HIGH DENSITY PLASMA EFFECTS ON ATOMIC AND IONIC SPECTRA

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HIGH DENSITY PLASMA EFFECTS ON ATOMIC AND IONIC SPECTRA

D. Burgess, R. Lee

To cite this version:

D. Burgess, R. Lee. HIGH DENSITY PLASMA EFFECTS ON ATOMIC AND IONIC SPECTRA.

Journal de Physique Colloques, 1982, 43 (C2), pp.C2-413-C2-432. �10.1051/jphyscol:1982232�. �jpa-

00221844�

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JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-413

HIGH DENSITY PLASMA EFFECTS ON ATOMIC AND IONIC SPECTRA

D.D. Burgess and R.W. Lee

Blackett Laboratory, Imperial College of Science and Technology , London SW7 2BZ, U.K.

Résumé

La compression d'un plasma par laser produit des champs électriques et magné- tiques d'une intensité exceptionnelle. Les perturbations des spectres sont telles qu'il reste peu de niveaux liés, même pour des émetteurs multiplement ionisés avec z > 10 - et la physique même du plasma devient complexe à cause d'interactions fortes entre les espèces qui le composent ou encore à cause d'effets de dégénéres- cence.

La physique de base des plasmas ultra-denses se distingue de celle qu'on sup- pose couramment dans la spectroscopie des plasmas, tant en astrophysique qu'en la- boratoire. Nous passons en revue les problèmes théoriques que pose l'interaction émetteur - plasma dans un milieu dense et nous donnons quelques exemples d'intérêt expérimental et théorique actuel.

Abstract

Laser compressed plasmas are sources of exceptionally intense electric and magnetic fields. Consequent perturbations of spectra are such that even for high ionization stage emitters with z > 10 only a few bound states may remain - and the physics of the plasma itself is complicated by the strong interactions amongst the constituent species and by near-degeneracy.

The basic physics of such ultra-dense plasmas is contrasted with that usually assumed in longer established areas of laboratory and astrophysical plasma spectroscopy, theoretical problems in treating emitter-plasma interactions in dense plasmas are surveyed, and some examples are given of problems of current experimental and theoretical interest.

1. Introduction

This paper discusses what are probably the most severe examples of the modification of atomic and ionic spectra by strong fields yet realized in the laboratory. Laser compressed plasmas presently under laboratory study reach densities of hot, highly-ionized matter measurable in grams or tens of grams cm-3, i.e higher than solid, with electron densities of interest being in the range 10 a 4

-1025cm-3- The corresponding plasma microfield due to the inter-particle electric fields is therefore such that even for highly-ionized emitters (with, say, z > 10) only the very lowest bound states retain any real identity with those in isolated ions. Since in such plasmas the particles are strongly interacting (i.e. compared with thermal energies), the calculation of the statistical properties of the microfield, allowing not only for its instantaneous magnitude but its fluctuations

(which are often more crucial than instantaneous field strength in determining effects on emitted spectra), is a formidable statistical-mechanical problem, even before the problems of the response of an atom/ion to the field, or the reaction of the emitter state back on the microfield are approached.

Complex as it is, this 'strongly-perturbed' spectroscopy of these very dense +

Presently on leave of absence at Lawrence Livermore National Laboratory, Livermore, California, U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982232

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C2-414 JOURNAL DE PHYSIQUE

plasmas is o f much more than j u s t academic i n t e r e s t . Laser-compressed plasmas are p r e s e n t l y t h e most f r e q u e n t l y s t u d i e d example o f t h e so-called I n e r t i a l Confinement (ICF) r o u t e t o f u s i o n power. Unlike t h e s i t u a t i o n i n t h e longer-established Magnetic Confinement programme, spectroscopy p l a y s a fundamental r o l e i n both t h e d i a g n o s t i c study and t h e underlying ph i c s o f ICF plasmas. For a l a s e r plasma a t an e l e c t r o n d e n s i t y , Ne o f ,say, 1 8 cm-3, t h e e l e c t r o n plasma frequency, wpe

,

a l r e a d y l i e s i n t h e s o f t X-ray s p e c t r a l region.

Consequently, laser-based d i a g n o s t i c s such a s Thomson s c a t t e r i n g , well e s t a b l i s h e d i n magnetic confinement work, a r e impossible, and t h e prime q u a n t i t a t i v e d i a g n o s t i c technique is X-ray emission spectroscopy ( t o g e t h e r , i n some c a s e s , with a b s o r p t i o n work based on back-lighting from s e p a r a t e l y generated heavy-element laser plasma X-ray s o u r c e s ) . Also, r a d i a t i o n t r a n s p o r t may play a s i g n i f i c a n t r o l e i n t h e physics o f t h e plasma a s a whole, f o r i n s t a n c e i n t h e compression dynamics v i a preheating o f m a t e r i a l ahead o f t h e converging shock f r o n t , and t h e r a d i a t i o n t r a n s p o r t i n t u r n is c o n t r o l l e d by processes such as p r e s s u r e broadening, s e r i e s merging and i o n i z a t i o n p o t e n t i a l depression. I n both c o n t e x t s it is c r u c i a l t o develop an understanding o f t h e s t r o n g p e r t u r b a t i o n s o f atomic and s p e c t r a l p r o p e r t i e s t h a t o c c u r , even though t h e complex n a t u r e o f t h e medium means t h a t t h e o r e t i c a l models a r e n e c e s s a r i l y l e s s e l e g a n t and more approximate than those used i n o t h e r problems i n v o l v i n g ' s t r o n g f i e l d s ' .

I n t e r e s t i n g l y , but o f less p r a c t i c a l importance, laser-generated plasmas a r e a l s o t h e sources o f probably t h e h i g h e s t magnetic f i e l d s y e t generated i n t h e l a b o r a t o r y . S c a l e l e n g t h s f o r e l e c t r o n d e n s i t y and temperature g r a d i e n t s i n t h e s e plasmas a r e s o s h o r t t h a t very i n t e n s e c u r r e n t s develop and l o c a l magnetic f i e l d s i n t h e megagauss range are generated. Whilst s p e a t r o s c o p i c e f f e c t s o f t h e s e B f i e l d s a r e u s u a l l y masked by t h e combined e f f e c t s o f both Doppler and p r e s s u r e broadening, some p o s s i b l e i n d i r e c t consequences w i l l be mentioned below.

I n t h i s paper t h e b a s i c p h y s i c a l s i t u a t i o n i n a l a s e r compressed plasma w i l l first be c o n t r a s t e d with t h a t i n more conventional magnetically contained and a s t r o p h y s i c a l plasmas. The observable e f f e c t s o f t h e plasma environment w i l l be surveyed, and a review w i l l be given o f some r a t h e r g e n e r a l problems encountered i n a t t e m p t s t o develop a sound t h e o r e t i c a l base f o r t h e spectroscopy o f t h e s e ultra-dense sources.

FUSION PLASMAS

Magnetic Containment (Tokamaks etc.) Inertial Confinement (Laser Compression etc.)

Te 10'

-

^{lo8 K} ^lo6- ¹⁰^'

l0lZ - ^lo1&^.m-3

Duration 0.1 - ¹⁰^s.

Size 1 - 1 0 m

I o n stages X - X X g

Kinetic pressure 1 bar

Statistics Debye ( n 1 3 1) Classical < 1) Ionization Non-LTE

Atomic Spectra Unperturbed Line Shapes Doppler Radn. Transport T<< 1

(except in far IR)

x- BX

1

-

^bar(Fusion lo6 a a r ) Strongly-coupled ( n i 3 y 1)

Near degenerate 1)

Saha fails

Strongly perturbed (nai 1) Pressure

Optically thick, even in continuum.

TABLE 1:Cmparison c f p h y s i c a l parameters i n low-density, magnetically and high-density, i n e r t i a l l y confined, f u s i o n plasmas.

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2. Canparison with t h e spectroscopy o f o t h e r l a b o r a t o r y plasma sources.

The r e c e n t advent o f laser-compressed plasmas r e p r e s e n t s perhaps t h e most r a p i d and extreme change i n t h e e n t i r e h i s t o r y o f l a b o r a t o r y plasma spectroscopy.

For t h e first time t h e physics o f t h e l a b o r a t o r y sources o f i n t e r e s t d i f f e r s i n fundamental r a t h e r than merely q u a n t i t a t i v e ways f r o m t h e t h e o r e t i c a l b a s i s assumed i n most o f t h e e s s e n t i a l l y a s t r o p h y s i c a l background o u t o f which t h e s u b j e c t grew. The n a t u r e o f t h i s physical change can be seen by c o n t r a s t i n g a l a s e r compressed plasma w i t h that generated i n a magnetically-contained f u s i o n device such as a Tokamak, Table 1. (For a more g e n e r a l survey o f o t h e r present-day plasma s o u r c e s , and t h e i r spectroscopy, s e e t h e r e f e r e n c e s by Burgess,l, and by Peacock and b g e s s , 2 ) .

Table 1 shows that l a s e r plasmas and Tokamaks a r e somewhat similar i n t h e i r e l e c t r o n temperature, T,, and i n that both a r e capable o f g e n e r a t i n g i o n s i n high i o n i z a t i o n s t a t e s . However, a t any deeper l e v e l t h e two sources d i f f e r s o g r e a t l y as t o almost r e p r e s e n t d i f f e r e n t s t a t e s o f m a t t e r .

The first p o i n t t o consider before d i s c u s s i n g s p e c t r o s c o p i c p r o p e r t i e s is t h e g e n e r a l s t a t i s t i c a l mechanical s t a t e o f t h e plasma. A Tokamak t y p i f i e s t h e t y p e o f plasma c l a s s i c a l l y encountered i n l t r a d i t i o n a l l l a b o r a t o r y and a s t r o p h y s i c a l spectroscopy i n that it is a h i g h temperature, low d e n s i t y system and is both c l a s s i c a l s t a t i s t i c a l l y ( i n t h e sense t h a t i t is very far from Fermi degeneracy) and is an i d e a l g a s (i.e. p = nkT) s i n c e t h e mean i n t e r p a r t i c l e i n t e r a c t i o n e n e r g i e s a r e much lower than kT

.

It is t h e r e f o r e 'weakly-interacting' i n t h e sense that:-

where R =

{L

is t h e mean i n t e r p a r t i c l e s e p a r a t i o n ,

o 4nN

1

(with T i n Kelvin)

Equivalently, ( a p a r t from a numerical c o n s t a n t ) t h e plasma is 'Debye' i n t h e sense that t h e Debye s h i e l d i n g l e n g t h i s much g r e a t e r than t h e i n t e r p a r t i c l e s e p a r a t i o n )

,

i .e. :

-

1 1

a = R /kD = 0.19 NeK

T ~ T

< 1 ( 2 )

This simply means that t h e r e is no b a s i c s t a t i s t i c a l mechanical problem i n working o u t t h e p r o p e r t i e s o f such a plasma a s t h e y p e r t a i n t o its spectroscopy. The 'macroscopic' behaviour o f a Tokamak plasma may be complex, e.g. i n its i n t e r a c t i o n with magnetic f i e l d s , t h e e x i s t e n c e o f c o l l e c t i v e plasma waves, e t c . , but t h e f r e q u e n c i e s o f such plasma modes a r e much lower than t h o s e involved i n most s p e c t r o s c o p i c p r o c e s s e s , t h e impact parameters o f i n t e r e s t i n excitation/de-excitation e v e n t s a r e much s m a l l e r than mean i n t e r p a r t i c l e r a d i i , and a s i n t h e c l a s s i c a l theory o f a s t r o p h y s i c a l spectroscopy, t h e s p e c t r o s c o p i c p r o p e r t i e s o f t h e plasma can be defined e n t i r e l y by a knowledge o f a s i n g l e o n e - p a r t i c l e parameter, namely t h e e l e c t r o n v e l o c i t y d i s t r i b u t i o n , p l u s a f e w atomic p r o p e r t i e s known a - p r i o r i

-

energy l e v e l s , binary c o l l i s i o n c r o s s - s e c t i o n s and t r a n s i t i o n p r o b a b i l i t i e s .

I n c o n t r a s t , a laser-compressed plasma with a k i n e t i c p r e s s u r e o f perhaps 10 atmospheres (and f o r laser f i s i o n 1 o1 ) ( i n p r e s e n t experiments t h e r a d i a t i o n p r e s s u r e due t o t h e l a s e r l i g h t a l o n e may reach lo6 atmospheres) may be near degenerate ( f o r ICF it w i l l have t o be c l o s e t o t h e Fermi a d i a b a t ) and it w i l l be

'strongly-coupled1 o r Inon-Debye', i . e . :-

I' ( o r a )

>

I ( 3 )

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JOURNAL DE PHYSIQUE

In consequence i t is a fnon-ideal' g a s , t h e p a r t i c l e v e l o c i t y and s p a t i a l d i s t r i b u t i o n s depend on t h e mutual i n t e r a c t i o n s o f t h e plasma c o n s t i t u e n t s (and near a z-times i o n i z e d e m i t t e r on t h e l o c a l emitter-plasma i n t e r a c t i o n , s e e Ref.l), and s p e c i f i c a t i o n o f t h e s p e c t r o s c o p i c s t a t e o f t h e medium i n t r i n s i c a l l y r e q u i r e s a treatment o f t h e c o r r e l a t i o n s between its c o n s t i t u e n t p a r t i c l e s . It is a l s o important t o n o t e that

r >

1 a u t a n a t i c a l l y i m p l i e s t h a t t h e r e w i l l be a s t r o n g coupling between a z-times i o n i z e d e m i t t e r and its l o c a l plasma environment, s o t h a t t h e plasma p r o p e r t i e s r e l e v a n t t o t h e determination o f t h e emitted spectrum are n o t n e c e s s a r i l y t h o s e t y p i c a l o f t h e plasma a s a whole, b u t i n s t e a d a r e t h o s e o f t h e s t r o n g l y - i n t e r a c t i n g l o c a l p o l a r i z a t i o n cloud s p e c i f i c t o t h e p a r t i c u l a r e m i t t e r . Thus t h e ' p e r t u r b a t i o n f due t o t h e plasma cannot be decoupled from t h e e m i t t e r ' s p r o p e r t i e s but w i l l c e r t a i n l y be dependent on t h e e m i t t e r ' s charge and perhaps a l s o on its atomic s t a t e . (As pointed o u t e.g. i n Ref.1, f o r large z t h i s problem can be important even when

r <

^1).

3. Mean plasma f i e l d s i n r e l a t i o n t o n u c l e a r Coulomb f i e l d s

It is now r e l e v a n t t o c o n s i d e r t y p i c a l values o f t h e m i c r o f i e l d , E

,,,

^{i n a}

laser compressed plasma i n r e l a t i o n t o t h e n u c l e a r f i e l d seen by bound e l e c t r o n s ,

%.

^{A s}^a^firstapproximation we c o n s i d e r t h e f i e l d to be s t r o n g i f : -

It i s important t o n o t i c e t h a t s p e c t r o s c o p i c e f f e c t s may be observable a t much s m a l l e r r e l a t i \ v a l u e s o f E

,

s i n c e e.g. f o r c l o s e l y spaced l e v e l s s t r o n g mixing, and hence t h e s. Iearance o f field-induced forbidden-lines, can occur a t r e l a t i v e l y low v a l u e s o f E

.

^Itis a l s o c r u c i a l t o n o t e t h a t a f r e q u e n t s i t u a t i o n i n a plasma is t h a t t h e f l u c t u a t i o n p r o p e r t i e s o f t h e f i e l d a r e more important than t h e e f f e c t s o f t h e i n s t a n t a n e o u s f i e l d i t s e l f , t h e s i m p l e s t (second-order) p i c t u r e being t h a t t h e h i g h frequency components o f t h e Fourier decomposition o f t h e f i e l d , E(k, w ) , induce off-resonant t r a n s i t i o n s e x a c t l y similar t o 2-quantum t r a n s i t i o n s i n t h e l a s e r physics case. If t h e f i e l d f l u c t u a t e s r a p i d l y enough, l i n e broadening w i l l be dominated by t h e frequency spread o f t h e f l u c t u a t i o n s r a t h e r t h a n any d i r e c t S t a r k s h i f t s due t o E i t s e l f .

.6

.4 P(Fl E

0

.2

FIG I

L

-NEAREST NEIGHBOUR --- HOLTSMARK

-

I I

J t I I I

0 1 2 3 4

THE MICROFIELD DISTRIBUTION P(E) VERSUS NORMALIZED FIELD STRENGTH FOR NEAREST NEIGHBOUR - AND HOLTSMARK---- THEORY

P(-) E Eo

FIG 2

-- - a = 0 = HOLTSMARK

E / E ~

THE MICROFIELD DISTRIBUTION VERSUS NORMALIZED FIELD STRENGTH FOR VARIOUS VALUES OF a = r0/hD

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With these r e s e r v a t i o n s , however, it is still useful t o consider t h e mean s t a t i c f i e l d i n a plasma of given density. Fig. 1 shows t h e predictions f o r t h e probability d i s t r i b u t i o n P(E) of t h e two simplest models i n which no account is taken o f p a r t i c l e c o r r e l a t i o n s , namely t h e nearest-neighbour approximation and t h e vector sum of t h e p a r t i c l e f i e l d s due o r i g i n a l l y t o Holtsmark (see e.g. G r i e m , Ref.3). Fig.2, based on t h e work of Hooper (4), shows changes i n t h e d i s t r i b u t i o n due t o perturber c o r r e l a t i o n s a s t h e parameter a , Eqn.(2), increases.

Weisheit and Pollock have shown computationally t h a t f o r very l a r g e t h e f i e l d d i s t r i b u t i o n is very c l o s e t o t h e simple n e a r e s t neighbour model. I n every case, t h e r e f o r e , t h e most probable f i e l d l i e s between t h e Holtsmark and s i n g l e n e a r e s t neighbour values, and even i n a very dense plasma a reasonable estimate of t h e mean f i e l d f o r present purposes is simply

With a l s o t h e value f o r E experienced by an o r b i t i n g e l e c t r o n of p r i n c i p a l quantum number n i n a hydrogen-like i o n o f charge z ,

z3

^e2

E, =

-

(6)

n* a2 we obtain:-

$

⁼E / E~ = 7.2 x 1 0 - 1 7 + i 7

z3

P (7)

The value of n f o r RE = 0.1 is plotted a g a i n s t N, i n Fig.3 f o r various values of z. The c o n t r a s t between a Tokamak and a l a s e r compressed plasma i n terms of t h e e f f e c t of t h e plasma on atomic s p e c t r a is then immediately evident. For an i o n of z=10 i n a Tokamak, RE = 0.1 only occurs f o r n >loo, s t a t e s which w i l l be q u i t e unobservable s i n c e t h e Doppler broadening w i l l have caused l i n e s t o merge f a r much smaller n. However, f o r a l a s e r plasma a t lo2* cm-3, t h e equivalent value a f n i s n=3

-

i n o t h e r words t h e e n t i r e energy l e v e l s t r u c t u r e of a hydrogenic i o n with z=10 w i l l c e r t a i n l y be s t r o n g l y changed by t h e surrounding plasma environment, and a t most only one o r two excited s t a t e s w i l l be observable a s d i s t i n c t e n t i t i e s .

1 0 ' ~ 1018 10" N , c m 3 70"

F I G 3 PLOT OF PRINCIPLE QUANTUM NUMBER AT WHICH THE HOLTSMARK MEAN FIELD I S 10% OF THE NUCLEAR COULOMB FIELD VERSUS ELECTRON DENSITY N, AND EMITTER CHARGE Z

4. m e t i c Fields

In present day Tokamak plasmas, t h e confining magnetic f i e l d s t y p i c a l l y l i e between 1-10 Tesla, and thus r a t e a s r a t h e r moderate f i e l d s by t h e standards o f

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t h i s Conference. From t h e p r a c t i c a l s p e c t r o s c o p i c viewpoint, t h e t o t a l magnitude o f t h e f i e l d is a c t u a l l y n o t o f t o o much i n t e r e s t , s i n c e t h e f i e l d is p r i m a r i l y e x t e r n a l l y a p p l i e d , and t h e only p a r t one would want t o measure is that generated by t h e plasma c u r r e n t , t y p i c a l l y between 1-10% o f t h e t o t a l . A s w i l l be shown below, i n p r a c t i c a l terms t h e Zeeman e f f e c t o f t h e o v e r a l l f i e l d i s normally already swamped by Doppler broadening. Although given s e n s i t i v e enough p o l a r i z a t i o n dependent techniques one could envisage a measurement o f t h e a b s o l u t e f i e l d magnitude i t would normally be hard indeed t o determine t h e plasma induced component and hence t h e p i t c h a n g l e from Zeeman s p l i t t i n g measurements. Attempts t o d e v i s e means o f measuring t h e p i t c h a n g l e o f t h e t o t a l f i e l d (an important q u a n t i t y i n s t a b i l i t y c o n s i d e r a t i o n s ) have t h e r e f o r e concentrated on l a s e r s c a t t e r i n g techniques; s e e t h e d i s c u s s i o n i n r e f e r e n c e (2).

Laser-generated plasmas on t h e o t h e r hand are s o u r c e s o f very i n t e n s e magnetic f i e l d s indeed spontaneously g e n e r a t i n g f i e l d s i n t h e range o f 100-1000 Tesla (1-10 Megagauss), Ref. (11 ). These f i e l d s a r i s e from t h e very s h o r t s c a l e l e n g t h s f o r d e n s i t y and temperature g r a d i e n t s i n such plasmas, t h e most important c a s e perhaps being when e l e c t r o n d e n s i t y and temperature g r a d i e n t s a r e orthogonal.

The r e s u l t a n t

" v

n,VTrr magnetic f i e l d s t h u s e s s e n t i a l l y arise a c c i d e n t a l l y , crossed d e n s i t y g r a d i e n t s o f this type u s u a l l y r e s u l t i n g from l a c k o f uniformity i n t h e l a s e r i r r a d i a t i o n of t h e t a r g e t .

A rough estimate o f t h e s i z e o f t h e B f i e l d s generated from d e n s i t y g r a d i e n t s can be obtained from:-

where Te is t h e e l e c t r o n temperature i n Kelvin, A and Z a r e t h e atcmic weight and t h e mean charge s t a t e of t h e t a r g e t m a t e r i a l r e s p e c t i v e l y and L is t h e s c a l e l e n g t h f o r t h e g r a d i e n t s measured i n microns. With L t y p i c a l l y t e n microns o r l e s s and t h e r a t i o of A t o Z being roughly u n i t y , v a l u e s o f B reaching one thousand Tesla a r e e a s i l y a t t a i n e d .

S p e c t r o s c o p i c a l l y , one then first has t o compare t h e Zeeman effects o f t h e f i e l d s i n t h e two t y p e s of plasma w i t h those o f Doppler and p r e s s u r e broadening, and secondly t o e n q u i r e whether any f u r t h e r e f f e c t s a r e caused, e.g. on t h e mechanisms underlying p r e s s u r e broadening.

Canparison o f Zeeman s p l i t t i n g and Doppler broadening is s t r a i g h t f o r w a r d and with T i n Kelvin y i e l d s : -

A rough e s t i m a t e o f t h e Zeeman s p l i t t i n g t o t h e l i n e a r S t a r k e f f e c t i n t h e Holtsmark mean f i e l d s i m i l a r l y y i e l d s

- Avz

⁼8.109 B(Tesla)(Z/n)N (10)

where n is t h e p r i n c i p a l quantum number o f t h e s t a t e o f i n t e r e s t , and N is t h e e l e c t r o n d e n s i t y i n cm-3.

It then follows t h a t i n a Tokamak, t h e Zeeman and S t a r k e f f e c t s a r e roughly canparable; but both a r e normally e n t i r e l y masked by t h e Doppler broadening which t y p i c a l l y w i l l be about one thousand ( o r more) times l a r g e r f o r t h e s o f t X-ray t r a n s i t i o n s ( 100

1

and below) o f prime i n t e r e s t .

I n a laser-generated plasma on t h e o t h e r hand very i n t e n s e magnetic f i e l d s w i l l be generated i n t h e s u r f a c e regions near t h e c r i t i c a l d e n s i t y l a y e r (where t h e l a s e r frequency is e q u a l t o t h e e l e c t r o n plasma frequency), i.e. a t e l e c t r o n d e n s i t i e s t y p i c a l l y o f t h e o r d e r o f 1C2i-1022cm-3. With v a l u e s o f B o f 100-1000 Tesla, Doppler broadening and Zeeman s p l i t t i n g may w e l l be comparable even f o r s o f t X-ray t r a n s i t i o n s . On t h e o t h e r hand both w i l l normally be exceeded by pressure ( 'Stark ) e f f e c t s on t h e l i n e width.

D i r e c t (Zeeman s p l i t t i n g ) e f f e c t s o f magnetic f i e l d s are t h e r e f o r e o f t e n masked by o t h e r e f f e c t s i n both Tokamaks and laser plasmas, d e s p i t e t h e very l a r g e values of B that may be reached i n l a s e r plasmas. It should however be noted that

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i n lower density, lower temperature plasmas simultaneous magnetic f i e l d and pressure e f f e c t s on s p e c t r a l l i n e shapes may be observable both a s modifications of l i n e shapes as a whole, and i n causing p o l a r i z a t i o n e f f e c t s . These have been considered f o r hydrogen Balmer-alpha by Drawin and h i s co-workers, Ref. .(g).

Similarly, i n t h e s u b - c r i t i c a l density l a y e r o f laser plasmas (Ne <loz1 cm-3) magnetic f i e l d e f f e c t s on s p e c t r a of highly ionized e m i t t e r s may be important and t h e canbination of these with pressure e f f e c t s has been considered i n some d e t a i l by Nguyen-Hoe, 10.

There may, a l s o , be a number of subsidiary e f f e c t s o f magnetic f i e l d s which a l s o need consideration. These include t h e curvature o f perturber o r b i t s relevant t o pressure broadening c a l c u l a t i o n s , see Ref.10, t h e f a c t t h a t i n very s t r o n g magnetic f i e l d s t h e e l e c t r o n v e l o c i t y d i s t r i b u t i o n as a whole may becane s t r o n g l y a n i s o t r o p i c , and f i n a l l y t h a t t h e e f f e c t s o f c o l l e c t i v e plasma waves on observed l i n e shapes (see below) may be modified by t h e presence of e l e c t r o n cyclotron and hybrid modes.

5. Observable e f f e c t s on s p e c t r a i n dense plasmas

The r a t h e r simple considerations already given e s t a b l i s h t h e d i f f e r e n c e between laser canpressed plasmas and o t h e r sources i n two impartant respects.

F i r s t l y t h e e x t e n t t o which t h e plasma perturbs t h e p r o p e r t i e s of bound a t m i c l e v e l s is g r e a t e r i n a l a s e r compressed plasma than perhaps i n any o t h e r source, even when t h e e m i t t e r charge is high. Secondly, t h e strongly-interacting

(non-Debye) nature of t h e medium poses s t a t i s t i c a l mechanical problem which a r e of themselves of g r e a t i n t e r e s t (these l a t t e r properties can, however, be simulated i n lower density, lower temperature sources given t h e n, and T, dependence of

r

).

Although t h e present paper is not primarily devoted t o experimental problems it is worth noting t h e d i f f e r e n c e s , Table 1 , between l a s e r plasmas and Tokamaks i n this r e s p e c t also. Whilst a Tokamak is r a t h e r u n i n t e r e s t i n g a s a source of s t r o n g f i e l d e f f e c t s , it is nevertheless c l o s e t o being a s p e c t r o s c o p i s t ' s dream source i n p r a c t i c a l terms i n t h a t t h e plasma is very long-lived, l a r g e , and quasi-stationary ( i n r e s p e c t e.g. t o p r o p e r t i e s such a s i o n i z a t i o n equilibrium, e t c . ) . I n c o n t r a s t a laser-canpressed plasma is something o f an experimenter's nightmare and f o r t h i s p r a c t i c a l reason camparison of experiment and theory is still a t a r e l a t i v e l y very e a r l y stage. The plasma is only microns r a t h e r than metres i n s i z e , and l a s t s f o r picoseconds r a t h e r than t h e seconds c h a r a c t e r i s t i c o f modern l a r g e Tokamaks, s o t h a t t r a n s i e n t e f f e c t s on i o n i z a t i o n balance etc.

need consideration. Only t h e extreme brightness of these very small plasmas makes them observable a t a l l !

Q u a l i t a t i v e l y , t h e observable changes i n t h e s p e c t r a from laser canpressed plasmas correspond i n general terms t o those studied f o r many y e a r s i n low d e n s i t y plasma spectroscopy, t h e d i f f e r e n c e being f i r s t l y t h a t t h e s e e f f e c t s a r e observable f o r t h e f i r s t time f o r high z ions, secondly t h e s e v e r i t y of t h e perturbation, and f i n a l l y t h e s t a t i s t i c a l mechanical problems o f t h e strongly-interacting medium. I n general spectroscopic t e r n s , t h e e f f e c t s which need consideration are:-

( 1 ) Line-broadening / s h i f t

( 2 ) Field-induced 'forbidden ( AL=O, + 2 ) t r a n s i t i o n s ( 3 ) 'Plasmont s a t e l l i t e s and o t h e r plasma wave induced f e a t u r e s ( 4 ) Ionization P o t e n t i a l Depression ( a s d i s t i n c t from s e r i e s merging) (5 ) Doppler narrowing.

These e f f e c t s w i l l be returned t o a f t e r a b r i e f survey of t h e present s i t u a t i o n i n regard t o c a l c u l a t i o n of t h e plasma microfield and t h e e m i t t e r response thereto.

6. General Canments on Calculation of t h e Plasma Microfield and Emitter Response.

There a r e two basic l e v e l s a t which t h e emitter-perturber coupling can be considered.

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JOURNAL DE PHYSIQUE

Qle, appropriate a t moderate d e n s i t i e s and t y p i c a l of c l a s s i c a l line-broadening t h e o r i e s (see e - g . Griem (Ref.3), can be l o o s e l y c a l l e d ' p e r t u r b a t i v e f i n t h a t t h e plasma i s regarded a s t h e source of a f l u c t u a t i n g perturbing f i e l d , which can be calculated independently of any knowledge of t h e atomic s t a t e of t h e emitter. I n more formal language t h e system d e n s i t y matrix can be factored i n t o a plasma ( ' b a t h ' ) part and a term describing t h e emitter.

Within t h i s b a s i c approach it is possible t o account f o r some emitter-perturber c o r r e l a t i o n s , f o r instance i n t h e well-known e f f e c t s on t h e s t a t i c p a r t of t h e microfield d i s t r i b u t i o n i f t h e e m i t t e r is charged (which can be r a t h e r a c c u r a t e l y handled, s e e Refs.4, 5, 6 and 71, o r even i n approximate attempts t o t r e a t screening e f f e c t s on t h e bound electron-nuclear i n t e r a c t i o n due t o penetration of plasma p a r t i c l e s , usually r e f e r r e d t o a s l p o l a r i z a t i o n s h i f t s ' (see e.g. Refs.1 and 3 f o r general references and Skupsky, Ref .8, f o r a s p e c i f i c t h e o r e t i c a l treatment). Tnis type of theory of l i n e broadening and r e l a t e d phenomena i n moderate density plasmas has been i n t e n s i v e l y developed

-

^and ^{i n} ^low

temperature/low d e n s i t y sources experimentally checked

-

over many y e a r s , and f o r this reason has formed t h e b a s i s of most t h e o r e t i c a l p r e d i c t i o n s y e t used i n diagnostic s t u d i e s on very dense, hot plasmas. However, t h e e x t e n t t o which such predictions f o r high z-emitters i n hot, dense matter have y e t been checked experimentally i n s i t u a t i o n s i n which d e n s i t i e s and temperature a r e known independently a - p r i o r i is s t i l l extremely limited.

A t t h e highest plasma d e n s i t i e s of c u r r e n t i n t e r e s t such a ' p e r t u r b a t i v e ' approach c l e a r l y f a i l s f o r two s e p a r a t e reasons. F i r s t l y , even i f t h e s t a t i s t i c s of t h e plasma remain c l a s s i c a l t h e basic assumption t h a t t h e plasma * p e r t u r b a t i o n f can be t r e a t e d a s independent of t h e d e t a i l s of t h e e m i t t e r s t a t e f a i l s i f t h e emitter-plasma coupling becomes s t r o n g and comparable t o thermal energies.

Secondly, a s t h e plasma becomes p a r t degenerate, and a s i n t e r a c t i o n d i s t a n c e s amongst t h e plasma p a r t i c l e s become comparable t o atomic r a d i i (which they c l e a r l y do a t n

>

1024m-3), t h e e f f e c t s of the Pauli p r i n c i p l e become important both on the lperturber-perturber' i n t e r a c t i o n s and because t h e s p e c i f i c occupany of 'bound' s t a t e s i n a given e m i t t e r may strongly a f f e c t t h e s t a t i s t i c s of t h e ' f r e e r ( l p e r t u r b i n g V ) p a r t i c l e s . I n such a regime t h e only way forward i s t o attempt a self-consistent 'first p r i n c i p l e s ' s o l u t i o n of t h e physics of t h e system o f nucleus and 'bound' and ' f r e e ' e l e c t r o n s a s a whole. Such treatments, s e e below, do g i v e some u s e f u l information on properties such a s mean i o n i z a t i o n s t a t e

(relevant e.g. t o conductivity c a l c u l a t i o n s ) , e f f e c t i v e i o n i z a t i o n p o t e n t i a l depression, and even some (mean) opacity information, but a t present cannot provide t h e d e t a i l e d spectroscopic information on e.g. s h a p e s l s h i f t s o f s p e c i f i c l i n e s needed f o r diagnostic purposes.

7. 'Moderate d e n s i t y plasmas 7.1 General conlments

Provided t h e 'system' density matrix can be factored i n t o ' e m i t t e r ' and 'plasma' parts and

r

< 1 (and even i n some cases i f

r

> 1 ), then it is a r e l a t i v e l y t r a c t a b l e plasma physics problem t o c a l c u l a t e e i t h e r t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e instantaneous e l e c t r i c f i e l d , P(E), o r t h e Fourier components of t h e f i e l d a s a whole, E(k, w). Unfortunately, t h i s s t i l l leaves us a long way from being a b l e t o c a l c u l a t e a p r a c t i c a l quantity such say as a line-shape

-

o r even perhaps from making contact with t h e r e s t of t h i s conference where mostly

* s t r o n g f f i e l d s a r e e i t h e r s t a t i c o r have j u s t one o r two well defined Fourier components! The problem i n t h e plasma case is that c a l c u l a t i o n of t h e atomic response r e q u i r e s fundamentally d i f f e r i n g approaches f o r t h e d i f f e r e n t frequency ccmponents of E ( w ) , combined with t h e f a c t t h a t t h e atomic response i s 'non-linear' i n t h e sense t h a t t h e various frequency components cannot be t r e a t e d independently. The plasma case corresponds t o a f i e l d which is ( a ) l s t r o n g * , ( b ) has a broad frequency spectnun and ( c ) has superimposed strong peaks a t well-defined frequencies, corresponding t o t h e underlying c o l l e c t i v e plasma modes.

The problems (which i n t h e plasma c a s e have been encountered f o r 20 years and

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more) a r e t h e r e f o r e i d e n t i c a l t o those i n attempting t o t r e a t t h e response of atoms t o strong multimcde l a s e r f i e l d s i n which t h e modes a r e not n e c e s s a r i l y randomly phased. For t h e purposes of t h i s Conference it seems worth i l l u s t r a t i n g t h e consequent d i f f i c u l t i e s from first p r i n c i p l e s , s i n c e t h e extent t o which t h e problem is simplified by t h e assumption of a f i e l d t h a t i s s t a t i c , o r has only a few Fourier components may not otherwise be evident.

F i r s t , l e t u s note t h e obvious f e a t u r e t h a t t h e difference i n response of a system of atoms/ions t o a f l u c t u a t i n g f i e l d E( w) compared t o t h a t i n a d i s t r i b u t i o n of s t a t i c f i e l d s is not just computational but is observable s p e c t r a l l y , Fig. 4, i n t h a t t h e f l u c t u a t i n g f i e l d may generate s p e c t r a l i n t e n s i t y a t frequencies i n t h e l i n e p r o f i l e where no i n t e n s i t y can ever be caused by purely s t a t i c e f f e c t s .

A ) TRANSITIONS

UNPERTURBED STATIC FIELD FLUCTUATING FIELD

7

B l SPECTRA

NO FIELD DISTRIBUTION DISTRIBUTION OF STATIC FIELDS OF FLUCTUATING FIELDS

FIG

+*

COMPARISON OF STATIC AND FLUCTUATING FIELD EFFECTS ON A 3- LEVEL ATOM

7.2 Treatments t o second order i n E

Fig. 4 suggests t h a t t o some extent t h e l i n e p r o f i l e 'mapst t h e Fourier components of t h e f i e l d , E(w). I f second-order perturbation theory were adequate ( i . e . f o r a l l components of E ) , t h i s would in f a c t be t r u e rigorously, a s can be shown by writing down t h e l i n e shape s t a r t i n g e i t h e r from t h e time-dependent

(auto-correlatiion f u n c t i o n ) approach (see e.g. Refs.3,12) o r a suit;able s p e c t r a l representation (see e.g. Ref.13). I f one then makes (say) an S-matrix type expansion i n t h e perturbation, t h e r e s u l t is t h a t t h e l i n e shape, L(n ), is very simply r e l a t e d t o t h e so-called 'dynamic s t r u c t u r e f a c t o r t , S(k, w ), (which a l s o describes t h e l i g h t s c a t t e r i n g p r o p e r t i e s of t h e medium) and thence t o t h e s p e c t r a l d e n s i t y of t h e f i e l d E ( w ) . I n f a c t f o r hydrogenic t r a n s i t i o n s (see below f o r o t h e r species):-

with

n

= ^{0 -}4

where i s t h e line-centre frequency

and y(n) ^a I dk S

*

(k n)

( t h e k Z a r i s e s from t h e transform of t h e r

-*

dependence of t h e e l e c t r i c f i e l d )

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JOURNAL DE PHYSIQUE

and S ( k l Q ) = 4:-

SD

^I ^m1 ^m

where E (k, n ) is t h e d i e l e c t r i c function of t h e medium (which is e a s i l y calculable provided t h e random phase approximation a p p l i e s , i.e.

r <

I ) , N is t h e d e n s i t y , and k~ is t h e i n v e r s e of t h e Debye length. (For l i n e s from non-hydrogenic s p e c i e s t h e approach is marginally more complicated i n i n t e r p r e t a t i o n i n t h a t t h e relevant frequencies i n S ( k ,

n )

a r e s h i f t e d from t h e detuning (w- q,) by amounts depending on t h e d i f f e r e n c e frequencies of t h e atomic s t a t e s coupled by E(

n

) )

.

To t h e e x t e n t t h a t second-order perturbation theory is v a l i d , t h i s treatment then provides an elegant formulation of t h e e n t i r e line-shape problem i n which c o l l e c t i v e (possibly non-thermal) properties of t h e plasma a r e incorporated.

Fig.5 shows a representation of y ( n ) f o r a plasma i n which e l e c t r o n and i o n c o l l e c t i v e modes are somewhat enhanced (e.g. by a non-Maxwellian e l e c t r o n v e l o c i t y d i s t r i b u t i o n ) . The l i n e profile-represented by L(

n

) then has t h e following general f e a t u r e s :

-

( a ) An o v e r a l l Lorentzian shape f o r small (w-

4.

( b ) Discrete s t r u c t u r b representing t h e i o n and e l e c t r o n plasma modes a s represented by t h e frequency dependence of t h e generalized width, y

(n

).

( c ) A f a l l - o f f i n y ( Q ) a t l a r g e (w- ub) because of t h e corresponding decrease i n S(k,

n )

f o r frequencies l a r g e compared t o t h e e l e c t r o n plasma frequency.

I

\

0 1 1 1

LOG ( o / w p e )

FIG 5 THE WIDTH FUNCTION d AS A FUNCTION OF FREQUENCY THE CASE SHOWN I S FOR A NON THERMAL PLASMA WHICH HAS PLASMON ENHANCEMENT B)AND ION ACOUSTIC MODE ENHANCEMENT A) THE THERMAL CONTRIBUTlONS FOR ELECTRONS ONLY ARE SHOWN WITH A DOTTED LINE

The d i f f i c u l t y with t h i s type of approach taken on its own (even i n low d e n s i t y plasmas) a c t u a l l y results from ( c ) above. For l a r g e Q = (w

-

^a,^), ^L(0 ) becomes dominated by t h e s h i f t operator, A (

n ) ,

r a t h e r than by y( n). A(n) i n turn depends on a Cauchy type i n t e g r a l over y ( Q ) , and hence on a l l frequency ccmponents i n t h e f i e l d E(

n).

Examination o f t h e S-matrix type expansion f o r y(

n ) shows t h a t second-order is never adequate f o r t h e low frequency part. In f a c t t o properly recover t h e ordinary quadratic Stark s h i f t induced by a s t a t i c f i e l d one has t o go t o i n f i n i t e order i n t h e S-matrix type expansion.

The point of t h i s comnent is t o note t h e d i f f e r e n c e with t h e ' i d e a l i z e d ' cases o f s t a t i c o r monochromatic f i e l d s (whether weak o r stqong). I n both t h e s e cases one has t h e b a s i c information t o sum t h e expansion t o a l l orders i n E if need be. I n t h e s t a t i c case one recovers t h e ordinary theory of t h e Stark e f f e c t , and i n t h e monochromatic o s c i l l a t i n g l a s e r f i e l d case t h e complexity is simply that higher order terms i n t h e S-matrix may generate a wid= spectrum of atomic sum and d i f f e r e n c e frequencies. In c o n t r a s t , t h e d i f f i c u l t y i n t h e plasma case is t h a t higher order terms i n t h e S-matrix expansion coupke more and more frequency components of E

,

E(n), E ( n 8 ) , etc. We can c a l c u l a t e t h e whole o f t h i s s e r i e s only if t h e f l u c t u a t i o n p r o p e r t ~ e s represented by E(k,n' ) have r a t h e r s p e c i a l

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properties.

A t present t h e r e a r e t h r e e main ways of proceeding beyond second order i n E(k,n):-

1 ) F a t r i e r Decomposition of E i n t o 'low ( s t a t i c ) and 'high1 ( 'impact ) frequency components

.

2 )

'

Unified ( c o l l i s i o n a l ) Theories 3 ) 'Model Microfield' Theories.

7.3 Fourier Cecomposition Theories

'Ihe approach of t h e o r i e s under this heading

-

which cover a wide v a r i e t y of d i f f e r e n t physical approaches, see 3,9,10,14,

-

^ist o d i v i d e t h e e f f e c t s of t h e Fourier components of E i n t o 2 c l a s s e s , those frequency) that can be t r e a t e d simply as ' s t a t i c , f o r which t h e a t m i c response can then be calculated t o any l e v e l required, and those ( *high1 frequency) f o r which t h e f l u c t u a t i o n p r o p e r t i e s a r e c r u c i a l (which a r e then normally only t r e a t a b l e t o second order i n E ( n ) ) . The approach is then t o 'renormalize' t h e atomic p r o p e r t i e s perturbed by a s p e c i f i c s t a t i c f i e l d , E

,

t o use t h e perturbed s t a t e s as a b a s i s t o c a l c u l a t e t h e dynamic response o f this s p e c i f i c e m i t t e r t o t h e f l u c t u a t i n g high frequency components E (k,n) through second order (for emission/absorption processes t h e k dependence w i l l be l o s t i n t h e i n t e g r a t i o n , but t h i s is not n e c e s s a r i l y t r u e i n some schemes involving l a s e r s c a t t e r i n g o f f t h e perturbed atoms. 'Ihese, however, are f e a s i b l e only i n low d e n s i t y plasmas). m e r e s u l t a n t spectrum f o r a p a r t i c u l a r s t a t i c f i e l d E is then averaged over t h e p r o b a b i l i t y d i s t r i b u t i o n P(E

). Dependent on the plasma, the p a r t i c u l a r t r a n s i t i o n studied i n a given species, o r t h e p a r t i c u l a r region of t h e l i n e p r o f i l e of i n t e r e s t , t h e spectrum may be dominated by t h e s t a t i c p a r t alone, by t h e high frequency f l u c t u a t i o n s alone, o r by a combination of both.

h e g r e a t merit of this type o f approach is t h a t t h e s t a t i c f i e l d d i s t r i b u t i o n P(E ) can be accurately predicted even i n very dense plasmas, i.e.

with

r

> 1. There a r e physical, mathematical and computational reasons f o r t h i s . Physically P(E) i s well-behaved a s t h e plasma d e n s i t y (and hence

r

) increases.

The doninant contribution t o t h e low frequency components o f E is due t o t h e i o n s , and increasing

r

corresponds t o increasing ion-ion repulsion, and hence increasing s h o r t range order with t h e separation of a l l n e a r e s t neighbour i o n s becoming r e l a t i v e l y constant (and hence c l o s e t o R,, ). P(E) t h e r e f o r e l i e s scmewhere between t h e extreme low-density (uncorrelated) Holtsmark l i m i t and t h e simple n e a r e s t neighbour approximation and (when compared with o t h e r t h e o r e t i c a l and experimental u n c e r t a i n t i e s ) these do not d i f f e r a l l t h a t g r e a t l y , ( p a r t i c u l a r l y in so far a s t h e value of t h e most probable f i e l d is concerned), s e e Fig.1. Secondly, c l u s t e r - i n t e g r a l type t h e o r i e s

-

s e e i n p a r t i c u l a r Hooper, Ref.4

-

can include ion-ion and electron-ion c o r r e l a t i o n s a n a l y t i c a l l y very successfully and, as indeed proved by comparison with Monte Carlo type c a l c u l a t i o n s , these a n a l y t i c predictions remain good t o s u r p r i s i n g l y high values of

r .

^Finally,

molecular dynamicsf codes a r e capable of d i r e c t l y predicting t h e P(E) due t o a l a r g e number of i n t e r a c t i n g p a r t i c l e s r e g a r d l e s s of I'

,

see Weisheit and Pollock, and these show t h e c l o s e agreement of P(E) with t h e simplest n e a r e s t neighbour model a s

r

becomes l a r g e .

Theories with t h e e f f e c t s of t h e o v e r a l l E f i e l d s p l i t i n t o s e p a r a t e s t a t i c and f l u c t u a t i n g c o n t r i b u t i o n s , which s t i l l form t h e majority of computations a r e , of course t o some extent t y p i f i e d by t h e old ' s t a t i c i ~ n ' - ~ e l e c t r o n impact' t b e o r i e s of t h e type formulated e a r l y i n t h e development of l i n e shape theory i n p l s m a s , e - g . by Baranger, 12, G r i e m and Kolb, 14, etc. However, we have d e l i b e r a t e l y avoided using t h e ' ~ t a t i c ~ / ~ i m p a c t ' terminology as f a r a s possible above s i n c e t h e h i s t o r i c a l b a s i s of such t h e o r i e s tended t o be t i e d t o a s p e c i f i c a l l y t c o l l i s i o n a l (one-perturber ) approach t o t h e high frequency component, and t o t h e view t h a t a given perturber species can only c o n t r i b u t e t o one o r o t h e r component of t h e o v e r a l l perturbation (hence 'static ion-electron i m p a c t t ) an approach which has been somewhat superceded a s p a r t i c l e c o r r e r e l a t i o n s have become o f i n t e r e s t .

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JOURNAL DE PHYSIQUE

The formal b a s i s f o r i n c l u s i o n of e l e c t r o n c o r r e l a t i o n s i n t h e high frequency component a s w e l l a s i n t h e s t a t i c contributions t o t h e l i n e shape was derived v i a a v a r i e t y of approaches r e l a t i v e l y e a r l y , ( 13, ,15,16,17 ) and before t h e advent o f i n t e r e s t i n u l t r a dense laboratory plasmas, and l e a d s d i r e c t l y t o t h e type o f approach already outlined i n Section 7.2. However r a t h e r more controversey has surrounded t h e inclusion of i o n i c contributions t o t h e dynamical p a r t of E i n a d d i t i o n t o t h e i r usual incorporation v i a t h e static theory.

The d i f f i c u l t y here r e a l l y stems from t h e whole b a s i s of dividing t h e t o t a l e f f e c t s o f t h e f i e l d i n t o low and high frequency p a r t s , i . e . i n deciding whether t h e f i e l d itself o r its f l u c t u a t i o n p r o p e r t i e s cause t h e dominant e f f e c t s . Tne 'obvious1 way t o c a r r y out t h i s d i v i s i o n is t o use t h e Fourier theorem and t o compare t h e time v a r i a t i o n of t h e f i e l d with t h e ' l i f e t i m e 1 of t h e atomic s t a t e s of i n t e r e s t , i.e. t o consider:-

and then t o i d e n t i f y E ( t ) a s being due t o a s p e c i f i c s e t of perturbers (e-g. i n t h e ' s t a t i c ion'-'electron impact t h e o r i e s as due t o t h e o v e r a l l ion f i e l d (dressed by t h e screening due t o t h e e l e c t r o n s ) i n t h e case of t h e s t a t i c p a r t , and t o t h e e l e c t r o n s i n t h e dynamic p a r t ) .

The c l a s s i c approach is then simply t o w r i t e t h e l i n e shape i n terms of t h e o v e r a l l dipole a u t o c o r r e l a t i o n function f o r t h e system a s a whole, and t h u s i d e n t i f y ( ~ t ) - l as simply R

,

t h e frequency separation from line-centre, s e e e.g. Bararger, 12, and a l s o Fig.6.

Lower bound on v a l i d i t y af impact approx.

i s O p e ( e l e c t r o n ) o r lJpi ( i o n s ) .

& be v a l i d over much wider range o f t i g h t l y bound atoms.

J i\ ;;;;:,,2::::ves,

I I

A w Small Large

A t Long . ⁴Short

Stat.ttech.:- Impact L i n i t (Ilarkav approx.) Atonic Physics:- I n t e r a c t i o n may sometines

be t r e a t e d i n Born approx.

(2nd order perturb.the0.)

S t a t i c L i m i t I n t e r a c t i o n strong.

Born approx. f a i l s Close encounters.

This l e a d s t o t h e almost universal i d e a t h a t f o r l a r g e Q

,

i-e. i n t h e f a r l i n e wings t h e s t a t i c theory automatically becomes v a l i d r e g a r d l e s s o f t h e d e t a i l s of t h e t r a n s i t i o n concerned and t h a t f l u c t u a t i o n s can then be ignored. From t h i s approach derive t h e still very widely used s t a t i c ion, e l e c t r o n impact t h e o r i e s , s i n c e f o r fi values of most i n t e r e s t i n l i n e s from low i o n i z a t i o n s t a g e s p e c i e s i n low density plasmas, Eqn. (161, t u r n s out t o be such t h a t almost always t h e o v e r a l l i o n f i e l d (dressed by t h e screening of t h e e l e c t r o n s ) can be t r e a t e d a s purely s t a t i c and t h e e l e c t r o n c o n t r i b u t i o n s as purely dynamic ( i . e . dominated simply by t h e f l u c t u a t i o n p r o p e r t i e s ) .

This approach, however, is nowhere l i k e a s c e r t a i n f o r t h e very d i f f e r e n t s i t u a t i o n s of high z-emitters i n h o t , dense matter. I n this case t h e c h a r a c t e r i s t i c f l u c t u a t i o n t i m e o f t h e 'low frequency' p a r t of t h e microfield ( t h e ion plasma frequency) is almost always much l a r g e r r e l a t i v e t o t h e o v e r a l l linewidth, and from (16) dynamical e f f e c t s of f l u c t u a t i o n s i n t h e i o n microfield a r e r e l a t i v e l y much more important, see e .g. Lee, 18, and a l s o Lee and Freeman,

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Reference 19.

However, even before t h e complications of dense high z plasmas a r e encountered, considerable caution needs t o be adopted before using Eq.(16) a s a valid b a s i s f o r dispensing with t h e dynamical contributions of a p a r t i c u l a r cmponent of t h e f i e l d , whether low o r high frequency. F i r s t l y , a s pointed out several times by one of t h e present authors (see e.g. Burgess, Ref. 1 ), widely exploited a s Eqn. (16) is, it is s t i l l wrong! Large n simply does not eutomatically imply t h a t a s t a t i c f i e l d c o n t r i b u t e s t o t h e p r o f i l e a t a l l a t t h a t point, and consequently dynamical c o n t r i b u t i o n s may well dominate even when t h e nean f l u c t u a t i o n time of t h e f i e l d is long. 'his is evident i n Fig.&, where t h e r e would be no s t a t i c contribution t o t h e p r o f i l e between t h e p a i r of unperturbed s t a t e s , s o t h a t dynamic c o n t r i b u t i o n s a r e dominant regardless of t h e value o f

(and of whether Eqn.(l9) p r e d i c t s t h e s t a t i c theory t o be ' v a l i d ' ) .

A second complication of f a i r l y widespread p r a c t i c a l consequence a r i s e s from t h e f a c t t h a t t h e time of i n t e r e s t can only be defined i n terms of t h e frequency separation from l i n e c e n t r e i n t h e case o f a l i n e o f simple form (e-g. a Lorentzian). However, once d i s c r e t e wing s t r u c t u r e s , such a s field-induced forbidden components, become important t h i s d e f i n i t i o n f a i l s . (This was f i r s t pointed out by Burgess,20, and t h e consequent f a i l u r e of s t a t i c i o n t h e o r i e s demonstrated experimentally by Burgess and Cairns,21, and incorporation of t h e previously neglected 'ion dynamic7 contributions has been t r e a t e d i n a s e r i e s o f papers by Lee,18,22, a s well a s by many o t h e r authors, e.g. 23,24). This r e s t r i c t i o n on s t a t i c t h e o r i e s is a l s o important i n high density high z plasmas ( a ) because of t h e s i g n i f i c a n c e of allowed-forbidden l i n e p a i r s a s u s e f u l d e n s i t y diagnostics and ( b ) because t h e same consideration a p p l i e s near t h e d i s c r e t e s t r u c t u r e s corresponding t o t h e e l e c t r o n and ion plasma frequencies i n Fig.5.

Since t h e r e a l i z a t i o n t h a t Eqn. (16 ) was not a s u n i v e r s a l l y v a l i d a s implied i n t h e o r i g i n a l treatments such a s t h a t of Baranger,Ref.l2,several approaches have been adopted i n attempts t o incorporate ion dynamic e f f e c t s , including e f f e c t s of ion-electron c o r r e l a t i o n s on t h e ' e l e c t r o n t microfield, w h i l s t r e t a i n i n g as much a s possible of t h e low frequency contributions t o L ( n ) i n an e s s e n t i a l l y s t a t i c treatment. One approach due t o Kogan ,25, and a l s o discussed by Hey,26, is t o consider t h e e f f e c t s of a time-varying f i e l d on a s i n g l e Stark component, and then model t h e s e v i a a modified f i e l d d i s t r i b u t i o n . The r e s u l t obtained is:-

Hol tarnap M

P(E/& ) = P(E/Q A

+

A (17)

jr

where A and A t are flvlctions vary& w i t ; f i e l d , M and Mr a r e t h e perturber and r a d i a t o r masses, and V F / V ~ is t h e r a t i o of t h e f i e l d P f l u c t u a t i o n frequency t o t h e s t a t i c s h i f t . ?his approach then allows some modelling o f t h e e x t r a broadening due t o slow f l u c t u a t i o n s i n t h e s t a t i c f i e l d , but is e s s e n t i a l l y based on a semi-collisional model and does not form a systematic basis f o r treatment o f ion-ion and electron-ion c o r r e l a t i o n s .

On t h e o t h e r hand, Lee,18,22, has demonstrated t h e a p p l i c a t i o n of a formalism o r i g i n a l l y due t o Dufty,27, t o t h e p r a c t i c a l inclusion of i o n i c c o n t r i b u t i o n s i n t h e c a l c u l a t i o n of t h e e f f e c t s of t h e 'high frequency' ( f l u c t u a t i n g ) p a r t of t h e f i e l d . This approach c a l c u l a t e s t o second order t h e dynamical e f f e c t s of a l l Fourier components of t h e f i e l d ( i - e . a l l c o n t r i b u t i o n s t o S(k, CJ) regardless o f frequency) whilst a l s o r e t a i n i n g t h e f u l l e f f e c t s of t h e s t a t i c f i e l d v i a P(E).

The treatment can be shown t o be formally c o r r e c t through second order i n t h e f l u c t u a t i n g p a r t of t h e f i e l d , and has been used t o consider field-induced forbidden l i n e s i n low d e n s i t y plasmas, Lee ,22, and W o n Lee and Wlrgess ,28, and r e c e n t l y by Lee t o t r e a t t h e broadening of l i n e s of high z s p e c i e s i n dense l a s e r plasmas, Ref.18,19. A p a r t i c u l a r l y i n t e r e s t i n g e f f e c t is t h e discovery ( t h e o r e t i c a l l y ) of enhancements of broadening i n plasmas of high mean perturber z due t o increased electron-ion c o r r e l a t i o n s , an e f f e c t c l o s e l y r e l a t e d t o predicted enhancements i n Thomson s c a t t e r i n g in high man z plasmas,29. The same formalism i s a l s o well adapted t o c a l c u l a t i o n s o f spectroscopic e f f e c t s of enhanced e x c i t a t i o n of e l e c t r o n plasma waves i n plasmas containing non-thermal l e v e l s of f a s t e l e c t r o n s , o r of i o n modes i n plasmas with d i f f e r i w i o n and e l e c t r o n

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temperatures.

7 - 4 'Unified

'

c o l l i s i o n a l t h e o r i e s

If t h e ccmplete microfield E can be approximated a s due t o binary encounters with s i n g l e perturbers occurring i n i s o l a t i o n , then s u f f i c i e n t s p e c i f i c a t i o n of t h e f i e l d f l u c t u a t i o n s e x i s t s t o allow y ( Q ) t o be calculated t o a l l orders i n an S-matrix expansion. I n o t h e r words, t h e atom-plasma i n t e r a c t i o n is described simply in terms of a binary c o l l i s i o n T-matrix, and t h e ccmplete l i n e shape from l i n e c e n t r e t o l i n e w i n g s follows d i r e c t l y from Eqn.(ll). The theory is t h u s ' u n i f i e d ' i n t h e sense t h a t i t avoids s p e c i f i c a l l y making any static/dynamic division.

'Unified' t h e o r i e s can be derived e i t h e r c l a s s i c a l l y o r quantum mechanically ( t h e c l a s s i c a l approach being much more cumbersome!) and c l a s s i c a l path approximation c a l c u l a t i o n s o f l i n e shapes have been performed p a r t i c u l a r l y by Vidal

,

Cooper and Smith ,30, and by Voslamber ,31. A t low d e n s i t i e s t h e t h e o r i e s work f o r e l e c t r o n broadening s i n c e t h e region i n t h e l i n e wing a t which t h e second order expansion is inadequate already corresponds t o very l a r g e k i n S ( k , n ), and thus t o very c l o s e and hence purely binary encounters. However, such a binary c o l l i s i o n a l approach is not v a l i d f o r t h e low frequency f i e l d components dominated by t h e ions.

In general, ' u n i f i e d t t h e o r i e s a r e of l i t t l e , i f any, use i n t h e a r e a s under discussion i n t h e present paper, s i n c e under high density conditions l i n e shapes of i n t e r e s t a r e dominated by those regions of S ( k , n ) (and hence r ( n ) ) i n which p a r t i c l e c o r r e l a t i o n s play an important and possibly c r u c i a l ) r o l e .

7.5 'Model Microfield

'

Theories

The remaining a l t e r n a t i v e , developed o r i g i n a l l y by Frisch and Brissaud,32, and a l s o used and discussed i n p a r t i c u l a r by Seide1,33,34, attempts t o circumvent t h e d i f f i c u l t y of t h e higher order moments o f E( n ) by a purely model/computational solution. That i s , an attempt is made t o f i n d a model of t h e f l u c t u a t i n g microfield which has t h e r i g h t s t a t i s t i c a l p r o p e r t i e s a t both high and low frequencies and allows a complete c a l c u l a t i o n of t h e atomic response. The model microfield adopted is chosen s o that it i s completely s p e c i f i e d by t h e instantaneous p r o b a b i l i t y d i s t r i b u t i o n , P(E) and t h e second-order c o r r e l a t i o n function, i . e . <E(t ).E(o)> ( o r equivalently t h e s p e c t r a l density function EL ( n ) ) . (Tnis amounts t o a statement that t h e exact d e t a i l s of t h e way t h e f i e l d changes do n o t matter). Since t h e model f i e l d is e n t i r e l y s p e c i f i e d by these parameters there a r e no d i f f i c u l t i e s involving t h e computation of higher order terms involving s e v e r a l s e p a r a t e f i e l d components E(n), E( n l ) e t c . For t h e random (close encounter) p a r t of t h e f i e l d t h i s approximation may be reasonable, although r a t h e r few d e t a i l e d c a l c u l a t i o n s have y e t been made. However, it appears much l e s s c l e a r whether t h e method can handle c o l l e c t i v e e f f e c t s i n non-thermal plasmas, p a r t i c u l a r l y f o r i n s t a n c e those involving both e l e c t r o n and ion modes.

I n t e r e s t i n g discussions of t h e r e l a t i o n of t h e Model Microfield theory t o more orthodox techniques have been given by Seide1,33, and by h f t y , 3 5 .

8. Fields i n Very Strongly Coupled Plasmas, (

r>>

1 ).

In those plasmas where t h e r e is l e s s than one p a r t i c l e per Debye sphere w e may need t o reconsider t h e basic approach presented above. We have d e a l t with t h e change i n t h e low frequencies microfield due t o strongly coupled systems but we have not discussed t h e b a s i c assumption that t h e plasma f i e l d can be t r e a t e d a s a perturbation t o t h e r a d i a t o r s t a t e . I n a s t r o n g l y coupled system this p e r t u r b a t i v e approach becomes suspect.

The only a l t e r n a t i v e is t o attempt t o compute from first p r i n c i p l e s a self-consistent s e t of r a d i a t o r s t a t e s i n t h e presence of t h e plasma, i.e. t o account f o r t h e plasma by a complete renormalization. The major d i f f i c u l t y with this method is t h a t t h e plasma is t r e a t e d a s e s s e n t i a l l y static and we t h u s o b t a i n a frequency independent s e t of r a d i a t o r e n e r g i e s and wavefunctions. Clearly this