JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Excitation and Ionization Rates of Neon Ions in a Stellarator DischargeEINAR HINNOV

Plasmna Physics Laboratory, Princeton University, Princeton, New Jersey 08540(Received 4 March 1966)

The intensities of various resonance multiplets of neon ions, from Ne II to Ne vIII, emitted by an ohm-ically heated hydrogen discharge of the C stellarator, with about 3% neon added, were measured with acalibrated grazing-incidence monochromator. The electron densities (N0 .101s cm-3) were determined bymeans of 4-mm microwave interferometers, and the electron temperature (Te-10-50 eV from the electricalconductivity of the plasma. The neon-ion concentrations at the different states of ionization were calcu-lated from the light intensities by means of an assumed energy dependence for the excitation cross sections,the magnitude of the cross sections being adjusted to make the sum of the calculated ion concentrationsagree with the known total neon concentration. Atomic oscillator strengths deduced from the excitation-ratecoefficients thus obtained, were found to be in reasonable agreement with various calculations. Ionization-rate coefficients were deduced from the time behavior of the ion concentrations and the known particle-confinement time in the discharge. As compared with expected values, the measured rate coefficients appearto be increasingly too large with successive states of ionization. The most likely reason for the discrepanciesis a possible deviation from Maxwellian electron-energy distribution in the discharge, to which these ratecoefficients would be particularly sensitive.INDEX HEADINGS: Neon; Source; Spectra.

I. INTRODUCTION

THERE exist a considerable number of measure-ments of cross sections of excitation and ioniza-

tion of atoms by electron collisions, and of radiativetransition probabilities. Many of these measurementshave achieved a very respectable degree of reliabilityand precision. However, practically all such measure-ments have been performed on neutral atoms and mole-cules, with only a few dealing with ionized atoms. Toour knowledge, there have been no measurements ofionization cross section, or resonance excitation proba-bilities of multiply ionized atoms. Furthermore, it doesnot seem likely that such methods as crossed atomicbeams could be applied to multiple ions in the nearfuture. Therefore, even relatively rough measurementsthat provide quantitative information about thesequantities are of interest. On the other hand, measure-ments of radiative transition probabilities of certaintransitions in multiply ionized atoms have become feasi-ble by means of the recently developed foil-excitationtechniques. Radiative lifetime measurements, such asthose of Berkner el al.1 for the resonance transitions oflithiumlike ions, would be a very valuable check on thesomewhat debatable assumptions made in the presentwork on the relationship between radiative and colli-sional transition probabilities.

In the past, radiation by various ions, especially He+and the ions of oxygen and carbon, has been used in theC stellarator, as well as many other high-temperatureplasma devices, in attempts to determine various plasmaparameters, especially the electron or ion temperature,or the energy loss by radiation, using a variety of cal-culated cross sections and transition probabilities. Inthe present paper we try to reverse this procedure, andattempt to deduce the probabilities of excitation and

' K. Berkner, W. S. Cooper, III, S. N. Kaplan, and R. W. Pyle,Phys. Letters 16, 35 (1965).

ionization of a measured amount of neon introducedinto the discharge of the C stellarator, while it is beingionized by electron collisions through the successivestates of ionization. Such measurements can yield onlythe rate coefficients for the processes under considera-tion, rather than the cross sections. Nevertheless, eventhe rate coefficients could be useful for determining thegeneral magnitude of the corresponding cross sections,and especially their scaling from one state of ionizationto the next.

We present first the experimental data, then the as-sumptions made for the purpose of interpretation of thedata, and finally, the results of the interpretation.

II. EXPERIMENTAL RESULTS

Description of the Discharge

Various aspects of the C stellarator characteristicsand the behavior of the discharges have been publishedrecently.2 We therefore do not describe the stellaratorin detail, but only mention such aspects of the operationas appear to be directly relevant to the present study.

Neon was continually leaked into the 20-cm-diamvacuum vessel of the stellarator on the side opposite thedivertor, at a rate sufficient to keep the neon pressureconstant at 1t-5 torr, against the vacuum pumps at thedivertor, the neon pressure being measured by cali-brated Westinghouse WL-5966 BA ion gauges. About70 msec before the start of each discharge, a puff ofhydrogen was added, also on the side opposite the di-vertor, through a fast-acting valve. The 70-msec delaywas experimentally determined to be just sufficient forthe pressure to begin to rise at the divertor. The opentime of the fast valve (-20 msec), and the back pres-

2 A. S. Bishop, A. Gibson, E. Hinnov, and F. W. Hofmann,Phys. Fluids 8, 1541 (1965); R. M. Sinclair, S. Yoshikawa, W. L.Harries, K. M. Young, K. E. Weimer, and J. L. Johnson, Phys.Fluids 8, 118 (1965).

1179

VOLUME 56, NUMBER 9 SEPTEMBER 1966

1180EINAR HINNOV

FIn. 1. Time behavior of the ohmic heating current, the electrondensity, and the conductivity temperature in the discharge.

sure of hydrogen were adjusted to give a peak electrondensity -3 X 1013 cm-

3 in the discharge.After a short rf pre-ionization pulse (not shown), the

programmed ohmic heating current, shown in the insetof Fig. 1, ionized and heated the gas to densities andtemperatures shown in Fig. 1. The electron density ismeasured by means of 4-mm microwave interferometers.The electron temperature is calculated from the meas-ured electrical conductivity, according to Spitzer.3 Thedashed curve shows the conductivity temperature forassumed Z= 1, i.e., for pure hydrogen plasma; the solidcurve takes into account estimated amounts of multiplyionized neon, and other impurities in the discharge.

The discharge was confined in a longitudinal mag-netic field of 35 kG, with a 450 rotational transform4

(the main purpose of which is to counteract the chargeseparation in the U bends caused by the curvature ofthe field) provided by l= 2 helical windings. The diam-eter of the discharge channel, determined by the mag-netic configuration of the divertor, was 11 cm. Underthese conditions the particle confinement time in thedischarge, which is at least approximately independentof the ion charge and mass, is given by5

r 0 43/T, msec, (1)

where T0 is in electron volts. Thus the particle con-finement time, which is also the average time availablefor production of the various states of ionization,changes from >2 msec early in the discharge to <1msec near the end.

Since the volume of the discharge is only about a thirdof the total volume of the vacuum vessel, the maximumdensity of neon in the discharge would be about threetimes the initial density of -3X 1011 cm-3 . Because theconfinement of neon ions is similar to that of hydrogenions, we can assume that the total neon-ion concentra-

:L. Spitzer, Jr., Physics of FTlly Ionized Gases (lntersciencePublishers, Inc., New York, 1956), 2nd ed., p. 138.

L. Spitzer, Jr., Phys. Fluids 1, 253 (1958).E E. Hinnov and A. S. Bishop, Phys. Fluids 9, 195 (1966).

tion in the discharge is very similar to the electron den-sity, as shown in Fig. 1, in its time behavior, and hasa maximum value approximately 7.5X 1011 cm-3 , i.e.,about 4 of the theoretical maximum. It seems unlikelythat this assumption is in error by more than about30%; however, the error could be substantially moreserious locally in the early part of the discharge. Be-cause of the axial hydrogen-density gradient at the be-ginning of the ohmic heating current, the plasma den-sity does not increase uniformly at all axial locations.As a result, longitudinal density oscillations are set up.As the plasma temperature increases, these oscillationsare damped out, and after the density peak, the axialdensity distribution is practically constant. However,most of the Ne ii and some of the Ne iII ionization occursbefore the density peak, and consequently their con-centrations (especially Ne ii) deduced from local ob-servations may be somewhat more in error than theconcentrations at later times. (The density oscillationsin a helium discharge are described, i.e. in Ref. 2. InFig. 1, for simplicity, we show only the averagedensity.)

Radially, we have assumed the distribution of elec-tron densities and temperatures to be represented bya uniform cylinder with a diameter equal to that of thecurrent channel (11 cm). While this is, of course, notexactly true, it appears to be a sufficiently accurate ap-proximation for our present purposes. We discuss theradial characteristics of the discharge further in con-nection with the measured light intensities.

Finally, there is the problem of the reliability of theelectrical conductivity as the temperature indicator, or.more specifically, the existence of true electron tem-perature in the stellarator discharges. This problem hasbeen debated for many years, and no very definite con-clusions have been reached. It is certainly possible toproduce discharges in which most of the current iscarried by nonthermal "runaway" electrons.6 It is alsopossible to produce "well-behaved" discharges, of whichthe present example is reasonably representative, inwhich several independent temperature indicators arein reasonable agreement with each other and with theconductivity temperatures Nevertheless, even in suchwell-behaved discharges there is usually some x-rayemission in the several keV range indicating the pres-ence of probable excess of high-energy electrons (i.e., anexcess over what would be expected from the conduc-tivity temperature). To summarize, the presently avail-able evidence indicates that most of the current in well-behaved discharges is carried by thermal electrons nearthe conductivity temperature, but the high-energy tailof the electron distribution may be considerably dis-torted. A more quantitative statement is not possiblFat present.

6 W. Stodiek and D. J. Grove, Bull. Am. Phys. Soc. 8, 155(1963).

7 E. Hiinnov, A. S. Bishop, A. Gibson, and F. W. Hofmann,Bull. Am. Phys. Soc. 9, 320 (1964).

1 180 Vol. 56

September1966 EXCITATION AND IONIZATION RATES OF NEON IONS 1

Measurement of the Neon Multiplet Intensities

A 1-m grazing incidence (830) monochromator, builtby Spex Industries, using a Bendix magnetic electronmultiplier with a tungsten photocathode as a detector,was attached to the stellarator on the side opposite tothe divertor. The sensitivity of the monochromator hasbeen calibrated, both in the laboratory before attach-ment to the stellarator, and in situ, by means of simul-taneous measurements of the intensities of line pairsoriginating from the same upper level, one of the linesbeing in the visible, the other in the ultraviolet region.Details of the monochromator and the calibration pro-cedure have been described elsewhere.8

The sensitivity calibration in si/n, besides allowingperiodic checking of the calibration, has producedanother, somewhat unexpected advantage. It was notedin Ref. 8 that it is not possible to use the resonanceseries of He i for the calibration, because of the uncertainlosses through resonance scattering between the sourceand the monochromator slit. However, in the C stel-larator (and presumably in other plasma sources witha high degree of ionization) the helium ions are rapidlyheated to a temperature much higher than the tem-perature of the discharge vessel. As a result, most ofthe neutral atoms in the discharge, including those newlyarriving from the outside, are also heated to comparableenergies, apparently through resonant charge exchange.(This has been directly observed in the C stellarator9

although all the data have not yet been quantitativelyanalyzed.) Consequently, only a relatively small frac-tion of the emitted He i resonance light is lost throughscattering, and, e.g., the X 537-5015 A pair can be usedfor sensitivity calibration at 537 A with reasonable ac-curacy. The results thus obtained have confirmed theremarkably flat sensitivity of the grating and detectorcombination used, as found in Ref. 8.

The various resonance multiplets of neon present aconsiderable variety of multiplet separations. In orderto avoid changing the slit widths during the measure-ments, we have assumed that the relative line strengthsin a given multiplet are those of theoretical multipletsin L-S coupling. The slits of the monochromator werechosen to give a (nearly) trapezoidal instrumental widthof -1.1 A, the exact size and shape of which were de-termined experimentally by scanning various lines ina dc discharge. This instrumental width is large com-pared to the intrinsic line widths (i.e., Doppler width)of the neon resonance lines. Since the spacings of themultiplet components are known experimentally,'"-1 2 it

I E. Hinnov and F. W. Hofmann, J. Opt. Soc. Am. 53, 1259(1963).

9 S. G. Hirschberg and E. Hinnov, Bull. Am. Phys. Soc. 11, 564(1966).

'o C. E. Moore, Atomic Energy Levels, NBS Circular 467,Washington, D. C., 1949.

11 B. C. Fawcett, B. B. Jones, and R. Wilson, Proc. Phys. Soc.(London) 78, 1223 (1961).

12 K. Bockasten, R. Hallin, and T. P. Hughes, Proc. Phys. Soc.(London) 81, 522 (1963).

3.0TIME (insect

FIG. 2. Time behavior of the intensities of resonancemultiplets of the neon ions.

is then a simple matter to relate the maximum inten-sity (of a wavelength scan of a neon multiplet) to thetotal multiplet intensity. In such cases, where the finestructure could be resolved and the components meas-ured separately, the assumption of theoretical intensityratios was found to be generally well justified.

The total intensities of the strongest multiplets ofthe neon ions from Ne II to Ne viii thus measured areshown in Fig. 2, the multiplets being labeled by theirapproximate wavelengths in angstrons. These multipletscorrespond to the following transitions'"-1 2 : Ne II 461,2s2 2p 'E 0P-2s2p 5 2S; Ne iII X490, 2s2 2p4 3P-2s 2P5 3P0 ;Ne iv A470, 2S2 2pI 'D'-2s 2p4 2D; Ne v X483, 2s' 2p2 3P-2s 2p1 'Po; Ne vi X402, 2s2 2p 2P0 -2s 2p2 'p; Ne viiX465, 2s2 1S-2s2p 'P0 ; and Ne viii X774, 2s 2S-2p 2P0 .Thus all these multiplets result from a 2p-2s transition,with the other n= 2 electrons successively removed.

In addition to the multiplets shown in Fig. 2, the in-tensities of several other multiplets arising from thesame configurations were measured. However, beforediscussing their relative intensities, or attempting tointerpret the measured intensities in terms of atomicparameters, we must review the problems of excitationand ionization of the ions by electron collisions.

III. EXCITATION AND IONIZATION OF NEONIONS BY ELECTRON COLLISIONS

Survey of Energy Levels

A simplified energy-level diagram of the neon ions,showing only the energy levels relevant to our presentwork, is given in Fig. 3. The multiplets represented inFig. 2 are shown in solid lines; other multiplets whoseintensities were measured are shown in dashed lines.The dotted lines represent still other multiplets betweenthe same configurations, whose intensities were notmeasured at this time.

We want to stress two aspects of the resonance levelsof these ions. One of these is the obvious fact that theresonance energy levels for the 2s-2p transition do notvary strongly with the state of ionization (we label the

1181

1 EINAR I IINNOV V 5

Nel Ne 11 Nel lI Ne LV Ne V Ne VI NeC V NeW VI

2sZ2p 2s 2s2 4 2s 222272 2s22p 2s5 ? S ,

z

o 200

0

= 150

CD

C)1-4

LU 100

=

50zCD-or.Z0

0

I -

F

2P

Is

p ---- IS

'S4

-/p

FIG. 3. Simplified energy-level diagram of the neon ions, showingthe average ionization potentials for the 2s and 2p electrons, andthe excitation potentials for the resonance multiplets measured inthe present work.

state of ionization, or the excess positive charge of theion, by j in the subsequent discussion), i.e., the transi-tion is roughly independent of the presence of other n= 2electrons. The other important aspect is that the elec-tron temperature, or the average electron energy, inthe discharge under consideration, is comparable to orgreater than the resonance excitation potentials of theseions.

In contrast, the ionization potentials increase rapidlywith increasing j, and are generally substantially largerthan the average electron energies in the discharge atthe appropriate times. For example, at 3 msec (Fig. 2)where Ne4+ is the dominating neon ion species, the elec-tron temperature is -30 eV while the ionization poten-tial of Ne4+ is about 130 eV. Thus, while most of theelectrons can excite these ions, further ionization is pro-duced only by the high-energy tail of the electron-energy distribution.

The ionization potentials for the 2s and the 2p elec-trons are shown separately in Fig. 3, and both are shownas weighted averages over the final configurations. (Tobe consistent, the ground levels should also be taken asaverages over the ground configurations, although theseare not shown in the figure.)

Evidently, in the lower states of ionization the differ-ence between the ionization potentials of the 2s and the2p electrons are nearly comparable with the magnitudeof the latter. Since the number of 2p electrons in these

states of ionization is larger than the number of 2s elec-irons, the rial e of ionization of the latter is nearlvnegligible. However, with increasing j the relative im-portance of the s-electron ionization grows rapidly, andit is comparable to the p-electron ionization in Ne viand Ne vii (in the ionization of the 2s2p 3P metastablestate in the latter case).

Finally, it should be noted that neither the concen-tration of ions at a state of given j, nor the total rateof ionization from j to j+ 1 can be uniquely determinedwithout knowing the relative populations of the variousmetastable states. Under the prevailing discharge con-ditions, it may not be assumed that the relative popu-lations are given by the statistical Boltzmann equilib-rium (although this condition is likely to provide agood upper limit for the metastable populations), norare the metastable populations negligible. In fact, in thecase of Ne vii, the metastable population is typicallylikely to be substantially larger than the ground-statepopulation. This circumstance constitutes an inherentuncertainty in some of our results that appears to beresolvable only by substantial variation of dischargeconditions (i.e., electron densities and temperatures) infuture work.

Excitation Cross Sections

In order to interpret the measured intensities of theneon resonance multiplets, we need to know the ap-propriate excitation cross sections. In particular, weneed the excitation cross sections for optically allowedtransitions in ions by electron collisions. At the presenttime there appear to be no direct measurements of crosssections for this type of transitions.

According to Bethe13 the excitation cross sections foroptically allowed transitions should be proportional tothe optical oscillator strengths for the transitions, andat high energies (electron energy E>>excitation potentialE.) the energy dependence of the cross section shouldbe lnE/E. It has been pointed out on several occasions(e.g., by Post14) that this same energy dependence is alsoreasonably good at low energies, i.e., that the excitationcross section could be written as

aa= oe ln U/ U, (2)

where we have written the energy U= E/E4 in units ofthe excitation potential. The factor e, the base of naturallogarithm, is introduced to normalize the U-dependentterm to unity at its maximum, which occurs at U= e.If -0o is expressed in atomic units (lrai= 8.8X 10-17 cm2),its appropriate value appears to be

au= f(EHIE.)', (3)

where E,, is the Rydberg (13.6 eV), and f the oscillatorstrength for the transition. [Equation (3) would repre-sent the maximum cross section for the transfer of

311 H. Bethe, Ann. Physik 5, 325 (1930).4 R. F. Post, Ann. Rev. Nucl. Sci. 9, 367 (1959).

1182 Vol. 56

September1966 EXCITATION AND IONIZATION RATES OF NEON IONS 1

energy E. in a classical electron collision if f were to begiven its classical interpretation as the number of elec-trons active in the transition.]

Whatever reservations we may have about the man-ner of arriving at the Eqs. (2) and (3), they do appearto represent tolerably well the excitation cross sectionsof allowed transitions both in magnitude and in shape(energy dependence), wherever reasonably reliablemeasured values are available, i.e., in neutral atoms.However, in ions it is to be expected that Eq. (2) seri-ously underestimates the cross section near the thresh-old (U-1), because it does not take into account thefocusing effect of the excess ion charge on the incomingelectron. We therefore modify this expression somewhat,to

a, = uoe ln2 U/ U (4)

with ao still given by Eq. (3).Theoretically, the most studied transition of this type

is the ls -* 2p transition in He+. Probably the mostaccurate calculation of this cross section is the recentresult of Burke, McVicar, and Smith,15 which we com-pare with Eq. (4) in Fig. 4, along with semiempiricallyobtained "recommended" cross sections by Seaton,"'van Regemorter,7 Percival,'8 and a scaled Born-approximation calculation (for Z= 1) by Omidvar.'9 Forcomparison, we also show the Eq. (2), and, at left, the"maximum" cross section ao [Eq. (3)].

The remarkable agreement between the results ofBurke el al. and Eq. (4) does not of course guaranteethe correctness of Eq. (4) or its applicability in othercases. However, at least two independent observationstend to support its applicability. Measurements of theexcitation rate of the n= 2 state in He+ in the C stel-larator" ' 0 are apparently in very good agreement withEq. (4) (and therefore also with the calculations ofBurke et al.). Second, measurements of the excitationrate of the 6s-6p transition of Ba+ in a low-temperature(-3000 0 K) highly ionized barium plasma 21" 2 are alsoconsistent with Eq. (4). Although in both cases we aredealing with rate coefficients rather than cross sections,so we cannot say anything definite about the detailedenergy dependence of the cross section, the energy de-pendence close to the threshold cannot differ too dras-tically from the assumptions.

The factor 2 in the logarithm in Eq. (4) is empirical,and should not be interpreted as Z of the ion. In fact,with further increase in Z, the shape of the cross section

15 p. G. Burke, D. D. McVicar, and K. Smith, Proc. Phys. Soc.(London) 83, 397 (1964).

16 M. J. Seaton, in Atomtic and Molecular Processes, D. R. Bates,Ed. (Academic Press Inc., New York, 1962), p. 414.

17 H. van Regemorter, Astrophys. J. 136, 906 (1963).'S I. C. Percival, Nucl. Fusion 6 (1966).19 K. Omidvar, Phys. Rev. 140A, 38 (1965).20 A. S. Bishop and E. Hinnov, Bull. Am. Phys. Soc. 10, 196

(1965).21 E. Hinnov, J. G. Hirschberg, F. W. Hofmann, and N. Rynn,

Phys. Fluids 6, 1779 (1963).2" L. C. Johnson, E. Hinnov, F. W. Hofmann, and N. Rynn,

Bull. Am. Phys. Soc. 9, 508 (1964).

h 015

10

051:

!a

Gv .0

ft

tx

Or

1.0 2.0 3.0 U=E/Ex 40 . . 6.0

FIG. 4. Comparison of the empirical excitation cross sectionsused in the present work, with various calculations. The abscissais the incident energy of the electron in units of the excitationpotential.

probably does not change very much.'8 We thereforepropose to use Eq. (4) as the excitation cross section forall of the ions under consideration.

Excitation-Rate Coefficient

The main motivations for using Eq. (4) for the exci-tation cross sections are that it gives explicit instruc-tions for scaling to different transitions, and that it canbe folded analytically into a Maxwellian electron-energydistribution to give an excitation-rate coefficient. Forthe latter, we obtain, with y=EX/kTd,

S. croe(8E/,7rm)' (y),

where

cD(y) = yl In2 exp, (- y) + ti-1 exp(- t)dt; .

(5)

(6)

The maximum value of 4'(y) is -0.8; therefore themaximum value of S. is about 2.2 aUOv, where v. is themean speed of electrons at a temperature (kT ) equalto the excitation potential.

The temperature dependence of the excitation-ratecoefficient Eq. (5) is shown in Fig. 5, together with thetemperature ranges, from Figs. 1 and 2, over which thevarious multiplets were measured. Clearly, the tem-perature dependence of the rate coefficient is very weak,at most slightly stronger than linear, so that moderateerrors in the temperature or in the detailed energy de-pendence of the excitation cross sections do not affectthe results very strongly. In particular the presence ofa moderate number of nonthermal high-energy (e.g.,several-keV) electrons does not produce a particularlynoticeable effect on the light intensities.

Ionization-Rate Coefficients

Because the rate coefficient, i.e., the product of crosssection and electron velocity averaged over a Maxwel-lian velocity distribution, is very insensitive to the pre-

Burke Milicar and Smaith--- Seatonivon Regemorter

A Born approx.Z lscoled)--- Percival-x- (re JnUllJ

-+- ee Wn2UIU

** sx -K---e- --

1183

1 EINAR HINNOV

kTe/Ex

FIG. S. Temperature dependence of the excitation-rate coefficientand the temperature ranges relevant to the present paper.

cise energy dependence of the cross section, we cannothope to say anything definite about this energy de-pendence from our present measurements. The best wecould do would be to obtain the general magnitude ofthe ionization-rate coefficients, and perhaps their tem-perature dependence over a limited temperature range.This information might enable us to determine the gen-eral magnitude of the cross sections near threshold, andperhaps their scaling from one state of ionization to thenext.

We therefore simply assume that the ionization crosssection has a shape not too different from Eq. (2), anduse the latter to compute rate coefficients for comparisonwith experimental results. Instead of Eq. (3), we use asthe maximum cross section, again in units of 7ra2,

uot= hq(EH/E1) 2, (7)

where Ei is the weighted-average ionization potential,different for the 2s and the 2p electrons as seen in Fig. 3,q is the number of equivalent electrons in the initialconfiguration, and b an undetermined constant, hope-fully of the order of unity. The lower curves in Fig. 6compare this cross section for the ionization of He+,with b= 0.8, with the cross section measured by Dolder,Harrison, and Thonemann?3

The ionization cross section of Ne+ has also beenmeasured,2 4 and is shown in Fig. 6. Its energy depend-ence near threshold is substantially different from thatof He+, and is, in fact, more similar to the ionizationcross section of neutral helium 25 also shown in Fig. 6.(The ionization cross section of Li+ ions has also beenmeasured,20 and shows an energy dependence inter-

23 K. T. Dolder, M. F. A. Harrison, and P. C. Thonemann, Proc.Roy. Soc. (London) A264, 367 (1961).

24 K. T. Dolder, M. F. A. Harrison, and P. C. Thonemann,Proc. Roy. Soc. (London) A274, 546 (1963).

25 D. Rapp and P. Englander-Golden, J. Chem. Phys. 43, 1464(1965).

26 W. C. Lineberger, J. W. Hooper, and E. W. McDaniel, Phys.Rev. 141, 151 (1966).

mediate between those of He+ and He, but somewhatcloser to the former.)

In addition, there is shown in Fig. 6 a modified versionof our empirical cross section, which we have used inthe past for calculation of ionization coefficients forneutral atoms, namely,

cir= u0e(InU/U) (1- U-2 ) (8)

with &oo given by Eq. (7) (with b=0.7). This equationis shown for the p-electrons only (q= 5, Ej= 3 .1EH) andfor both p and s electrons (q=2, Ei=5.0EH for thelatter).

Compared with Eq. (2), Eq. (8) yields a rate coeffi-cient that is smaller at given Te, and rises more rapidlywith increasing T, (we are at present concerned withonly kT,<EO). However, in the limited temperaturerange over which each ion is being ionized in our presentdischarge, the difference in the slope is not likely to bemeasurable, so that even such differences in c-(U), atbest, show only as differences in the undetermined con-stants b. For comparison with experimental results, wehave therefore used the Eq. (2) shape throughout, ex-cept in the case of Ne+ ionization where Eq. (8) isknown to b& a better approximation. The rate coefficientcorresponding to Eq. (2) is the same as the second termin Eq. (5), i.e., for the ionization of the jth state

sgi+= a IeY) l h-1 exp (-t)dt,."3 Is

(9)

where yi=.E/kT 6 . The superscript in the ionization-rate coefficient is used to distinguish it from the excita-tion-rate coefficients in specific cases.

IV. EVALUATION OF EXCITATION-RATECOEFFICIENTS

Concentrations of Neon Ions

Under the discharge conditions depicted in Fig. 1, itappears to be safe to assume that the observed multiplet

FIG. 6. Comparison of measured ionization cross sections forHe+ and Ne+ with empirical formulas used for evaluation of ioni-zation rates of neon.

1184 Vol. 56

September1966 EXCITATION AND IONIZATION RATES OF NEON IONS

intensities are very closely equal to the correspondingrates of excitation by electron collisions from the lowerterm of the multiplet. In view of the fEc-2 scaling of theexcitation cross section, cascading from higher excitedstates of a given ion, or direct excitation from the pre-vious state of ionization are clearly negligible. At thelow densities and relatively high temperatures there isno radiation trapping, or electron-ion recombination ofany consequence, nor is there any likelihood of colli-sional de-excitation of the short-lived excited states ofthese transitions. Neither is there any prospect of ex-citation by ion collisions, since the ions have energiescomparable to the electrons, and therefore much smallervelocities. The only type of transition that could pos-sibly cause some difficulties in this respect is collisionalexcitation through optically forbidden transitions, es-pecially in cases like Ne vii, where a metastable state ofconsiderable population has an energy about half theenergy of the resonance level. However, such transitionsinvolve a change in multiplicity (i.e., a spin change inthe bound electrons), and their rate should be substan-tially smaller than the direct excitations of opticallyallowed transitions, in spite of the smaller energychanges involved.

We therefore assume that the measured multipletintensities are given by

1j= NVj1 'N0 Sj photons/cm *sec, (10)

where N/' is the concentration of the j-times-ionizedneon in the lower multiplet term of the transition. Inorder to find the total concentration, we must estimatethe relative populations of the ground and metastableterms.

Comparison of Figs. 1, 2, and 3 shows that the en-ergy separations of the metastable and ground termsare generally small compared to the temperature at thetimes most of the radiation of the jth ion is emitted,and that the temperature does not vary much duringsuch times-i.e., exp(-Em/kTj) is nearly constant andof the order of unitv. We mav therefore considerBoltzmann-equilibrium populations corresponding tothe temperatures at the times of the intensity peaks,as first approximations to the total ion densities. Theratios of total densities Nj to the N1 ' corresponding tothe multiplets described in Fig. 2 that result from thisassumption are

N2 /l 212t 1.5; N3/N3'> 2.0 ;N4/N 4'> 1.6;

NI/NaZ'2.3; N 6/1N61 Z7. (11)

N1 and X7 do not have metastable terms, so N11'= 1 ,

and N 7 '= N7. (There is the 2s22p43s4P term in Ne ii, butit is 27 eV above the ground term, and apparently hasa relatively short radiative lifetime.)

We can obtain a different estimate of these ratios bycomparing the emitted intensities corresponding to tran-sitions with the different multiplicities, and assumingappropriate ratios of the transition probabilities, whichwe discuss later. By this means we find that the ratios

80-

- 7.0 -

5 6.0

- 4.0-

2.0-

1.0 -a 10tC cr3

itu

'U .

, N, N , "

0 1.0 2.0 3,0TIME (maeC)

-0- Total estimated frominitial amount

* ZNj 2

'U

N6 a ,

.- X-xN. - x..-" N, -~X

4.0 ao

FIG. 7. Neon-irn concentrations deduced from the measuredlight intensities, compared with total neon density estimated fromthe initial density and the time behavior of the discharge.

given in Eq. (11) are generally too great, although notvery much for the first three. From these considerations,it seems better to take for our actual experimentalconditions

N 2/Ni2 '= 1.3; N0 /N3'= 2.2; N4 /N4'= 1.3;NA/2 5'= 1.6; NA7 /N6'=3, (12)

i.e., the metastable populations appear to be roughlyone-half of what would be expected at thermal equi-librium. It is also apparent that only in the case of N6is there any likelihood of a considerable error resultingfrom the uncertainty of the relative metastable popula-tion. We can now calculate the sum of the neon-iondensities at any time in the discharge from the Eqs.(3), (5), (10), and (12), adjusting the f values of Eq.(3) so that the sum is equal to the total expected neondensity at that time. The latter quantity, we recall, wasto be -7.5X10 11 at the peak of the electron densityand follow the time dependence of the electron densityreasonably closely. The result, for a certain set of fvalues, is shown in Fig. 7.

Before discussing the f values, let us briefly considerthe significance of the results in Fig. 7. These results arelikely to be most reliable for N 3, N4 , and N5. At thetime these ions dominate, the total estimated neon den-sity is probably correct to within -30%, which is com-parable to the estimated absolute accuracy of the light-intensity measurements. The results are not sensitiveto errors in temperature measurements or to the detailsof the excitation cross-section energy dependence. Norare there likely to be substantial errors in the relativemetastable populations of these ions. It would seemthat the absolute accuracy in the concentrations of theseions (and by implication in the corresponding f values)should not exceed a factor 2 in these cases, and the re-lative accuracy from one ion to the next should be sub-stantially better.

In the results for N 1, N2 , N6, and N7 the errors maybe somewhat more serious; in case of the first two be-

1185

a

TABLE I. Comparison of absorption oscillator strengths deduced from the measured multiplet intensities with various calculations.

B., L., L.aB., L., Extrapolated

Ion X Transition f (a) (b) Varsavsky" Couiombd

Ne II 461 2p0-2S 0.035 0.33 0.038

Ne III 490 'p-'p° 0.11 0.46 0.14Ne IV 470 2D"-

2D 0.15 0.135358 2D-2p (0.21) 0.21544 4

S0-

4P [ <0.42] 0.49 0.22

Ne V 483 3p-3po 0.14 0.11 0.14 0.093572 'P-3D0 0.12 0.15 0.11 0.20 0.12359 3P-3S' (0.16) 0.20 0.14 0.19416 'D-ID' [>0.17] 0.37 0.28 0.40

Ne VI 402 2p0

-2p 0.35 (0.32)b (0.23)b 0.36

563 2p'-2D (<0.17) (0.11)b (0.087)b 0.17 0.13434 2p'2S (0.07) (0.033)b (0.060)b 0.036454 4P-4S' [>0.04] 0.09

Ne vii 465 'S-1P' -0.60 (0.39)b (0.27)b 0.63 0.56Ne VIII 774 2S-

2p° -0.15 0.155 0.152

a See Ref. 27.b Extrapolated from Bolotin and Jucys, see Ref. 28.-See Ref. 29.d See Ref. 30.

cause of the relatively unsettled axial uniformity of theplasma conditions, in the case of AN6 because of the un-certainty in the metastable population, and in the caseof N7 because of the relative smallness of this concen-tration combined with the uncertainty in NG. Never-theless, even in these cases it seems unlikely that theerrors should be in excess of a factor -3.

Oscillator Strengths

In the preceding section we regarded the f values ofEq. (3) merely as adjustable constants. However, ac-cording to the discussion in Sec. III these constantsshould be equal to the optical oscillator strengths of themultiplets. In Table I, we compare the f values usedin computing the ion densities of Fig. 7, and a numberof other f values obtained from measurements of rela-tive intensities of these multiplets in the same discharge.with various calculated absorption oscillator strengths.These f values are given, in the fourth column for themultiplets identified in the first three columns.

The values in parentheses in the fourth column aregiven for multiplets that were not spectrally resolvedfrom other neon lines, or from the lines of oxygen,carbon, or nitrogen impurities. The intensities of thesemultiplets were detesmined from the time behavior ofthe composite signal, compared with the time behaviorof the resolved multiplets of the same state of ionization.In most cases the interfering multiplet intensities wererelatively low, so the errors relative to the values notin parentheses probably do not exceed 430%. In onecase, the 2 P-2 D ?563 doublet in Ne vi, which overlapswith the resonance triplet of Ne vii of rather uncertainintensity, the f value given in the table is known to besomewhat too high, as indicated. In other cases, theymay be either high or low.

The values in square brackets are those for differentmultiplicity from the "primary" multiplets on which

the ion-density determinations were based. The valuesgiven in the table correspond to assumed relativemetastable-population densities as they would be atthermal equilibrium. Since the actual metastable popu-lations are probably somewhat smaller, the values inthe square brackets are probably too small, as indicated(too large in the case of the quartet in Ne iv since theIS is the lowest energy term in this ion).

The fifth column of Table I contains the oscillatorstrengths for Ne v calculated by a self-consistent-fieldmethod by Bolotin, Levinson, and Levin." The calcula-tions under b take into account the perturbation of theground configuration by the configuration 2p4 ("doubleconfigurational approximation"), which is neglected inthe calculations under a ("single configurational ap-proximation"). Thus the values in the 56 column arepresumably more accurate. A similar calculation hasbeen performed earlier by Bolotin and Jucys2 8 for theberyllium and boronlike ions up to 0 v and F v, respec-tively. The values in parentheses for Ne vi and Ne viiare estimated by extrapolating the results of Bolotinand Jucys to higher Z.

The sixth column gives the oscillator strengths cal-culated by Varsavsky29 by his screening method. Theseventh column is obtained by extrapolating the tablesof Coulomb-approximation transition integrals of Batesand Damgaard"0 to smaller effective quantum numbers,it,?. These last columns are quite comparable at largej, but diverge very noticeably at small j. It is also in-teresting to note that the Coulomb-approximationvalues are rather similar to the single-configurationalself-consistent field calculations, and that the configura-

2? A. B. Bolotin, I. B. Levinson, and L. I. Levin, Soviet Phys.-JETP 29, 391 (1955).

28 A. B. Bolotin and A. P. Jucys, Soviet Phys.-JETP 24, 537(1953).

29 C. M. Varsavsky, Astrophys. J. Suppl. 6, 75 (1961).'" D. R. Bates and A. Damgaard, Phil. Trans. Roy. Soc.

(London) A242, 101 (1949).

1186 EINAR HINNOV VO1. 56

September1966 EXCITATION AND IONIZATION RATES OF NEON IONS 1

tion interaction modifies this agreement noticeably,especially in the relative strengths of some of themultiplets.

Comparing the calculated oscillator strengths withour present results, it is immediately apparent that ingeneral magnitude they are in agreement with theCoulomb approximation and the self-consistent fieldcalculations, certainly within experimental error. In Neii and Ne iII our values are considerably lower than theresult of Varsavsky's calculation, by an order of mag-nitude in the first case. In Ne v and Ne vi the relativestrengths within the supermultiplets appear to be indecidely better agreement with the double-configura-tional approximation. (This result is qualitatively con-sistent with our previous measurements3 ' of relativemultiplet strengths in 0 iv.) Although, in the measure-ment of relative excitation of different multiplets in thesame supermultiplet, most of the possible sources oferror that beset any individual measurement are elimi-nated, there are still sufficient experimental uncertain-ties to prevent us from deciding at present whether theremaining discrepancies are real or not.

In the case of multiplets of different multiplicity ina given ion, such as X470-X544 in Ne iv, the measuredintensity ratios yield the ratios of products of f valueswith lower-term. populations, e.g., f3 44N(4 S)/'f470N' (2 D),the latter being the N3' of the previous section. As men-tioned above, in these cases the f values in Table I (insquare brackets) were obtained by assuming the ther-mal-equilibrium values for the population ratios. Com-parison of the f ratios thus obtained with the ap-propriate calculated f ratios indicates that the actualmetastable populations should probably be smaller thanthe equilibrium values. The differences between Eqs.(11) and (12) were deduced from this consideration.

Although there is considerable room for improvementin the accuracy and detail of measurements, the gen-erally satisfactory agreement between our deduced fvalues and the calculated oscillator strengths could betaken as at least a provisional indication of applicabilityof the assumption made earlier about the excitationcross sections and rate coefficients. In this respect, it isfortunate that our measurements are least subject toexperimental error at the intermediate j, Ne v and Nevi, where the available calculated oscillator strengthshave the best claim for validity. This agreement alsoimparts some credence to the results at lower j, wherethe calculated oscillator strengths are subject to con-siderable uncertainties.

However, before leaving this topic, we must mentiona recent measurement of the radiative lifetime of the2s2p6 2S term of the Ne ii by G. MI. Lawrence and J.Hesser at the Princeton Observatory (private com-munication). Although the measurement did not yielda definite value for the lifetime, it did indicate a valuecertainly less than 2X 10-10 sec. The oscillator strength

31 E. Hinnov, Phys. Fluids 7, 130 (1964).

given in Table I implies a lifetime of 3 X 10-t° sec. Thereis therefore an indication of a definite discrepancy in thiscase, but whether it is caused by an error in the initialspatial distribution of neon, or in the assumed excitationcross section, or in our light-intensity measurements,can be decided only with further experimental data. Thefact that our result is in agreement with the extrapolatedCoulomb approximation is not particularly significant,as the Coulomb approximation cannot be expected to bevalid in transitions involving several equivalentelectrons.

In the case of the resonance transition of the lithium-like Ne viii, everybody appears to be in good agreement,including the above-mentioned lif etime measurement ofBerkner et al.,1 and a Hartree-Fock calculation byWeiss.3 2 Unfortunately, this is one of the cases where theuncertainties in the ion density determinations are re-latively large in the present experiments, so the agree-ment may still be somewhat fortuitous.

V. IONIZATION-RATE COEFFICIENTS

The rate of ionization of the jth neon ion in the dis-charge may be written as

d11AUj/dt = IN1 NeS'-A T3 *N hSj+ 1'-AN;l/7 , (13)

where -r is the particle confinement time given in Eq.(1). In this relationship it is assumed that there is nosubstantial spatial separation of the two ion species.Experimental evidence3 indicates that the jth ion hasa reasonably uniform radial distribution at all times, ifthe last term at the right in Eq. (13) is larger than themiddle term, i.e., if the confinement time is shorter thanthe lifetime with respect to further ionization. IfNSji+l> ,r-', the ion radial distribution is still uniformpast its peak density (referring to Fig. 7), but somewhereon the downward slope (--1.8 msec for N 1, 2.0-2.2msec for N 2) the radial distribution develops into a cyl-indrical shell. (Unfortunately the thickness of the shellis difficult to measure with precision, and, when it be-comes too thin, there is considerable doubt about thelocal electron temperature and density. However, roughestimates of these quantities indicate that the shellthickness is not inconsistent with the picture of ionsdrifting radially across the confining field with a velocity-rolir, where ro is the radius of the discharge.)

Avoiding such times when the spatial distribution isnot uniform, there is still sufficient overlap in time of theNj to attempt solution of the system of Eq. (13) for theSj1+', using the N1 in Fig. 7. To simplify this process,wve have regarded Si2 as known (determined from thecross section given by Eq. (8), with b=0.7), and as-sumed that the middle term in Eq. (13) is negligiblecompared to the last term for j= 7. We can then startat j= 7 and work to lower j using the Sji+l determinedin one step to find Sj4-i in the next; or we can start

32 A. W. Weiss, Astrophys. J. 138, 1262 (1963).

1187

EINAR HINNOV

KTe (Wv)

FiG. 8. Comparison of the ionization-rate coefficients calculatedfrom assumed cross sections, with the rate coefficients deducedfrom the time behavior of the ion concentrations.

with j= 2 and work similarly to higher j. In principleboth ways should yield the same results.

In practice, of course, they do not. There is con-siderable ambiguity in determination of the slopes of theNS, not to mention the uncertainties of the 1Vj valuesthemselves. Nevertheless, a moderately self-consistentpicture can be determined, and is shown in Fig. 8. Theionization-rate coefficients determined from the experi-mental Nj fit into the shaded areas designated by Sji+lwith probable values indicated by solid curves. Forcomparison, we have plotted the S1j+l calculated fromEq. (9), with all b= 1. The numbered squares indicateanother type of rough estimate that is sometimes usedfor the ionization-rate coefficient

Sjj+l1 [N,3(1j)Azj1_-l (14)

where Atj is the time interval between half-intensitypoints in Nj light, e.g., in Fig. 2, and tj is the time of thelight maximum. These squares are plotted for tempera-tures at tj.

Evidently, there is a rough agreement with the cal-culated values for S2

3 , but for larger j the "measured"values are increasingly too great. Furthermore, the dis-crepancy for a given j increases with decreasing tem-perature. One possible explanation of this is, of course,that the actual near-threshold ionization cross sectionsare larger than the values assumed in the calculation.However, the deviation seems somewhat too great to bereal: for Se,' b 8 would be required.

A more likely explanation is that the problem lieswith the electron temperature measurements. Thus it ispossible that the electron temperatures are actuallysomewhat smaller than those indicated in Fig. 1, andthe high conductivity is produced by an excess of high-energy nonthermal electrons. Because the ionizationpotentials are large compared to the average electronenergy, the ionization rates of higher j would be par-

ticularly sensitive to such distortions of the electronenergy distribution. Rough estimates indicate that the"measured" and "expected' rate coefficients could bebrought into agreement if 2%-8%, the percentage in-creasing with time in the discharge, of the electrons hadenergies larger than 300 eV. Although the presence ofsuch a large fraction of high energy electrons appearsimprobable, it cannot be ruled out on existing evidence.

Another possibility is that the electron temperature isactually somewhat larger than assumed, i.e., that theactual plasma resistivity is increased by some unknownprocess. An increase of T6 by a factor about 1.4 fort>2 msec in Fig. 1 would be required to eliminate thediscrepancies in Fig. 8. Again, this explanation appearsimprobable but not impossible.

V. CONCLUSIONS

We have attempted to determine the rate coefficientsof excitation and ionization of the various neon ionsfrom a relatively simple set of measurements in a high-temperature plasma device, with the hope that theseresults may enable us to determine the appropriate crosssections or transition probabilities. In the case of theionization cross sections, the results appear the moredoubtful, probably as a consequence of inadequateknowledge of the electron energy distribution in the dis-charge. On the other hand, in the case of excitationprobabilities, which do not depend critically on the details of the electron energy distribution, as long as theaverage energy is comparable to or larger than the ex-citation potential, the results seem at least reasonable.More detailed experimental work, especially at thehighest and the lowest states of ionization treated inthis paper, is necessary to modify or confirm the presentresults on the excitation rates.

For reliable measurements of the ionization-rate co-efficients it is necessary to establish the true electronenergy distribution in more detail. It appears that some-what higher electron densities would be more appropri-ate for these measurements, even though the measure-ment of the densities may be less convenient. On theother hand, the temperature range of the stellaratordischarges seems to be well suited for this purpose, be-cause the strong dependence of the rate coefficients ontemperature in this range offers the possibility of ac-curate measurements.

ACKNOWLEDGMENTS

The author is indebted to Dr. F. W. Hofmann andDr. A. S. Bishop for substantial contributions in theexperimental part of this investigation.

This work was supported under the auspices of theU. S. Atomic Energy Commission, Contract No.AT(30-1)-1238.

1188 Vol. 56