7a will also be taken to 'be that given by (511o5) in order to take the nonadiabatic broadening into account approximatelyo With the following relations obtained earlier,.3/2 4N ( )3/2 ax = 4.21 Ni (AXa )3/ 7aao 16 Ne (AXaO)2 (2.36 - 2 n m ) 3 Ve (51-7) 'ac = 1 80 x 10-6 Xa T V7e = 6.215 x 105 NTJ 5 e2 ao A = one fhes dtr to e on the far wing) becomes (5-ll.8) I (jAo~ ) " 2 aPc a2u 71a06 _Ni T i511.8) Ia(^C ) - 2 l~~~ae aM*i A9,9 a 9 &D~uo 2 ) =1,, 2a I (~O)2 LaWNe (Xao Aa) )b(2,36.,n,) The first term in the above expression depends only on the electron density, The second term is the ratio of the wing intensity due to ion broadeTirg to the wing intensity due to electron broadening. Note that for given frequency and temperature this ratio is independent of the ion -au density when Ne = Ni, except for the slowly varying dependence of ~n 6m on Neo This is of importance in the application of this line broadening theory to the study of Balmer line profiles in stellar atmospheres since the wing distribution can generally be used with s-fficient accuracy to calculate the profileso For given Aua)~ and temperature the relative im

1.36 portance of ion and electron broadening is; essentially independent of the ion densityO Therefore, even for the low ion densities encountered in stellar atmospheres, one cannot neglect the electron broadening compared to the ion broadening in calculating the shape of the wide Balmer lines that are observed. The general impression that one gets from a survey of the astrophysical literature is that the electron broadening can be neglected at low ion densities. This conclusion is not justified by the calculations presented in this dissertationO For example, for a gas whose temperature is 15,000~K the second term in (5 11o8) is about lo5 0 at 30A from the line center for the H Balmer line. Therefore, the ion and electron broadening are comparable under these conditions for this lineo For the higher series members of the Balmer series the XaU (see Table I) are larger than for H7. Therefore, the electron broadening becomes more important since Xaa appears in the denominator of (5-11.8), which involves the ratio of ion to electron broadening on the line wingo It can be seen from the Holtsmark ion broadening theory (5"10c8) -5/2 that the absorption coefficient behaves like Awj on the wing of the line. Therefore, a log-log plot of the absorption coefficient vs frequency should yield a straight line: (5411 9) i(log Ia)ions - 2 log ~aW + constant On the other hand, the theory involving electron broadening alone predicts that the absorption coefficient behaves like Aw2 on the wing and yields a slope of two on a log-log plot: (5all lO) ) (log Iaa)electrons = 2 log Noa a constant t The combined theory of electron and ion broadening predicts that the slope

i57 of log-log plot of the absorption coefficient vs frequency shoeuld be intermediate between 2 and 2.5. This behavior is borne out experimentally 64 and was also indicated by work of Miss Underhill, who studied the frequency dependence of Balmer line wings observed in Bttype stars. In these studies it was found that the slope was generally near 2 and in no case greater than 253. From these observations it was concluded that the Holtsmark theory does not adequately represent the process of line formation in B-type starso The astrophysical results are not to be regarded as a proof of the invalidity of the Holtsmark theory, but serve to confirm the discrepancy between theory and laboratory data. 12. Broadening of the Balmer Lines by Nonadiabatic Electron Collisions The theoretical results of Section IIIol5 for transitions between nearly degenerate states are complicated by the presence of exponential factors exp iciabT in the phase integrals Cba (see 3-15.11). In this section criteria for neglecting these exponentials will be obtained. The Cba involve integrals of the type 00 0 (5-12!.1) / dT [Hj(T)] e abT I -00 For the electron broadening of hydrogen lines, Hj(T) is folund from (55-33)~ ear ( 5212 0H2?j ( T) = (V.T2+p2 )3/2 (Pjkj+Vk[lkT) k j substituting this expression for Hj(T) into (5-12.o) yields the following two integrals: m25a G (~avbT (5=12o3a) d e <Loo (vk+pj)P

138 (5-12.3b) dT.r.e- r '-...oo (v T +p~) The second integral is identically zero for 0b = 0. The integral (5-12.3a) has been evaluated by Foley32 in terms of Hankel functions, Hi(iK), of the first order with an imaginary argument (5-124a) f dT -iabT o H (5-12 4a) (V2kT2+p)3/2 = j Vk vk Similarly, the integral (5-12.5b) is given by dT-e ^ -1~i d o P j3a (5-12.4-b) 0/ d 2 Te-i3 = 2 7 Jj b H 1 (] J- k (vkktp3) P vk d" lb Lk For large p jb/vk, the Hankel function becomes* (5-12.5 ) Hl(ie) ~ e; K 1 j-m-U vk:,2 Therefore, the nonadiabatic (off-diagonal matrix elements vanish exponentially with increasing Pjlb/vk. However, for sufficiently high velocities, the collision time, rd ^P/v, is small compared to the rate (b) and one may approximate the exponentials exp leLbT in Cba by unity. If the splitting ab is due to static ion fields, then {,; is of the order (5-12.6) ob Xab = 4.52 XabN 2/3 where again Xab is the Stark shift in units of circular frequency that corresponds to the Holtsmark mean field strength (see 4-1.23). The maxi*Jahnke-Emde, Tables, pages 137-138, Ref. 65.

159 mum value of K for a given density and temperature is of the order (5-12.7a) r = aab Pm = 562 Xab - Ni2/3 (5-12~7a) Km e me -Ni' 7' v" e N' where Pm is the Debye screening length (5-5o5), v is the average electron velocity, me and e are the electronic mass and charge, and Ne and Ni are the electron and ion densities. The Xab can be calculated witli the definition (4-1.6). For equal ion and electron densites, Km becomes -5 1/6 (5-12.7b) Km = 355 x 10 Xab N For N = 1016 ions/cm3, m =.016 Xab. Therefore, K is small compared to unity for Xab = 1-20 and for impact parameters less than the Debye shielding distance. Thus, one may neglect the exponentials exp iabT in calculating the electron broadening of the Balmer lines, This simplifies the numerical work immensely since one must average the line profile due to electron broadening over the static ion field splitting. This would involve averages of the Hankel function with the Holtsmark probability distribution if K were not small compared to unity. 135 Comparison of the Theory with Experiment 0 Am experimental study of the broadening of the Balmer line Hp (4861A) 12 2 was carried out by Turner ' and Doherty in the shock-tube laboratory at The Universality of Michigan. These experiments were described briefly in Cnapters II and IVo The Balmer line Hp was chosen for study for the following reasons: 1.o The intensity of the HE line was high enough so that time-resolved spectra could be obtained, but not so intense as to result on self-absorption. The HQ line was not selected because of self-absorptiono

140 2o The line is isolated from the other Balmer lines so that there is no overlappingo The higher series members of the Balmer series tend. to broaden and overlap at high ion densities. 35 There is no central component for Ho (the central components do not exhibit a static Stark effect and our theory for electron broadening is uncertain at the line center)o 4o There are relatively few Stark components in H, so the theoretical line profiles are comparatively easy to compute 5. The Hp line lies in a convenient region of the spectrum (4861A)! The line shape was calculated with the aid of the series expansion (5-10c.7) for A0a greater than the Hp half-width. For frequencies smaller than the half-width it was necessary to compute the intensity by numerical integrations of the basic integral expression for Ia6( o)) 'The result of these calculations for the wing of HE is shown in Figure 8, where the theory for ion and electron broadening is compared with the Holtsmark distribution for ion broadeningo It is seen that the intensity on the far wings is appreciably greater if the electron broadenilng is taken into accounto The temperatures and ion densities were chosen to conform to typical conditions encountered in the shock tube These calcullations showed that the shape of the distribution is not very sensitive to the temperature In Figure 9, where we have compared the Holtsmark theory with a typical Hp profile obtained in the shock tube,* it is shown that the Holtsmark distribution yields too narrow a liine, while our theory yields an additional broadening that is of about the right order of magnitudeO Further experiments are needed to determine more accurately the wing distri*Private communication with-Eo Bo Turner and Lo Doherty,

141 ic/)~-~ \ \.~ ~ IONS + w ELECTRONS z 0 N- T 5\ \ x 101 Cm'3 11,000 OK HOLTSMARK (IONS) 4\ \ x 10 12,500 8\ x I6 13,500 2 x 1017 15,000 LOG __ LOG n --- Figure 80 Comparison between the Holtsmark theory and the theory for broadening by both ions and electrons illustrating the density dependence~

142 I- L- I 1 THEOCRY - 7 /HOLTSMARK THEORY \^0: = 11,500~ 11. _ __100,10= (9.8~ 15) CM ~ - - 16 -ION AND ELECTRON- / THEORY HOLTSMARK THEORY -xA X (ANGSTROMS) Figure 9o Comparison between the present theory and the Michigan experiments~ and. the Michigan experiments.

143 bution of Balmer lines other than Hi in the shock tube and to correlate this information with the ion densities and temperatures obtained from hydrodynamic considerations. The good agreement between the theory for ion and electron broadening and the existing experimental data is, however, encouragingo It appears, therefore, that serious errors are introduced into calculations of Balmer line absorption coefficients at all densities by neglecting electron broadening compared to ion broadening. 14. Critique; Some Unsolved Problems The present theory of hydrogen line broadening by both ions and electrons is not valid near the line center. There are two reasons for this. The nonadiabatic contribution to the electron broadening has been calculated with a perturbation approximation that is not valid for large times and small frequencies, AM. This approximation restricts the applicability of the theory to frequencies that are large compared to the half-width due to electron broadening. A second source of difficulty in calculating the line profile near resonance is the failure of the statistical theory for small frequencies. Corrections to the statistical theory which take into account the motion of the ions have been calculated by Mrs. Krogdahl.59 However, it now appears that the statistical assumptions that are implicit in the Krogdahl formalism are probably not correcto Furthermore, the Krogdahl approximation was based on the adiabatic assumption. It is not clear that one can neglect nonadiabatic effect3 in calculating corrections to the statistical theoryo The theory for electron broadening also suffers from the ad hoc manner in Twhich the close collisions were treatedo In reality there is a gradual transition between the strong and weak collisionr approximations. A theory which takes this transition region into account properly must

144 avoid the assumption that the amplitudes ICba(t) 12 are small compared to 'unity Another interesting problem is the simultaneous consideration of both the firsts and second-order Stark effects. The quadratic Stark effect will tend to shift the Balmer lines and will probably cause some asymmetry in the frequency distribution.

APPENDIX A DERIVATION OF THE INTENSITY DISTRIBUTION I(to) Let Ho be the Hamiltonian of an unperturbed atom or molecule and let On and E~ be the wavefunctions and energies of the unperturbed eigenstates n of Hoo The 0n and En then satisfy the stationary Schrodinger equation (A1o) Ho n = En n o The Hamiltonian which describes the interaction of the radiating system with the assembly of perturbers is written (Ao2) H(t) = Ho + Hi(t), where Hi(t) contains the position coordinates of the perturbers and is time dependent because of their motion. The Schrodinger equation corresponding to H(t) is (A.3) i-ftn(t) = H(t)fn(t), where the n(t) are a set of wavefunctions which describe the time development of the perturbed system0 A set of initial conditions can now be chosen such that at some time t=to the set of wavefunctions [nL(t)] reduces to the unperturbed set [0n] (Ao4) %-n(to) = n Since the on are a complete orthonormal set of functions, it follows at once that the Y~n(t) also form an orthonormal set of functions~ To show this, consider two solutions, )l(t) andl2(t), which satisfy the 145

146 time-dependent SchrSdinger equation (Ao3)o By the Hermitian character of the Hamiltonian it is clear that (Ao5) it(1,2) = ('T7,H7) - (H,1t 2) = -it(A,) o This is equivalent to writing (A.6) a (Y1 2) = O so that the complete set of solutions which are orthonormal at to stay orthonormal for all timeo Now let us investigate how the perturbations due to the surrounding assembly of particles affect the optical transition probabilities of the radiating systemo The interaction of the radiating atom with the electromagnetic field is (A.7) HI(t) = -. * E cos (ct + a) = -iEt cos e cos (at + a), where ei is the dipole moment operator, Fw is the amplitude of the electric field vectors and a is the phase of the light waveo The total Hamiltonian which now includes the radiating system, perturbing particles, and electromagnetic field is (Ao8) HT(t) = H(t) + H~(t) = Ho + EH(t) + HI(t), and the wavefunction r(t) which describes this system satisfies the timedependent Schrodinger equation (Ao9) inr(t) = HT(t)a(t) o It is now convenient to expand f(t) in terms cf the set of functions* Xn(t)~ *These wavefunctions are referred to as "collision smeared" wavefunctions in the terminology of Bloom and Margenau 13

147 (A.10) 4(t) = An(t)n(t) o n Substitution of this expression for r(t) into the Schrodinger equation (Ao9) yields an equation for the expansion coefficients (A.ll) t) i (H, An(t) = (t) where (HI) - [rjn(t), H (t)m(t)]. The superscript c refers to matrix elements between the "collision smeared" states 'n(t) (not the stationary states On)O An(t) can now be expanded by the well-known iteration procedureo Integrating (All) yields (A.12) An(t) = An(to) + 1 dt Z (H)cm (t1) to Similarly, 1 t) C (A.13) Am(ti) = Am(to) +1 dt2 F (H) Ap(t) Putting Am(ti) into the right-hand side of (A012) gives directly An(t) = An(to) + i dtl [H(tl)]c A(t) to (Ao 14) 1 t1 rF~t~c [HA(t2)c + " ( iS t mdtlZ [ d tI2j [ (t) 2)] Ap(t2) 2 Mih72O P MP to 0 Let A(ts) and Hj(t ) be matrices whose elements are An(ts) and [He(ts)]cm respectivelyo In matrix equation form (A.14) becomes (t A(t) dt H(tA(t) + dt H(t )A(to) to (Ao15) + 1 f dt1 1 dtz H(t)HQ(t2)A(t2) )to to

148 Repeating this iteration procedure finally yields 00 (Aol6) A(t) = [1 + E Rs(t)]A(to) s=l or 00 An(t) = An(to) + Z Rs (t) Am(t), s=l m nmn where the time-development matrix [1 + Z R (t)] is defined by s (A17s) + R t) t t s tli c (A 7) 1 R (t) = 1 dts dt[H(ts)... H(t) to t to The probability that at time t the radiating system is in a collision smeared state described byln(t) is given by the absolute value squared of the quantum mechanical amplitudes An(t). From (A1o6) these probabilities are 2 A n(t)2 = I IAn )(to)) l+ ns(t)) n~ttil 0 s n~to[ nn nn + Z lAM(t o)12 RnI (t) Rr*(t) m 0 rs nm Ao l8) + m (to)A(to) (t) +Z 7Z A*(t )An(t ) R)s*(t) s m m o o + A Z A*(to)Am(to) R (t) R (t) m p P rs F The first sum in this expression can be simplified by making use of the unitary property of the time-development matrix [1 + Z RS(t)]: s [1 + RS(t)]t (1 + R) = 1, s r so that (Aol9) (Rt + RS) = -ZR R o s rs Then, for the diagonal elements one has

149 (A.20) s(Rt + R ) = R Rhmn S nn!~ nn nm or (A. 21) Z(Rt + R) = -Z R s rs m: mnf mn Using this relation (Ao18) becomes iAn(t)12 _ i( Z An (to ) 12 E Rs (t) Rr*(t) m o rs nm nm (Ao22) -E IA (t )12 RS*(t) Rr (t) m n rs mn mn + nondiagonal terms in A*'(o)Am(to)o Now RS(t) is of order s in H2(t) according to (Ao17), so it is also of order s in the electric field strength E.o Therefore, up to order Er, (Ao22) involves only Rh(t) and reduces to iAn(t)|2 |An(to)12 = IAm(to)12 IR'(t)12- IA(t )12R1 (t)12 (Ao23) -+ Z [A(to)Am(to)R1m(t) + An(to)An(t,)Rm(t)] nm o nm +Z Z A*(to)Am(t )iRl |2 m p p o nm With the usual assumption that the phases of the qiuantum mechanical amplitudes Am(to) of the unperturbed eigenvalues of Ho are randomly distributed in a thermal ensemble, it follows that the density matrix An(to)Am(to) nm(to) is diagonal when averaged over the ensemble (Ao24) Pmn(to) = Pn(to) omn Anderson, Bloom, and Margenau et alo take Pn(to) to be the Boltzmann distribution corresponding to the unperturbed eigenstates of H0:

150 E0o e -E/kT (A.25) Pn(to) = Z(T) Z(T) e This assumption leads to certain difficulties which are examined in Chapter III, Section 8. With the assumption of random phases, the transition probabilities per unit time are now (A.26) An(t) 12 - |An(to) 2 1 E [P(t o) ] jRn(t) 2 t-t= t-t'o m where, according to (A.17) and (A,7), R1m(t) is given by Jto where [iC(t) = (I(t),7m(t)). Let us now fix our attention on a particular term in the expression (A.26) for the transition probabilities. For matter in thermal equilibrium with a radiation field, the averaging process indicated by the double bar includes an average over the random phase angle a, the amplitude squared of the electric field vector E.,2 and the square of the direction cosine, cos2G. This averaging process yields Now for a thermal radiation field the energy density B,(T) is given by (A29) 0 (T = IIt2 + l _ l2 I l2 Bp cos2(Qt+a) (A.29) BJT)=4 - 4 8= i where Be(T) is given by the Planck distribution,

151 (A. 30) B (T) = t3- k 2c3 (exp kTi ) The total transition probability, using equations (A.29), (Ao30), (A,28), and (Ao26), is then IAn(t)12 - |An(to)2 _ 2 o)3 1 1 t-to 3 ItC3 lo (t-t ) - c3 c (exp - 1) L 0 Pto (Ao31) Zp t p dt1[r..m(to) e 0ic - ct dtl~Cm(tl) e i7. + ~ _ eti jAvg The energy radiated and absorbed per second I( ) due to transitions which populate and depopulate the state n is given by the product of the photon energyfiw and the total transition probability (A531) after a time long enough so that a transition has taken place (Ao32) I(c) = to lim 1 [jAn(T)2 - |An(0)|2], T+oo T or _ W3)4, PXC3 (dholkT1)3~sre1 C" I~,.,,,, lim 1 ( [Pm(O) Pn()] T+ T 35Tc3 (e-/kT~) (Ao33) r - c - ilt /p c +igt i (t) e dt + cm( e dt|j nm Avg Avg

BIBLIOGRAPHY 1o Eo Bo Turner, The Production of Very High Temperatures in the Shock Tube with an Application to the Study of Spectral Line Broadening, AFOSR TN 56-150, ASTIA Document No. AD 86309, Univ. of Mich. Engo Res. Inst. (1956); Dissertation, Univo of Mich. (1956) 2. Eo Bo Turner and Lo Doherty, Astrono Jo, 60, 158 (1955). 35 R. N. Hollyer, Jro, A. C. Hunting, Otto Laporte, E. H. Schwartz, and E. B. Turner, Phys. Rev., 87, 911 (1952)o 4, R. N. Hollyer, Jr., A. C. Hunting, Otto Laporte, and E. B. Turner, Nature, 171, 395 (1953). 5o Jo Holtsmark, Physiko Z,, 20, 162 (1919); ibido, 25, 73 (1924); Anno der Physik, 58, 577 (1919); ibid., 25, 73 (1924). 6. S. Verweij, Publ. Astron. Insto Amsterdam, No~ 5 (1936). 7. P. Schmaljohann, Unpublished State Examination Work, Kiel,1936. See Ao Unsl1d, Physik der Stern-Atmospharen, Berlin, 1938. 8. G. I;ste, J. Jugaku, and L. Ho Aller, Publo of the Astron. Soc. of the Pacific, 68, 23 (1956). 90 Lo Spitzer, Jr., Phys. Rev., 55, 699 (1939); ibid.o 56, 39 (1939); ibi do., 348 (1940)o 10. A. Uns'old, Vjschr. Astron, Ges, 78, 213 (1943). 11. H. Griem, Z Physik, 137, 280 (1954)o 12. D. R Inglis and. E Teller, Astrophys. J, 90, 439 (1939). 135 So Bloom and Ho Margenau, Phys. Rev., 90, 791 (1.953) 1.4 Po Wo Anderson, Phys. Rev., 76, 647 (1949); Dissertation, Harvard Univo (1949), 15. Bo Kivel, So Bloom, and Ho Margenau, Physo ReVo, 98, 495 (1955). 152

153 BIBLIOGRAPHY (Continued) 16. D. Bohm and L. H. Aller, Astrophys. J,, 105, 131 (1947). 17. R. G. Fowler, J. S. Goldstein, and B. E. Clotfelder, Phys. Rev., 82, 879 (1951). 18. R. G. Fowler, W. R. Atkinson, and L. W. Marks, Phys. Rev., 87, 966 (1952). 19. R. G, Fowler, W. R. Atkinson, B. E. Clotfelder, and R. J. Lee, Research on Radiation Transients in Gas Discharges, Univ. of Okla. Res. Inst., Final Report to Office of Naval Research, Project NR 072-221/ 9-21-49, Contract No. N9 onr 97700, Univ. of Okla. Res, Inst. (1952). 20. W. Atkinson, One-Dimensional Fluid Flow Produced by Confined Sparks, Technical Report to Office of Naval Research, Project NR 061 087, Contract Nonr 982(02), Univ. of Okla, Res. Inst. (1953). 21. B. E. Clotfelder, Experimental Studies of Transport Phenomena in Highly Ionized Gases, Technical Report to Office of Naval Research, Project No. NR 061 087, Contract Nonr 982(02), Univ. of Oklao Res. Inst. (1953); Dissertation, Univ, of Okla. (1953). 22. W. R. Atkinson and W. R. Holden, Ionized Gas Flow in Electrically Energized Shock Tubes, Technical Report to Office of Naval Research, Project NR 061 087, Contract Nonr 982(02), Univ. of Okla. Res. Inst. (1954). 235 A. C. Kolb, Naval Research Laboratory, Unpublished (1956). 24. G. E. Seay and L. B. Seely, Jr., Bull. Am, Phys. Soco, II, 1, 227 (1956). 25. H. Maecker, Z. Physik, 129, 108 (1951)o 26. W. Lochte-Holtgreven and W. Nissen, Z. Physik, 133, 124 (1952), 27. G. Jurgens, Z. Physik, 134, 21 (1952). 28. Th. Peters, Z. Physik, 135, 573 (1953). 29. W. D. Henkel, Z. Physik, 137, 295 (1954). 50. W. Nissen, Z. Physik, 139, 638 (1954).

154 BIBLIOGRAPHY (Continued.) 31o Ao Co Kolb, A preliminary account of this research was presented to the American Physical Society, Bullo Am, Phys o Soco II,, 22 (1956) and to the American Astronomical, Society, Astron Jo 60, 6 1.66 (1955), 352 Ho M, Foley, Phys Revo,, 69, 628 (1947); Dissertation, TJnivo of Micho (1942)o 335 Ho Margenau and. Ro Meyerott, Astrophyso, J,, 121, 194 (1955)Q 354 E o Schrodi:nger, An?1 der Physik, 80, 457 (1.926 ) 355 Po So EIptein, Py o Re vo 28, 695 T(926? ) 360 Eo Uo Condon and Go Ho Shortley, The Theory of Atomic Spectra, Cambridge tUniv.o Prebs (195.11)o 57 io Lo Landau and, Lo Lifshitz, The las i- Theo.ry of Field.s pp 125, 195, Addidson-'Wesiley PreSs - Ino, Cambridge, esSO (1951) 58. 0o Kelen, Py, Zi Py 41, 207 (1927)o 59, W'o Pauli, '-. dbiluclh der PLys:ik 2ri d E, VJol. 2, Part 1, Jo Ho Edw'ards, Azin Arbor `iL946 ) 40o Jo Ho Van V iek and Hio Margenau P. y o P Rev o 76, 1211 (1949)o 4. Ho Kumh and F' London, Phil, ag, 1go, 8, 983 (-954); Ho Kuh:n, Proc. Roy o coc, A18, 987 (193 4) 42 Ho rgenau, Pyo Rev o, 82., 156 ( 951) ibid 40, 87 19)2) ibido, 487, 755 (i935)j ibido,O 76, 121. (1i949) 4 H Margenau and WO W, Watson, Rev, Mod, Phys, 8, ~22 (1956 ) 44 E.o 'Lind.ho.lm, Arkivo f Fy,_o, 3 A, No o 17 (1I 945) 45. A. Ao Markoff WahLrscheir lich keitsrecir.mg, elipzi.g (1914)o 46 O C:handra'Sektar, Rev, Modo Phys 15, 753 (1945 )O 470 To Hoietein, Phy.so Revo, 79, 744 (1.950)o 48. Jo van Kranendonk, Dissertation, UJniv o of Amsterdam (1952),

155 BIBLIOGRAPHY (Concluded) 49~ Ho Ao Lorentz, Proc, Amsto Acad,, 8, 591 (1906); The Theory of Electrons, Teubner, Leipzig (1916). 50. Jo Ho Van Vleck and Vo Fo Weisskopf, Rev. Mod. Physo, 17, 227 (1945)o 51. Jo Stark, Be.rl Akad. WissO, 40, 932 (1913)o 52. Oo Struve, Astrophyso Jo, 69, 173 (1929); ibid.,,2 85 (1929). 553 Co To Eivey, Astrophys. J, 69, 237 (1.929). 54. G. Jt Odgers, Astrophys JO, 116, 444 (1952 ) 55. Ao Underhill, Astrophys J., 116, 446 (1952). 56. Lo H Aller, Astrophys. J., 109, 27 (1949). 57 o Jo McDonald, Publo Dominion Astrophys Obs., 9, 269 (1953). 58. Co de Jager,?Utrecht Researches, 13, 1 (1.952 ) 59. Mo Ko Krogdahl, Astrophys. J,, 1.L10, 355 (1949). 60O, Oniv:ersity of Pittsburgh Conference on Line Broadening (Sept. 1955), 61. V. Weisskopf, Physkik., Z 54, 1 (19533)o 62. E. Schrodinger, Arin der Physik, 80, 457 (192,6), 635 Lo Spitzer, Jro, Ro So Cohen, and P, McR, Routley, PLhySo RevO, 8L, 230 (1950)o 64 Ao UJnderhill, Astrophys o J, 107, 349 (1948 ) 65~ Jahnmke-Emde, Tables of Functions, Dover Publications, New York (1945) 66. W. Grobner and No Hofreiter, Integraltafel, Springer-Verlag (1949),