Kinetic theory of spectral line broadening in a nonequilibrium plasma

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Kinetic theory of spectral line broadening in a nonequilibrium plasma

Yu. Klimontovich

To cite this version:

Yu. Klimontovich. Kinetic theory of spectral line broadening in a nonequilibrium plasma. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-113-C7-126. �10.1051/jphyscol:19797434�. �jpa-00219437�

JOURNAL DE PHYSIQUE Colloque C7, supplkment au no 7, Tome 40, Juillet 1979, page C7-113

Kinetic theory of spectral line broadening in a nonequilibrium plasma

Yu. L. Klimontovich

Moscow State University, USSR

1. Introduction. - The theory of spectral line broa- microscopic phase densities in the sixdimentional dening has been developed for many years. On can space of each component of the plasma

find an account of its recent achievement in a number

of monographs and reviews [I-71. Up till now, howe- Na(x, t) =

c

^{6(x -} xia(t)), x = ( r , ~ ) ver, most attention was focused on the problems of the ¹ C i C N ,

radiation of ideal and equilibrium plasmas. Relatively (2.1)

few papers have been published, where nonideal and nonequilibrium broadening effects in plasmas were considered. Among them there are papers in which the influence of Debye screening [8, 91, of turbulence [lo-131, of the nonequilibrium velocity distribution of charged particles [14] are investigated.

The complexity of such problems is obvious.

No attempt is made here to give a complete survey on the recent achievements in the theory of spectral line broadening in nonideal and nonequilibrium plasmas ; this is impossible in a short report.

The main purpose of this report is to show how the theory of spectral line broadening in nonequilibrium plasmas can be included into the modern kinetic theory of partly ionized plasmas.

The partly-ionized plasma is an example for a many component system with chemical reactions. Therefore the kinetic theory of this system is very complicated.

We have no possibility here to discuss all processes.

We shall restrict ourselves to ones which can be described by taking into account the long range interactions between the particles and the interaction with the electromagnetic field.

For this reason we shall first consider kinetic equations the corresponding cross sections of which are calculated in the polarisation approximation.

This means that the cross sections are determined in the Born approximation, but with the dynamical polarization taken into account. At the end of the report we shall briefly discuss the possibility of simultaneous inclusion of both the strong interac tions at small distances (to leave the Born approxima- tion) and long range collective interactions. .

The list of references contains, mainly, books and reviews, while original articles were included only where they were more or less used in the preparing this report.

and the microscopic electric and magnetic fields EM, BM. These equations are [I 5-1 71

1 dEM 4 n

rot BM = - - _{c a t .}

+

- _c

C

_n e,

S

vNa(r, p, t) dp

,

div BM = 0

,

For given experimental conditions these functions are random functions.

In the case of the Coulomb plasma the system of equations for the random functions N,, EM is simpler, namely

2.2 THE MONOATOMIC GAS. - In the statistical theory of a monoatomic gas it is possible to use, as a starting point, the exact equations for the microscopic phase density

N(x, t) =

1

^6(x - xi(t)) , x = (r, p) (2.4)

l $ i $ N

and microscopic force FM(r, t)

a N a N a N

2. The basic microscopic equations. - 2.1 THE

- +

v - + F M ( r , t ) - = 0 ,

a t ar ap

FULLY IONISED PLASMA. - In the statistical theory of (2- 5 ) a fully ionised plasma it is possible to use as the pM(r, t) = - gradr

1

^{+(I r -} r' 1) ~ ( r ' , pl, t) dpl dr' starting point the closed system of equations for the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797434

C7-114 YU. L. KLIMONTOVICH

With this method of description the statistical theory of nonequilibrium processes in plasmas can be reduced to determining the first, second and higher moments of the random functions Na, EM, BM.

It is possible to connect these moments with the distribution functions f,, fz, f,,

...

Therefore the B-B-G-K-Y chain if equations can be obtained with the help of eq

.

^(2.3).

2.3 THE BASIC EQUATIONS FOR A PARTLY-IONISED PLASMA [15-171. - A fully ionised plasma and a monoatomic gas are two limitting cases of the partly- ionised plasma.

The kinetic theory of a partly-ionised plasma is much more complicated. This is due to the need to take into account intramolecular motions and ionisa- tion and recombination processes.

In the simplest case the partly-ionized plasma consists of three components : electrons, singly- charged ions, and atoms. The first two components

will be designated by the indeces a e, b

=

^{i, and the}

third by the double index ab The charged particles of components a, b will

-

^ei.

be called fvee and those of the component ab will be called bound. The system as a whole is neutral, so that the total number of negatively charged particles (both free and bound) is equal the total number of positively charged particles. We denote this number by N. Here (in this respect) we restrict our discussion to the case of a Coulomb plasma.

We can use now as the starting point the equation for the microscopic phase density of pairs of charged particles in the twelvedimentional space

In the classical theory the microscopic density Nab is given by definition

The corresponding quantum function is the operator of the phase density. It is defined by

Here

@,,

is the corresponding operator density matrix.

Instead of the variables r,, rb,'it will be more convenient to use

The equation for the operator density matrix bab(R', r', R , r", t) takes the form

+

[($ab(l r'

1) +

~ub@', r', t)) - (4ab(l

0 +

~ a b ( ~ " , rN, t))] b n b

.

(2*9) The operator

Oub

is determined by the operator of the electric potential @ by the relation

The equation for the electric potential $3 can be represented in the form

The eqs. (2.9)-(2.11) are the basic microscopic (operator) equations for a partly-ionized plasma. It is possible to obtain with the help of this system the eq. (2.5) (for the fully ionized Coulomb plasma).

In order to separate the free and the bound states and to describe transitions between them, we shall use eigen- functions of the energy operator of an individual atom. They are defined by the equation

We can represent now the density operator in the form

fiab(R1, r', R , r", t) = ((2

- 1

^P.,(P1, P", t) Pa(rl) Y J(rl'). YP.(Rf) Y$@") dP1 dP"

.

(2.13)

KINETIC THEORY O F SPECTRAL LINE BROADENING IN A NONEQUILlBRlUM PLASMA C7-115

From (2.9)-(2.13) we obtain a system of equations for the operator bap(P1, P", t )

a b a p

iFi

-

(P', P , t) = (E,

+

^{E p f - ED -} Epn) jap(P1, P", t )

+

^----

at (2 n m 3 ~ [ [ o a y ( p ' , p , t ) ~ ~ p ~ , p , t ) y -

- l j a , ( P 1 , P , t ) I j , p ( ~ , P , t ) ] d P . (2.14) The matrix element

uap

is defined by the expression

The values a = n,

P

⁼ m correspond to the discrete spectrum, while a = p', = p" correspond to the conti- neous spectrum.

We have thus reduced the eq. (2.9) to a system of equations for four operator density matrices

p^,,(P1, P", t ) ; @,,,,,,(P1, P", t ) ; pap@', P", t ) ; pP,p(P', PU7 t )

.

(2.16) The first two describe the bound and free states of charged particles respectively, and the last two describe transitions between the free and bound states.

3. Kinetic equations for the distribution functions of the electrons, ions and atoms. - We present the deri- vation of the kinetic equations in two stages. In the first we obtain kinetic equations for the distribution functions of pairs of charged particles. From this equation we obtain afterwards the equations for the distribution func- tions of the electrons, ions and atoms. We confine ourselves here to the case a spatially homogeneous plasma.

Under this condition we have

Here f,(P1, t ) is the distribution function ofpairs of free (a = p') and bound (a = n ) charged particles,

After averaging the system (2.14) we obtain under the condition (3.1) a system of two equations for the functions

2

v

( P I , t ) = - i; S I m ( 6baP(P1, P", t ) 6 o p a ( ~ " , P', t ) )

-

d P ^E Za(P1, t)

.

dt B (2 n k ) 3

Here Ia(P1, t ) are the corresponding collision integrals. They are determined by the correlation of the fluctuations 6pap, 6UUp. It is possible to write the collision integral I, in the polarization approximation in the form

Za(Pf, t) =

- 1

dm dK dP"

/

^Par(K)

l2 -

¹ 6(kK - (P' - P")) 6 ( l w - (E,

+

^{Ep, - Ep - Ep.,)) x}

( 2 7113 fi p lYZ

The matrix elements Pap are determined by the expression

P a p ( K ) = ![% exp (i

2

^Kr)

⁺

^{e, exp}

(-

ⁱ

3

^Kr)] Y $ ( r ) Y p ( r ) dr ,

E(O, K) is the corresponding dielectric constant of the partly-ionized plasma. It is determined by the formula

4 n n V

&(a, K ) = 1

+ 1

dP' dP1' S(hK - (P' - P"))

1

^a

1 L '

^-

P " 1 ^.

^{( 3 6)}

K ( 2 nh) ap A(w

+

^{i d ) - (E,}

+

^{Ep, - ED - EpTr)}

C7-116 Y U . L. KLIMONTOVICH

The sum in the expressions (3.2)-(3.6) has the meaning

a

It follows from (3.6), (3.7) that the polarizability of a partly-ionized plasma consists of four parts

Theindices f and b denote the free and bound states.

The spectral density of the field fluctuations in the expression (3.4) is given by the formula

x 6(hK - (I?' - P ) ) 6 ( h - (Ea

+

^{E p - Ep - E p ) )}

¹

^Pap(K)

I2

K2

I

^{E ( W ,} K) ^{( 2 ' (3.9)} The spectral density of the field fluctuations (as does the polarizability) consists of the four terms

Now we can make the second step in the derivation of the kinetic equations for the distribution functions of the electrons, ions and atoms.

The free charged particles remain most of the time at distances such that for them the eigenfunctions of the continuous spectrum can be replaced by plane waves. In this approximation the matrix element

1

^Pprpt,(K)

l2

of the free particles is given by

To describe the motion of free particles it is more _{a f a}

convenient to use the variables pa, p,. Then we have (pa, t) = - (2 ~ h ) ~ kab(Pa3 Pb, d ~ b

"J n

- (P, t ) = In(P, t)

.

Note that the function Nf (pa, p,, t) determines the dt mean number of the pairs of free charged particles

with momenta pa and p,. In order to take into account The distribution functions of the electrons, ions and the possibility of the formation of atoms from free atoms are normalized in the following way

particles of

arbitrary

pairs, we make the following V Na substitutions in the collision integral term (3.4), which

(~;;i;ji

Sf.(Pa, t) dp. = C. ⁷ describe the transitions from the free states to the

bound states and vice versa ^Nab

(, nvfi)3

Sf,@,

^{t) d p =}

^-

^{N c a b}

^.

^(3.15)

Nf ha, Pb, t ) ^-+ Nfa,(~a, t) fb@b, t ) (3.12)

Here Ca, Cab are the electronic, ionic and atomic A similar substitution is made in expressions (3.6), concentrations respectively. They satisfy the condi-

(3.9). tions

Then, taking into account that C a + C a b = l , C a = C b . (3.16) In the eauilibrium state the distribution functions

S

^V are given by the expressions

f a = f,bba, pb, t,

(2

d ~ b 7 (3. 3,

Pa - E p ,

.

Nfa = exp[ kT

1'

we obtain from (3.3), (3.4) the following system of

the kinetic equations for the distribution functions of ^- En - Ep

electrons, ions and atoms

1.

^(3.17)

KINETIC THEORY O F SPECTRAL LINE BROADENING IN A NONEQUlLlBRlUM PLASMA C7-117

Where the chemical potentials p,, ,ub, pa, are given by C,Cb-n,n, (m,m, k ~ ) " " 1 p a + P b = k b ; - - - =

- -

^-

Cab nab M 2nh2 Z '

(3.19) (3.18)

We can consider now the structure of the collision integrals; the collision integrals for the electrons and ions can be represented as the sum of two terms and the concentrations n,, n,, n,, satisfy the ionization-

equilibrium condition (the Saha formula) Ia(~a, t ) = [IaI 1

+

^11af2

-

^(3.20)

The first term is given by the expression

e2 I pi2

[Ia(pA, t ) ] = --!-{dp: d o dK

-

^d(hK - @A

-

^pi))

^S (hw

-

(

- -

))

^x

(2 n)3 fi K2 2m 2ma

For the fully ionized plasma (Ca = C, = 1, Cab = 0) 4. pi

+

^{ml P';-p:}

+

^nl

Pi

this term coincides with Balescu-Lenard collision

integral. For the partly-ionized plasma the collision is the inelastic scattering process.

integral (3.21) consists of four parts which describe In all these processes the number of the particles

four processes with momentum p, remain unchanged. Therefore the

collision integral (3.20) has the property

1.

PA +

^{~ ; ' b @} P:

+

^{~ ; b}

is the elastic scattering process.

2. pi

+

m, P;'-p:

+

^{p i ,}

+

^{p i ,}

is the (in the direction from left to right) ionization

of an atom by collision with an electron (a = or In order to obtain an expression for the second term ion (a = i). in (3.20) it is necessary to put in (3.4) a = n,

fi

= m to make the substitution p',

P'

-, pi, p; and to inte- 3. pi

+

pya

+

pYb *pi

+

^nl

^P;

grate with respect to pl,. As a result we obtain the is the process inverse to (( 2 )). following expression

After substituting the functions (6E dE),,, &(a, K) this integral breaks up in turn into four parts, which describe the following processes :

5 . p,:

+

^p;

+

^{pia C* mPW}

+

^pia

.

6. p:

+

^p;

+

^{ml P;' tt m P}

+

^pia

+

^{p i b .}

(3.24) 7. p : + p l , + p ' ; , + p ' ; , o m P " + n l P ; .

8. p:

+

^pl,

+

^{m1 Py t,mP"}

+

^{n, Pi.}

Here the processes (( 5 >), ^<( 7 >), ⁽⁽ 8 ^>) are recombination and ionization processes which change the number of the free charged particles. The process cc 6 ⁾⁾ is the inelastic scattering process accompanied by particle exchange. In all this processes the number of particles with momentum changes. Therefore the collision integral [I,(pA, t)], has the property

C7-118 YU. L. KLIMONTOVICH

The collision integral In(P1, t) in the kinetic equa- but for the second term tion for atoms also can be represented in the form of a

sum of two terms

z 1

[ I n p l y ~ ) I Z d P ' f 0 . (3.29)

In(P', t) = [In(Pf, t)] ¹

+

[In@", t ) ] ~

.

^(3.26)

In the case of a zero degree of ionization, there The properties of collision integrals are such that remains in the collision integral In(P1, t) only one they ensure conservation of the total (free and bound) term, describing the process number of Charged particles, total momentum and

energy.

"' + m1 P; * mP" ⁺ n l

.

(3.27) Let us consider the state of a plasma in which the It is elastic (n = n, ; m = m,) and inelastic scattering non-equilibrium character is due only to the fact that

of the atoms. the concentrations na, n,, nab do not satisfy the ioniza-

Each term in (3.26) describes four processes. For tion equilibrium condition and the concentrations the first term we have are functions only of the time.

Using the kinetic equations for the distribution [ ~ ~ ( p f , t)

1 v

d p = 0

,

(3.28) functions of electrons, ions, atoms (see (3.14)) we

n (2 nfi) obtain equations for the concentrations n,, n,, nab

-

dna _dt ^{= (xna nab - pn? nb)}

+

(or, n:b -

p,

n, n, nab)

+

^{(a2 n i -}

p,

n,2 n,2)

+

Here cr is the impact-ionization coefficient,

/?

is the triple-recombination coefficient, etc. The coefficients a, /3 are connected by the general relation

It is possible using the developing method to obtain the kinetic equations, in which not only Coulomb interactions are taken into account, but also a trans- verse electromagnetic field. The kinetic equations have the form (3.14), but now the expressions for the collision integrals I,, I,, In have the additional parts.

For example

The collision integral

IF)

also can be written as the sum of two terms

This collision integrals take into account not only all the usual processes of photoionization, photore- combination, emission, absorption, etc., but also all so-called anomaleous effects (Cherencov effect, the anomalous Doppler effect, etc.).

The spectral density of the field fluctuations :

The polarization of the plasma is defined by the two functions d l ( a , K), &'(a, K).

4. Kinetic theory of spectral line broadening [15]. -

For the calculation of spectral line broadening it is possible to use the equation for the component of

polarization vector Pnm(R, t) which corresponds to the n - m transition. This function is connected with the off-diagonal density matrix f,,(R, P, t) by a relation :

Thus the task is reduced to the investigation of the kinetic equation for the function fnm(R, P, t). The density matrix is connected with operator

Fn,

^as

The operator density matrix

fin,

obeys the equation analogous to (2.14). With help of this equation we can obtain following kinetic equation

I t is obvious now, that we need an expression for collision integral In, to solve the problem of spectral line broadening.

Now we present the results obtained with the follow- ing simplifying approximations.

KINETIC THEORY OF SPECTRAL LINE BROADENING IN A NONEQUILIBRIUM PLASMA C7-119

I. Corcelations of fluctuations, which determine the collision integrals are calculated, neglecting the action of the mean electromagnetic field.

2. All calculations are performed in the polarization approximation.

3. The dipole approximation is used.

The corresponding collision integral is given by expression

and can be represented in the form of a sum

(Induced) + I(spontaneous)

Znrnm, P, t) = Inrn ^{n rn} (4.4) Here we present, as an example, the expression for the induced part of the collision integral which is proportional to the spectral density of field fluctua- tions

:So

a, ^dr J,&I+e-*r+i(w-.:m-Knr

I$:*'(CO, R, P ) = - -

x [.I 1 , d n n l ~ @ m ~ r - ( d n n ~ ( d r n m ) j ] ( 6 E j 6 ~ i ) - r , K f n r n ( R , P , ~ ) + n - m , * , - o . (4.5)

The line width and frequency shift of the spectral line are determined from the following equation

Using (4.5), (4.6) we obtain the following expression for the spectral line width

The spectral density of field fluctuations is determined by expression (3.32).

Neglecting the Doppler effect and at o - on, = 0 :

We see that the y,, is determined by two different contributions. The first one is determined by the spectral density of the field fluctuations a t frequencies close to the transition ones of free atoms. The second contribution is deter- mined by the spectral density at low frequencies close to the zero.

The corresponding expression for the frequency shift Awn, is given by

Neglecting the Doppler effect we can obtain the following simplified form

The expression for the quadratic Stark effect in the 1 AEn =

-

field with spectral density (6E 6E), follows from (4.10).

C

^Idnnl

I2

^x

6 A . + , I

For the monochromatic field

.[

^{o n n . - 0}

⁺

^{Wnn1 1 + a}

(6E 6E), =

f

IF,,

l2

2 n(6(w

-

o , )

+

6(0

+

o,))

.

(4' Let us consider now the more general expressions The corresponding expression for the energy shift of for the spectral line width taking into account both

the level n is the induced and spontaneous fluctuations.

C7-120 YU. L. KLIMONTOVICH

5. Broadening determined by the induced and spon- Here the folloying notations for the transition pro- taneous processes. Transition probabilities [I 51. - The babilities are used

integral In in the kinetic equation for the distribution

function of atoms

f,

can be represent in the form Wnm = ~ ( 1 1 ) nm

+

^Wg)

.

(5.2) (see (3.31), (3.20))

Where the two contributions denote the transition In@', t) = I: qI" t)

+

ZiL)(P, t) probabilities determined by fluctuations of Cou- lomb ((( // D) and transverse (((

I

>>) field respecti-

=

C

[ Wmn fmP,

'1

- Wnm fn(P,

'11 .

^{(5 1) vely} :

m

Here the conditions dm = 0, dm, = 0 are used.

Dissipative matrix ynm which determines the spectral line broadening, is expressed in this approximation as

We see that the transition probabilities are deter- mined by the functions (SE SE),,, &(a, K), which have the structure (3.8), (3.10). Therefore both

Y!m and

yA

can be represented as a sum of the four terms.

For the radiation region

The expressions for the broadening and shift of the spectral line presented above are also valid in nonequilibrium states. In this case the functions ynm, Amnm are determined by the distribution func- tions of electrons, ions and atoms which obey the kinetic eq. (3.14).

Therefore it is possible to determine the spectral line broadening for the nonequilibrium processes provided the solution of the kinetic equations are known. It seems natural that such a solution can be obtained practically only using the simplified (model) collision integrals.

For the radiation region the problem is reduced to the calculation of nonequilibrium spectral density of field fluctuations. This problem is related with the problem of spectral line broadening in turbulent plasma.

.

^(5.6) Let us consider now the particular applications of the general formulae.

In the equilibrium state the right hand side of this 6. Broadening by electrons [3-7, 151. - We shall expression can be rewritten using Einstein coeffi- determine the atomic spectral line broadening with

cien ts the following assumptions :

W& = B; p&, -t- A : . (5.7) 1. The concentration of atoms is small.

2. There is local equilibrium of electrons in the It follows from (5.5), (5.7) that the term determined velocity space.

the spontaneous processes leads to the

3. The dynamical polarisation is into account expression for the line width :

by using the effective potential.

y: 1

, =

2 [ Z

A:

+ 1

A:]

..

^(5.8) This is equivalent to the substitution

n> n l m z n l

4 n

-

¹ + - ^{471 r 2 K 2}

For the partly-ionized plasma the most general K2

I

^{&(a, K)}

l2

^{K2 1}

+ r i

^{K~ ' (6.1)}

transition probabilities are determined by the colli-

sion integrals I,@', t) (see eq. (3.4)). Four dissipative Let us devide the expression for ynm in two parts

#matrix ynm, y,,,,, y,,,, y,,,,,, can be expressed via these according to the structure of the general expres- transition probabilities. sion (4.7). The first part is determined by the high-

KINETIC THEORY OF SPECTRAL LINE BROADENING IN A NONEQUILIBRIUM PLASMA C7-121

frequency (w z on,) field fluctuations. After some calculations one obtains

Here rmin is defined as a maximum of the following three values :

rmax is defined as a minimum of two values :

rmax = mm

_. (

r,,

- 2:).

This result differs from the well known one obtained by Vlasov and Fursov [18, 21 only by the numerical coefficient (& instead of 4 3 ) . In (7.2) only the long- range (r > p,) interactions are taken into acount.

The inclusion of short-range interactions (r < p,) leads to the following result [14] :

The resonance spectral line broadening, caused by the resonance interaction of atoms, considered above is an inelastic process. We see that in this case the broadening is growing proportional to the gas pressure.

For the second contribution to which is determined 8. spectral line broadening by col~isions [19, by the low frequency field fluctuations One obtains 4, 151. - When the elastic collisions are taken into the following expression : account the backward process is possible : the pres- 4

6

^{(dnn -} dmm12 e2 n rmax sure growth causes narrowing of the spectral lines.

Ynm =

-

_h2

-

^{ln 4}

- .

(6.5) The significance of collisions in the elastic pro-

3 v ~ e rmin cesses depends upon the relation of Doppler

width K V ~ , the frequency of elastic collisions

;

^and

Where rmin is determined by (6.3) and rmax is defined

by the line width ynm.

It was assumed above that the velocity distribution

VTe

)

^(6.6) function of atoms is fixed. This is justified if two / w - % m l inequalities are satisfied :

r , is Lewis parameter.

Similar calculations have been also performed for the nonequilibrium distribution functions (see for ex. [14]).

7. Resonance spectral lme broadening by collision between atoms. - Let us consider another limiting case when the broadening is determined by atomic collisions.

The main contribution to the expression (4.7) can be obtained assuming that dnn = 0, dm, = 0 and considering only n - m transitions. Then the expres- sion for ynm can be written as

Let us single out the contribution to (3.9) which corresponds to the discrete spectrum (cc = n , ,

p

= m,) and put ~ ( w , K) = 1 because the polariza- tion effects are negligible in this approximation.

Finally let us single out the contribution of the resonance levels. In order to do this it is necessary to put n = n,, m = m, in the expression for the spectral density of field fluctuations.

After this we obtain the following expression for the line width (see [14]) :

It follow from (8.1) that the pressure dependence of y,, is not the only reason of the change of the spectral line profile. The change of the velocity of the atoms due to the elastic collisions is also signi- ficant.

Practically the relatively simple model of collision integrals are used when one wants to take into account the elastic collisions.

For example in the model of strong collisions [19,4]

.the collision integral has the form

Here v is the effective frequency of collisions, f(P) is the Maxwell distribution function.

Summarising the results of [19, 41 it is possible to conclude that the elastic collisions cause the narrowing of the Doppler width while the pressure is growing.

This is reason for the narrowing of the spectral lines.

The result depends also upon the function ynm(p).

The calculations [19, 41 have shown that the quali- tative results does not depend upon the particular model.

9. Prisoning of radiation. - The prisoning of radia- tion is significant, for example, in the theory of gas lasers.

fnm is oscillator strength.

C7-122 YU. L. KLIMONTOVICH

Let us assume that cc a >> (upper level) and cc b )>

are two essentiel levels ; assume also that the transi- tion from the upper level to the ground state (( o >>

is possible. Then the effects of spontaneous emis- sion (at a-o transition) and the resonance absorption (on the o-a transition) are possible. This process of resonance absorption prevents the radiation from leaving the active volume. Thus the prisoning of radiation takes place.

The efficiency of prisoning depends upon the concentration of atoms in the ground state. The

prisoning is complete if 1,

<

L, where I, is the mean free path of a photon emitted at the a-o transition.

The theory of prisoning of radiation was first constructed on the phenomenological level [20,21].

Recently the microscopic theory have been also developed [22-27, 151.

The collision integral I,I(P, t), which is determined by the transverse field fluctuations describes naturally the prisonning effect. The expression for this collision integral is given by the following form similar to (3.21) :

Now we shall make the following simplifications in 5. The recoil effect is neglected in the description order to describe the prisoning of radiation with the of the spontaneous emission at the a-o transition.

help of the kinetic equation for the function f,(P, t ) : The ~ o p p l e r effect can be also neglected here. For 1. The equation is written for the two levels a, b. the region of radiation we obtain the following dis-

sipative contribution : 2. The Doppler effect and recoil are taken into

account only for the a-o and b-o transitions. - A," fa@', t)

.

(9

-

³⁾

3. Only the spontaneous emission is taken into

account for the transitions from the levels a, b to Here A," is the Einstein coefficient for the a-0 trans&

all other ones except the ground state. tion.

Let ~ bdenote the b dissipative cons- 6 . The recoil effect is neglected in the term which tants. Then

describes the induced emission at the a-o transition.

Y =

z

^{A; ; Y; =}

z

^A:

^.

(9.2) However the Doppler effect is important here.

a > m > O b l m r O

4. We assume that the populations of the levels a After this simplifications the collision integral in and b are much less than that of the ground state ; i.e. the kinetic equation for the distribution func-

fa,fb

< fo -

^{tion fa@,'} t) has the form

Here the following notation is used Thus the prisoning of radiation leads to the redis- tribution of atoms in momentum space. Finally the J6(Ko V - KO V? faP1,

r)

dPf Maxwell distribution is established (if external sources

K(KQ, KO V) = are absent). Substituting in (9.5) the Maxwell dis-

[6(Ko V

-

KO V') fop", t) dP" tributions

J (9.5) N V

-- " a , ~ P 2

where KO is the unit vector, dl2 is the element of the V (2 7 c m 3 f40m) = (2 7cMkn3I2

(- -)

spatial vector in K-space. (9.6)

If the velocity distribution function of atoms in

the ground state is fixed, one obtains the linear one obtains ^"a integral equation for the distribution functionL(P, t). K = ~ .

KINETIC THEORY O F SPECTRAL LINE BROADENING IN A NONEQUILIBRIUM PLASMA ^C7-123 It follows from eq. (9.4), (9.7) that the integral term Here n, m are parabolic quantum numbers, W(E) is equal to zero in eq. (9.4). Thus the collision integral the field strength distribution function of ions,

reduces to y, is the width of spectral line defined, by fast fluctua-

tions (see §§ 4-7) and the inequality/'(lO. 1).

I,@', 't) = - 7,: fa@', t )

.

(9.8) In the zero approximation when the motion of We can make the conclusion that fixing of equili-

brium momentum distribution for all levels corres- ponds to the condition of complete prisoning.

On the contrary, neglecting the prisoning one obtains the following expression for the collision integral :

The influence of radiation prisoning on the processes in lasers was investigated in [22-271.

10. Influence of the field of ions on the spectral l i e profile [3-7, 151.. -The expressions of @ 4-6 which determine the spectral line width containe the spectral densities of field fluctuations with short correlation times z,,.

This corresponding the following inequality z,,,

<< -

1

.

Ynm

The correlation times of ions and electrons are different due to the strong difference in their masses.

Introducing the notations T , for tliese correlation ~

times one obtains

ions are not taken into account the function W(E) is the Holtsmark distribution. More general results are obtained in the papers of Kogan [28], Beranger and Mozer [I, 63, Ecker and Schumacher [8] ; Kuri- lenkov and Filinov [9].

In the past, Stark broadening methods had most applications in equilibrium plasmas. This methods may apply also to nonequilibrium plasmas.

The influence of electric fields from plasma waves bn line profiles has only recently become the subject of experimental investigations.

For strong wave excitation or turbulence the col- lective contribution may dominate and the shapes may be very different from line profiles calculated by formula (10.5) (see § 14).

11. Influence of the electron field on the intensity of radiation on the wings of spectral lines [7, 151. - For investigation of the influence -of the electrons on the intensity of radiation on the wings of the spectral lines, in inequality (10.1) the parameter

1 1

Q - . I o - W n m I Ynrn

Then on the wings of the spectral lines when

Here pWesi are the corresponding Weisskopf radius.

From the formula (6.5) follows that

Then the inequalities (1 0.1) may by satisfied simul- taneously at the conditions

The inequalities are just only at very low pressure.

More usual situation is

In this case it is possible to use quasi-static approxi- mation for the ions. Then the shape of spectral line is defined by the formula

m

Inm(0) = 2

j'

₁ ^{Ynm X}

o (o - Wnm

+

- h (dm - dmm) E

Y + YL

the fluctuations of electron field are also quasistatic.

In order to obtain more general formula giving qualitative description of the intensity of the radiation for all frequencies in the expression (10.5)

Ynrn YAW,,

Ynm + Tnm = (11.2)

Ynrn

+

YAW,, ' where

We see that in the case

1

^o

-

on, (

<<

Ynm ^width

vnm

^{x ynm}

^.

^{(1 1.3)}

npwe In opposite case when

l o - 0 ~ ~ 1 % - Ynm the width y",, x yAWn, (1 1 .4) np&.

and the intensity of the radiation is described by the formula

C7- 124 YU. _L. KLIMONTOVICH

his

is the double of the intensity of radiation defined by the ion field.

Then at the condition (11.1) influence of electron field and the ion field are equal.

More details on can find in a number of reviews [l-71.

12. Simultaneous account of strong (pair) and weak (collective) interactions. - We can discuss now the possibility of simultaneous inclusion of both the strong interactions at small distances (to leave the Born approximation) and long range collective inter- actions.

The B o l t n a n n collision integral for a plasma contains a divergence at large distances and the Balescu-Lenard integral diverges at small distances.

In many papers (see

5

56 in [16]) different forms of the collision integral, which simultaneously take into account binary collision processes and polariza- tion processes, have been proposed.

The simplest form proposed is a combination of three integrals : the Boltzmann I, Landau I, and Balescu-Lenard I,-, integrals

I, = I : - I," + I : - = . (12.1) In this expression the integral I t compensates the divergence of the Boltzmann integral at large dis- tances and the divergence of the integral at small distances. Such a generalisation, although

attractive because of its relative simplicity, is not completely satisfactory since this approximation leads to incorrect expressions for the thermodynamic functions for the nonideal plasma (see

8

56 in [16]).

We shall consider here another model, in which the dynamical character of the plasma polarization is taken into account approximately.

The collision integral for the fully ionized plasma can be written in the form

If we substitute the solution of the equation for the correlation function gab in the polarization approxi- mation we obtain the Balescu-Lenard expression.

The expression in the integrand in this case is proportional to the square of (Pa,@). One of the factors (in the initial expression (12.2)) remains unchanged, while the second changes, when pola- rization is taken into account :

We can take the averaged effect of the dynamical polarization into account in the following way. In place of (12.3) we use following effective potential :

We use here the expression for the spectra1 density cides with the Debye potential. In same approxima- of the field fluctuations. tion the correlation function defined by expression

In the state of the local equilibrium

- -

1

s e - r l r g

.

(12.6)

fab = 1

+

^{gab = exp -}

J

Using the effective potential we can obtain the (' ' 5, following system of equations for the distribution We see that in this case the effective potential coin- functions fa, fa, for the fully ionized plasma [16]

KINETIC THEORY O F SPECTRAL LINE BROADENING IN A NONEQUILlBRlUM PLASMA C7-125

Corresponding collision integral has not the diver- gences neither at small distances nor at large dis- tances.

The discussed method of effective potential is used in paper [36] for the account of the spectral line broadening in the partly ionized plasma. In this paper we considered also the results of Capes, Vols- lamber papers [12].

13. Narrowing of spectral limes by cooling of the gas under electromagnetic radiation. - Now we consi- dere shortly the some results of the kinetic theory of the cooling the gas by the electromagnetic field 131-351.

We shall use again the kinetic equation for the distribution function of the atoms f,(P, t ) with the collision integral (9.1). Using the induced part of this integral in which the spectral density of the field fluctuation is :

(6E 6E),,, =

-

(2 n)4 ₂

[I

E,,., 126(w - w,) 6(K - KO)

+

^{w + - w, K + - K] ,} (13.1) we can obtain the following equatlon for the temperature of the gas

Here we use the notation for the imaginary part of the polarizability ~ ( 0 , K, V) of atoms with velocity V

We see that the temperature of the gas becomes lower if o - wQb < 0.

If the width of the resonance is not zero then in the expression (1 3.3)

n 6(o - wab - KV) -, ^Yab

( o - wab - KV)'

+

^y$'

(From the (1 3.3), (1 3.4) it is not difficult to find that

The corresponding time of cooling

The situation considered here is, of course, the simplest. For some other details it is necessary to take into account

in

the real situation.

14. Influence of turbulence on the broadening of spectral lines. - The investigation of the influence f of the turbulence on the radiation spectra is one of the problems of the kinetic theory of fluctuations in nonequilibrium plasmas (see [lo-121 and the refe- rences quoted therein).

The kinetic fluctuations are these, the correlation time of which is larger or of the same order as the relaxation times of the kinetic equations. In other words the kinetic fluctuations are the fluctuations of

the distribution functions of the electrons, ions and atoms.

In strongly nonequilibrium systems the collision integrals and consequently all kinetic characteristics are drastically changed on account of the kinetic fluctuations. This leads also to an essential change of the shape of the spectral lines. The interest in this problem is governed by the development of the experimental investigations of turbulent plasmas.

We mention here only some of the recent papers on the theory of the line broadening in turbulent plasma.

In paper [12] a method was developed for the inves- tigation this problem, which is similar to the method of the kinetic theory of fluctuations [15- 171.

In papers [lo, 111 the investigation of the most specific problem is worked out. Especially the influence of the Langrnuir turbulence on the Stark component is investigated. In [ l l ] the corresponding theory is developed without application of the perturbation theory; in this way the region of applicability of theory is enlarged, and one may hope that experi- mental data may be explaned.

15. Conclusion. - In the theory of the broadening of spectral lines there is yet a lot of unsolved problems, which are connected with the influence of collective processes in the nonequilibrium plasma.

The investigation of this problems is an actual but very difficult task, the solution of which needs col- lective efforts of researchers of different countries.

YU. L. KLIMONTOVICH

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