4. Atomic Processes

4. ATOMIC PROCESSES" 4. I. Introduction While a plasma is often thought of and defined in a pure sense as a quasineutral ensemble of electrons and completely stripped nuclei (such as protons), there is of course no such ideal plasma fulfilling this definition. First of all, 100% ionization is an asymptotic approximation, and second, there continually exist with some orbiting electrons heavy impurity ions that exhibit atomic bound state characteristics and undergo atomic processes. Also, most laboratory produced plasmas have a beginning as a neutral atomic or molecular gas and they exist in various ionic states for finite periods of time until the ultimate degree of ionization is achieved. It is the purpose of this part to provide guidelines and examples for determining the dominant atomic processes occurring in and affecting a particular plasma. T h e most commonly used models for analysis are described in Section 4.2 and the methods of analysis leading to a knowledge of either plasma parameters or basic atomic information (depending on the point of view) are exemplified by a plasma generated in a theta-pinch device. Such a plasma has cylindrical and near-axial symmetry and has served to provide basic data on atomic processes occurring in trace amounts of elements present. Such data are of wide interest to astrophysicists as well as plasma physicists. The importance of a knowledge of the correct velocity distribution function (usually assumed to be Maxwellian) for the plasma particles and the experimental methods available for measuring this function are discussed in the same section. It is the presence and interaction between ions, neutrals, electrons and photons that lead to the atomic processes which both affect the plasma and provide information on the plasma state. T h e general processes involved are excitation, deexcitation, and ionization of atoms, recombination of free electrons with ions, and transitions of free electrons between continuous energy states. T h e perturbing effect of neighboring particles, manifested through the 'broadening of energy levels and spectral lines, is another process which may be important. I n Section 4.3 these broad categories are subdivided according to specific mechanisms and even finer distinctions are made in some cases, e.g., between ionization processes

* Part 4 by R. C. Elton. 115

116

4. ATOMIC

PROCESSES

involving ions (or atoms) in a relaxed state and those in excited states. In this delineation, the process rate is used wherever possible as a basis for comparison. 4. I. 1. Emphasis

In this part an attempt has been made to emphasize methodology in contrast to technology which has been adequately covered in other volumes (see Section 4.4.8).There would of course be no need for either if the effects of the atomic processes were not observable, and it is just such effects that have become powerful diagnostic methods for studying high-temperature plasma. This application (diagnostics) is summarized in Table I. As knowledge of atomic processes is extended to the ionic phenomena occurring in high-temperature plasmas, the inadequacies of the existing TAULE I. Plasma Parameters Measurable from Atomic Processes Parameter" Process

EED IED E T

Line emission

x

Continuum emission

X

IT E D

IL

MF EL C T

x

x

X

X

x

x

x

x x

Spectral line shapes Spectral line shifts

x

Spectral line reversal

X

x

x

x

X

x

x

Zeeman effect

X

Faraday rotation

X

Particle analysis

x x

Radiative absorption

Light scattering

ID

x

x

x

x

x

x

Key: EED: electron energy distribution; IED: ion energy distribution; E T : electron temperature; IT: ion temperature; ED: electron density; ID: ion density; IL: impurity level; MF: magnetic fields; EL: energy losses; C T : confinement times.

4.1.

INTRODUCTION

117

theories (as usually derived for neutral atoms) and the insufficiencies in experimental data become apparent. I t is here that the plasma spectroscopist is able to contribute to the knowledge of basic physicaI properties through the determination of rate coefficients, oscillator strengths, line shapes, etc., using the plasma as a source of known or independently determined properties. The methods involved here also constitute a major emphasis of this part. There is no doubt that the major stimulation of atomic processes in plasmas is due to collisions of charged particles rather than radiation. Thus collisional excitation and ionization, followed by radiative and collisional recombination and decay, are the processes most responsible for controlling the densities of ionic species, the population of bound states and the type of equilibrium obtained. It is, therefore, these processes which will be emphasized in the following sections. Other processes, of less significance to the analysis of high-temperature plasmas, are included in less detail. (It is hoped that an abundance of references will compensate for any incompleteness due to the condensation necessary in some areas.) 4. I .2. Limitations

The initial ionization of a neutral gas is generally easy to achieve in the laboratory and, when there are sufficiently strong electric fields present, the breakdown process is often adequately described as a Townsend avalance initiated by radiative or collisional ionization processes. When only low-energy photons are present with insufficient energy to produce ionization in a single interaction (as in the case of laser-induced spark discharges), more complicated mechanisms such as multiple photon absorption must be invoked to understand the breakdown process. Since the present part is restricted to a discussion of atomic phenomena playing a significant role in the heating, confinement and analysis of plasmas, the details of the breakdown phase will not be further pursued. Likewise, molecular processes will also be omitted from the discussion. Finally, it might be noted that most of the formulas given in the following sections for estimating the process rates were originally derived for the one-electron hydrogen atom (and hydrogenic ions) for which the theory is most precise and for which experimental results most often exist for comparison. Wherever possible, the formulas have been generalized by the insertion of actual excitation and ionization energies and the use of the effective charge z to replace the nuclear charge 2. For example, X’X, -+ x and E, = XHlnZ--t x1x2n2 = E, where x and E are the ionization and binding energies, as generalized from the hydrogen values, xH and E H*

118

4. ATOMIC PROCESSES

For processes involving only higher bound and free states (which are nearly hydrogenic anyway) appreciable reliability can be expected within the limits of precision of the original formulism. For processes involving lower energy states, semiempirical methods are sometimcs available, such as the use of effective Gaunt factors obtained from experimental comparisons. Caution is in order in applying these formulas to general cases; nevertheless, those given are considered sufficient for first estimates, beyond which the references cited should be sought for limitations in applicability and for more detailed analytical methods. 4. I .3. General References

The subject of collision processes is covered by Bates,’ Bederson and Fite,’ Massey and B ~ r h o pMott , ~ and Massey4 and Hasted.s The relevant atomic spectroscopy is given by Bethe and Salpeter,6 Bond et ~ 1 . Born , ~ and Wolf,* Candler,’ Condon and Shortley,” EdlCn,” Griem,” Heitler,13 H e r ~ b e r g , ’Kuhn” ~ and Shore and Menzel.I6 D. R. Bates, ed., “Atomic and Molecular Processes.” Academic Press, New York, 1962. B. Bederson and W. L. Fite, eds., “Methods of Experimental Physics-Atomic Interactions,” Vol 7. Academic Press, New York, 1968. H. S. W. Massey, E. H. S. Burhop and H. B. Gilbody, “Electronic and Ionic Impact Phenomena,” 2nd ed. Oxford Univ. Press, London and New York, 1969. N. F. Mott and H. S.W. Massey, “The Theory of Atomic Collisions.” Oxford Univ. Prcss, London and New York, 1965. J. B. Hasted, “Physics of Atomic Collisions.” Butterworth, London and Washington, D.C., 1964. H. A. Bethe and E. E. Salpeter, “Quantum Mechanics of One- and Two-Electron Atoms.” Academic Press, New York, 1957. Published originally in “Handbuch der Physik” (S. Fliigge, ed.), Vol. XXXV. Springer, Berlin, 1957. J. S. Bond, K. M. Watson and J. A. Welch, “Atomic Theory of Gas Dynamics.” Addison-Wesley, Reading, Massachusetts, 1965. * M. Born and E. Wolf, “Principles of Optics.” Pergamon Press, Oxford and New York, 1957. C. Candler, “Atomic Spectra.” Van Nostrand, Princeton, New Jersey, 1964. l o E. U. Condon and G. H. Shortley, “The Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York, 1935. l 1 B. EdMn, in “Handbuch der Physik” (S. Fliigge, ed.), Vol. XXVII, p. 81. Springer, Berlin, 1964. l 2 H. R. Griem, “Plasma Spectroscopy.” McGraw-Hill, New York, 1964. l 3 W. Heitler, “The Quantum Theory of Radiation,” 3rd Ed. Oxford Univ. Press (Clarendon), London and New York, 1954. G. Herzberg, “Atomic Spectra and Atomic Structure.” Dover, New York, 1944. K. G. Kuhn, “Atomic Spectra.” Academic Press, New York, 1962. l 6 B. W. Shore and D. H. Menzel, “Principles of Atomic Spectra.” Wiley, New York, 1968.

4.2.

EXPERIMENTAL METHODS OF PLASMA ANALYSIS

119

Spectroscopic methods are described by Harrison et al.,” Samson” and Sawyer,” as well as in Volume 4 of this series” on methods in atomic and electron physics. Methods for optical diagnostics of plasmas are included in Part 11 of Volume 9B, as prepared by Jahoda and Sawyer. Plasma diagnostic techniques as distinguished from methods are intentionally omitted here and are described by Griem,12 Huddlestone and Leonard2’ and Lochte-Holtgreven.

4.2. Experimental Methods of Plasma Analysis This section is devoted to a discussion (with examples) of: (a) effects of atomic processes which are observable in plasmas, (b) how such observations can be utilized for analysis in terms of specific plasma parameters and (c) what information concerning fundamental atomic processes can be obtained from a plasma which has been well diagnosed by other means. The important equilibrium models which have evolved are reviewed and the formulation of the rate equations required for certain models is discussed. Examples of the usefulness of such an analysis in determining heating rates as well as radiative energy losses are given. Finally, the dependence upon a particular particle velocity distribution function is discussed. 4.2.1. Rate Coefficients in Plasmas

Fundamental to most of what follows in this part are the reaction rates for the important atomic processes. For spontaneous transitions these are given simply by the transition probability. For collisional processes, the rates are dependent upon the densities of the colliding species. Consider binary reactions in a unit volume of a system containing N , particles (or photons)/cm3 of one species moving with a constant velocity o in a stationary lattice of particle density N,. If o(o) is the cross section for a particular reaction at this relative velocity, I’G. R. Harrison, R. C. Lord and J. R. Loofbourow, “Practical Spectroscopy.” Prentice-Hall, Engiewood Cliffs, New Jersey, 1948. I s J. A. R. Samson, “Techniques of Vacuum Ultraviolet Spectroscopy.” Wiley, New York, 1967. R. A. Sawyer, “Experimental Spectroscopy.” Dover, New York, 1963. 2o V. W. Hughes and H. L. Schultz, eds., “Methods of Experimental PhysicsAtomic and Electron Physics,” Vol4. Academic Press, New York, 1967. z 1 R. H. Huddlestone and S. L. Leonard, eds., “Plasma Diagnostic Techniques.” Academic Press, New York, 1965. 22 W. Lochte-Holtgreven, ed., “Plasma Diagnostics.” North-Holland Publ., Amsterdam, 1968.

4.ATOMIC

120

PROCESSES

the number of target particles per centimeter susceptible to reaction is given by N , o . Since the distance traveled per second by all projectiles in a unit volume is N l v , the reaction rate/cm3 is given by the product ( N 2 0 ) ( N l u )When . a distribution of relative vclocities (often assumed to be Maxwellian) is present, as in a high-temperature plasma, the product au must be averaged as ( a u ) , which is known as the rate coefficient for the process. 4.2.2. Statistical Models

The method of analysis chosen to describe the distribution over atomic states of a particular plasma depends upon the most appropriate equilibrium model. These models and the regions of applicability are described in detail e l ~ e w h e r e ’ ~and ~ ~ will ~ - ~be ~ outlined only briefly here for guidance and completeness. 4.2.2.1. Local Thermodynamic Equilibrium (LTE) Model. When a plasma is in local thermodynamic equilibrium there exists a unique temperature which determines the velocity distribution function for the species with the dominating reaction rate (usually the electrons). If such equilibrium exists, the analysis of the state of the plasma is particularly simple, since it is only such local plasma parameters as electron density, electron temperature and composition that determine the relevant populations. Statistical mechanical relations and a knowledge of the transition probabilities suffice to analyzc the emitted atomic radiation. When LTE is known to exist for the electronic bound states, the population densities N , and N b of any two energy states a and b of energy E, and E, respectively are related by the Boltzmann formula

N,/Nb

=

(w,/wb)exp( - A E a b / k T ) ,

(4.2.1)

where AE,, = E, - E,; w , , are the respective statistical weights, k is erg/”K) and T is the electron the Boltzmann constant (1.38 x temperature. For any level 6 , the population density Nb is related to the total density N by

(4.2.2) 2 3 L. H. Aller, “Astrophysics-The Atmospheres of the Sun and Stars”, 2nd Edition. Ronald Press, New York, 1963. 24 G. Elwert, 2. Natuvforsch. 7a,432 (1952). 2 5 R. W. P. McWhirter, Spectral intcnsitics. In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S. L. Leonard, eds.), p. 201. Academic Press, New York, 1965. 2 6 R. Wilson, J. Quant. Spectr. Radiative Transfer 2,477 (1962).

4.2. EXPERIMENTAL

METHODS OF PLASMA ANALYSIS

121

where the subscript 1 refers to the ground state and P(T ) is the partition f ~ n c t i o n ’3~* 2*7 ~ w j exp( -AE,,/kT).

P(T) =

(4.2.3)

j

Also, the densities of successive ionic species are related through the Saha equation (neglecting certain high density corrections’ 2 , 2 7 * 2 81 exp( - xz - /kT ) = 3.0 x 10’’

2P’( T ) ----(kT)3/2 Pr-’( T )

exp( - x Z - l / k T )

(~m-~), (4.2.4)

for kT in eV (1 eV = 1.6 x lo-’’ erg) and where c( is the degree of ionization N’/(N’ + NZ-’), x Z - l is the ionization energy, z (superscript) is an index designating a particular ionic species, h is Planck’s constant, rn is the electron mass and N , is the electron density. T h e Saha function has been tabulated by Drawin and F e l e n b ~ k , ’and ~ AllerZ3has also listed partition function ratios of astrophysical interest. The Kirchhoff relation for the emission density E , and the absorption coefficient K , 8, = @“(TI, (4.2.5) also holds for plasma in LTE, where By(T ) is the Planck function. L T E CRITERION. T o obtain local thermodynamic 4.2.2.1.1. TOTAL equilibrium, the reverse of all fast processes must be maintained and exact balancing of total rates for complimentary processes must be allowed to take place. Also, the relaxation times (reciprocal of the rates) for the important processes must be shorter than the characteristic times for significant variations in local plasma conditions. Since most plasma of interest are optically thin to internal radiation (except perhaps for the resonance lines), collisional processes are usually more important in establishing LTE than radiative processes. Consequently, collisional deexcitation rates must exceed radiative decay rates for true LTE.29,30 Thus, at a sufficiently high electron density and a sufficiently small energy separation between levels, collisional L T E can be achieved, at least for higher bound states. With an assist from radiative trapping in the

’’ ’’

11. W. Drawin and P. Felenbok, “Data for Plasmas in Local Thermodynamic Ecluilibrium.” Gauthier-Villars, Paris, 1965. H. R. Griem, Phys. Rev. 128,997 (1962). *’ H. Van Regemorter, Astvophys. J . 136,906 (1962). 30 H. R. Griem, Phys. Rev. 131,1170(1963).

122

4.

ATOMIC PROCESSES

resonance lines, the minimum level for which LTE can be assumed to exist can be lowered, i.e., the minimum electron density required for complete L T E in all levels may be reduced. A necessary (although not sufficient) criterion for the existence of LTE in a steady, homogeneous plasma has been derived"S2 5 , 2 6 * 3 0 as

N , ;1- C(kT)'I2x3,

(4.2.6)

where C is a constant equal approximately to 1.4 x l O I 3 assuming complete trapping of the resonance lines and to 1.4 x loi4 assuming no trapping whatsoever.12 [Here, kT and x (the ionization potential) are in electron volts.] Griem" has given an additional criterion for determining when the assumption of resonance radiation trapping is valid (assuming Doppler broadening to dominate and no multiplet structure which would reduce the trapping effect)

Nd 21.1 x 1 0 ~ ~ ( k T / p ) ~ / ~ /(cm-2). f3L

(4.2.7)

Here d is the layer thickness in cm, KT is in eV, p is the mass number of the element in question, A is the wavelength of the resonance line (A), N is the density of absorbing ions ( ~ m - and ~ ) f is the absorption oscillator strength for the transition. PLASMA CRITERION. For transient 4.2.2.1.2. 'TRANSIENT HOMOGENEOUS homogeneous plasmas, a necessary criterion' ' v 3 O for the existence of L T E is that the relaxation time for the slowest process must be shorter than the characteristic time for significant variations in the plasma parameters; e.g., the local electron temperature. The longest such relaxation time for a particular ionic species is usually (N,X12 ) - 1 , i.e., the reciprocal of the collisional excitation rate from the ground state 1 to the first excited state 2 [with XIz given in Eq. (4.3.5)]. This criterion often leads to marginal L T E in pulsed laboratory plasma, at least for low lying levels. This estimate again is somewhat pessimistic, since the excitation rate may often be increased through resonance absorption of external radiation from optically thick lines. Also, not all ground state electrons need be excited (or ionized) to reach LTE, i.e., the relation (N,Xl 2)-1 should be multiplied by the fraction (at equilibrium) W + ' / ( N 2+ W " ) for ionization from state z to x + 1. For higher levels the relaxation times given by Eq. (4.3.5) are very short, so that for excited states partial L T E is rarely, if ever, limited by this transient criterion. 4.2.2.1.3.INHOMOGENEITY CRITERION FOR STATIONARY PLASMAS. When inhomogeneities exist (such as in stationary arcs) and are of such magnitude that the local plasma conditions change significantly in distances comparable to the diffusion lengths of the neutral atoms or ions during characteristic relaxation times, the local conditions are no longer well defined.12

4.2. EXPERIMENTAL METHODS OF PLASMA

ANALYSIS

123

The criterion for establishment of LT E here requires that spatial variations in plasma temperature over a diffusion length be very small. 4.2.2.2. Corona Equilibrium Model. When electron densities are too low for the establishment of LTE, it is still possible to obtain equilibrium whereby the collisional excitation and ionization which does occur (radiative excitation and ionization still being negligible for optically thin plasmas) is balanced by radiative decay and recombination respectively." , 2 4 - 2 6 , 3 1 Providing the criteria for transient and inhomogeneous plasmas given above for L T E are fulfilled (if relevant), the fractional ionization is now given by the rate equation for the population density N" for species z dN"ldt = N""N,R - W N , I (4.2.8) which for a steady-state plasma is identically zero yielding a formula analogous to the Boltzmann relation

N'"lNZ

=

IIR.

(4.2.9)

Here I and R are the ionization and total recombination rate coefficients given below in Eq. (4.3.25)and Eqs. (4.3.35)through (4.3.43))respectively. For the processes considered, this ratio is independent of electron density; and the dependence upon temperature provides a convenient method for estimating the electron temperature in a plasma from the population ratios of ionic species (see Section 4.2.3.1). In contrast to the L T E model, the corona model depends critically on a knowledge of the rate coefficients. For bound states of a particular species, the primary process balance may be expressed by

N,N,X,,*

=: A,*,N,,,

(4.2.10)

where 111 designates a further excited state (relative to n) and A,,, is the transition probability for spontaneous radiative decay from state n* to n. The emission rate per unit volume b,,,, from an optically thin line follows as (4.2.11) b,., = N,N,X,,*h v,;, , with v,," the frequency of the emitted radiation. 4.2.2.3. Modified Corona (Collisional-Radiative) Model. For higher excited states with closer spacings, collisional deexcitation becomes more important and radiative decay less, with the effect that the pure corona equilibrium model is not always strictly applicable. In this regime, intermediate between the corona and L T E models, the so-called collisional31 R.v.d.R. Wooley and D. W. N. Stibbs, "The Outer Layers of a Star." Oxford Univ. Press (Clarendon), London and New York, 1953.

124

4. ATOMIC

PROCESSES

radiative model2’ *32-34 becomes useful, for which quasi-steady-state solutions of the detailed rate equations involved are sometimes possible. 4.2.3. Rate Equations with Examples

For a specific energy level, the population density N required in an analysis based on either the corona or the modified-corona model is determined by all of the collisional and radiative processes affecting that level. The combined effect may be described analytically by equating dN/dt to the sum of the individual process rates, each multiplied by the population density of the initial state. [A simple cxample of this was given in Eq. (4.2.8) for ionization and recombination.] I t is clear that such a differential equation is required for each level involved and that these equations must be solved simultaneously. This is often a formidable numerical task and is part of the reason for seeking analytical expressions for the rate coefficients involved. Such rate equations have been solved numerically, often coupled with magnetohydrodynamic relations to yield the atomic behavior as a function of time for specific plasmas, and the results have been compared with the time measured line emission where a ~ a i l a b l e . ~ ~ In . ~ ~principle, -~* history of the electron temperature can be thus determined by trial and error, providing the density and chemical composition are known. 4.2.3.I, Application: Quasi-Steadystate Plasma Diagnostics. Fortunately it is often possible to make certain approximations to simplify the analysis, One of the simplest is to assume a steady-state or quasi-steady-state model in which dN/dt = 0 [as in Eq. (4.2.9) above for the ground state of an ion], a method which has been employed for some time in solar corona analyses.23 T o assure the appropriateness of this assumption, it is sufficient to show that for each process the time for relaxation to a steady-state condition is less than the characteristic time for changes in plasma conditions, Such relaxation times may be estimated from the corresponding process rate, the reciprocal of which represents the e-folding time for essentially com3 2 ID. R. Bates, A. E. Kingston and H. W. P. McWhirter, Proc. Roy. SOC. (London), Ser. A267,297 (1962). 3 3 D. R. Bates, A. E. Kingston and R. W. P. McWhirter, Proc. Roy. SOC. (London), Ser. A 270,155 (1962). 3 4 A. Burgess and H . P. Summers, Astrofihys. J. 157,1007 (1969). 3 5 I,. M. Goldman and K.W. Kilb, Plasma Phys. (/. Nucl. Energy, Pt. C ) 6,217 (1964). 3 6 G. I).Hobhs, R. W. P. McWhirter, W. G. Griffin and T. J. L. Jones, Proc. Intern. Conf. Ionization Phenomena Gases, 5th Vol. 2 , p. 1965. North-Holland Publ., Amsterdam, 1962. 3 7 A. C. Kolb and R. W. P. McWhirter, Phys. Fluids7, 519 (1964). 38 U. Uiichs and H. H. Gricm, Phys. Fluids9,1099 (1966).

4.2. EXPERIMENTAL

METHODS OF PLASMA ANALYSIS

125

pleting the process to the succeeding state. Plasma conditions are often such that the fraction of total atoms or ions affected by the process is and the relaxation to a steady-state corona or L T E population ratio will occur in a shorter time, given approximately by the same reciprocal rate multiplied by this f r a ~ t i o n . ~ ’ An application of this approximation for a transient laboratory plasma is described in the appendix of a paper by Kolb et aL4’ for the early heating period in a theta-pinch device, where at low temperatures the ionization and recombination relaxation times were of the order of a microsecond (comparable at least to typical rise times in the experiment). By assuming equal populations between stages of ionization x and x + 1 [i.e., N’ = Nr+‘ in Eq. (4.2.9)] at the time of peak resonance line emission for species x (which is justified by a delay in the onset of collisional excitation), a temperature history consistent with higher values at later times was obtained. This is mentioned as an example of the simplifying assumptions possible in the analysis of corona-like plasmas. 4.2.3.2. Application: Non-Steady-State (Transient) Plasma Diagnostics. Taking the other extreme, where ionization times are longer than characteristic times for parameter variations and recombination times are much longer by orders of magnitude {as is often the case at high temperatures because of the inverse temperature dependence o f the recombination rate coefficient [see Eq. (4.3.35)]}, it is also possible4‘ to simplify the rate analysis by associating the measured rise of resonance line emission with the ionization relaxation time (discussed in the preceding section) for the next lower species, since the excitation times are usually much shorter than ionization times. Similarly, it is possible to relate the emission decay rate to the further ionization rate of the radiating species.42 I n either case, average temperatures for the time interval spanned are obtained from the collisional ionization rate coefficients used (see Section 4.3.3.1). This is a further application of approximate solutions to the rate equations for pulsed plasmas. 4.2.3.3. Application: Excitation Processes. In the above two examples, attention was focused on the rate of development of ionic species and little attention was given to excitation except in a gross manner. That approach was sufficient for such applications as the determination of an overall heating rate associated with a temperature rise. However, in 3 9 T. F. Stratton, X-ray spectroscopy. In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S. L. Leonard, eds.), p. 391. Academic Press, New York, 1965. 4 0 H. R. Criem, “Plasma Spectroscopy,” p. 153. McGraw-Hill, New York, 1964. 4 1 A. C. Kolb et al.,Plasma Physics Contr. Nucl. Fusion Res., Proc. Conf., Culham, Engl. 1965 1, 261 (1966). 4 2 H. J. Kunze, A. H. Gabriel and H. R. Griem, Phys. Fluids 11,662 (1968).

126

4. ATOMIC PROCESSES

order to study in detail the excitation of discrete levels, rate equations must be written in terms of the population densities of the particular levels involved. Fortunately the excitation and deexcitation rates are often sufficiently high (compared to the population density fluctuation rate dN/dt)to assume dN/dt = 0 in solving the equations. Of special analytical interest are the hydrogenic and helium-like ions, particularly the latter since the interaction of the singlet and triplet systems embody many of the processes cataloged below, including spinforbidden transitions. Resonance level excitation in helium-like ions has been investigated in detail in laboratory theta-pinch plasma^^^'^^ (see Fig. 1) and will serve as an example of the method of application of a 2

2

3

CURRENT COLLECTOR

MONOCHROMATOR

FIG.1. Schematic diagram of a typical theta-pinch experiment as used for rate coefficient determinations. The numbers refer to the capacitor bank discharge sequence. The laser is used for diagnostic measurements of scattered radiation.

quasistationary solution of the excitation rate equations to the experimental determination of uncertain rate coefficients from measured line emission. The reverse of this problem is the estimation of the extent of line emission (for radiative energy loss analysis) from known rates. 4 3 R. C. Elton and W. W. Koppendorfer, Phys. Rev. 160, 194 (1967); see also Proc. Intern. Conf. Phenomena Ionized Cuses, 8th, Vienna, 1!36Y, p p . 137, 451. Springer, Vienna, 1967. 44 H. J. Kunze, A. H. Gabriel and H. R. Griem, Phys. R e v . 165,267 (1968).

4.2.

127

EXPERIMENTAL METHODS OF PLASMA ANALYSIS

For a deuterium plasma43 at a measured electron temperature of = 250 eV (from the distribution of X-ray continuum e m i ~ s i o n ~ ’ - ~ ~ ) , an electron density of 6.2 x 10l6 c m - 3 (from visible continuum emission), and a measured oxygen content of 0.674, the effects of ionization and recombination on the detailed level populations for helium-like OVII are negligibly small. The rate equations for the population densities of the 1 ‘S ground level (Ni), the 2 ‘S level (N2) and the 2 ‘P level ( N 3 )of helium-like ions may be written in terms of collisional rates N,X and transition probabilities A as

kT

dNl/dt

=

dN2ldt

=

and dN3/dt

N3A31

- N1Ne(x12

+

NlNeX12 - N2(NeX23

= N1NeX13

x13), N2‘NeX2,2 + N3(A32 $-

NeX22’

+ N2NeX23

+

+ NeX32)

(4.2.12) (4.2.13)

- N3(NeX32 + A32 + A 3 1 ) ,

(4.2.14)

where the processes proceed from the first to the second index and the primes refer to similar levels in the triplet system. Additional singlettriplet P-P and s-P exchange transitions are of lesser importance3 for high-x ions and have been omitted here for simplicity, as have collisional deexcitation rates leading to the ground state. By calculating the required rates (as detailed below) and by measuring the emissions from the 2 ‘P and 2 3P states in decay to the ground state (which are directly related to N3 and N3,respectively), it was possible to obtain estimates for the magnitudes of X , , + X13 and X12.+ Xi38 within a factor-of-two reliability. The results are in agreement with Eq. (4.3.5) for the X , transition, assuming XIJX1 M 3 from statistical weights. No adequate theory exists for triplet excitation in OVII. In similar experiment^^^ principally on helium-like CV with improved temperature measurements (using laser ~ c a t t e r i n g ~ ~ ”the ~ * ) OVII , results 4 5 F. C. Jahoda, E. M. Little, W. E. Quinn, G. A. Sawyer and T. F. Stratton, Phys. R e v . 119,843 (1960). 4 6 H. R. Griem, “Plasma Spectroscopy,” p. 285. McGraw-Hill, New York, 1964. 4 7 R. C. Elton, E. Hintz and M. Swartz, €’roc. Intern. Conf. Phenomena Ionized Gases, 7th Vol. 3, p. 190. Gradevinska Knjiga Publ., Belgrade, 1966. 4 8 R. C . Elton, U.S. Nav. Res. Lab. Rept. 6738 (1968); see also R. C. Elton and A. D. Anderson, U.S. Nav. Res. Lab. Rept. 6541 (1967). 49 H. J. Kunze, in “Plasma Diagnostics” (W. Lochte-Holtgreven, ed.), p. 550. NorthHolland Publ., Amsterdam, 1968. S. A. Ramsden, Laser scattering. I n “Physics of Hot Plasmas.” Oliver & Boyd, London (in press).

#

See also Part 3 of this volume.

128

4.

ATOMIC PROCESSES

were also confirmed. In the CV experiment it was possible to simplify the analysis by assuming an n = 2 population distributed according to statistical weights between the 2s and 2P states for each system. It was also possible in the analysis for CV to include ionization rates from the ground state and from the n = 2 triplet states. These rates were estimated thcoretically and the rate of ionization from the ground state was also measured from the inverse of the procedure described in Section 4.2.3.2 above, since the temperature was known. In the emission analysis of high-z helium-like ions, questions still remain concerning the population density of the 2s levels. Since twophoton emission originating in the 2 ‘s level has been observed in a thetapinch p l a ~ m a , ~it’ now appears possible to extend the analysis to details of the 2S, 2P relative populations. 4.2.3.4. Radiative Energy Losses. The amount of energy radiated away by a plasma during the critical heating phase is an important consideration in the design of a high-temperature plasma device.35* 3 7, 5 2 - 5 6 Resonance line radiation from impurity ions is the chief concern in a (mostly) hydrogen or deuterium plasma, and there is general optimism toward reducing this problem with improved vacuum systems. Nevertheless, impurities such as oxygen continue to exist in small quantities (tenths-of-percent). These originate from container materials such as quartz and from possible water vapor formed during a discharge. ‘I’he radiant energy loss rate can be estimated to the extent that one knows the atomic processes occurring and the appropriate rate coefficients. However, reliable estimates are not always possible, particularly for the intermediate stages of ionization. For this reason, the early stages of ionization are often ignored on the assumption that the losses from the higher stages (such as OVII and OVIII) at higher temperatures will dominate. This assumption is justified only in very rapidly heated discharges, as will be shown in the example described below. In particular, lithium-like ions such as OVI with a low resonance excitation energy may be expected to release significant energy through rapid collisional excitation and radiative decay. As an example of the importance of radiant energy losses during the early heating phase, consider the measurement of resonance line emission R. C. Elton, L. J. Palumbo and H. R. Griem, Phys. Rev. Lettevs20,783 (1968). S. Glasstone and I<. H. Lovberg, “Controlled Thermonuclear Reactions.” Van Nostrand, Princeton, New Jersey, 1960 5 3 K.F. Post, Plasma Phys. (J.Nucl. Energy, Pt. C ) 3,273 (1961). 5 4 11. R. Griern, “Plasma Spectroscopy,” Chapter 8. McGraw-Hill, New York, 1964. s 5 T. S . Green, D. L. Fisher, A. €3. Gabriel, F. J. Morgan and A. A. Newton, Phys. Flitidc 10,1663 (1967). sf, R. C. Elton, Bull. Am. Phys. SOC. 12,787 (1967). 51

52

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EXPERIMENTAL METHODS OF PLASMA ANALYSIS

129

from oxygen ions 0111 through OVIII along the axis of a theta-pinch device (see Fig. 1) operated at a deuterium fill pressure of 60 mTorr with 0.6% oxygen as an intrinsic impurity. T h e techniques used have been described elsewhere4’ * 5 6 - - 5 * and the results are shown in Fig. 2, where the

FIG.2. Measured heating and radiative (0111-OVIII) cooling rates for a theta-pinch device operated at capacitor bank energies of 640 kJ and 1300 kJ. Upper level uncertainties due to a large optical depth for spectral lines from low stages of ionization are indicated by short horizontal lines.

heating rate as determined from the measured rate of temperature rise is compared to the radiative cooling rate obtained from the measured line emission, summed over the resonance lines for each ionic species. Cases are shown for two capacitor bank energies, the higher corresponding to the more rapid heating rate. I n this experiment there remains a large uncertainty in the upper limit of the radiant cooling rate for the low (0111-OVI) stages of ionization, due to the uncertain optical depth of the lines (measured axially), and this i s indicated in Fig. 2 by the short horizontal lines. T h e significant contribution to the radiant cooling rate by emission from many intense resonance lines in low ionization stages remains (for a relatively low heating rate). These data on radiant cooling for the case of minimum (intrinsic) impurities were integrated and found to represent about 10% of the total 57

R. C. Elton and N. V. Roth, Appl. Opt 6,2071 (1967). R. C. Elton, J . Quant. Spectr. Rudiative Transfer& 393 (1968).

130

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ATOMIC PROCESSES

energy per electron (3KT/2).At an intentionally increased oxygen concentration (by a factor of -5), a significant decline in the measured peak temperature was observed. This was attributed to the radiant cooling rates at early times becoming comparable to the heating rates. 4.2.4. Particle Diagnostics

The diagnostics of a plasma by the detection and energy analysis of charged particles is often subclassified into passive and active methods. I n the former, it is the escaping charged particles generated in the plasma proper that are detected; while in the latter, beams of particles are intentionally injected into the plasma and the emerging beam is analyzed for the effects of beam-plasma interactions. T h e methods and techniques have been reviewed recently by Afrosimov and Gladkovskii5 with extensive references. Electron beams may also be used to enhance the temperature in a plasma through beam-plasma interactions.60 4.2.4. I. Passive Diagnostics. Through an analysis of the corpuscular emission from a plasma, it is in principle possible (and perhaps the best available method) to determine the ion energy distribution and lifetime, and the plasma energy losses by escaping particles. There remains of course the question as to whether the energy spectrum of escaping particles (particularly neutrals) actually represents the conditions in the plasma proper, For example, the distribution deduced may be affected by the confining magnetic field which the observed particles have by some means bcen able to penetrate, 4.2.4.2. Active Diagnostics. Plasma can be probed with beams of charged or neutral particles to obtain information on the average density, temperature, and fields along the beam direction. The use of ion beams is often limited by the ability to penetrate the confining magnetic field. T h e method, if feasible, offers the possibility of obtaining the neutral particle density through charge transfer with the ions in the beam (see Section

4.3.5).

Neutral beams are the most widely used as p r ~ b e s . ~ lGenerally -~~ beams with keV energies (greater than the ion thermal energies in the 5 9 V. V. Afrosimov and I. P. Gladkovskii, Soviet Phys.-Tech. Phys. (English Transt.) 12, 1135 (1968); see also V. V. Afrosimov, Proc. Intern. Conf. Phenomenu Ionized Gases, 7 t h Vol. 3, p. 3. Gradcvinska Knjiga Publ., Belgrade, 1966. 6 o I. Alexeff, R. V. Neidigh and W. F. Peed, Phys. Rev. 136, A689 (1964). b 1 R. J. Kerr, Nucl. Fusion3,197 (1963). 6 2 H. P. Eubank, P. No11 and F. Tappert, Nucl. Fusion 5,68 (1965). G 3 V. V. Afrosimov, R. A. Ivanov, A. I. Kislyakov and M. P. Petrov, Soviet Phys.Tech. Phys. (English Transl.) 11,63 (1966).

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METHODS OF PLASMA ANALYSIS

131

plasma) are used to overcome scattering effects. Particle losses from the beam are then attributed to charge transfer in collisions with plasma ions, stripping by heavy particles, and to impact ionization due to plasma electrons. Only small fields are necessary to deflect the resulting charged particles, so that single collisions are considered to be sufficient to account for observed losses from the beam. T h e plasma density N is given approximately by59

(4.2.15) where Fl and F z represent the beam fluxes passing through a plasma region of length d with and without the plasma, respectively, oCtis the charge transfer cross section, u o is the beam particle velocity and (oionvU,)E I is the average of the product of the electron impact ionization cross section and the relative velocity (see Section 4.3.3.1). When oCt dominates in the denominator as, for example, is the case for resonance charge exchange between hydrogen beam atoms and a hydrogen plasma, the electron density is determined uniquely by Eq. (4.2.15). On the other hand, conditions can be chosen such that the electron impact ionization term dominates, in which case the electron temperature is found by using relations such as Eq. (4.3.25),for a known density. By probing with different neutral beams, it is therefore in principle possible to obtain for a particular plasma both the density of ions and the electron temperature. T h e magnitude of local electromagnetic fields may also be deduced from the further deflection of a beam of charged particles subsequent to penetrating the confining field. 4.2.5. Maxwellian Velocity Distribution 4.2.5. I. Analytical Criteria. Vital to atomic processes is a knowledge of impacting particles in velocity space. cribed below) it is possible to assume velocity u ) as defined by

a plasma analysis involving the the statistical distribution of the Under certain circumstances (desa Maxwellian distribution (for the

dN/N = (2/n)'/z(rrr/kT)3~/zvz exp( -mv2/2kT) d v ,

(4.2.16)

in which case the particles will have a mean kinetic energy of 3kT/2. The rate coefficient formulas which follow in this part are based on this assumption. In principle the same velocity distribution must be used for all impacting particles, including both free electrons and (heavier) free ions and atoms.

132

4. ATOMIC PROCESSES

The collisional effects of the electrons are usually the more important, however, because of their higher velocity. It is this feature which encourages the rapid establishment of an equilibrium velocity distribution for the free electrons. It has been pointed out by Wilson26 that there are at least three criteria to be satisfied if the free electrons in a plasma are to have a Maxwellian velocity distribution. These are tee

4 tff,

teh, tpart

Y

(4.2.17)

where tee is the energy relaxation time for colliding electrons. For a specific experiment, it must be much less than: (a) i f f , the energy decay time for free-free processes, (b) teh, the characteristic electron heating time, and (c) tparl,the characteristic containment time for each particle. The first condition is met for all feasible temperatures independently of density, and it remains to compare teegiven by SpitzeP4 for electrons as tee = (3.3 x 1O4)(kT)”’/Ne (sec),

(4.2.18)

(assuming kT in eV and In A w 10 for the slowly varying dimensionless impact parameter A) with characteristic times for individual experiments. ~ is , -0.1 psec, For a temperature of 1 keV and a density of 10I6 ~ m - tee so that it is generally safe to assume that a Maxwellian velocity distribution exists for free electrons in a typical high-temperature plasma. This assumes that there exist no collective effects originating from waves and turbulence capable of producing such anomalies as suprathermal electrons superitnposed on the Maxwellian di~tribution.~ 5 * 6 6 These latter concerns are the reason for various validity tests (outlined below) for the Maxwellian assumption. To these three criteria might be added a fourth specifying that tee be less than the relaxation time for electron impact induced atomic processes such as excitation, ionization, etc., as delineated in Section 4.3. Of these processes, excitation usually proceeds the fastest [as ( N e X ) - ’ ] and using Eqs. (4.2.18) and (4.3.5), the condition for a Maxwellian electron velocity distribution becomes (using rates per unit volume)

(Ne/NZ)(AE/kT) exp(AE/kT) % 0.05,

(4.2.19)

where N’ is the target ion density in the plasma. For this purpose the product f(j) in Eq. (4.3.5) was taken as 0.1. This condition is clearly 64 L. Spitzer, Jr., “Physics of Fully Ionized Gases,” 2nd Ed. p. 133. Wiley (Interscience), New York, 1965. 6 5 H. J. Kunze and H. R. Griem, Phys. Rev. Letters21,1048 (1968). 6 6 W. S. Cooper, 111 and H. Ringler, Phys. Rew. 179,226 (1969).

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OF PLASMA ANALYSIS

133

satisfied for all practical cases since the ratio N e / N zis high whenever AE is small compared to kT. When ions are the impacting particles, tee is replaced by t i i (the ion-ion relaxation time). This is the time necessary for the establishment of an ion thermal distribution in velocity space. Disregarding the effect of electron-ion interactions which might serve to further the process when electron and ion temperatures are comparable in magnitude, the time t i i will be greater than tee by a factor of ( m i / m e ) 1 / 2 / ~or 4 approximately 432/(p)/z4 for mass number p. For z = 1, p = 2 (deuterons) and for the remaining plasma parameters as given above for t e e , tii approximately equals 6 psec, again satisfying criteria (b) and (c) for most cases of interest. When further ionization is possible, the time t i i should also be compared with the ionization relaxation time tion = ( N J ) - ' due to electron impacts [derived using Eq. (4.3.25) below]. For ions of effective nuclear charge number z (kT/X 4 l),the condition tii < tionbecomes

(4.2.20) For X/KTz 5 and q [the number of outer shell electrons-see Eq. (4.3.25)] equal to unity, this reduces to z-4

~1lj2

< 1o3(Ni/Ne),

(4.2.21)

which is clearly satisfied for all ions of interest. 4.2.5.2. Methods for Testing Validity. While it is often suitable for optical-atomic methods to assume that the free impacting particles in a plasma have reached a Maxwellian distribution of velocities at a given local temperature, there is always the concern of a superposition of anomalies as discussed above, and therefore the methods listed below have been devised for determining the velocity distribution function in a particular plasma. 4.2.5.2.1.PARTICLE DETECTION. In principle, the most direct method is to measure the velocity of particles leaving the plasma. For this purpose, particle detectors have been developed;67 however it is often difficult to extrapolate over all directions from a measurement confined to an exit port in the plasma device. There also remains the questions of charge exchange and sheath effects at and beyond the plasma boundary that could distort the energy spectrum. The measurement of radiation scattered 4.2.5.2.2.LIGHT SCATTERING. from plasma particles is a powerful diagnostic method (see Table I) which has been reviewed in this volume (Part 3) and e l s e ~ h e r e . ~ ~ ~ ~ ~ A feature of this method is that for wavelengths of the incident radiation

'' H. P. Eubank and T. D. Wilkerson, Rev. Sci. Instr. 34,12 (1963).

134

4.ATOMIC PROCESSES

much less than a Debye length for a particular plasma, the scattered line radiation has a spectral distribution which is characteristic of the random thermal motion of the electrons and which is Gaussian for a hlaxwellian velocity distribution function, Similarly, for wavelengths much greater than a Debye length, the spectral distribution of the scattered line emission may be characteristic of the ion velocity distribution function, depending on the detailed spectrum of plasma waves.

4.3. Specific Atomic Processes in Plasmas I n the sections which follow, the various atomic processes occurring in and influencing a plasma are defined, and where possible convenient formulas are given for estimating the relevant process reaction rates.

4.3. I. Excitation 4.3. I,I . Electron Impact Excitation. The transition of a bound orbital electron in an atom of species S to a higher energy bound state by the absorption of kinetic energy in an inelastic collision with a free electron is shown by

e + S + S* + e (4.3.1) where S" designates the excited species. T h e rate for this process in a of the plasma is proportional to the cross section r~ and the product N", densities of the target particles and the colliding electrons and to the electron velocity w (see Section 4.2.1), i.e.,

dN"*/dt = -dN"/dt = N , N ' ( ~ J v ) . (4.3.2) Here the averaging of the velocity times the velocity-dependent cross section is carried out over an appropriate velocity distribution function, often assumed to be Maxwellian (see Section 4.2.5). The solution of such rate equations will yield the population of excited states as a function of time (see Section 4.2.3 for examples), to which more directly observable effects such as the radiant energy density hvnI,A,,,N* due to spontaneous decay may be related. (Here A,,, is the transition probability and h v,,, is the energy difference between states n* and n.) The remaining requirement here is a suitable value for the excitation cross section 0 , to which the following is devoted as is a recent review article by Moiseiwitsch and Smith6* and a section by Harrison69 in this series. 68*69

Seep. 135 for references.

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

135

4.3.1.1.1. ALLOWEDTRANSITIONS-SEMIEMPIRICAL METHOD. Calculations and experiments can provide accurate information for specific cases, but they are often difficult and time consuming. It is therefore important to supplement them if possible by approximate general formulas which permit quick estimates to be made for a large number of cases. Most satisfactory would be a suitable analytical expression which could be conveniently integrated over the relevant velocity distribution function. Such expressions have been sought for some time now and the derived results have been compared with observations, notably on neutral atoms. The Born approximation approach, later modified by Bethe7’ for higher energies, has yielded results in good agreement with experiments for electron energies well above the threshold energy for excitation, where short-range interactions are negligible. This is of limited usefulness however in plasmas, where the energy level spacings of principal importance are often equal to or greater than the mean thermal energy of the electrons. Seaton71 and Van RegemorterZ9have successfully developed a modified Coulomb-Born formula for electric dipole transitions, which covers the threshold regions as well as higher energies. Bethe-type corrections are first included through a Kramers-Gaunt g-factor, which varies with energy. Substitution of an effective Gaunt factor g, determined empirically from comparisons with available excitation cross-section data’ 9 . 7 1 (mostly for 1s-2p transitions in hydrogen) and from quasiclassical line broadening theory for ion lines72 leads finally to a cross section o;,* for allowed transitions from level n to n* as

(4.3.3) for excitation of allowed transitions between some initial level n to final excited level n*. With E and AE in eV, this reduces numerically to o:,,*

=

2.3 x 10-’3fnn.g/~ AE (cm’).

(4.3-4)

In these formulas xH is the ionization potential of hydrogen (13.6 eV), j,,,is the absorption oscillator strength for the allowed transition, a , is the B. L. Moiseiwitsch and S . J. Smith, Rev. Mod. Phys. 40,238 (1968). M. F. A. Harrison, Electron impact ionization and excitation of positive ions. In “Methods of Experimental Physics-Atomic Interactions” (B. Bederson and W. L. Fite, 68

69

eds.), Vol. 7A, p. 95. Academic Press, New York, 1968. 70 H. A. Bethe, Ann. Physik 5,325 (1930). 71 M. J. Seaton, The theory of excitation and ionization by electron impact. In “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 374. Academic Press, New York, 1962. 7 2 H. R. Griem, Phys. Rev. 165,258 (1968).

4. ATOMIC

136

PROCESSES

Bohr radius, E and AE are the initial electron kinetic energy and the excitation energy increment respectively, and 2 is given in Fig. 3 for atoms7 .74 and ions.29 , 7 3 , 7 4 For ions, 2 is approximated by a constant value of 0.2 near threshold and approaches the values for neutral atoms with increasing electron kinetic energy. Griem,72 in deriving similar (except at the rather impractical high energy limit) values for 2 from quasiclassical line broadening

1.0

0.01

I

F I G . 3. The as a function of

10


100

effective Gaunt factor g for excitation of neutral atoms and positive ions the ratio of incident electron kinetic energy E to the excitation energy AE.

theory, has suggested an additional and somewhat weak dependence upon the Coulomb parameter x(xH/c)1/2and the parameter 2(n")'/(z + 1). The excitation rate coefficient (av) = X required in Eq. (4.3.2) (where the average is over a Maxwellian velocity distribution) is given by29973

(4.3.5) 73 C. W. Allen, "Astrophysical Quantities," 2nd Ed. The Athlone Press, Univ. of London, 1963. 74 M. Blaha, Astrophys. J. 157,473 (1969).

4.3. SPECIFIC ATOMIC

PROCESSES IN PLASMAS

137

where both AE and kT are expressed in electron volts. T h e effective Gaunt factor averaged over a Maxwellian velocity distribution (g) is given" in Fig. 4. T h e semiempirical relations in Eqs. (4.3.3) through (4.3.5) apply to many allowed dipole transitions' (including some higher multipole interaction^)^' with a reliability of a factor of t ~ 0 , or~ better ~ * at ~ ~ intermediate energies. 72 They also hold for 1s-2p transitions in high-z ions, as has been recently verified e ~ p e r i m e n t a l l y .However, ~ ~ , ~ ~ the full generality of the formulation is not yet e ~ t a b l i s h e d . ~ ~ ' ~ ~

The above formulas can also be applied to innershell excitation, providing appropriate oscillator strengths are available. TRANSITIONS--COLLISION STRENGTH.For esti4.3.1.1.2. FORBIDDEN mating the rate of population of levels through collisional excitation by optically forbidden (in emission) transitions, approximate formulas as convenient as those of Eqs. (4.3.3) through (4.3.5) for allowed transitions are not presently available. It would be a mistake to assume, on the basis 75

0. Bely, Proc. Phys. SOC. (London) 88,587 (1966).

138

4.

ATOMIC PROCESSES

of relative oscillator strengths, that the rates for such transitions are negligible. This has been established by a limited number of experiments and calculations (mostly on neutral atoms).76 On the other hand, rates for nearby higher energy dipole transitions are in general sufficiently high so that excitation followed by cascading is usually the dominant process in populating upper levels of forbidden transitions. For optically forbidden transitions not involving a change in spin, the Born approximation is sometimes adequate.76 However many forbidden emission lines of interest do involve intercombination transitions with a change of spin, for which the Born approximation is useless since an overlap of wave functions for the electron and target atom or ion is required. The exchange contribution for such transitions is included in the CoulombBorn-Oppenheimer approximation through first-order perturbation theory, and practical calculations of total cross sections first became possible for neutral atoms through further approximations proposed by Ochkur7’ and R ~ d g e Such . ~ ~ calculations are tedious and have been carried out for only a few specific cases. Bely7’ has recently reviewed the problem and extended these approximations to positive ions. The agreement with experimental results reported for total cross sections at low electron energies may not necessarily hold at higher electron energies and may in fact be fortuitous. This has been pointed out by Miller and Krauss” from gross disagreement between their theoretical results and recent experimental results of Vriens et d8’for 1 IS-2 3S differential cross sections in neutral helium. For an order of magnitude estimate, one may turn to the collision strength SZ defined here specifically in terms of the “forbidden” excitation cross section o ; , . by76

d n * = (R~H/mn&)na,~,

(4.3.6)

where w, is the statistical weight. This collision strength C2 is a dimensionless parameter identical for excitation and deexcitation and is somewhat analogous to the line strength in radiation analyses. I t varies in magnitude from zero at threshold to the order of unity at a Rydberg (13.6 eV) above threshold for many common neutral atoms,74 and often has an average *~~,~’ value of the order of unity for low stages of i o n i z a t i ~ n . ~ ~(This should not be considered a general statement, particularly when branching M. J. Seaton, Rev. Mod. Phys. 30,979 (1958). V. Ochkur, Sow. Phys. JETP (English Transl.) 18, 503 (1964). M. R. H. Rudge, Proc. Phys. SOC. (London) 85,607 (1965). 7 9 0. Bely, Nuovo Cimento ILB,66 (1967). K. J. Miller and M. Krauss, J . Chem. Phys. 48,2611 (1968). L. Vriens, J. A. Simpson and S. R. Mielczarek, Phys. Rev. 165,7 (1968). 8 2 C. B. Tarter, Astrophys. J . Suppl. Ser. 1 8 , l (1969). 76

77

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

139

is present leading to l2 < 1.) l2 also varies as approximately 2-’ along an isoelectronic sequence,74 where Z is the nuclear charge (at least for excitation of ions in apq c ~ n f i g u r a t i o n ~ ~ ) . The corresponding excitation rate coefficient is again found, by integrating over a Maxwellian velocity distribution function [assuming a constant to be average collision strength (Q) M Q (thre~hold)~’], Xin*=

(OSV) =8 x

10-8[(12)/wn(kT)1’’] exp( -AE/kT)

(cm3 sec-I),

(4.3.7)

with AE and k T again expressed in electron volts. 4.3.1.1.3. METHODS OF CROSS-SECTION MEASUREMENT. T h e most direct experimental approach for obtaining o(u) is to direct a beam of monoenergetic electrons into a target gas, beam, or plasma and observe the rate of excitation. T h e cross section is deduced from a measurement of the spontaneous decay radiation emitted from a specific vohme, knowing the density of scattering centers in the target region. With alternate processes such as population by cascading from higher bound states (as well as the continuum) accounted for, this principally straightforward approach has been successfully used to obtain cross sections reliable to -10% for neutral atoms. For ions the method has been somewhat limited by a low signal-to-noise ratio. 8 3 T h e techniques involved here have been described in detail by Fitea4 and are reviewed for laser transitions in ions by Bennett. While a direct measurement of the excitation cross section for multiply charged ions is very difficult by the monoenergetic beam technique, it is sometimes possible to measure the excitation rate coefficient (av) defined in Eq. (4.3.2). Such measurements are possible in transient plasmas where excitation rates are often much higher than ionization rates and therefore can be studied for a particular ion species in a transient state. I n such a method, the rate of excitation is in principle determined from the observed radiative decay of the excited state. Often there are competing rates involving the same level that must be included in the analysis, so that a set of differential equations evolves for the relevant levels (see Section 4.2.3). Consideration of competing collisional processes is particularly 8 3 R. T. Brackman and W. L. Fite, “Experimental Research on Collisions of Heavy Particles at Energies Up to 2 MeV.” U.S.A.F. Weapons Lab. Rept. AFWL-TR-68-96, 1968. 8 4 W. L. Fite, The measurement of collisional excitation and ionization cross sections. In “Atomic and Molecular Processes” (D. R . Bates, ed.), p. 421. Academic Press, New York, 1962. 8 5 W. R. Bennett, Jr., Collision processes in the argon ion laser. In “Invited Papers from the 5th Intern. Conf. on Physics of Electronic and Atomic Collisions” (L. M. Branscomb, ed.). JILA, Boulder, Colorado, 1967.

140

4. ATOMIC PROCESSES

important in laboratory plasmas, in contrast to the analysis of low density solar plasmas. This method of obtaining rate coefficients from plasmas of known physical characteristics has been illustrated above (Section 4.2.3.3), and has been ~ s e d ~ successfully ~ . ~ ~ for , ~selected ~ * allowed ~ ~ and forbidden transitions of CV, NV, OVI, OVII and Ne 11-VIII. 4.3.1.1.4. DATA.T h e data available on excitation cross sections due to electron collisions are by now quite extensive, particularly for neutral and singly-ionized atoms, and have been summarized in numerous review articles and books. A recent summary has been provided by Seaton.88 Previous reviews have been prepared by Massey and Burhop,3 Fite,84 Hasted5and as a bibliography by Kieffer.go Seaton, 7 1 , 7 The theoretical methods employed in calculating these cross sections are often quite involved, and no attempt will be made to review and evaluate the various approaches here. Such has been provided in some of the above review articles as well as by Belygl and also in the proceedings of the International Conferences on Physics of Electronic and Atomic Collisions, the most recent of which (6th) was held in Boston’’ in 1969. ‘9”

4.3. I.2. Heavy Particle Impact Excitation. T h e effects of inelastic collisions between heavy particles are not generally considered to be significant atomic processes in plasmas, because of the low velocity of the impacting particle compared to electrons. That heavy particle excitation is not completely lacking is borne out by evidenceg3 of large proton-induced excitation rates between closely spaced hydrogen levels (2s-2p) in the solar chromosphere. (A similar situation could exist in hydrogenic ions in high temperature laboratory plasmas.) The study of inelastic interactions between heavy particles is complicated by alternative processes not present in the electron impact case. Charge transfer effects enter (see Section 4.3.5)involving excited final states (as E. Hinnov, J . Opt. SOC. Am. 56,1179 (1966). R.W. P. McWhirter, B. C. Boland, F. C. Jahoda and T. J. L. Jones, Intern. Conf., Phys. Electron. Atomic Collisions, 4th, Quebec. Science Rookcrafters, Hastings-on-Hudson, New York, 1965. M. J. Seaton, Reports on astronomy, Trans. Intern. Astron. Union, 13A, 238 (1968). 8 9 M. J. Seaton, Electron collisions with positive ions, Proc. Conj. Atomic Phys., New York, 1968. Plenum Press, New York, 1969. L. J. Kieffer, Bibliography of low energy electron collision cross section data, Natl. Bur. Stds. Misc. Publ. 289, March 10,1967. 9 1 0. Bely, Electron collisions with positive ions. JILA Rept. No. 89, Univ. of Colorado, 1967. 9 2 Abstracts from the 6th International Conference on the Physics of Electronic and Atomic Collisions. M.I.T. Press, Cambridge, Massachusetts, 1969. 9 3 E. M. Purcell, Astrophys. J . 116,457 (1952). 86

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PROCESSES I N PLASMAS

141

distinguished from direct excitation); and for collisions between complicated ions one is concerned with the internal states of the interacting particles both before and after the collision, as well as with the Coulomb repulsion between the ions.94 Fite,84 Hasted5 and De Heer9’ have summarized the progress on heavy particle impact excitation studies and available experimental data has been listed by Thomas. 9 6 For low velocities of the impacting heavy particle, an adiabatic conditiong7 is approached and the target particle has sufficient time to adjust to the perturbation, with a large probability of remaining in its initial state. Therefore, only when the time a/u (spent by a heavy particle of velocity u within an interaction distance a) becomes comparable to or less than hJAE will there exist a considerable probability for an induced transition, i.e., u

2 a AElh,

(4.3.8)

where a has been found97for neutral target atoms to be an atomic dimension of order lo-’ cm. For ions of effective nuclear charge x, a may be expected to scale as z-’, i.e., a’ M lO-’/z cm. This may be rewritten for ( l / u ) = (2ptMp/nkT)’’2as (neglecting Coulomb deceleration effects)

AE(p/kT)’/’ 6 0.52,

(4.3.9)

where AE and k T are expressed in eV, M p is the proton mass, and p is the atomic mass number of the impacting particle. Thus, for hydrogen ( z = 1) at k T = 1 eV, AE must be less than 0.5 eV, such as is the case for the nearly degenerate 2s and 2p levels mentioned above. Another case of interest is in helium-like (impurity) ions such as OVII, where according to Eq. (4.3.9), at a temperature of k T = 250 eV, proton induced transitions between levels with spacings less than 50 eV (such as 2S-2P with AE x 7 eV) may become likely. 4.3. I .3. Photoexcitation. Photoexcitation, as the name implies, involves the absorption of radiation by an atom or ion of species S and the excitation to a higher energy state shown by hv

+s

-b

S”.

(4.3.10)

94 K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28, 432 (1956). 9 5 F. J. De Heer, Experimental studies of excitation in collisions between atomic and ionic systems. In “Advances in Atomic and Molecular Physics” (D. R. Bates and I. Estermann, eds.), Vol. 2, p. 327. Academic Press, New York, 1966. 9 6 E. W. Thomas, A listing of available experimental data on the formation and destruction of excited states by collisions between atomic systems, Tech. Rept. No. ORO2591-22, Atomic Energy Commission, Oak Ridge, Tennessee. H. S. W. Massey, Rept. Prog. Phys. XII, 248 (1948-1949).

’’

142

4. ATOMIC PROCESSES

I n laboratory high-temperature plasmas, the absorption cross section is too low and the continuum emission is too weak to produce observable continuum absorption spectra. I t is only in the cooler regions of the solar photosphere and chromosphere that the continuum absorption spectrum is dominant. That does not mean that absorption spectra cannot be generated in the laboratory, particularly for neutral gases at high pressures irradiated externally. However, it is not a process that is of importance in the analysis of the plasma proper. Photoexcitation by the absorption of resonance line radiation is of greater interest, due to a much enhanced absorption coefficient (which is dependent on the broadening of the line) compared to that of the continuum. When detailed balancing can be invoked for rates of reverse radiative and collisional processes between levels 1 and 2 of a two level atom, for example, it follows that the importance of photoexcitation relative to collisional excitation depends on the magnitude of the spontaneous transition probability A,, as compared to the collisional deexcitation rate N , X 2 , . 4.3.1.3.1. OPTICAL DEPTH.The extent of reabsorption of resonance line emission in a plasma may be estimated from the optical depth defined by z = rcd (rc being the absorption coefficient). This quantity determines the amount of absorption occurring in a path length d through the LambertBeer law given in Eq. (4.3.30) below. When z approaches or exceeds unity, photoexcitation may become significant, For Doppler broadened spectral lines, the optical depth z at the central wavelength A, in a plasma of depth d may be obtained fromg8 z = (e2/mc2)(~Mc2/2kT)’/21?Aofd,

(4.3.11)

where f is the absorption oscillator strength (which may be taken as approximately 0.5 for many resonance lines), M is the atomic mass, m is the electron mass, 1, is the center wavelength and N is the absorbing particle density. This reduces to

z

=

5.5 x 10-”10Nd(p/kT)”2,

(4.3.12)

with p the atomic mass number, and for k T (eV), N ( ~ m - ~ A,) , (A), d (cm) andf = 0.5. 4.3.1.3.2. SELF-ABSORPTION CORRECTIONS, A knowledge of the extent of self-absorption of line radiation in a plasma (and the resulting photoexcitation) and/or reabsorption in cooler boundary layers is of prime importance in any analysis involving spectral line radiation. If the center of a line under study is optically thick, that is, the absorption length is 98

H. R. Griem, “Plasma Spectroscopy,” p. 197. McGraw-Hill, New York, 1964.

4.3. SPECIFIC

ATOMIC PROCESSES IN PLASMAS

143

appreciably less than the total depth of the emitting layer, corrections must be applied for the reabsorption. Account must then be taken of the mechanisms of radiative transfer occurring in the plasma. This can become extremely complicated, particularly for nonhomogeneous layers as has been shown by the many models proposed for astrophysical interpretation of observed emission from optically dense sources. The simplest method of accounting for self-absorption is to solve for the internal specific intensity I , in the basic equation of radiative transfer:

dI,/dx = 8,- K,I,,

(4.3.13)

assuming a homogeneous layer of self-radiant plasma of (constant) temperature T , depth d , emission flux density 8,and absorption coefficient K, . This yields a relation

1, = (&,/.,)[l

-'

exp( - K , 4 l

,

(4.3.14)

where, for local thermodynamic equilibrium (LTE), &,/ic, = F (the generalized source function) is identical to the Planck blackbody function B,( T )through Kirchhoff's law [see Eq. (4.2.5)]. For low absorption Eq. (4.3.14)reduces to

I,

=

K,Fd = ic,B,(T)d

(for LTE),

(4.3.15)

and for high absorption the intensity I , approaches the blackbody limit B,( T )(in the case of thermodynamic equilibrium).

One method of correcting for self-absorption using these simple relations invoIves determining experimentally the intensity which is approached asymptotically at increasing optical depth. This may be achieved, for example, by varying the concentration of absorbers beginning in an optically thin region. T h e absorption coefficient K , may then be determined from the measured intensity relative to this limit using Eq. (4.3.14). From this an equivalent optically thin intensity given by Eq. (4.3.15) may be obtained. This method is valuable, for example, in correcting for self-absorption on the peak of a measured line p r ~ f i l e . ~ ~ . ~ ~ Although in the strictest sense, Eq. (4.3.14) is only valid for an ideal plasma, it is a valuable approximation of some utility for: (a) ion lines originating in hot regions of the plasma and not affected by reabsorption in cooler outer layers (where such ions do not exist), or for (b) the wings of spectral lines broadened beyond the narrower absorption bands occurring in outer layers of lower density. 4.3.1.3.3. NONEQUILIBRIUM RADIATIVE TRANSFER. Often radiating ions are produced in plasmas not in LTE and the source function F is not 99

R. C. Elton and H. R. Griem, Phys. Rev. 135, A1550 (1964).

144

4.

ATOMIC PROCESSES

given simply by the Planck function. Furthermore, it is not always safe

to assume that an upper limit on observed emission is determined by the

peak intensity of optically thick resonance lines. This has been demon~ t r a t e d ~O 0~in . ' a cylindrical theta-pinch device radiating adjacent allowed dipole and intercombination resonance lines of helium-like OVII. These lines appeared in the expected emission ratio of 2: 1, respectively, when viewed (radially) through an optically thin (for both lines) layer; however, the optically thin intercombination line dominated over the optically thick allowed resonance line when the plasma was viewed longitudinally. In the latter case, the allowed line emission was depleted by resonance scattering through optically thin layers not parallel to the axis of observation, so that radiative equilibrium was prohibited by the dimensions of the plasma. In a case such as this, the method for correcting for reabsorption based on Eq. (4.3.14) must be used with caution, i.e., the exponential dependence of emission on optical depth must be checked in detail experimentally over a range of absorber concentrations. 4.3.2. Deexcitation

4.3.2.I , Electron Impact. T h e process of electron impact deexcitation involving species S is described by e + S*-t S + e , (4.3.16) (where the excess energy is carried away by the electron in a radiationless transition). A knowledge of the rate for this process is particularly important for determining the proper equilibrium model for plasma analysis as discussed in Section 4.2.2. I t is, for example, this process that often requires one to use the collisional-radiative model intermediate between the L T E and corona regimes. The deexcitation rate coefficient X,., is related (for thermodynamic equilibrium) to the excitation rate coefficient X,,* through detailed balancing, i.e., lo' N,N,X,,* = N,N,,X,,*,. (4.3.17) The Boltzmann distribution relation

N,,*w, = N,o,, exp( - A E J k T ) ,

(4.3.18)

when inserted into Eq. (4.3.17) leads to a relation having general validity:

X,,,,,

=

X,,,,( u,,/wnr) exp(AE,,,,/kT).

(4.3.19)

l o o W. W. Koppendorfer and R . C. Elton, Transfer of axial radiation through a long cylindrical plasma. In Proc. Intern. Conj. Phenomena Ionized Gases, 8th, Vienna, 1967, p. 451. Springer, Vienna, 1967. O 1 M. J. Seaton, Rew. Mod. Phys. 30,979 (1958).

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

145

Except for n = 1, usually AE,,, 5 kT and the rate for collisional deexcitation is the same order as for excitation. 4.3.2.2. Spontaneous Radiative Decay

4.3.2.2.1. ALLOWED TRANSITIONS. For large energy differences between bound levels, the greatest probability for deexcitation of a species S* is by spontaneous radiative decay, with the emission of a photon of energy hv according to S”

+

s + hv.

(4.3.20)

The intensity of the emission is proportional to the density of excited atoms or ions. The probability of a downward transition n* -+ n is usually written in terms of the absorption oscillator strength fnn, (which is the effective number of electrons per atom) as

A,,,

= (Snze2/mc3)v Z ( ~ , / ~ , . ) f n = n *6.67 x 1 0 1 5 ( ~ n f , , * / ~ , J 2 )(sec-’)

(4.3.21)

for 1 (A),and increases rapidly with energy according to v 2 = (c/A)’. Transition probabilities are useful for comparison with alternate (collision induced) rates and for analysis of L T E plasmas, as discussed elsewhere in this part. Two critical compilations of transition probabilities for neutral and all ionic species of hydrogen through neon and sodium through calcium ’~ and colleagues103are Also have been published by Wiese et ~ 1 . “ Wiese currently studying systematic trends in transition probabilities, from which quite accurate values may often be obtained by extrapolation. T h e results are to be published in graphical form. For other transitions, there is the compilation of The Coulomb approximation method of calculation due to Bates and Damgaardlo4 has been worked out by Griem’ for many lines common to laboratory and astrophysical plasmas and the method has been extended to multipole transitions recently by Oertel and S h ~ m o . ’ ~ ’

4.3.2.2.2.INTERCOMBINATION TRANSITIONS. For the high-z ions often found in high temperature plasmas, it is not at all uncommon to observe intense intercombination resonance lines, where a change in spin is l o * W. L. Wiese, M. W. Smith and B. M. Glennon, “Atomic Transition Probabilities,” Vol. I, “Hydrogen through Neon,” NSRDS-NBS-4. U.S. Gov. Printing Office, Washington, D.C. (1966) ; Vol. I1 “Sodium through Calcium,” NSRDS-NBS-22 (1969). l o 3 W. L. Wiese, Private communication, 1968; see also W. L. Wiese and A. W. Weiss, Phys. Rev. 175,SO (1968). l o 4 D. R. Bates and A. Damgaard, Proc. Roy. SOC. (London), Ser. A 243, 101 (1949). l o 5 G. K. Oertel and L. P. Shomo, Astvophys. J. Suppl. Sev. 16,175 (1968).

146

4.ATOMIC PROCESSES

present in the t r a n s i t i ~ n . ~ ~ ~This ~ ~is" due ~ ~ in - ~part ' ~ to a rapid rise in the intercombination transition p r ~ b a b i l i t y ' l ~ , "with ~ 2 (approximately as 2'' for moderate 2) and also to a population of upper levels by collisional excitation at a rate comparable to that for the allowed dipole 4.2.3.3). Such transitions must be considered t r a n ~ i t i o n(see ~ ~ 'Section ~~ in radiative energy loss estimates and are of additional value in diagnostics, due to the small optical depth. For example, such optically thin resonance lines are useful for measuring impurity concentrations and for absolute intensity measurements by a technique (branching ratio) of relative intensity measurements of lines with a common upper level and located in widely different spectral regions. 3 , 1 The currently available data on transition probabilities for forbidden transitions are given by Wiese et al.'" and for gaseous nebulae lines by Garstang. 'I7 sl'

4.3.2.2.3. MULTI-PHOTON EMISSION. T h e decay of energy states by the emission of two or more photons"a,i19 is normally considered a small effect. However, for certain metastable states such as the 2s states of hydrogenic and helium-like ions, two-photon transitions to the 1S ground state may be the dominant mode of decay; and there exists evidence of the effect in planetary nebulae.'20.'21 For hydrogenic ions, the transition B. EdlCn, Physica 13,545 (1947); Ark. Fys. 4,441 (1951). B. C. Fawcett, A. H. Gabriel, W. G. Griffin, B. B. Jones and R. Wilson, Nature 200,1304 (1963). l o * G. A. Sawyer, A. J. Bearden, I. Hennins, F. C. Jahoda and F. L. Ribe, Phys. Rev. 131,1891 (1963); see also F.C. Jahoda, F. L. Ribe, G. A. Sawyer and R. W. P. McWhirter, Proc. Intern. Conf. Phenomena Ionized Gases, 6th Vol. 3, p. 190. Gradevinska Knjiga Publ., Belgrade, 1966. I o 9 R. L. Blake, T. A. Chubb, H. Friedman and A. E. Unzicker, Astrophys. J . 142, l(1965). 1 1 0 G. G. Fritz, R. W. Kreplin, J. F. Meekins, A. E. Unzicker and H. Friedman, Astrophys. J. Letters 148, L133 (1967). K. Evans and K. A. Pounds, Astrophys. J . 152,319 (1968). 112 H. R. Rugge and A. B. C. Walker, Space Res. 8,439 (1968). l1 R. C. Elton, Astrophys. J . 148,573 (1967). 114 G. W. F. Drake and A. Dalgarno, Astrophys. J. 157,459 (1969). 115 W. G. Griffin and R. W. P. McWhirter, An absolute intensity calibration in the vacuum ultraviolet. Proc. Conf. O p t . Znstr. Tech., London, 1961, p. 14. Wiley, New York, 1963. 1 1 6 E. Hinnov and F. W. Hofrnann, J. Opt. SOC. A m . 53,1259 (1963). R. H. Garstang, Transition probabilities for forbidden lines. In "Planetary Nebulae" (D. E. Osterbrock and C . R. O'Dell, eds.). Springer, New York, 1968. 1 1 8 M. Goppert-Mayer, Ann. Physik 9,273 (1931). 1 1 9 G. Breit and E. Teller, Astrophys. J . 91,215 (1940). l Z o L. Spitzer and J. L. Greenstein, Astrophys. J . 114,407 (1951). 12' M. J. Seaton, Monthly Notices Roy. Astron. SOC.115,279 (1955). Io6

lo7

4.3.

SPECIFIC ATOMIC PROCESSES IN-PLASMAS

147

probability is givenlZ2by 8.226 Z 6 sec-l and for helium-like ions by123 16.4 (2-0.8)6 sec-’, where 2 is the nuclear charge. Evidence for the coincident emission of two-photons from hydrogenic He I1 in a beam has been reported,”” and the broad continuum feature (only the sum of the photon energies is fixed by the 2 s to 1s energy difference) has been observed for helium-like Ne IX in a theta-pinch plasrna.’l The usefulness of this emission lies both in diagnostics, particularly of solar and astrophysical plasma, and in obtaining a better basic understanding of the (almost) metastable 2 ‘S state population, as detailed absolute measurements of the X-ray continuum emission from high-Z elements are completed. 4.3.2.3. Induced Emission. Enhanced radiative decay of an excited level can occur in an intense radiation field through the process of induced emission. Einstein originally postulated a transition probability proportional to the radiant flux and similar to the probability for photoexcitation (see Section 4.3.1.3). For this process the absorption oscillator strength fnnl [used in calculating absorption coefficients for formulas such as Eq. (4.3.13)] is replaced by an emission oscillator strength f,*,.These are related by statistical weights w, and w,*: fn*n

=

(mn/Wn*)fnn**

(4.3.22)

The effect of induced emission on the radiation emitted from a plasma may be found by inserting a negative absorption term in the equation of radiative t r a n ~ f e r ’ ~(see ” ~ Section ~ 4.3.1.3.2). From this may be derived an effective absorption coefficient (related to the emission coefficient through the Kirchhoff equation) which is the normal absorption coefficient multiplied by (1 - wN*/w*N). Using Eq. (4.2.1), this reduces to [l - exp( -hv/kT)] for L T E plasmas. Plasma absorption and emission relations must include this factor, which significantly deviates from unity only when hv becomes small compared to kT. 4.3.3. Ionization 4.3.3. I. Electron Impact Ionization. A neutral atom or an ion of species S may become further ionized by the impact of a colliding electron whose kinetic energy is greater than the binding energy of a particular orbital J. Shapiro and G. Briet, Phys. Rev. 113,179 (1959). G. W. F. Drake, G. A. Victor and A. Dalgarno, Phys. Rev. 180,25 (1 969). 1 2 4 M. Lipeles, R. Novick and N. Tolk, Phys. Rev. Letters 15, 690 (1965); see also R. Marrus and R. W. Schrnieder, Bull. Am. Phys. SOC.11, 15, 794 (1970). l z 5 G. V. Marr, “Photoionization Processes in Gases.” Academic Press, New York, 1967. lZ2

*

4.ATOMIC PROCESSBS

148

electron, i.e., the threshold energy for ionization. This process may be written as e t- S -+ S + + 2 e , (4.3.23) where the energy of the colliding electron has been reduced in magnitude by the binding energy and kinetic energy of the liberated electron. The rate of generation of ion-electron pairs in a plasma by this process is given by dN"+'/dt = -dN'/dt = N,N'(crv), (4.3.24) where N , and N' are the colliding electron and the target particle densities, u is the relative velocity and cr is the cross section for the reaction. The average indicated is performed over the velocity distribution function, usually assumed to be Maxwellian. The theory of ionization of atoms by electron impact has recently been reviewed by Rudge.'26 FROM THE GROUND STATE.Similar to the situation 4.3.3.1.1. IONIZATION for collision induced excitation (of which ionization can be considered an extension to free electron states beyond the ionization limit), accurate ionization cross-section calculations for specific cases are tedious and time consuming. It is again desirable to have simple expressions of general value which are perhaps not the most accurate for every case, but which are analytically convenient in deriving ionization rates for a wide range of cases of interest. Some of the most reliable experimental data7' exist for ionization from the 1s state of hydrogen and hydrogenic ions.'27 Probably the most reliable theoretical results to date are based upon a Coulomb-Born calculation'28~'29and are in close agreement with the H e + data.'28 For a useful general expression it is appropriate then to turn to an extension originally made for higher-2 helium-like ions.44 T h e resulting ionization rate coefficient becomes 2.5 x q(kT/x)"2 exp( -z/KT) (cm3 sec-'), (4.3.25) I = x3/2 1 47 where x is the ionization energy in eV (as is k T ) . T h e factor q denotes the number of equivalent outer shell electrons (e.g., q = 2 for helium). T h e x-dependence is accounted for by using actual ionization potentials. A further correction (increasing) for step-wise excitation and ionization is possible3' and is particularly significant for low states of ionization. Kunze et al.44 point out the similarity of such a formula with the semiempirical excitation rate formula in Eq. (4.3.5) for helium-like ions using

(Wx)

M. R.H. Rudge, Rev. Mod. Phys. 40,564(1968). I<. T. Dolder, M. F. A. Harrison and P. C. Thonemann, Proc. Roy. SOC. (London), Ser. A264,367 (1961). 128-129 Seep. 149 for references. lZ7

4.3. SPECIFIC ATOMIC

PROCESSES I N PLASMAS

149

an effective oscillator strength of 1.25. Furthermore, using an asymptotic expression for the oscillator strength, Griem, has recently shown'30 that a (numerically) similar expression derives from the excitation rate formula, except with 1 + (kT/x) replaced by 1 + (3 k T / ~)in the denominator. Equation (4.3.25)is also in substantial agreement with classical and semiempirical relations discussed by M ~ W h i r t e r . ' ~ Such formulas for I should be scaled upward by factors of two to three to account for other than electric dipole transition^.'^' A semiempirical formula with three free parameters suggested from calculations by Bethe7' and Rudge and Schwartz12' has been used by Lotz'31-134 to predict and tabulate cross sections and rate coefficients for electron impact ionization from the ground state of atoms and ions ranging from hydrogen through zinc. Work on innershell ionization has been limited so far to neutral atoms, but an extension of collisional excitation formulas with appropriate oscillator strengths (see Section 4.3.1.1.1)should be possible as described above. 4.3.3.1.2. IONIZATION FROM AN EXCITED STATE. I n the consideration of corona equilibrium (see Section 4.2.2.2), it is the ionization rate from the ground state that is of major importance; however, ionization from upper states is often relevant in the analysis of (non-LTE) excited state populations and for estimating near-LTE ionization relaxation times. T h e use of Eq. (4.3.25) for such cases could result in an overestimate of the ionization . that ~ for ~ typical plasmas, KT is much greater rate. Kunze et ~ 1argue than the ionization energy xx = x - E" of an excited state E", and exchange effects could be small. By extending the semiempirical excitation rate formula of Eq. (4.3.5), they arrive at an expression for the ionization rate coefficient I" from the n = 2 states of helium-like ions given by

I*

M

9 x 10-7[P/f(kT)"2] exp( -X"/kT),

(4.3.26)

for X* and kT in electron volts. P is a correction factor analogous to that used in line broadening c a l c ~ l a t i o n s ' ~for quadrupole interactions and other high-order effects (not included in Kunze et u Z . ~ ~ given ) by the larger of ( 1 -t 2kT/X") or 3. Close agreement is found when Eq. (4.3.26) M. R. H. Rudge and S. 13. Schwartz, Proc. Phys. SOC. (London)88,563 (1966). I. L. Beigman and L. A. Vainshtein, Soviet Astron. A J (English Transl.) 11, 712 (1968). 130 H. R. Griem, Private communication, 1970. 1 3 1 W. Lotz, Astrophys. J. Suppl. Ser. X I V , 207 (1967). 32 W. Lotz, 2.Physik 206,205 (1967). 1 3 3 W. Lotz, 2.Physik 216,241 (1968). 3 4 W. Lotz, Z . Physik 220,466 (1969). 1 3 5 H. R. Griem, Phys. Rev. 140,A1140(1965). 12*

Iz9

150

4. ATOMIC PROCESSES

is applied to hydrogenic CVI ions and compared with the results of BornCoulomb calculations by Beigman and Vain~htein'~'for ionization from the 2p level. I n this case (for ionization from excited states) no definitive experimental data are available for comparison with calculations for excited state ionization. Still, this formula is probably reliable to within a factor of two-to-three. 4.3.3.1.3. DATA.The majority of the calculations on electron impact ionization have been restricted to neutral atoms, where data exist for comparison. Since it is not the purpose of this part to survey the extensive efforts in this field, the reader is referred to review articles by S e a t ~ n , Massey ~ ~ ' ~ ~and B ~ r h o p ,Hasted' ~ and to the continuing reference bibliography available from Oak Ridge National Laboratory.' 3 6 Experimentally, the efficiency of ionization by electron collisions has been studied since 1924 by a rather straightforward method (at least in concept) which involves the directing of a nearly monoenergetic beam of electrons into a target gas (or beam) and measuring the ion current generated. The ion current ir+l produced is found by multiplying Eq. (4.3.24) by the area A of the beam, the depth d of the absorbing layer, the resulting state of ionization 5 of the target material [not to be confused with a in Eq. (4.2.4)]and the electron charge e

i z f l = tAed(dN'+'/dt)= ateN,vAN'd.

(4.3.27)

Since the electron current is given by e N p A , this becomes

i Z + l= ieaNz(d.

(4.3.28)

A knowledge of the target density and the two currents involved is therefore sufficient for obtaining a measurement of the electron impact ionization cross section 0. Discussions of the techniques involved in this method have been given by FiteE4and more recently by Harrison6' in Volume 7A of this series. A critical evaluation of all the meaningful experiments and data has been completed by Kieffer and D ~ n n . ' ~ ~ I n plasma physics, it is generally the transition through successive stages of ionization (due to electron impact ionization) that is of interest, rather than solely the initial ionization from a neutral atomic state. In the solar corona, impact ionization rates are also involved in determining the equilibrium conditions. Measurements of the impact ionization cross 136 Bibliography of Atomic and Molecular Processes for July-December 1968 (compiled by Atomic and Molecular Processes Information Center). Oak Ridge National Laboratory Rept. ORNL-AMPIC-12, June 1969. 13' L. J. Kieffer and G . H. Dunn, Rev. Mod. Phys. 38,l (1966).

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

151

section by the direct electron beam method is very difficult with an ionized gas as a target, and to date there have been only three successful experimentsl 2 7,138.1 39 involving the ionization of He 11, Ne I1 and N 11, respectively. While the fundamental quantity involved here is the cross section for the interaction, what is frequently required in plasma and solar physics applications is the product (av) in Eq. (4.3.24).This is the rate coefficient for the process and, when multiplied by the density of colliding electrons, leads to the ionization probability per ion. It is therefore suggestive that a measurement of the rate of ionization in a laboratory plasma could be used to determine the ionization rate coefficient at a particular electron temperature. This is true providing the electron density is known, as well as any spatial variations in density and temperature. This method is of course not capable of yielding a unique energy dependence for the cross section, and in that sense is no substitute for direct cross section measurements where they are possible and required. Ideally, for the purpose of measured ionization rates, one would like to obtain a step function in electron density and temperature and determine the rate of ionization by the more slowly rising ion spectral intensity. Although difficult to achieve, these conditions have been approached in at least two e ~ p e r i m e n t s ~where j , ~ ~ the temperature rose at a sufficiently rapid rate (compared to the lower ionization rate for helium-like ions of OVII and CV) that in at least one case (for CV) where an independent temperature measurement existed, a value for the ionization rate from the ground level was deduced and compared with theory. 4.3.3.2. Heavy Particle Impact Ionization. As was the case with excitation by inelastic impacts by heavy particles, any study of ionization by a similar process is complicated by a large variety of charge exchange and ionization processes involving various possible states of the colliding particles both before and after the collision. Whereas there remained a possibility for a significant contribution to excitation of closely spaced levels by this process, ionization may be considered completely negligible relative to electron collisional ionization in a plasma with equal densities and approximately equal temperatures for the two classes of perturbers. There is evidence5 that the adiabatic maximum rule of Massey9’ given in Eqs. (4.3.8) and (4.3.9) above might be applicable for predicting the energy of the maximum cross section for heavy particle ionization. 13*

K. T. Dolder, M. F. A. Harrison and P. C . Thonemann, Proc. Roy. SOL.(London),

139

M. F. A. Harrison, K. T. Dolder and P. C. Thonemann, Proc. Phys. Soc. 82, 368

Ser. A 274,546 (1963).

(1963).

152

4. ATOMIC PROCESSES

This subject and, in particular, the experimental aspects are discussed by Fite84,140and De Heer.95 4.3.3.3. Photoionization. Photoionization involves the absorption of a photon whose energy is greater than the lowest binding energy of the target atom or ion (of specks S). This absorption results in an ion S + in a higher state of ionization and an electron with kinetic energy equal to the excess photon energy. ‘l‘he process is indicated by hv

+ S + S + + e.

(4.3.29)

Except for the breakdown and prebreakdown (precursor) phase of plasma physics,lZ5photoionization due to inherent radiation is of minor consequence to the physical state of most plasmas. This follows from the fact that the plasma is generally optically thin (except at the center of the resonance lines), i.e., both the photon “density” and the cross section for continuum absorption are low at the photoionization threshold energy. The usual method for measuring the photoionization cross section is to determine the attenuation of a beam of radiation of density go passing through a gas absorption cell of length d and known pressure. The attenuation factor is then given by the Lambert-Beer law as &/&, = exp( - o N d ) ,

(4.3.30)

where 8, is reduced to d after passing through a layer of thickness d with Nabsorbers per em3, each of total cross section g. The methods and techniques involved in the experimental determination of photoionization cross sections and the available data (primarily for ground state neutral atoms) have been reviewed by Wei~sler,’~’ Samson,’42 MarrLZ5 and by thc first two of these authors in Volume 7A of this series.’43 For neutral atoms the density of absorbers is known from the pressure of the neutral gas at ambient temperature. For ion photoionization measurements, an adequate determination of the absorbing ion density is much more d i f f i ~ u 1 t . Il ~n ~a recent experi140 W. L. Fite, J. D. Garcia, E. Gerjuoy and J. A. Peden, “Collisions of Heavy Particles at High Velocities,” Air Force Weapons Lab. Rept. AFWL-TR-69-26, 1969. 1 4 1 G. L. Weissler, Photoionization in gases and photoelectric emission from solids. In “Handbuch der Physik” (S. Fliigge, ed.), Vol. 24, p. 304. Springer, Berlin, 1956. 142 J. A. R. Samson, The measurement of the photoionization cross sections of the atomic gases. In “Advances in Atomic and Molecular Physics” (D. R. Bates and I. Estermann, eds.), Vol. 2, p. 177. Academic Press, New York, 1966. 1 4 3 J. A. R. Samson and G. L. Weissler, Absorption, photoionization and scattering cross sections. In “Methods of Experimental Physics-Atomic Interactions” (B. Bederson and W. L. Fite, eds.), Vol. 7A, p. 142. Academic Press, 1968. 144 H. E. Blackwell, G. S. Bajwa, G. S. Shipp and G. L. Weissler, J. Quant. Spectr. Radiatizte Tvansfer 4,249 (1964).

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PI
153

ment reported by Forman and Kunkel,14’ the density distribution of neutral hydrogen in a plasma was studied using ultraviolet radiation and a known photoabsorption cross section. Photoionization cross sections may be calculated by a general formula’46 recently revised by Peach. Additional recent calculation^'^^^ 147a are also available for light atoms and ions of astrophysical interest. I t is often convenient to study the inverse process which occurs in plasmas, namely radiative recombination. The corresponding cross section is related to that of photoionization through detailed balancing. In fact, in high-temperature, non-LTE plasmas, it is the usual predominance of radiative recombination over three-body recombination (see Section 4.3.4) that prohibits the determjnation of electron impact ionization rates from the inverse (three-body) recombination process-just as radiative decay of excited levels often predominates strongly over collisional deexcitation for ions in plasmas. 4.3.3.4. Autoionization. I n the one-electron atom, the discrete energy levels are followed by a continuum of levels above the ionization limit. I n a configuration in which either an inner electron is excited or two electrons are simultaneously excited to discrete bound levels, some of the energy levels may lie above the first ionization limit, i.e., in the continuum. When a configuration interaction between the quasibound levels and the continuum states exists, one of the electrons can undergo a transition to an orbit which extends to infinity, i.e., the atom can undergo ionization. This process is called autoi~nization.’~’The excess energy is carried away by the free electron in a radiationless transition, leaving the atom or ion in the ground state. T h e process may be described for species S by S** -+ S + + e , (4.3.31) where S** represents the intermediate state. Formation of the autoionizing state may result either from photoexcitation or from particle impact excitation. The effect has appeared most dramatically as resonances in photoionization absorption ~ p e c t r a . ’ ~ ~ ~“Collisional” ’ autoionization has been invoked by astroP. R. Forman and W. B. Kunkel, Phys. Fluids 11,1528 (1968). A. Burgess and M. J. Seaton, Monthly Notes Roy. Astron. SOC. 120, 121 (1960); also G. Peach, Mem. Roy. Astron. SOC. 71,13 (1967). 1 4 7 M. H. Hidalgo, Astrophys. J . 153, 981 (1968); 157, 479 (1969). 147a E. J. McGuire, Phys. Rev. 175,20 (1968). A. G. Shenstone, Phys. Rev. 38,873 (1931). 145

146

‘49 “Autoionization-Astrophysical, Theoretical and Laboratory Experimental Aspects” (A. Temkin, ed.). Mono Books, Baltimore, 1966. I 5 O W. R. S. Garton, Spectroscopy in the vacuum ultraviolet. In “Advances in Atomic and Molecular Physics” (D. R. Bates and I. Estermann, eds.), VoI. 2, p. 93. Academic Press, New York, 1966.

154

4. ATOMIC PROCESSES

physicists' 5 1 i 1 5 2 in the analysis of ionization equilibrium in low density plasmas, using the semiempirical formula of Eq. (4.3.5)and certain selection r ~ l e s ' ~ for , ' ~autoionization. ~ These authors' 52 found the allowance for increased ionization to have little effect on the equilibrium concentrations of light elements such as oxygen and silicon, but a noticeable effect for iron ions. 4.3.3.5. Lorentz Ionization. A promising method for creating plasma in a magnetic trap utilizes a neutral beam that will penetrate the confining magnetic walls, and subsequently ionize without collisions within the trap. This process is known as Lorentz i o n i ~ a t i o n . l ~ ~ -The ' ~ ionization occurs because of the Lorentz force F = e(v x B)' acting on the electrons and nuclei of excited neutral atoms or molecules traversing a magnetic field B at a high velocity v. I n effect, the equivalent electric field causes a lowering of the ionization potential for the atom, resulting in a tunneling of electrons into free states. The relevant cross section increases rapidly with degree of excitation, since the ionization energy for highly excited states varies approximately as n-'. This method is effective only if the incoming beam has a substantial degree of excitation, which may be generated by collisions in the neutral gas or in thin foils. ''3'

4.3.4. Recombination

of a free electron with a positive ion of species S + may proceed by three processes described by radiative :

e

+ S + -+

S

+ hv,

(4.3.32)

+e

(4.3.3 3)

collisional (three-body) : 2e

+ S + -+

and dielectronic recombination : e

+ S + -+

S** -+ S

S

+ hv, + h v ~ .

(4.3.34)

T h e first of these is always present regardless of density and is the reverse process of photoionization, while the second process becomes significant I,. Goldberg, A. K. Dupree and J. W. Allen, Ann. Astrophys. 28,589 (1965). J. W. Allen and A. K. Dupree, Astrophvs. J . 155,27 (1969). D. R. Sweetman, Nitcl. Fzrsim Szippl. Pt. 1,279 (1962). 1 5 4 J. R. Hiskes, Nzrcl. Fusion2, 38 (1962). 1 5 5 C. C. Damm, J. H. Foote, A. H. Futch, A. L. Gardncr and 11. F. Post, P ~ ~ Rev. Js. Letters 13,464 (1964). D. R. Bates and A. Dalgarno, Electronic recombination. Irt "Atomic and Molecular Processes" (D. R. Bates, ed.), p. 245. Academic Press, New York, 1962. lS2

4.3. SPECIFIC

155

ATOMIC PROCESSES I N PLASMAS

(and perhaps dominant) at high electron densities. T h e third is related to the inverse process of autoionization. it is not strictly correct to treat radiative and As has been pointed collisional recombination as individual processes to be added linearly, since there are other processes integrally involved; and the subject may be properly treated as a “collisional-radiative” recombination process. Nevertheless, for the sake of a magnitude estimation of the effects in plasmas, they will be treated separately here. There exist both direct data157-162concerning the rate coefficient for collisional-radiative recombination, and also an abundance of information12 on the continuum emission from high temperature plasmas. T h e continuum emission often includes both free-free bremsstrahlung and a large free-bound recombination contribution, particularly near a series limit. Comparisons with theoretical predictions, using independently measured plasma conditions, have fostered substantial confidence in this formulation, which is the basis for the recombination rate coefficients given here. 4.3.4. I . Radiative Recombination. A convenient formula for the rate coefficient for radiative recombination into all levels of hydrogenic ions has been derived by S e a t ~ n and ’ ~ ~should be accurate to a factor of two or three. It is given by

R, = 5.2 x 10-14~(~/kT)1/2[0.429 + 3h(X/kT)

+ 0.469(~/kT)-l’~]

[cm’ sec-’1,

(4.3.35)

where z is the effective charge (acting on the free electron) of the ion onto which recombination takes place. I n the growth stages of hightemperature pulsed plasmas, recombination is often of minor significance numerically (see example in Section 4.2.3), since the plasma does not reach a state of equilibrium in the short times involved; thus the accuracy of an estimate based upon Eq. (4.3.35)may be sufficient. This may not be the case however during the decaying stages. For a more precise estimate, a summation over the bound states is involved; this is included in a formula derived by Griem165 from the continuum emissivity. This treatment also allows for some deviation from hydrogenic behavior. E. Hinnov and J. G. Hirschberg, Phys. Rev. 125,795, (1962). F. Robben, W. Kunkel and L. Talbot, Phys. Rev. 132,2363 (1963). 1 5 9 J. Y. Wada and R. C. Knechtli, Phys. Rev. Letters 10,513 (1963). 6 o C. B. Collins and W. W. Robertson, J. C‘hem. Phys. 40,2202 (1964). 1 6 1 L. P. Harris, J. Appl. Phys. 36,1543 (1965). 1 6 2 E. Hinnov, Phys. Rev. 147,197 (1966). 163 J . C. Morris, R. U. Krey and R. L. Garrison, Phys. Reo. 180,167 (1969). 1 6 4 M. J. Seaton, Monthly Notices Roy. Astron. SOC. 119,81 (1959). 6 5 H. R. Griem, “Plasma Spectroscopy,” p. 161. McGraw-Hill, New York, 1964. Is’

15*

4.

156

ATOMIC PROCESSES

4.3.4.2. Three-Body Recombination. In the case of collisional (threebody) recombination, the excess energy released in the inelastic process is carried off by the second free electron, as described in Eq. (4.3.33). Also, because of the three-particle interaction, the rate will scale as the product of the target density and the square of the free electron density; and the rate coefficient will scale as the electron density. Therefore, this process 10

i

1

1

I1

iz

13 14 15 LOG10 N B Z - ~[CM-’1

I

1

I

I

I

16

17

18

9

8

7

6

5

n‘ 4

3

2

I

0

10

19

FIG.5. The quantum number n’ of the collision limit (k., the level from which radiative decay is about as probable as collisional excitation to higher levels) versus N.z- for various values of x/kT. See Eq. (4.3.37).

4.3. SPECIFIC ATOMIC

PROCESSES I N PLASMAS

157

may predominate at high densities. The rate coefficient per ion is given by Griem’ 6 6 as

R, z 1.4 x 10-31~-6N,(n’)6(~/kT)2 exp[x/(n’

+ 1)2kT],

(4.3.36) where z is the effective nuclear charge and where n’ is the quantum number of the so-called collision limit,26or the level from which radiative decay is about as probable as collisional excitation to higher levels. At higher levels collisional excitation and ionization dominate, and at lower levels radiative decay is more likely. This collision limit is obtained by iteration from“j6 72’ 1.26 x 102xl4/’7N-2/17(x/kT)-”17 exp[4~/17(n’)~kT], (4.3.37) and is plotted in Fig. 5. The ratio of A, to R,obtained from Eqs. (4.3.35) and (4.3.36) is plotted in Fig. 6 versus N p - ’ for values of the parameter x/kT. From this the relative importance may be determined for a particular plasma. 4.3.4.3. Dielectronic Recombination. In addition to radiative recombination and (at higher densities) collisional three-body recombination, there is an additional process of particular astrophysical interest referred to as dielectronic recombination. This process may be most significant at low densities and in dilute radiation fields34 for any ion with at least one remaining bound electron, and is the inverse process of autoionization discussed above. It is a double process shown by Eq. (4.3.34), where a free electron attaches itself to an ion in a radiationless transition to give a doubly excited quasibound state, which may be transformed through radiative decay to a stable state of sufticiently low energy that the reverse process will not take place. T h e importance of this process in the establishment of equilibrium among ionic species in the solar corona was recognized by Burgess,167 who calculated a rate coefficient for Fe XVI to be about 20 times larger than that previously estimated. This process may also be involved’68 in producing a preferential population of stellar atomic levels of high total quantum number, with possible consequent enhanced stimulated emission. The process of dielectronic recombination is not easily described by a general analytical formula. Burgess’ 6 y has nevertheless derived a useful expression whose limitations have yet to be fully examined by detailed calculations on specific cases. The total rate coefficient Rd may be written (4.3.38) 166 67 16B

H. R. Griem, “Plasma Spectroscopy,” p. 160-162. McGraw-Hill, New York, 1964. A. Burgess, Astrophys. J. 139,776 (1964:)). L. Goldberg and A. K. Dupree, Nature 215,41 (1967).

4. ATOMIC PROCESSES

158

10

12 14 LOGlo N , z - '

16 [CM-3]

FIG.6. The ratio of collisional (R,) to radiative (R,) recombination rate coefficients versus N.z-' for various values of x/kT. See Eqs. (4.3.35) and (4.3.36).

where the summation is over those excited bound states S T z of the ion species S+' for which radiative decay to the ground state S t z is possible, i.e., AEjl > 0. The total rate coefficient is given by Burgess'69as

R~ = 2.4

x

10-9(k~)-3/2~(4

x Cfi,A(x) exp[ - E I j / a k T ] (cm3 sec-'), i 169

A. Burgess, Artrophys. J. 141,1588 (1965).

(4.3.39)

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

159

for El and k T in eV, and where

x = Elj/13.6(x + l),

+ B ( z ) = ~ ‘ / ~+( l)5/2(z2 z + 13.4)-’12 a = 1 + 0.015~~(2: + 1)-’.

A(x) and

=

+

~ l ’ ~ / ( l0.105~ 0.15~’),

(4.3.40) (4.3.41) (4.3.42) (4.3.43)

The limitations x > 0.05, x 5 20 and E l j / a k T 5 5 are not particularly restrictive for the relevance of this process; e.g., for El j / a k T > 5, radiative recombination will most likely dominate. Except for certain Gases [such as Ca’ + e where this formula (4.3.38) overestimates the recombination rate by a factor of 1.5 due to strong coupling effects’ 69], agreement within 20% with detailed calculations is found. Equations (4.3.38) through (4.3.43) should therefore suffice to determine the magnitude of this process in a low density plasma and in a weak radiation field. More detailed calculations are presently underway.”O 4.3.5. Charge Transfer The transfer of charge between heavy particles may occur through several processes, the most common of which is charge exchange indicated by s,++ sz + s, + sz+, (4.3.44) for the impact of species S, + onto species S 2 . 4.3.5.1. Plasma Effects. The charge exchange effect of most concern in plasma physics is the loss from a region of magnetic confinement of energetic charged particles due to neutralization by slow neutrals that enter the region from the outer layers of cool gas. The influence of this effect on the cooling of high-temperature plasmas is discussed by A r t s i m ~ v i c h , ’who ~ ~ finds that under certain conditions (e.g., a surrounding ambient pressure of lo-’ Torr at 300°K) a high temperature (lo8 OK) pure hydrogen plasma suffers a loss of energy by charge exchange comparable to that of bremsstrahlung emission. Charge exchange may play a more positive role as a plasma heating process when fast neutral atoms are injected through a confining magnetic field into a region of cold plasma with a high number density of ions. 170 B.

W. Shore, Astrophys. J . 158,1199(1969). L. A. Artsimovich, “Controlled Thermonuclear Reactions.” Oliver & Boyd, London, 1964. 71

4.ATOMIC PROCESSES

160

Charge exchange may then result in the formation of a high-temperature plasma by the replacement of slow ions with fast ones within the trap. At low ion temperatures (and correspondingly low velocities), the cross sections for charge exchange are particularly high for resonance reactions, i.e., between ions and atoms of the same element. Thus, in a low-temperature (few eV} plasma, it is reasonable to expect equal ion and neutral temperatures. 4.3.5.2. Cross Sections. T h e experimental determination of charge transfer cross sections is generally carried out with an ion beam impinging on either a volume of neutral gas or a crossed neutral beam. I n determining the charge transfer cross section, the products of the reaction (slow ions and fast neutrals), or the attenuation of the fast incident ion beam are detected. Difficulties in interpretation arise from the possibility of the presence of alternate and additional reactions such as

(a) general electron transfer : SI+

(b) impact ionization : or

+ SZ

S1+

S,+

+ Sz

+

+ Sz”+ + (n - I)e,

S1

--*

-+

S2 +

SI+

S,”+

+ Sz + (n - l ) e ,

(c) charge stripping of product S1 : S1

+ Sz

+

+ Sin+ + ne,

S1+

+ SZ + e ,

(4.3.45) (4.3-46)

(4.3.47)

(d) negative ion formation of product S, :

s1 + sz + s1- + s z + , as well as (e) ion-atom interchange in molecules

s,++ szs3 + SISZ+ + s3,

(4.3.48) (4.3.49)

in gas mixtures. A knowledge of the states of the initial and final particles (also of importance in interpretation) is difficult to obtain. Most of the available data are for light elements, as summarized by H a ~ t e d and ~ ” Fite.’40*’73 ~~ More recent data on the effects in atmospheric gases by ions of similar specie^'^^.'^* and by heavier ions’76 of aluminum and iron are now available, as are data on certain alkali elements.177 1 7 2 J. B. Hasted, Charge transfer and collisional detachment. In “Atomic and Molecular Process” (D. R. Bates, ed.), p. 696. Academic Press, New York, 1962. 7 3 W. L. Fite, Ann. Geophys. 20,47 (1964). R. F. Stebbings, B. R. Turner and A. C. H. Smith, J. Chem. Phys. 38,2277 (1963). 7 5 R. F. Stebbings and A. C. H. Smith, J. Chem. Phys. 38,2280 (1963). 7 6 J. K. Layton et al., Phys. Rev. 161,73 (1967). J. Perel, R. H. Vernon and H. L. Daley, Phyr. Reo. 138, A937 (1965).

4.3. SPECIFIC ATOMIC PROCESSES IN

PLASMAS

161

There is also evidence from a survey of available data by Hasted’ that the adiabatic maximum relation of Massey9’ (see Section 4.3.1.2) is applicable for predicting the energy of maximum cross section for charge exchange, providing the parameter a in Eq. (4.3.8)is set equal to 7 x lo-’. This is consistent also with the more recent data of Layton for heavy ion collisions in atmospheric gases, et A detailed analysis of the interaction of ions with heavy atoms and partially stripped ions (which are often found as impurities in plasmas) is very difficult at present, due to the lack of cross-section data. It is conceivable that proton-ion charge transfer processes could compete with electron impact i ~ n i z a t i o n ‘and ~ ~ thus warrant consideration in plasma experiments designed to determine the magnitude of such processes. 4.3.6. Free-Free Transitions

A further atomic process involves the transition of a free electron through a continuum of free states (Lea, free-free transitions), as contrasted with transitions such as ionization and recombination involving free and bound states. Such changes in the energy of the free electron occur in the vicinity of other particles, and in plasmas it is usually the electron-ion dipole interaction which predominates, i.e., electron-electron quadrupole effects’ 7 8 are proportionally smaller as (v/c)’ and usually negligible (except for relativisitic electrons). I n calculating the release of energy in free-free transitions, a Maxwellian velocity distribution is usually assumed for the plasma particles. T h e energy is radiated as bremsstrahlung continuum emission and the process involving ion species S + is shown as e* + S + e + S+ + hv. (4.3.50) The emission density per unit wavelength interval is estimated for hydrogenic ions by the semiclassical formula”

&;f =

CNeNiz2gf,c exp( -hc/Ak T ), R2(kT)“’

with c the velocity of light, and

C = 32n2e6/32/(3)c3(2nrn)3/2= 6.36 x Equation (4.3.51)may be rewritten as

&if= 1.9

(4.3.51) (cgs).

(4.3.52)

x 10-26NeNiz2gffexp( -hc/RkT)/A2(kT)1/2 (W/cm3 - lOOA),

(4.3.53) J. M. Dawson, Radiation from plasma. In “Advancesin Plasma Physics” (A. Simon and W. B. Thompson, eds.), Vol. 1, p. 1. Wiley (Interscience), New York, 1968.

162

4. ATOMIC

PROCESSES

Here, lowering of the ionization potential” is neglected andg,, is a Gaunt correction factor’79 (of order unity), hc/A and kT are in electron volts and A is in angstroms. The total bremsstrahlung emission is found from Sf‘dA to be

Jr

8‘‘ = 1.5 x 10-3’NiNez2g,,(kT)1/2 (W/cm3), (4.3.54) where again kT is in eV, and N, and Ni are in cm-3. A discussion of finer points of this effect in plasmas, particularly for frequencies near the plasma frequency, has been presented by D a w ~ o n ”and ~ Dawson and Oberman.’’’ Also, free-free absorption coefficients for nonhydrogenic atoms have been calculated by Peach.’81 Bremsstrahlung emission is a valuable indicator of plasma density,” since it varies as the product of N , and Ni, usually considered equal. The weak dependence upon temperature is also an asset. T h e magnitude is often comparable to radiative recombination free-bound emission of frequency v mentioned above, which may be obtained from

~ the order unity) is averaged where g,, , the free-bound Gaunt f a ~ t o r ”(of over the sublevels 1 of each level n > n’, with n’ given by Eq. (4.3.37) above and Fig. 5. For high-z ions, recombination radiation may dominate, as is suggested experimentally by sharp edges in the continuum spectrum which correspond to the onset of the recombination transitions from the continuum to additional lower levels. Total (including free-free and freebound transitions) continuous absorption coefficients have been calculated for light atoms by Peach.’*’ Continuum emission from electron interactions (both bremsstrahlung and recombination) with neutral atoms and molecules, capable of forming negative ions, is possible but is expected to be significant” only at low temperatures (less than 1 eV for hydrogen) and low degrees of ionization. Interactions with molecules and molecular ions are not considered here because of their usually low abundance in a plasma, 4.3.7. Spectral Line Broadening

The additional broadening of spectral lines (over the negligibly small natural width) emitted from a plasma is a function of the local density and W. J. Karzasand R. Latter, Astrophys. J . Suppl. Ser. 6 , 167 (1961). J. Dawson and C. Oberman, Phys. Fluids 5,517(1962). 181 G.Peach, Mem. Roy. Astron. SOC. 71,l (1967). G. Peach, Mem. Roy. Astron. SOC. 71,29 (1967). 180

4.3.

SPECIFIC ATOMIC PROCESSES IN PLASMAS

163

temperature, and therefore serves as a powerful tool in determining local plasma conditions without further perturbation. Of equal importance is the requirement of accurate line shapes for an analysis of radiative transfer through plasma. T h e dominant line broadening processes in plasmas are the Doppler effect due to the thermal motion of the radiating particles, and the Stark effect on the responsible energy levels from the microfields of nearby charged particles (including electrons). T h e former process has been well understood for some timez3while for the latter it is generally within the last ten years that meaningful calculations and experiments have been completed. Reviews of line broadening have been prepared by Baranger,lB3 Griem,” W i e ~ e ’and ~ ~’ r r a ~ i n g . ~ T ~ h’ e~ present ’ ~ ~ status of pressure broadening effects on isolated ion lines are reviewed in a recent paper by Griem.72 The effects of the various broadening mechanisms may be convoluted to predict the line shape expected from a particular plasma, providing the processes are statistically independent. For example, the dispersion profile resulting from electron Stark broadening may be conveniently folded with a Gaussian profile due to Doppler broadening, using wellknown Voigt function^.'^^"^^ 4.3.7. I. Doppler Broadening. I n a plasma, the random motion of the radiating particles leads to a temperature dependent broadening of the emitted spectral line. The full width at half intensity is given’89 for a Maxwellian velocity distribution function by

AA = 7.7 x 1 0 - 5 / 2 ( ~ ~ 7 4 ~ / 2 ,

(4.3.56 )

for A 1 and 1 in A and kT’in eV. Here p is the atomic mass number. Thus it is at high temperatures and for light elements that Doppler broadening becomes significant. T h e temperature T’ used here is that for the radiating particle, atom or ion, and not the electron temperature T as used in the other formulas in this part. It is this fact, along with the independence of Doppler broadening on density, that makes the 1 8 3 M. Baranger, Spectral line broadening in plasma. In “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 493. Academic Press, 1962. 184 W. L. Wiese, Line broadening. In ‘‘Plasma Diagnostic Techniques” (R. H. Huddlestone and s.L . Leonard, eds.), p. 265. Academic Press, New York, 1965. G. Traving, “Uber die Theorie der Druckverbreiterung von Spektrallinien.” Karlsruhe, Braun, 1960. l S 6 G. Traving, Interpretation of line broadening and line shift. In “Plasma Diagnostics” (W. Lochite-Holtgreven, ed.), p. 127. North-Holland Publ., Amsterdam, and Wiley, New York, 1968. H. C. van de Hulst and J. J. M. Reesinck, Astvophys. J . 106,121 (1947). J. T. Davies and J. M. Vaughn, Astrophys. J. 137,1302 (1963). l E 9W . L. Wiese, Line broadening. In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S. L. Leonard, eds.), p. 269. Academic Press, New York, 1965.

4.

164

ATOMIC PROCESSES

measurement of Doppler broadened line widths in a high-temperature plasma a powerful diagnostic tool for determining the ion kinetic temperature and for indicating any gross deviations from a Maxwellian distribution for the ion velocities. 4.3.7.2. Pressure (Stark) Broadening. Because of the high density of charged particles present in a plasma, the main broadening process due to neighboring particles (pressure broadening) is due to the Stark effect of the microfields present. T h e calculations are complicated and usually involve a mixture of a quasistatic formulism (applicable mainly to the heavier ions) and an impact theory for the faster electrons, The broadening is generally insensitive to temperature and therefore also to the assumption of thermal equilibrium and varies with an approximately linear to a two-thirds power dependence upon density. The determination of density from measured widths of Stark broadened lines in low temperature, high density plasmas thus becomes possible with a minimal knowledge of the electron temperature. As listed in Table I, the shifting of spectral lines due to the Stark effect is also a function of the electron density. However the theory is not nearly as precise as for the widths,12 and further uncertainties are often introduced by such effects as Zeeman shifting in the presence of a magnetic field. For some spectral lines the measurement of the shift-to-width ratio can become a rather sensitive function of the electron temperature in certain ranges‘”; however, the method is still subject to the rather large uncertainties in the theory required for the line shifts and is therefore only useful when calibrated against other methods.

4.4. Useful Supplementary Material Critical compilations of data listing atomic energy levels and also wavelengths of spectral lines are vital to a plasma analysis based upon the processes described above. T h e following references are quite general and are supplementary to the useful compilations of data on specific processes referenced in the preceding sections. 4.4. I. Atomic Energy Levels

A compilation of available data on atomic energy levels of atoms and ions (in intermediate stages) for elements of astrophysical interest has lg0

D. D. Burgess, Phys. Letters 10,286 (1964).

4.4.

USEFUL SUPPLEMENTARY MATERIAL

165

been published by Moore191 with revisions for Si I-IV.19’ Levels are given in cm-l(K) as measured from the ground state for the particular species. A tabulation of energy levels for the X-ray region has been prepared by Bearden et a1.193i194 Grotrian diagrams of some neutral and ionized atoms of particular astrophysical interest have also been prepared by Moore and Merrill19’ and for many neutrals by Candler.9 4.4.2. Multiplet Tables

Available data on wavelengths of multiplets have been tabulated by Moore for lines of astrophysical interest in both the visible196 and ultraviolet’97 regions. A tabulation in this form is most convenient for the identification of groups of lines within a given multiplet structure. Revisions are in preparation and are presently available19’ for Si I-IV. 4.4.3. Wavelengths

A brief but often convenient tabulation of wavelengths for strong lines in the arc and spark spectra of neutral and singly ionized atoms is given in the “Handbook of Chemistry and The MIT tables’99 list wavelengths for the visible spectrum for similar species of most elements. Also, the multiplet tables in the preceding section (4.4.2) are useful for i91

C. E. Moore, “Atomic Energy Levels,” Vols. 1-3. Natl. Bur. Stds. Circular 467

(1949-1 958).

l g 2 C. E. Moore, Selected Tables of Atomic Spectra-Atomic Energy Levels and Multiplet Tables, Section 1-Si I, and Section 2-Si 11, Si I11 and Si IV, NSRDS-NBS 3. U.S. Gov. Printing Office, Washington, 1).C., 1965. 1 9 3 J. A. Bearden, X-Ray Wavelengths and X-Ray Atomic Energy Levels, NBSNSRDS 14. U.S. Gov. Printing Office, Washington, D. C., 1967. 9 4 J. A. Bearden and A. F. Burr, Atomic Energy Levels, Rept. NYO-2543-1. U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1965. l g 5 C . E. Moore and P. W. Merrill, Partial Grotrian Diagrams of Astrophysical Interest, NSRDS-NBS 23. U.S. Gov. Printing Office, Washington, D.C., 1968. i 9 6 C. E. Moore, A Multiplet Table of Astrophysical Interest. Natl. Bur. Stds. Tech. Note 36 (1945). l g 7 C. E. Moore, An Ultraviolet Multiplet Table, Vols. 1-5. Natl. Bur. Stds. Circular 488 (1950-1962). I g 8 “Handbook of Chemistry and Physics (1970)”. The Chemical Rubber Co., Cleveland, Ohio. I Q 9 G. R. Harrison, “Wavelength Tables with Intensities in Arc, Spark, or Discharge Tube of More than 100,000 Spectrum Lines between 10,000-2000 A.” Wiley, New York, 1939, 1969.

166

4.

ATOMIC PROCESSES

line identification. T h e visible solar spectrum is listed according to wavelength by Moore et ~ 1 . ’ ~ ~ For optical transitions in the vacuum ultraviolet and soft X-ray spectral regions below 2000 A, a recent critical compilation of wavelengths, intensities and term designations for all stages of ionization of hydrogen through argon has been prepared by Kelly2’’ and is currently being extended to krypton. This compilation is also available2” on magnetic tape for rapid sorting and processing of spectral data. A somcwhat incomplete but useful tabulation of wavelengths for atoms and ions of elements of particular interest to plasma physicists and astrophysicists has been prepared for the ultraviolet through infrared regions by Striganov and Sventitskiy.’ O 3 For X-ray transitions, a wavelength tabulation has been compiled by Bearden.’93.’04 4.4.4. Ion izat ion Potentials

Ionization potentials of atoms and ions are given in some of the above tabulations’91~’92~’96~’97~zoi~2’3 for most elements of interest and are also listed by Allen73and by Lotz.’05 4.4.5. Transition Probabilities

Oscillator strengths and transition probabilities are tabulated according to multiplet in critical compilations by Wiese et uZ.102 Also, Allen73 has compiled data for many lines of astrophysical interest. Systematic trends are described by Wiese and Weiss.’ O 3 A bibliography, continually updated with revisions, is also available,’06 as are recent summaries of theoretical and experimental methods respectively by Layzer and GarstangZo7and (in Volume 7 of this series) by Wiese.208 C. E. Moore, M. G. J. Minnaert and J. Houtgast, “The Solar Spectrum 2935-8770

A.” Natl. Bur. Stds. Monograph 61 (1966).

2 0 1 R. L. Kelly, Atomic Emission Lines Below 2000 Angstroms-Hydrogen through Argon. U.S. Naval Res. Lab. Rept. 6648 (1968). ’O’ R. L. Kelly, Naval Postgraduate School, Monterey, California (private communication). ’03 A. R. Striganov and N. S. Sventitskiy, “Tables of Spectral Lines af Neutral and Ionized Atoms.” Atomizdat, Moscow, 1966; Plenum Press, New York, 1968. ’04 J. A. Bearden, X-Ray Wavelengths, Rept. NYO-10586. U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1964. ’05 W. Lotz, J. Opt. SOC. Am. 57,873 (1967). 2 0 6 I3. M. Miles and W. L. Wiese, Bibliography on Atomic Transition Probabilities. Natl. Bur. Stds. SpecialPubl. 320 (1970). ’07 D. 1,ayzer and R. H. Garstang, Ann. Rev. Astron. Astrophys. 6,449 (1968). 2 0 8 W. L. Wiese, Transition probabilities for allowed and forbidden lines; lifetimes of excited states. I n “Method of Experimental Physics-Atomic Interactions” (B. Bederson and W. L. Fite, eds.), Vol. 7A, p. 117. Academic Press, New York, 1968.

4.4.USEFUL

SUPPLEMENTARY MATERIAL

167

4.4.6. Information Centers

An Atomic and Molecular Processes Information Center has been established at Oak Ridge National Laboratory in collaboration with the Office of Standard Reference Data at the National Bureau of Standards (NSRDS). I t presently publishes bibliographic material semi-annually as AMPIC reports. Two monographs are also in preparation entitled “Ion-Atom Interchange Reactions” and “Ionization, Excitation and Dissociation by Heavy Particles.” The Joint Institute for Laboratory Astrophysics (JILA) in Boulder, Colorado also maintains an information center for cross-section data for low energy electron and photon collisions with atoms and molecules. This information is made available through journal publications, and NSRDS and JILA reports. A two-part report on low energy electron collision cross-section data has just been published.209 4.4.7. O t h e r Useful Tabulations

Drawin and FelenbokZ7have tabulated Saha and partition functions, ionization potential lowerings, Debye radii, continuum emission coefficients and Inglis-Teller relations, especially useful in analyzing L T E plasmas. A similar set of tabulations restricted to hydrogen has been prepared by Eberhagen and Lunow”’ who have aIso included listings of carbon arc intensities, partial densities of neutrals and protons, relative Balmer line intensities, and Balmer line-to-continuum intensity ratios. has listed semiempirical ionization cross sections and Lotz’ 3 2 , 1 rate coefficients for hydrogen through zinc. Blackbody functions for a range of temperature and photon energy are -” as are conversion tables from wavelength to electron available, For use in X-ray crystal spectrography, tables of Bragg angles 343205

2 7 3 2 ’

’ 0 9 L. J. Kieffer, “Compilation of Low Energy Electron Collision Cross Section Data Parts I, 11, JILA Info. Center R e p . Nos. 6, 7 (1969); see also Atomic Data 1, 19, 121 (1969). A. Eberhagen and W. Lunow, Tabellen zur Auswertung von Intensitats-Messungen an Wasserstoffplasmen. Institut fur Plasmaphysik Rept. IPP 1-23, IPP 6-20 (1964). ’I1 Srnithsonian Physical Tables, Vol. 120, p. 79-86. Smithsonian Institution, 1959. M. Pivovonsky and M. R. Nagel, “Tables of Blackbody Radiation Functions.” Macmillan, New York, 1961. C . C. Ferriso, Blackbody Radiation Tables, Space Science Lab. Rept. AE62-0862, 1962. 2 i 4 T. R. Bowen, Blackbody Radiation Tables, Tech. Note 1, Rept. No. TN-AMSM1RNR-1-63. Advanced Research Projects Agency, May 1963. ’I5 G. T. Stevenson, Blackbody Radiation Functions. Nav. Ordn. Test Sta. NavWeps Rept. 7621, May 1963. 2 1 6 J . A. R. Samson, A Conversion Table for Wavelengths to Electron Volts. Geophys. Corp. of Am. Rept. 61-5-N, April 1961.

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for certain crystals are available in terms of characteristic line wavelengths and relative intensities. Tables for converting wavelength to wave number have also been 4.4.8. Techniques

In addition to specific references in the text and the preceding data sources, it is perhaps worthwhile to list some additional surveys of experimental techniques of particular value in applying the methods and processes described to the analysis of specific plasmas. For general optics there is a volume by Jenkins and White,’19 and for general spectroscopy the books by Harrison et a1.l’ and by Sawyer” are of value. For vacuum ultraviolet spectroscopy a survey of techniques by Samson” is available. Spectroscopic techniques in the X-ray region are reviewed by Blokhin.’” There are three books on plasma diagnostic techniques that cover these areas and more: one by Griem” and two collections edited by Huddlestone and Leonard2’ and by LochteHoltgreven.2

2 1 7 X-Ray Emission Line Wavelength and Two-Theta Table. ASTM Data Series DS-27, Philadelphia, Pennsylvania, 1965. 2 1 8 C. D. Coleman, W. R. Bozman and W. F. Meggers, Table of Wavenumbers. Natl.

Bur. Stds. Monograph 3,1960. 219 F. A. Jenkins and H. E. White, “Fundamentals of Optics.” McGraw-Hill, New York, 1957. 2 2 0 M. A. Blokhin, “Methods of X-Ray Spectroscopic Research.” Pergamon Press, Oxford and New York. 1965.