MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS

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MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE

PLASMAS

P. Jaeglé, G. Jamelot, A. Carillon, A. Sureau

To cite this version:

P. Jaeglé, G. Jamelot, A. Carillon, A. Sureau. MAIN ASPECTS OF ATOMIC PHYSICS

IN DENSE PLASMAS. Journal de Physique Colloques, 1978, 39 (C4), pp.C4-75-C4-85.

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

MAIN ASPECTS OF ATOMIC PHYSICS

IN DENSE PLASMAS

P. JAEGLE, G. J A M E L O T , A. C A R I L L O N a n d A. S U R E A U Laboratoire d e Spectroscopie Atomique et Ionique d u C.N.R.S.,

Universite Paris-Sud, BBt, 350,91405 Orsay, France a n d

G r o u p e d e Recherches Coordonnees d u C.N.R.S. sur 1'Interaction Laser-Matiere, Ecole Polytechnique, 91 120 Palaiseau, France

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

Une etude detaillee du transfert de rayonnement en milieu collisionnel inhomogene s'impose pour pouvoir interpreter les observations experimentales des processus atomiques dans ces plasmas, de m&me que pour l'evaluation des bilans d'energie. Nous presentons ici les principaux aspects de cette etude avec des exemples numeriques compares A des resultats experimentaux. Le cas des inversions 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 a u - d e l des limites d'ionisation dans l'ttablissement des populations de nombreux etats excitts des ions, rBle dQ i la presence d'electrons libres en grande densite. Les ttats autoionisants, etats discrets diZu-6~ dans le continuum, produisent des recombinaisons rksonnantes dont il faut tenir compte pour expliquer la composition du plasma en ions mais aussi les populations des niveaux excitts. La recombinaison diklectronique est au nombre de ces resonances ; il est possible d'autre part de presenter des indications preliminaires au sujet des resonances dans la recombinaison

A

trois corps. Nous developpons enfin la theorie des etats autoionisants pour parvenir i l'haluation de la perturbation de la densite d'ktats dans le continuum au voisinage des niveaux autoionisants.

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

A detailed study of radiative transfer in an unhomogeneous medium dominated by collisions is 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 here with 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 ions are of a great importance for the population rates of many excited levels because of the large density of free electrons occupying these states. Discrete levels, diluted in the continuum owing to autoionization process, induce resonances in recombination which must be taken into account for explaining ion abundances as well as excited level populations. Dielectronic recombination and preliminary indications on resonance in three-body recombination are presented. The theory of autoionizing states is developed with a view to estimate the perturbation of density of states in the continuum close to autoionizing levels.

1. Introduction. - Besides being of astrophysical interest, atomic physics applied t o multiply charged ions is of a growing importance for plasma studies related t o the thermonuclear research. Within this scope, plasmas submitted t o magnetic confinement in T o k a m a k s a n d plasmas produced i n laser implosion experiments, giving rise t o inertial confinement, exhibit each other a large difference in density scales

a n d in gradients of density a n d temperature. T h a t fact leads t o a substantial discrimination between the requirements of atomic d a t a for both types of investigations. Calculations of wavelengths a n d oscil- lator strengths o f lines f o r m a n y ions is necessary for the study of relatively low density plasmas confined in Tokamaks. But f o r understanding the processes occurring in very dense plasmas produced in laser

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P. JAEGLE, G. JAMELOT, A. CARILLON AND A. SUREAU

FIG. 1. - Spectra of aluminium laser-produced plasma at high density (lower curve) and a t low density (upper curve) ; changes in intensity and line width are clearly seen.

experiments such more data are required. Here we will concentrate the attention on features resulting from high density.

Evidences of such features are easily found in the ultraviolet or X-ray spectrum of the ions. An example is given on figure 1 which showes, in the soft X-ray range, a high density spectrum (lower curve) as compared to a low density spectrum (upper curve) for an aluminium plasma produced by laser impact. The highest density is .of about 102' electrons/cm3 and lowering the density is obtained by shifting the 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 lines are very affected by changing the density and, to some extent, the temperature of the plasma.

Generally speaking, the broadening of the lines by collisional and quasistatic Stark effect is quite significant at high density. Moreover a challenging problem is still to find an evidence of Zeeman effect since large self-generated magnetic field has been claimed in laser-produced piasmas [l]. However, the intensity and the profile of the line is dominated, in many cases, by the reabsorption of the radiation in the dense core of the plasma as well as in external cold shells [2,3]. As a consequence, the interpretation of the spectra with regard to line shape and intensity requires the calculation of the radiative transfer in taking into account features as plasma unhomogeneity and Doppler shift due to ion radial expansion. This will be the object of the second section.

On the other hand, because of the large density of free electrons and the presence of ions of various ionization stages, the continua of states above ionization limits are of a great importance in atomic processes occurring in dense plasmas. The role of autoionizing levels of the ions is extensively studied. Indeed such levels, corresponding to a two or more electron excitation, consist in a mixing of discrete and continuous states of the ions ; due to their location in the energy range of the free electrons these levels yield channels for recombination mechanisms able to populate some particular excited levels or to modify the total population of an ion species. The enhancement of density of states in a small energy interval arround the level and the ability of an autoionizing state to interact with discrete states more than a pure continuous state does make these mechanisms to be very efficient. Thus the third section will be devoted to the role of autoionizing levels, especially in phenomena such as dielectronic recombination or resonant three-body recombination.

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MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS C4-77

measurement, the reabsorption could give rise to a large inaccuracy. Before to come into a detailed treatment of line shape formation in this case, let us derive a simple criterion of optical thinness that can be put on in dense plasmas of small volume.

Under the assumption that the plasma is homogeneous, the absorption coefficient for a line of energy E and width SE is given by :

where A,, is the spontaneous transition probability between levels 1 and 2, NI and NZ, the population densities of the levels, g, and g,, their statistical weights. A typical size of a laser-produced plasma is of 100 y ; then the optical thickness will be less than 0.5 if :

k < 50 cm-'

is satisfied. Under this condition the total reduction of line intensity will not exceed 40

%

and one can calculate that resulting additional broadening of the line will be approximately of 15

%.

Thus for avoiding 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 ground state 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 plasma in which the 1s-2p line of Mg XI1 exhibits a width of 3 eV ; we have E

--

1 500 eV and

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

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

It results from (3) that, for most of the ions whose the resonance lines are lying in the soft X-ray range, the reabsorption becomes signficant at density of 10'' cmw3.

2 . 2 TRANSFER OF RADIATJON IN A MEDIUM DOMI- NATED BY COLLJSJONS. - NOW we will start from the well 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 very small length travelled by the light in the plasma. To calculate I, from (4), we need suitable expressions of j, and k, accounting for all the properties of the medium at the frequency of interest. For an isolated line, at any frequency these coefficients depend on : i) the total transition probability of the line, ii) the profiles of spontaneous emission, stimulated emission and absorption, iii) the population densities of upper and lower levels, iiii) the contribution of the continuous spectrum.

From very general arguments the profiles of stimulated and spontaneous emission can be taken for identical [6] but they differ from the absorption profile. Considering moving ions, this is due to the scattering of radiation which gives rise to a small shift in frequency between absorbed and emitted photons when their directions are different. The profiles then are related by the so-called frequency redistribution function [7, 81. However, in dense plasmas, the typical time of radiative scattering process in many cases is larger than the time between two electron impacts and so emission and absorption profiles are independent each other. Furthermore, even in the X-ray range where radiative probabilities are large, the profiles are dominated by Stark effect due to near particles or otherwise by thermal Doppler effect, according to the values of the plasma parameters. Thus, one and the same profile is to be used for both emission and absorption.

On the other hand, in a general treatment of radiative transfer it is necessary to solve a system of coupled equations expressing simultaneously the radiation propagating at several wavelengths and the population rates of all the levels involved. Indeed, the dependence of absorption and stimulated emission on radiation intensity entails a reaction of the radiation on excited level populations. In dense plasmas, the role of collisions in dominating ion level excitation allows again a great simplification. For any two ievel transition, an equation of type (4) can be integrated separately. In homogeneous plasma, one is left to :

where X is the length and Z,(O) an incoming intensity ; j,/k, is often called the source function. Unfortunately

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

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

2.3 LINE PROFJLES. - Factors coming from different origins participate in making up the line shape exhibited by dense hot plasmas. Besides Stark and Doppler effects and absorption broadening, we must point out at least the role of the continuous spectrum, which often is significant in the extreme ultraviolet range, and the role of the ion expansion velocity giving rise to a non-thermal Doppler effect.

A proper account of the continuous contributions coming from free-free and bound-free transitions is of great importance for interpreting self-reversed and absorbed profiles. If emission and absorption of radiation of frequency v proceed from discrete transitions at once with continuous spectrum, the coefficients j, and k , are to be written :

where jL and k , are sums over all the cooperating discrete transitions, jc and kc being integrals over the space of free electron momenta. Substituating these expressions in (5) shows that discrete and conti- nuous spectra does not provide the emerging intensity with separate summed contributions, although this fact has been disregarded in many circumstances. That is why the transitions with free states take some part in the line shape formation.

For instance, it has been shown that a discrete line will exhibit an absorbed profile, even though the level populations are in statistical equilibrium, provided that the continuous absorption coeffi- cient kc is large enough [9, 121. But in case of weak continuous absorption, absorbed profiles are reliable evidences of underpopulation of the upper levels of the transitions under consideration. In fact, the continuous emission must also be large for an absorbed profile 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 the plasma. Let us consider a line which is self-reversed (intensity in the center less than in the wings) upon a continuous spectrum of intensity 0.1 B, and of optical thickness 0.29 (25 %-absorption) ; this line is still self-reversed for an optical thickness of 4.6 (99 %-absorption) if the emitted intensity does not increase; but it becomes absorbed (white line) as soon 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 the classical expressions of Bremsstrahlung [5] the coeffi- cients jc and kc of (6) can be written :

where N, and Te are the density and the temperature

of electrons, varying along the light path ; p and q are adjustable parameters.

As regards the ion velocity, it is to be considered as one of the most probable causes of profile asymmetry [10, 111. At a distance X into the plasma, the motion of ions removes the frequency v, of the top of the line profile in such a way that v, must be replaced by 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 expanding with velocity v, ; X, is the coordinate of the center of the plasma. Beside the asymmetry of the profile, that can be seen on figure 2, the optical thickness

FIG. 2. - Asymmetrical line shape interpretation. The lower curve is the densitogram of a photographic recording of AI4+ lines. The upper curve is the calculated line shape at the wavelength of

126.06 A for the expanding plasma model shown in inset.

is altered by the non-thermal Doppler effect. As we will see below, ion expansion is a limiting factor for the gain which could be expected in the occurrence of population inversion. This fact must be taken into account in experiments dealing with long plasmas in soft X-ray laser research.

Turning back to expressions (6), we will assume

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

paragraph 2.1, and putting B,, for Einstein's coefficient for absorption, we have :

Here @(v) is the profile function, common to emission and absorption, which we will briefly discuss in the present paragraph. Before all, let us point out the factor 1/6E = llh6v to replace the function 4(v) in a simplified expression like (l), what amounts to take in a rectangular profile. Although a rectangle is a poor approximation of line shape, it enables a satisfactory expectation of the general behaviour of the line, provided that the width is properly esti- mated. For instance, results reported in paper of reference number [l31 suggest that the effect of a Voigt profile instead of a Gaussian profile is due far less to the modification of analytical shape than to the account, in the first case only, of the collisional broadening which predominates in dense plasmas. As a matter of fact an exact expression of the profile function 4(v) can be found only when the thermal broadening exceeds largely any other contribution, but the natural width (if the radiative transition probability is very large). Under conditions giving rise 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 large inaccuracy will affect the theoretical prohles of non- hydrogenic lines. To combine ionic quasi-static effect with electronic impact effect is an actual difficulty [l 51. Moreover, quasi-static and impact approximations are questionable in many cases and must be replaced by more accurate intermediate calculations [16]. A moderate position, which is valid for densities near the critical shell of Nd-laser produced plasmas, consists in using a Lorentzian profile whose the width 6v accounts for the intensity of electron collision perturbation [17]. Thus we define @(v) as :

As far as all collisional rates are proportional to electronic density, the same dependence must be found in 6v. For the examples given in this paper we took :

where T, is the electronic temperature. The dependence against T,- ' l 2 has been set as a rough approximation

taking into account the fact the thresholds of the most important collision induced excitations to be very smaller than the electronic temperature : then their rate goes down slowly with increasing temperature [l 81.

In order to be ensured from an unexpected large effect of the form of 4(v) we performed also calculations using a T, - dependent Gaussian profile. On condition that the magnitude of the width 6v is the same as in (IO), the general behaviour of the line remains unchanged [12].

2.4 SHAPES OF OPTICALLY THICK LINES. - Expres- sions from (4) to (1 1) have been used for numerical calculations of line shapes in dense unhomogeneous plasma. A plasma model has been chosen so that it depictes a section of laser-produced plasma with a fairly homogeneous hot dense core, having a diameter of 100 p, and an external shell of conti- nuously decreasing temperature and density. The expansion velocity- has been deduced from fitting an experimental result as shown on figure 2. The results show, on figure 3, the role of the ionic density

FIG. 3. - Change in line shape versus ionic density Ni (the values are indicated for the core of the plasma). Ionic level populations are assumed 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 a constant proportion over all the frequencies covered by the line, and, on figure 4, the role of the collisional broadening which carries opposite effects on the center and on the wings of the line.

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

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

ponding to each profile.

reflected by the line shape will always require a careful examination of many informations of experimental and theoretical origin.

2 . 5 SPECTRAL FEATURES DENOTING UNBALANCED

EXCJTATJON AND POPULATJON INVERSION. - Owing to the development of dense plasma investigations for achieving soft X-ray amplification, a prime attention is lent to the spectral features resulting from unbalanced excitation, especially from population inversions [9, 191. Experimental results have been reported for some multicharged ions of carbon, for which a rather simple line intensity analysis seems able to prove weak population inversions because the plasma density is small enough to avoid all reabsorption effects and, on the other hand, the transition probabilities are accurately known for these ions [20, 21, 221. Here we report, on figure 5, the calculations concerning the spectrum of neon-like ions of aluminium in the neighbourhood of the critical density (10,' of a laser-produced plasma. The experimental spectrum can be also seen on figure 1 (4d 'P,, 3D1, 3P1) with a comparison between high and low density features. From figure 5, it results that both underpopulations of 'P, and 3 ~ levels ,

and population inversion between 3P1 and ground level account for experimental spectrum. No such

xi-

FIG. 5. - On the right : experimental spectrum. In the middle :

calculated spectrum in assuming 3P, population inversion and 'P,, 3D, under-populations as shown on the left part of the figure.

account can be found if balanced populations are assumed, whatever the plasma model we choose. Figure 6 reveals what may be an impediment in future experiments on soft X-ray amplification by plasmas. In homogeneous plasma model, of length 300 p, we calculated the maximum intensity and the width AV of emerging line, versus a population inversion N,/N,, for several expansion velocities; A,v is the width of the optically thin expansion-free profile. Obviously, the plasma expansion will have a damaging action on the line amplification. However, very few is known on expansion of long plasmas produced by cylindrical focusing of a laser beam.

FIG. 6. - If expansion velocity in cancelled, intensity I at the center of the line increases exponentially versus population inver- sion, whereas line width AV decreases (solid curves). Velocity expansion of 7.5 X 106 cm/s (dashed curves) - afterward 1.5 X 10' cm/s (dash-dotted curves) - brings down the gain and enlarges the line width (I = 300 p ; N , = 10'' ; N, = 3 X 10").

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MAIN ASPECTS OF ATOMIC PHYSICS IN DENSE PLASMAS

making up many transitions from discrete and continuous levels. From a general point of view, this problem has been often visited of late years for astrophysical and laboratory plasma studies [23, 241. However, the ability of generating very dense plasmas is a new fact in the course of investigations ; processes which were of little importance in earlier studies must be now considered for accurate calculations. This is especially true for the collisional processes involving more than one electron, since their frequency goes up rapidly when the density does so. Indeed, the two electron processes provide the plasma recombination with an alternative to the one-electron mecha- nism. This ought to lead to develop quantum mechanical calculation of electron impact ionization cross- sections, from which the three-body recombination can be computed owing to the detailed balance principle. As continuous states of ions are involved in these processes, we deal necessarily with the structure of the ion continua which are anything but smooth because of the presence of many autoionizing states.

In the first part of this section we will review briefly various atomic processes involving autoionizing states, investigated at the present time on account of their role in plasmas. The second part will contribute to the theoretical study of autoionizing states themselves in approaching the calculation of the perturbed density of states in the continuum surrounding an autoionizing level. This is done in view of future improvements in rate coefficients calculations for processes between free electrons and ions.

3.1 PROCESSES INVOLVJNG AUTOIONIZING STATES. - On figure 7 are shown resonances occurring in various processes in consequence of the excitation of an autoionizing level by electronic impact. Let Z be an ion of charge Z in its ground state. The stars will denote the excitation of one (*) or two (**) electrons. When the energy of a twice excited state is larger than the lowest ionization energy - which corres- ponds to the ion Z

+

1 in its ground state - this state is known to get a radiationless decay process by transition to the continuum (autoionization).

Figure 7a shows the pseudo-stationary state Z** appearing in the course of the scattering of a free electron by the ion Z

+

1, that is a radiationless capture followed by autoionization. The energies of impinging and ejected electrons can be different if a photon is emitted or absorbed. Then, the excitation of autoionizing level Z** is inducing a resonance in the Bremsstrahlung spectrum of the electron in the field of the ion Z

+

1 [25]. Figure 7d represents another process in which the resonant scattering of a free electron leads to a change of energy; but now this change is due to the excitation of the target. In fact, here we see that autoionizing resonances will give contributions 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 represent

respectively the ground levels of ions of charge Z and Z

+

1.

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

On figures 7b and 7c are represented two different mechanisms of resonant recombination. Both are initiated by the encounter of a free electron with an ion Z

+

1, resulting in a twice excited state 2""

of ion 2. ~ h e k , this last can be stabilized in a single excited state either by radiative transition (Fig. 7b) or by a new electronic collision (Fig. 7c). The first process is the well known dielectronic recombination, which has been studied for explaining the ion abundances in the solar corona [27]. In the second case, we have a resonance in the continuous spectrum of three-body recombination :

For a specific situation, figure 8 due to Landshoff et al. [28], shows the great role played by these resonances in laser-produced plasmas. It must be pointed out that the curve denoted by dielectronic on this figure represents in fact the total rate of both above- mentioned mechanisms. Until now, such calculations took only aim at a right expectation of ion abundances in plasmas, which requires to sum a large number of approximately known recombination rates, corresponding to all the resonances of the ion. However, in order to explain unbalanced excitation of ions in plasmas (see Sec. 2,

3

2.5), it will be necessary to perform detailed rate calculations for specified excited levels of ions.

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

DIE LECTRONIC

FIG. 8. - Rate of resonant recombination as compared to radiative and collisional non-resonant processes [28].

for a particular stabilizing transition, the recombination 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. S - ' if cgs units are used ; g,,, is the statistical weight of Z**, g,,, is the statistical weight of (Z

+

l)-ion ground state ; T is the temperature; E is the kinetic energy of the impinging electron before the capture; A , is the radiative transition probability from Z** to Z* and A, is the autoionization probability of Z**, which can be expressed under the form :

Here, o, is the cross-section for radiationless capture of an electron of energy E and velocity v ; g(E) is

the number of free electron states per unit volume with energy between E and E

+

dE. For a Maxwell distribution of free electrons, the probability of finding an electron with energy between E and E

+

dE in the volume of an ion is f (E) dE, with :

and :

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

Since the autoionization probability A, of relation (1 3)

depends on free electron state density, this treatment includes partially the feature of autoionizing level consisting in superposition of continuous states to a discrete one. For practical use of formula (16), E is replaced by the energy difference between two successive resonances. However, one can see that the 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) do not account for the modification of density of states at energies close to autoionization. This modification will be examined in the next section.

Before to come to the case where stabilization takes place without emission of photons, we must mention an important application of the above calculations. If dielectronic recombination is the only process yielding Z**, all radiative transitions having Z** as upper level will depend on the tempe- rature in accordance with a, in (16). Thus these lines, which are said dielectronic satellites of ( Z

+

1)- ion, can give a valuable temperature-jauge for plasma diagnostics [26, 321.

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

a free electron removing the energy difference between Z** and Z*. The inverse process is a resonance in ion-electron impact ionization. Thus, while understanding of dielectronic recombination depended on a development of photoionization theory, resonant three-body recombination requires improvements in electron impact ionization calculations, which are still more complex for quantum mechanics. Let

( voi ) be the electron impact ionization rate, after averaging over a Maxwell distribution of free electrons; from the detailed balance principle, the coefficient

X,

of three-body recombination will obey :

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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 autoionizing states [33]' results in a parametric expression of photoionization cross-section in the neighbourhood of the resonance. Let o, be the cross-section for transition between a discrete state and a pure continuous state of the ion. Near an autoionizing level, the photoionization cross-section may be written :

with :

E - E

E = -

r

being the half-width of the resonance and

E

a fixed energy which is little different from the energy of the pure discrete state involved in the autoionizing level. The shape of the resonance depends on the value of Fano's g-parameter. Figure 9 gives a practical

FIG. 9. - Autoionizing profile of photoabsorption observed at the wavelength of 84.37 A in an aluminium laser-produced plasma.

example of the photoabsorption profile due to such a resonance observed in the spectrum of AI3+-ions, in laser-produced plasma [34]. In fact, the rate of dielectronic recombination must be integrated over similar cross-section profiles instead of using a pure discrete transition probability like A , in (16).

The parametrization appearing in expression (1 8) can be extended to resonances in electron-impact ionization [35, 36, 371. In the work of Tweed [37] the definition of the g-parameter of Fano is extended to the range of validity of the first Born approximation, including exchange effects. The fact that two free electrons are present in the final state (initial state in the case of recombination !) leads to two possible expressions of differential cross-sections, according to the choice of the first angular integration. In assuming that a scattered electron, leaving the target immediately after the collision, is distinguishable from an ejected electron which is slightly delayed by

the life-time of the resonance, Tweed defines an ionization cross-section o;jected corresponding to a

fixed angle for the ejected electron and to integrated momenta over all angle of electron scattering. The parametric form of this cross-section is :

where a, b, aoi and a;jec'"* are functions of ejected electron momentum. The sum in expression (19) is over the values of the quantum numbers specifying the resonance, including the total spin S . The total cross-section appearing in (17) could be obtained by integrating parameters a and b, as well as the non- resonant part c,,, over the ejected electron momentum. However, as far we know, ab initio calculations of the resonant ionization cross-section oi have not been performed until now and a few neutral atoms or single ions have been experimentally investigated. Thus it remains difficult to include reliable values of resonant three-body recombination coefficients in plasma numerical models, although this process is likely of large consequence at high density.

3.2 DENSITY OF STATES IN THE NEIGHBOURHOOD OF AUTOIONIZING LEVELS. - In the last paragraph of this paper we wish to make use of the autoionizing state theory in discussing statistical aspects of the role of these states for the ions in a plasma. The start point of this discussion can be easily understood with rererence to the problem of Boltzmann's distribution for autoionizing levels. In a local thermodynamical equilibrium plasma, for an ordinary excited level the population ratio with respect to the ground level is given by :

But this relation cannot be employed directly for autoionizing levels unless the mixed discrete-continuous feature of these levels is neglected. Further- more a plasma has in it a gas of free electrons which are occupying the continuum of states of the ions. Thus free electrons take part to the population of autoionizing levels. We will first give an approximate expression of this part. Then, a more detailed study of the structure of the continuum of states near an autoionizing resonance will be presented. Let us consider the mixing of discrete and continuous configurations which results in the autoionizing state wave function :

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

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

Introducing now the statistical weight of free electrons surrounding (2

+

l)-ions :

and taking into account that :

we obtain the following expression of the statistical weight of the autoionizing state :

where g, is the statistical weight of the discrete state involved in the resonance. Making use of Boltzmann's law gives the population ratio :

gz+ l 8 n rn312(2 AE

+

( b 2 ) -

-

h exp

(-

g)

gz Nz+1

(26) which enables to express the population NZ,, of the autoionizing state according to the probabilities a2 and ( b2 ) that the system is in a discrete or a continuous state. If a2 + 0, in using Saha equation for the value of the ion density ratio N Z + ,/ N Z , (26) reduces to the fraction of free electrons of energy comprised between 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 autoionizing states, the wave functian on the left member of (21), which varies quickly in the energy range of the resonance, describes in fact an unbound state depending on E, namely the discrete state is diluted in the continuum. Calculation of coefficients a and b of the right remember of (21) leads to write 1+9, under the form :

Sin A cp.

-

cos A

.

i$,

* E =

where cp' is a modiJied discrete state, A is a phase shift in the continuum wave function caused by the

discrete-continuous interaction; VE is the matrix element ( cp

I X

I

4,

).

Now the probability to find a quantum-mechanical system in a given subspace equals the trace of the operator resulting from the product of the density operator and the projection operator in this subspace. We consider the subspace of energy states comprised between E and E

+

AE, in the volume A3r round the ion. We have the projection operators :

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

P=-

Q

with :

Q = Tr { e-3e1KT f

X being the Hamiltonien of the system. We have :

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

e - sin2 ~ A ~ ~ ~sin A

~ = e ~ ~ ~ y [ ~ ~

1 vE12

+{A3rd3r(m

X

From this expression, general feature of density of states in the neighbourhood of an autoionizing state is sbown on figure 10. The autoionizing state raises a distorded Lorentzian upon the exponential curve describing the density of states in the continuum of the ion. The ratio between the maximum d, of

Density of states

FIG. 10. - Structure of the continuum of states around an autoionizing level. E q is the energy of the pure discrete state cp. The mixing of cp with the continuum leads to a shift of the peak energy

(12)

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

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

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

sin A = 0 for the pure continuum. This yields

which represents the relative enhancement of density of states due to the presence of the discrete state cp

above the ionization limit. It must be pointed out that the maximum of the enhancement dM takes place generally at an energy slightly shifted with respect to the energy E,, of the discrete state cp.

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

We deduce from this study that the difficulties encountered in a so complex medium, for interpreting a collection of experimental data, are largely reduced by modelling the plasma properly for numerical computation ; moreover, quantitative balance prediction involving many particular processes will require to deal energies 'in developping quantum calculations, especially by building good approximations, suitable for practical purposes. In this way, dense plasma studies offer to atomic physics an attractive field of original works, emphasizing features like the role played by continuous-discrete mixed states in presence of a dense gas of free electrons.

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