P. Jaegle et al- Main Aspects of Atomic Physics in Dense Plasmas

8/3/2019 P. Jaegle et al- Main Aspects of Atomic Physics in Dense Plasmas

1/11

JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 7, Tome 39, Juillet 1978,page C4-75

MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMASP. JAEGLE, G. JAMELOT, A. CARILLON and A. SURE AULaboratoire d e Spectroscopie Atomique et Ionique d u C.N.R.S.,

Universite Paris-Sud, BBt, 350,91405 Orsay, Fra ncean d

Gro upe de Recherches Coordonnees d u C.N.R.S. sur 1'Interaction Laser-Matiere,Ecole Polytechnique, 91120 Palaiseau, Franc e

Resume.- a mise en evidence experimentale de l'influence de la densite sur l'tmission de rayons Xmous par un plasma produit par laser conduit d etudier les phenomknes qui ont une influence surl'intensite et la largeur des raies spectrales.

Une etude detaillee du transfert de rayonnement en milieu collisionnel inhomogene s'imposepour pouvoir interpreter les observations experimentales des processus atomiques dans ces plasmas,de m&meque pour l'evaluation des bilans d'energie. Nous presentons ici les principaux aspects decette etude avec des exemples numeriques compares A des resultats experimentaux. Le cas des inver-sions de populations, pouvant engendrer une amplification de rayons X mous, est envisagt.

Nous montrons d'autre part l'importance du rBle des continua d'ttats situts au-de l des limitesd'ionisation dans l'ttablissement des populations de nombreux etats excitts des ions, rBle dQi lapresence d'electrons libres en grande densite. Les ttats autoionisants, etats discrets diZu-6~ ans lecontinuum, produisent des recombinaisons rksonnantes dont il faut tenir compte pour expliquerla composition du plasma en ions mais aussi les populations des niveaux excitts. La recombinaisondiklectronique est au nombre de ces resonances ; l est possible d'autre part de presenter des indi-cations preliminaires au sujet des resonances dans la recombinaison A trois corps. Nous developponsenfin la theorie des etats autoionisants pour parvenir i 'halua tion de la perturbation de la densited'ktats dans le continuum au voisinage des niveaux autoionisants.

Abstract.- xperimental evidence of a strong dependence upon the particle density, of softX-ray features of laser-produced plasmas, leads to investigate several phenomena taking effect online intensities and line widths.

A detailed study of radiative transfer in an unhomogeneous medium dominated by collisionsis a primary necessity for interpreting experimental observations of atomic processes in plasmas,and for investigating the energy balance as well. Main aspects of such a study are presented herewith numerical examples compared with experimental results. The case of population inversions,able to produce soft X-ray amplification, is considered.

On the other hand, it is emphasized that the continua of states above ionization limits of ionsare of a great importance for the population rates of many excited levels because of the large den-sity of free electrons occupying these states. Discrete levels, diluted in the continuum owing to auto-ionization process, induce resonances in recombination which must be taken into account forexplaining ion abundances as well as excited level populations. Dielectronic recombination andpreliminary indications on resonance in three-body recombination are presented. The theory ofautoionizing states is developed with a view to estimate the perturbation of density of states in thecontinuum close to autoionizing levels.

1. Introduction. - esides being of astrophysicalinterest, atomic physics applied to multiply chargedions is of a growing importance for plasma studiesrelated to the thermonuclear research. Within thisscope, plasmas submitted to magnetic confinementin Tokamaks and plasmas produced in laser implosionexperiments, giving rise to inertial confinement,exhibit each other a large difference in density scales

and in gradients of density and temperature. Thatfact leads to a substantial discrimination betweenthe requirements of atomic data for both types ofinvestigations. Calculations o f wavelengths a nd oscil-lator strengths of lines for many ions is necessaryfor the study of relatively low density plasmas confinedin Tokamaks. But for understanding the processesoccurring in very dense plasmas produced in laser

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978410
http://www.edpsciences.org/http://dx.doi.org/10.1051/jphyscol:1978410http://dx.doi.org/10.1051/jphyscol:1978410http://www.edpsciences.org/


2/11

P. JAEGLE, G. JAMELOT, A. CARILLON AND A. S U R E A U

FIG.1. - pectra of aluminium laser-produced plasma at high density (lower curve) and at low density (upper curve) ; chang es in intensityand l ine width are clearly seen.experiments such more data are required. Here wewill concentrate the attention on features resultingfrom high density.

Evidences of such features are easily found in theultraviolet or X-ray spectrum of the ions. An exampleis given on figure 1 which showes, in the soft X-rayrange, a high density spectrum (lower curve) ascompared to a low density spectrum (upper curve)for an aluminium plasma produced by laser impact.The highest density is .of about 102' electrons/cm3and lowering the density is obtained by shiftingthe observed plasma shell far from the target. It'can be seen that the relative and absolute intensities,the widths and, in some cases, the shapes of the linesare very affected by changing the density and, tosome extent, the temperature of the plasma.Generally speaking, the broadening of the lines bycollisional and quasistatic Stark effect is quite signifi-cant at high density. Moreover a challenging problemis still to find an evidence of Zeeman effect sincelarge self-generated magnetic field has been claimedin laser-produced piasmas [l]. However, the intensityand the profile of the line is dominated, in many cases,by the reabsorption of the radiation in the densecore of the plasma as well as in external cold shells [2,3].As a consequence, the interpretation of the spectrawith regard to line shape and intensity requiresthe calculation of the radiative transfer in takinginto account features as plasma unhomogeneity andDoppler shift due to ion radial expansion. This willbe the object of the second section.

On the other hand, because of the large densityof free electrons and the presence of ions of variousionization stages, the continua of states above ioni-zation limits are of a great importance in atomicprocesses occurring in dense plasmas. The role ofautoionizing levels of the ions is extensively studied.Indeed such levels, corresponding to a two or moreelectron excitation, consist in a mixing of discreteand continuous states of the ions ; ue to their locationin the energy range of the free electrons these levelsyield channels for recombination mechanisms ableto populate some particular excited levels or tomodify the total population of an ion species. Theenhancement of density of states in a small energyinterval arround the level and the ability of an auto-ionizing state to interact with discrete states morethan a pure continuous state does make these mecha-nisms to be very efficient. Thus the third sectionwill be devoted to the role of autoionizing levels,especially in phenomena such as dielectronic recombi-nation or resonant three-body recombination.

2. Radiative transfer in discrete lines. - . 1 LIMITOF OPTICAL THINNESS IN LASER-PRODUCED PLASMAS. -In optically thin plasmas, the spectral line intensitiesdepend directly on transition probabilities and upperlevel populations. If reabsorption occurs, not onlythe intensity is modified but the width of the emergingline does no longer correspond to a pure theoreticalprofile. As far as Stark effect would be estimatedfrom the observed line width for particle density


3/11

MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS C4-77

measurement, the reabsorption could give rise to alarge inaccuracy. Before to come into a detailedtreatment of line shape formation in this case, let usderive a simple criterion of optical thinness that canbe put on in dense plasmas of small volume.

Under the assumption that the plasma is homoge-neous, the absorption coefficient for a line of energy Eand width SE is given by :

where A , , is the spontaneous transition probabilitybetween levels 1 and 2, NI and NZ, the populationdensities of the levels, g, and g,, their statisticalweights. A typical size of a laser-produced plasma isof 100y ; then the optical thickness will be less than0.5 if :

k < 50 cm-'is satisfied. Under this condition the total reductionof line intensity will not exceed 40 % and one cancalculate that resulting additional broadening ofthe line will be approximately of 15 %. Thus foravoiding larger errors we must fulfil :

where NI and N, are in cm-3, E and SE, in eV, A , ,in S-'. For most of the transitions from the groundstate of the ions, this can be replaced by :

where N is the population density of the ground state.As an example, let us consider a magnesium plasmain which the 1s-2p line of Mg XI1 exhibits a widthof 3 eV ; we have E -- 1 500 eV and

The reabsorption will have a negligible effect uponthe line width provided that :

while the 1s-5p line, at 1 900 eV [4], has a transitionprobability of 2.15 X 10" S- ' and allows densityas large as :

It results from (3) that, for most of the ions whosethe resonance lines are lying in the soft X-ray range,the reabsorption becomes signficant at density of10'' cmw3.

2 . 2 TRANSFERF RADIATJON IN A MEDIUM DOMI-NATED BY COLLJSJONS. OWwe will start from thewell known equation of radiative transfer [5] :

dIv = Ci, - k, I,) dx (4)where v is the .frequency, I,, the intensity, j, and k,,the emission and absorption coefficients, dx, a verysmall length travelled by the light in the plasma.To calculate I, from (4), we need suitable expressionsof j, and k, accounting for all the properties of themedium at the frequency of interest. For an isolatedline, at any frequency these coefficients depend on :i) the total transition probability of the line, ii) theprofiles of spontaneous emission, stimulated emis-sion and absorption, iii) the population densities ofupper and lower levels, iiii) the contribution of thecontinuous spectrum.

From very general arguments the profiles ofstimulated and spontaneous emission can be taken foridentical [6] but they differ from the absorptionprofile. Considering moving ions, this is due to thescattering of radiation which gives rise to a smallshift in frequency between absorbed and emittedphotons when their directions are different. Theprofiles then are related by the so-called frequencyredistribution function [7, 81. However, in denseplasmas, the typical time of radiative scatteringprocess in many cases is larger than the time betweentwo electron impacts and so emission and absorptionprofiles are independent each other. Furthermore,even in the X-ray range where radiative probabilitiesare large, the profiles are dominated by Stark effectdue to near particles or otherwise by thermal Dopplereffect, according to the values of the plasma para-meters. Thus, one and the same profile is to be usedfor both emission and absorption.

On the other hand, in a general treatment ofradiative transfer it is necessary to solve a system ofcoupled equations expressing simultaneously theradiation propagating at several wavelengths andthe population rates of all the levels involved. Indeed,the dependence of absorption and stimulated emissionon radiation intensity entails a reaction of the radiationon excited level populations. In dense plasmas, therole of collisions in dominating ion level excitationallows again a great simplification. For any twoievel transition, an equation of type (4) can be inte-grated separately. In homogeneous plasma, one isleft to :

where X is the length and Z,(O) an incoming intensity ;j,/k, is often called the source function. Unfortunatelythe use of (5) is very limited for accounting for observedspectra because of the strong unhomogeneity ofthe plasmas in practical cases. Therefore equation (4)must be integrated numerically in order to include


4/11

(24-78 P. JAEGLE, G. JAMELOT, A. CARILLON AN D A. SUREAU

the effects of non-homogeneous plasma. This hasbeen done recently in a few works [9, 10, l l].

2 .3 LINEPROFJLES.- actors coming from diffe-rent origins participate in making up the line shapeexhibited by dense hot plasmas. Besides Stark andDoppler effects and absorption broadening, we mustpoint out at least the role of the continuous spectrum,which often is significant in the extreme ultravioletrange, and the role of the ion expansion velocitygiving rise to a non-thermal Doppler effect.

A proper account of the continuous contributionscoming from free-free and bound-free transitionsis of great importance for interpreting self-reversedand absorbed profiles. If emission and absorptionof radiation of frequency v proceed from discretetransitions at once with continuous spectrum, thecoefficients j, and k , are to be written :

where jLand k , are sums over all the cooperatingdiscrete transitions, jc and kc being integrals overthe space of free electron momenta. Substituatingthese expressions in (5) shows that discrete and conti-nuous spectra does not provide the emerging intensitywith separate summed contributions, although thisfact has been disregarded in many circumstances.That is why the transitions with free states takesome part in the line shape formation.

For instance, it has been shown that a discreteline will exhibit an absorbed profile, even thoughthe level populations are in statistical equilibrium,provided that the continuous absorption coeffi-cient kc is large enough [9, 121. But in case of weakcontinuous absorption, absorbed profiles are reliableevidences of underpopulation of the upper levelsof the transitions under consideration. In fact, thecontinuous emission must also be large for an absorbedprofile to occur in the case of statistical equilibrium.In order to give a numerical example, let B,, the black-body emission intensity at the temperature of theplasma. Let us consider a line which is self-reversed(intensity in the center less than in the wings) upona continuous spectrum of intensity 0.1 B, and ofoptical thickness 0.29 (25 %-absorption) ; this lineis still self-reversed for an optical thickness of 4.6(99 %-absorption) if the emitted intensity does notincrease; but it becomes absorbed (white line) assoon as the optical thickness reaches 2.5 (92 %-absorption) if the emitted intensity rises up to 0.7 B,.

For numerical integration of (4), in using theclassical expressions of Bremsstrahlung [5] the coeffi-cients jc and kc of (6) can be written :

where N , and Teare the density and the temperatureof electrons, varying along the light path ;p and qare adjustable parameters.

As regards the ion velocity, it is to be consideredas one of the most probable causes of profile asym-metry [10, 111. At a distance X into the plasma, themotion of ions removes the frequency v, of the topof the line profile in such a way that v, must be replacedby vo(l + v(x)/c). For a spherical expanding plasma,the component u(x) of the ion velocity along the propa-gation, axis can be approximately written as :

where R, is the radius of the plasma front expandingwith velocity v , ; X, is the coordinate of the centerof the plasma. Beside the asymmetry of the profile,that can be seen on figure 2, the optical thickness

FIG.2. - symmetrical line sha pe interpretation. The lower curveis the densitogram of a photographic recording of AI4+ lines.The upper curve is the calculated line shape at the wavelength of126.06 A for the expanding plasma model shown in inset.is altered by the non-thermal Doppler effect. As wewill see below, ion expansion is a limiting factorfor the gain which could be expected in the occurrenceof population inversion. This fact must be takeninto account in experiments dealing with long plasmasin soft X-ray laser research.

Turning back to expressions (6), we will assumethat only one discrete transition contribute to radiationof frequency v. In using the same notations as in


5/11

MAIN ASPECTS O F ATOMIC PHYSICS IN DENSE PLASMAS C4-79

paragraph 2.1, and putting B,, for Einstein's coeffi-cient for absorption, we have :

Here @(v) s the profile function, common to emissionand absorption, which we will briefly discuss in thepresent paragraph. Before all, let us point out thefactor 1/6E = llh6v to replace the function 4(v)in a simplified expression like (l), what amountsto take in a rectangular profile. Although a rectangleis a poor approximation of line shape, it enablesa satisfactory expectation of the general behaviourof the line, provided that the width is properly esti-mated. For instance, results reported in paper ofreference number [l31 suggest that the effect of aVoigt profile instead of a Gaussian profile is due farless to the modification of analytical shape thanto the account, in the first case only, of the collisionalbroadening which predominates in dense plasmas.

As a matter of fact an exact expression of the profilefunction 4(v) can be found only when the thermalbroadening exceeds largely any other contribution,but the natural width (if the radiative transitionprobability is very large). Under conditions givingrise to a dominant Stark broadening, as in laser-produced plasmas [14], it is not worth seeking sophisti-cated profiles for radiative transfer, so long a largeinaccuracy will affect the theoretical prohles of non-hydrogenic lines. To combine ionic quasi-static effectwith electronic impact effect is an actual difficulty [l 51.Moreover, quasi-static and impact approximationsare questionable in many cases and must be replacedby more accurate intermediate calculations [16].A moderate position, which is valid for densitiesnear the critical shell of Nd-laser produced plasmas,consists in using a Lorentzian profile whose thewidth 6v accounts for the intensity of electron collisionperturbation [17]. Thus we define @(v) as :

As far as all collisional rates are proportional toelectronic density, the same dependence must befound in 6v. For the examples given in this paperwe took :

where T, is the electronic temperature. The dependenceagainst T,- ' l2 has been set as a rough approximationtaking into account the fact the thresholds of the mostimportant collision induced excitations to be verysmaller than the electronic temperature : then theirrate goes down slowly with increasing temperature [l 81.

In order to be ensured from an unexpected largeeffect of the form of 4(v) we performed also calcu-lations using a T,- ependent Gaussian profile. Oncondition that the magnitude of the width 6v is thesame as in (IO), the general behaviour of the lineremains unchanged [12].

2.4 SHAPESF OPTICALLY THICK LINES. - xpres-sions from (4) to (1 1) have been used for numericalcalculations of line shapes in dense unhomogeneousplasma. A plasma model has been chosen so that itdepictes a section of laser-produced plasma witha fairly homogeneous hot dense core, having adiameter of 100 p, and an external shell of conti-nuously decreasing temperature and density. Theexpansion velocity- has been deduced from fittingan experimental result as shown on figure 2. Theresults show, on figure 3, the role of the ionic density

FIG. .- hange in line shape versus onic densityNi (the values areindicated for the core of the plasma). Ionic level populations areassumed to obey statistical equilibrium. N , is the electronic density,T,, the electronic temperature, V,, the front expansion velocity.

which puts the optical thickness up or down in aconstant proportion over all the frequencies coveredby the line, and, on figure 4, the role of the collisionalbroadening which carries opposite effects on thecenter and on the wings of the line.

The most pertinent indications infered from calcu-lations, performed under the assumption of statisticalequilibrium of excited levels, are : ) the enhancementof optical thickness by increasing the ionic densitydoes not generate absorbed profile but emphasizesthe features of self-reversed profile, the maximumintensity remaining almost constant and the widthgetting more and more large, ii) the lowering of themaslmum intensity joined with some smoothingof the shape reveals the collisional broadening toprevail in line shape. Nevertheless, it can be seenthat a detailed discrimination of all physical factors


6/11

C4-80 P. JAEGLE, G. JAMELOT, A. CARILLON AND A. SUREAU

FIG. 4.- he upper part of the figure shows a change in line shapeversus collisional broadening. The transition probability being keptconstant. The lower part shows the emerging line shape corres-

ponding to each profile.

reflected by the line shape will always require acareful examination of many informations of experi-mental and theoretical origin.

2 .5 SPECTRAL EATURES DENOTING UNBALANCEDEXCJTATJON AND POPULATJON INVERSION. - wing tothe development of dense plasma investigations forachieving soft X-ray amplification, a prime attentionis lent to the spectral features resulting from unba-lanced excitation, especially from population inver-sions [9, 191. Experimental results have been reportedfor some multicharged ions of carbon, for whicha rather simple line intensity analysis seems ableto prove weak population inversions because theplasma density is small enough to avoid all reabsorp-tion effects and, on the other hand, the transitionprobabilities are accurately known for these ions [20,21, 221. Here we report, on figure 5, the calculationsconcerning the spectrum of neon-like ions of alumi-nium in the neighbourhood of the critical density(10,' of a laser-produced plasma. The experi-mental spectrum can be also seen on figure 1(4d 'P,, 3D1, 3P1)with a comparison between highand low density features. From figure 5, it resultsthat both underpopulations of 'P, and 3 ~ ,evelsand population inversion between 3P1 and groundlevel account for experimental spectrum. No such

xi-FIG. 5. - n the right : experimental spectrum. In the middle :calculated spectrum in assuming 3P , population inversion and 'P,,

3D, under-populations a s shown on the left part of the figure.

account can be found if balanced populations areassumed, whatever the plasma model we choose.

Figure 6 reveals what may be an impediment infuture experiments on soft X-ray amplification byplasmas. In homogeneous plasma model, of length300 p, we calculated the maximum intensity and thewidth AV of emerging line, versus a populationinversion N,/N,, for several expansion velocities;A,v is the width of the optically thin expansion-freeprofile. Obviously, the plasma expansion will havea damaging action on the line amplification. However,very few is known on expansion of long plasmasproduced by cylindrical focusing of a laser beam.

FIG. 6.- f expansion velocity in cancelled, intensity I at thecenter of the line increases exponentially versus population inver-sion, whereas line width AV decreases (solid curves). Velocityexpansion of 7.5 X 106 cm/s (dashed curves) - fterward1.5 X 10' cm/s (dash-dotted curves)- rings down the gain andenlarges the line width (I = 300 p ; N , = 10'' ; N, = 3 X 10").

3. Resonances in recombination. - Among thecauses of spectral modifications appearing in relationwith plasma density, as shown on figure 1, we havenow discussed several factors taking place in radiativetransfer. Another class of phenomena, which has tobe investigated for understanding the experimentalobservations, regards the rates of populations, sincethe terms like N I and N , in expressions (6) of emissionand absorption coefficients are fixed by equations


7/11

MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS

making up many transitions from discrete and conti-nuous levels. From a general point of view, thisproblem has been often visited of late years for astro-physical and laboratory plasma studies [23, 241.However, the ability of generating very dense plasmasis a new fact in the course of investigations ; processeswhich were of little importance in earlier studiesmust be now considered for accurate calculations.

This is especially true for the collisional processesinvolving more than one electron, since their frequencygoes up rapidly when the density does so. Indeed,the two electron processes provide the plasma recombi-nation with an alternative to the one-electron mecha-nism. This ought to lead to develop quantum mecha-nical calculation of electron impact ionization cross-sections, from which the three-body recombinationcan be computed owing to the detailed balanceprinciple. As continuous states of ions are involvedin these processes, we deal necessarily with thestructure of the ion continua which are anythingbut smooth because of the presence of many auto-ionizing states.In the first part of this section we will reviewbriefly various atomic processes involving auto-ionizing states, investigated at the present time onaccount of their role in plasmas. The second partwill contribute to the theoretical study of autoionizingstates themselves in approaching the calculation ofthe perturbed density of states in the continuumsurrounding an autoionizing level. This is done inview of future improvements in rate coefficientscalculations for processes between free electronsand ions.

3.1 PROCESSESNVOLVJNG AUTOIONIZING STATES.-On figure 7 are shown resonances occurring in variousprocesses in consequence of the excitation of anautoionizing level by electronic impact. Let Z bean ion of charge Z in its ground state. The stars willdenote the excitation of one (*) or two (**) electrons.When the energy of a twice excited state is largerthan the lowest ionization energy - hich corres-ponds to the ion Z + 1 in its ground state - hisstate is known to get a radiationless decay processby transition to the continuum (autoionization).

Figure 7a shows the pseudo-stationary state Z**appearing in the course of the scattering of a freeelectron by the ion Z + 1, that is a radiationlesscapture followed by autoionization. The energiesof impinging and ejected electrons can be differentif a photon is emitted or absorbed. Then, the excitationof autoionizing level Z** is inducing a resonance in theBremsstrahlung spectrum of the electron in the fieldof the ion Z + 1 [25]. Figure 7d represents anotherprocess in which the resonant scattering of a freeelectron leads to a change of energy; but now thischange is due to the excitation of the target. In fact,here we see that autoionizing resonances will givecontributions to collisional excitation cross-sections.

FIG.7. - Resonances produced by autoionizing levels in ion-electron scattering : a) resonance in free-free transitions, b ) die-lectronic recombination, c) resonance in three-body recombination,d) contribution to excitation cross-section. Z and Z + 1 representrespectively the ground levels of ions of charge Z and Z + 1.

It has been shown [26] these contributions to be verysignificant for incoming electrons of energy closeto the excitation thresholds.

On figures 7b and 7c are represented two differentmechanisms of resonant recombination. Both areinitiated by the encounter of a free electron withan ion Z + 1, resulting in a twice excited state 2""of ion 2. ~ h e k ,his last can be stabilized in a singleexcited state either by radiative transition (Fig. 7b)or by a new electronic collision (Fig. 7c). The firstprocess is the well known dielectronic recombi-nation, which has been studied for explaining theion abundances in the solar corona [27]. In the secondcase, we have a resonance in the continuous spectrumof three-body recombination :

For a specific situation, figure 8 due to Landshoffet al. [28], shows the great role played by these reso-nances in laser-produced plasmas. It must be pointedout that the curve denoted by dielectronic on thisfigure represents in fact the total rate of both above-mentioned mechanisms. Until now, such calculationstook only aim at a right expectation of ion abundancesin plasmas, which requires to sum a large number ofapproximately known recombination rates, corres-ponding to all the resonances of the ion. However,in order to explain unbalanced excitation of ionsin plasmas (see Sec. 2, 3 2.5), it will be necessaryto perform detailed rate calculations for specifiedexcited levels of ions.

In the case of pure dielectronic recombination,


8/11

C4-82 P. JAEGLE, G. JAMELOT, A. CARILLON AND A. SUREAU

DIE LECTRONIC

FIG.8.- ate of resonant recombination as compared to radiativeand collisional non-resonant processes [28].for a particular stabilizing transition, the recombi-nation rate may be written [29] :cr, = A r A , g,** h exp (- EIKT)A , + A , 2 g,, , (2 n r n K ~ ) ~ ' ~

where a, is in cm3. - ' if cgs units are used ; g,,, is thestatistical weight of Z**, g,,, is the statisticalweight of (Z + l)-ion ground state ; T is the tempe-rature; E is the kinetic energy of the impingingelectron before the capture; A , is the radiativetransition probability from Z** to Z* and A , isthe autoionization probability of Z**, which canbe expressed under the form :

Here, o, s the cross-section for radiationless captureof an electron of energy E and velocity v ; g(E) isthe number of free electron states per unit volumewith energy between E and E + dE. For a Maxwelldistribution of free electrons, the probability offinding an electron with energy between E and E + dEin the volume of an ion is f (E) dE, with :

and :

Putting (15) in (13) and substituating in (12) weobtain the formula derived by Burgess [30] :

Since the autoionization probability A, of relation (13)depends on free electron state density, this treatmentincludes partially the feature of autoionizing levelconsisting in superposition of continuous states toa discrete one. For practical use of formula (16),E is replaced by the energy difference between twosuccessive resonances. However, one can see thatthe radiative transition probability A , appears in(16) as if both Z* * and Z* were pure discrete levels.A more general treatment can be found in 127, 311.On the other hand, expressions (14) and (15) donot account for the modification of density of statesat energies close to autoionization. This modificationwill be examined in the next section.

Before to come to the case where stabilizationtakes place without emission of photons, we mustmention an important application of the abovecalculations. If dielectronic recombination is theonly process yielding Z * * , all radiative transitionshaving Z** as upper level will depend on the tempe-rature in accordance with a, in (16). Thus theselines, which are said dielectronic satellites of ( Z + 1)-ion, can give a valuable temperature-jauge for plasmadiagnostics [26, 321.

Now, the resonant three-body recombination, repre-sented on figure 7c, can be expressed as :

a free electron removing the energy difference betweenZ** and Z*. The inverse process is a resonance inion-electron impact ionization. Thus, while under-standing of dielectronic recombination dependedon a development of photoionization theory, resonantthree-body recombination requires improvements inelectron impact ionization calculations, which arestill more complex for quantum mechanics. Let( vo i ) be the electron impact ionization rate, afteraveraging over a Maxwell distribution of free elec-trons; from the detailed balance principle, thecoefficient X, of three-body recombination will obey :

where E, is the ionization energy from Z* level.


9/11

MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS C4-83

Returning to the simpler case of photoionization,it is well known that Fano's theory of autoionizingstates [33]' results in a parametric expression ofphotoionization cross-section in the neighbourhoodof the resonance. Let o, be the cross-section for transi-tion between a discrete state and a pure continuousstate of the ion. Near an autoionizing level, the photo-ionization cross-section may be written :

with :E - EE = - r

r being the half-width of the resonance and E afixed energy which is little different from the energyof the pure discrete state involved in the autoionizinglevel. The shape of the resonance depends on thevalue of Fano's g-parameter. Figure 9 gives a practical

FIG.9.- utoionizing profile of photoabsorption observed at thewavelength of 84.37 A in an aluminium laser-produced plasma.example of the photoabsorption profile due to sucha resonance observed in the spectrum of AI3+-ions,in laser-produced plasma [34]. In fact, the rate ofdielectronic recombination must be integrated oversimilar cross-section profiles instead of using a purediscrete transition probability like A , in (16).

The parametrization appearing in expression (1 8)can be extended to resonances in electron-impactionization [35, 36, 371. In the work of Tweed [37]the definition of the g-parameter of Fano is extendedto the range of validity of the first Born approximation,including exchange effects. The fact that two freeelectrons are present in the final state (initial statein the case of recombination !) leads to two possibleexpressions of differential cross-sections, accordingto the choice of the first angular integration. Inassuming that a scattered electron, leaving the targetimmediately after the collision, is distinguishablefrom an ejected electron which is slightly delayed by

the life-time of the resonance, Tweed defines anionization cross-section o;jectedorresponding to afixed angle for the ejected electron and to integratedmomenta over all angle of electron scattering. Theparametric form of this cross-section is :

where a, b, aoi and a;jec'"* re functions of ejectedelectron momentum. The sum in expression (19) isover the values of the quantum numbers specifyingthe resonance, including the total spin S . The totalcross-section appearing in (17) could be obtained byintegrating parameters a and b, as well as the non-resonant part c,,, over the ejected electron momentum.

However, as far we know, ab initio calculationsof the resonant ionization cross-section oi have notbeen performed until now and a few neutral atomsor single ions have been experimentally investigated.Thus it remains difficult to include reliable values ofresonant three-body recombination coefficients inplasma numerical models, although this process islikely of large consequence at high density.

3.2 DENSITYF STATES IN THE NEIGHBOURHOOD OFAUTOIONIZING EVELS.- n the last paragraph of thispaper we wish to make use of the autoionizing statetheory in discussing statistical aspects of the roleof these states for the ions in a plasma. The start pointof this discussion can be easily understood withrererence to the problem of Boltzmann's distributionfor autoionizing levels. In a local thermodynamicalequilibrium plasma, for an ordinary excited levelthe population ratio with respect to the ground levelis given by :

But this relation cannot be employed directly forautoionizing levels unless the mixed discrete-conti-nuous feature of these levels is neglected. Further-more a plasma has in it a gas of free electrons whichare occupying the continuum of states of the ions.Thus free electrons take part to the population ofautoionizing levels. We will first give an approximateexpression of this part. Then, a more detailed studyof the structure of the continuum of states near anautoionizing resonance will be presented. Let usconsider the mixing of discrete and continuousconfigurations which results in the autoionizing statewave function :

We first consider an approximation in which theactual continuum of states involved in the integralon the right member of this expression is replaced


10/11

C4-84 P. JAEGLE, G. JAMELOT, A . CARILLON AND A. SUREAU

by a pseudo-continuum of discrete states. Then,substituting for b new coefficients correspondingto this approximation (dirnentionally different from b)and denoting by ( b2 ) the mean value of theirsquares in small energy range AE where the discrete-continuous interaction is meaningful, normalizationof (21) implies :

Introducing now the statistical weight of free electronssurrounding (2 + l)-ions :

and taking into account that :

we obtain the following expression of the statisticalweight of the autoionizing state :

where g, is the statistical weight of the discrete stateinvolved in the resonance. Making use of Boltzmann'slaw gives the population ratio :

gz+ l 8 n rn312(2 AE+ ( b 2 ) - - h exp(-g )z Nz+1 (26)which enables to express the population NZ,, of theautoionizing state according to the probabilities a2and ( b2 ) that the system is in a discrete or a conti-nuous state. If a2 + 0, in using Saha equation forthe value of the ion density ratio N Z + / N Z , 26) reducesto the fraction of free electrons of energy comprisedbetween E and E + AE in a Maxwell distribution.Instead, if ( b2 ) + 0, (26) is identical to Boltzmann's.distribution for ordinary excited levels.

However, in Fano's treatment of autoionizingstates, the wave functian on the left memberof (21), which varies quickly in the energy range ofthe resonance, describes in fact an unbound statedepending on E, namely the discrete state is dilutedin the continuum. Calculation of coefficients a and bof the right remember of (21) leads to write 1+9 underthe form :

SinA cp. - cos A . $,E =where cp' is a modiJied discrete state, A is a phaseshift in the continuum wave function caused by the

discrete-continuous interaction; V E is the matrixelement ( cp I X I4 ).

Now the probability to find a quantum-mechanicalsystem in a given subspace equals the trace of theoperator resulting from the product of the densityoperator and the projection operator in this subspace.We consider the subspace of energy states comprisedbetween E and E + AE, in the volume A3r roundthe ion. We have the projection operators :

and the density operator for a L.T.E. plasma ise- Je /KT

P=- Qwith :

Q = Tr { e-3e1KTX being the Hamiltonien of the system. We have :

Using the expansion (27) with the value of A calculatedin [33] and taking A3r the volume in which the ionicwave function is not negligible, (28) gives for thenumber of states in the energy interval AE [38] :

e - ~ ~ ~ ~in2A sinA~ = e ~ ~ ~ y [ ~vE12+{A3rd3r(From this expression, general feature of density ofstates in the neighbourhood of an autoionizingstate is sbown on figure 10. The autoionizing stateraises a distorded Lorentzian upon the exponentialcurve describing the density of states in the continuumof the ion. The ratio between the maximum d, of

Density of states

FIG. 10.- tructure of the continuum of states around an auto-ionizing level. E q is the energy of the pure discrete state cp. Themixing of cp with the continuum leads to a shift of the peak energyin the distribution of density of states.


11/11

MAIN ASPECTS O F ATOMIC PHYSICS IN DENSE PLASMAS C4-85

the irregularity and the normal value dc of the densitycan be deduced from (29) in putting :

cos A = 0at the top of the resonance, and :

sin A = 0for the pure continuum. This yields

which represents the relative enhancement of densityof states due to the presence of the discrete state cpabove the ionization limit. It must be pointed outthat the maximum of the enhancement dM takesplace generally at an energy slightly shifted withrespect to the energy E,, of the discrete state cp.

4. Conclusion. - Having in mind considerablevariations of spectra emitted by the ions of laser-produced plasmas, we have proceeded to a surveyof density-dependent processes concurring to suchvar~atlons. kadiatlve transfer, on the one hand,resonances in recombination, on the other hand,have been investigated in detail as two of the mostsignificant causes of observed features.

We deduce from this study that the difficultiesencountered in a so complex medium, for inter-preting a collection of experimental data, are largelyreduced by modelling the plasma properly for numeri-cal computation ; moreover, quantitative balanceprediction involving many particular processes willrequire to deal energies 'in developping quantumcalculations, especially by building good approxi-mations, suitable for practical purposes. In thisway, dense plasma studies offer to atomic physicsan attractive field of original works, emphasizingfeatures like the role played by continuous-discretemixed states in presence of a dense gas of free electrons.

References

[ l ] STAMPER,. A., RIPIN,B. H., Phys. Rev. Lett. 34 (1975) 138.[2] WEISHEIT, J. C., TAR TER , . B., SCOFIEL D,. H., RICHARDS,L. M., J . Quant. Spectrosc. Radiat. Transfer 16 (1976) 659.[3] CHASE,L. F., JORDAN,W. C., PEREZ, . D., PRONKO,. G.,Appl. Phys. Lett . 30 (1977) 137.[4] BAYANOV, . P., G ULID OV ,. S., M AK , A. A., PEREGUDOV ,G. V., SOBELMAN,. I., STARIKOV,. D., CHIRKOV,. A.,J.E.T.P. Let t . 23 (1976) 183.[5] BEKEFI, ., Radiation Processes in Plasmas (John Wiley andSons) 1966.[6] OXENIUS,.. J. Q uant. Spectrosc. Radiat. Transfer5 1965) 771.[7] IVANOV,. V., Transfer of Radiation in Spectral Lines, N.B.S.special publication 385 (1968).[8] RICHTE R, ., in Plasma Diagnostics, W. Lochte Holtgreveneditor (North-Ho lland) 1968.[9] JAEGLB,P., JAM ELOT, ., CA RILLON , ., SUREAU, . , LaserInteraction and Related Plasma Phenomena (PlenumPublishing Corporation, N.Y.) 1977, vol. 4, 229.[l01 JANNIT I, ., NICOLISI, ., TO NDELLO, ., G ARIFO,L., MAL-VEZZI, . M ., in the volume referenced unde r no 9.[l ] IRONS, . E., J. Phys. B 8 (1975) 3044 and 9 (1976) 2737.[l21 JAMELOT, ., T h b e , Universitt Paris-Sud, Orsay (1977).[l31 APRUZESE,. P., DAVIS,., WITNEY, . G., J. Qu ant. Spectrosc.Radiat. Transfer 17 (1977) 557.[l41 POQUERUSSE,., These, UniversitC Paris-Sud, Orsay (1975).[l 51 GRIE M,H. R., Spectral Line Broadening by Plasmas (AcademicPress, N.Y.) 975.[ l61 CABY-EYRA UD,., COULAUD, ., NGUYE N-HOE ,. Quant.Spectrosc. Radiat. Transfer 15 (1975) 593.[l71 BARAN GER, . , in Atomic and Molecular Processes, ed. byD. R. Bates (Academic Press, N.Y.) 1962.[l81 VAINSTEIN,. A., SOBELMAN,. I., YOUKOV, . A., SectionsEficaces de Collisions Atome-Electron et Ion-Electron,Phys. Math. (en russe), Moscou (1973).

[l91 JAEGLB, ., JAM ELOT,., CARILLON, . , SUR EAU, ., DHEZ, .,Phys. Rev. Lett. 33 (1974) 1070.[20] IRONS, . E., PEACOCK, . J ., J. Phys. B 7 (1974) 1109.[21] DEWHU RST, . J., JACOB Y, ., PERT,G. J. , RAMSDEN,. A.,Phys. Rev. Lett . 37 (1976) 1265.[ 2 2 ] D r x o ~ , . H., ELTON,R. C., Phys. Rev. Lett. 38 1977) 1072.[23] MCWIRTHER,n Plasma Diagnostic Techniques, ed. by R. H.Hud dleston e and S. L. Leonard (Academic Press, N.Y.)1965.[24] Atomic Processes and Applications, ed. by P. G . Burke andB. L. Moiseiwitsch (North-Holland, Amsterdam) 1976.[25] BELL,K . L., BURK E, . G., KINGSTON, . E., BERRINGTON,K . A., Abstracts of Papers at the Xth ICPEAC (Paris,July 21-27, 1977) 348.[26] PRESNYAKOV ,. P., Ysp. Fiz. Naouk 119 (1976) 49.[27] SEATON, . J. , STOR EY,. J., in the volume referenced underno 24.[28] LANDSHOFF,. K. , PEREZ, . D., Phys. Rev. A 13 (1976) 1619.[29] Electronic and Ionic Impact Phenomena, H . S. W. Massey,H. B. Gilbody (Clarendon Press, Oxford) vol. 4, (1974).[30] BU RGESS , ., Astrophys. J. 139 (1964) 776.[31] DU BAU , ., Thesis, University of Lo ndo n (1973).[32] BHALLA, . P., G AB RIEL , . H., PRESN YAK OV,. P , , Mon.Not. R . Astron. Soc. 172 1975) 359.[33] FANO,U,, Phys. Rev. 124 (1961) 1866.[34] JAMELOT, ., CARILLON, A., JAEGL E, ., SU RE AU , ., 1. Physi-que Lett . 36 (1975) L-293.[35] SHORE, . W., J.O.S.A. 57 (1967) 881.[36] BALASHOV, . V., LIPOVET SKY,. S., SENASHENKO,. S.,Sov. Phys. J.E.T.P. 36 (1973) 858.[37] TWEED, . J. , J. Phys. B 9 1976) 1725.[38] SUREAU,A., JAEGLE,P., IInd International Conference onInner Shell Ionization Phenomena, Contributed Papers(Freibu rg, Marc h 29-April 2, 1976) 97.

Documents

P. Jaegle et al- Main Aspects of Atomic Physics in Dense Plasmas