TUTORIAL REVIEW

On the atomic state distribution function in inductively coupled plasmas- II. The stage of local thermal equilibrium and its validity region

J. A. M. VAN DER MULLEN

Eindhoven University of Technology, Physics Department, P. 0. Box 513,5500 MB Eindhoven, The Netherlands

(Received 25 July 1988; in reuisedform 8 Moy 1989)

Ahstrac-This paper, the second in the series, deais with the study of the impact of different typcts of equiiibrium departures on the atomic state ~str~buti~n function in plasmas. Global trends are studied and attention is paid to the first stage if equilibrjum departure: the stage of local thermal equjIibrium (LTE).

Contents

1. 2.

3.

4.

5.

Introduction Different forces applied to different particles; violation of LIE The transport of radiation; violation of PE and disturbance of LBE The influence of transport; disturbance of LSE and LBE Concluding remarks

References

Appendix 1, Summary of the four types of balances Appendix 2. A list of abbreviations of the various types of equilibria

1. INTRODUCTION

TOSTUDY the relation between the atomic state distribution function (ASDF) and the plasma parameters we investigated in Part I 111 the state of the~odynam~c equilib~um (TE). Emphasis was put on the dynamic features of the elementary level. By classifying the elementary processes in balances of forward and corresponding backward processes a first insight into the hierarchy of processes was obtained. Four types denoted by Maxwell, Boltzmann, Saha and Planck balances were studied. By employing the principle of microscopic reversibility (MR) on these balances, it was shown possible to derive the corresponding TE distribution laws. One condition needed in all the derivations is that the balances are in equilibriums i.e. that the number of processes per unit time and volume for the forward process equals that of the corresponding backward process. An outline of this approach is given in Appendix 1.

It should be realized that the study of the ASDF is especially focused on the balances of Saha and Boltzmann; the state of equilibrium departure of these balances will determine the distribution function, However, owing to the coupling of the different types of balances it is possible that the deviations from ~uilibrium of the balances of Planck and Maxwell will affect the equilibrium state of the Saha and Boltzmann balances as well. It is the aim of this paper first, to classify features of equilibrium departure, to study how these work out in the distortion of elementary balances and how this will affect the distribution function of atomic states; second, to look for moderate conditions of equilibrium departure such that the

2 J. A. M. VAN DER MULLEN

atomic state distribution function can still be given by the Saha-Boltzmann relation. This stage of equilibrium departure is known as local thermal equilibrium (LTE).

It is well known that TE is a fiction, i.e. a theoretical concept which can only be approached in practice. In any real plasma deviations from TE can be found. This applies especially to the inductively coupled plasma (ICP). As long as we want to use the ICP as a spectroscopic tool, it is not realistic and even undesirable to require the ICP to be in TE for the simple reason that in that case the line radiation of the analytes would be drawn into the Planck radiation background. This makes the detection of the various elements fundament- ally impossible. Therefore the presence of a Planck radiation field is something we want to avoid.

Nevertheless it is useful to understand the state of TE as it serves as a frame of reference with which an actual thermodynamic behaviour can be compared. Especially the represen- tation as given in Part I (summarized in Appendix 1 of this paper) in which TE is regarded as the equilibrium state of a set of balances, is very instructive. The reason for this is that TE deviations can be classified in stages and that each stage can be characterized by the validity or violation of (some) balances of the Planck, Boltzmann, Saha and Maxwell type.

The TE deviations found in laboratory plasmas are mainly caused by the fact that these plasmas are created by external forces in a limited space. Three features must be considered here:

(1) external forces are needed to create, maintain and limit the plasma, (2) radiation will escape from the plasma owing to the spatial limitation, and (3) gradients of temperature and densities will be present.

Corresponding to these features there are three causes for the distortion of the balances of Boltzmann and Saha and thus the ASDF:

(Cl) Different kinds of particles respond differently to the applied forces so that a difference between the translation temperature of the various particles will be induced. Since the translation energy of colliding particles is related to the internal energy, the temperature inequality will affect the ASDF. (C2) The escape of photons from the plasma will severely affect the Planck balances which can imply a distortion of the Boltzmann and Saha balances as well. (C3) Transport of particles effectuated by the presence of gradients may locally induce imbalances in the various elementary processes.

However, if the interaction between material particles occurs frequently enough so that the balances of the Maxwell type remain locally in equilibrium (number of forward processes remains equal to that of the backward processes), then we can follow the proof given in Part I and find that the Maxwell distribution is still present. If the balances of Boltzmann and Saha remain locally in equilibrium as well, then, as proved in Part I, the ASDF will retain its TE form.

This stage of TE departure in which Planck balances are violated but all other balances are locally in equilibrium is denoted by local thermal equlibrium (LTE). In this stage moderate gradients of density and temperature can be permitted and the plasma parameters have to be specified locally. This definition of LTE can be depicted as

LTE = local (TE - PE) in which PE stands for Planck equilibrium. or LTE = (LIE + LME + LSE + LBE > in which the acronyms stand for:

LIE: local isothermal equilibrium: all the different material particles have the same translational temperature, LME: local Maxwell equilibrium: all Maxwell balances are in equilibrium, LSE: local Saha equilibrium: all Saha balances are in equilibrium, and LBE: local Boltzmann equilibrium: all Boltzmann balances are in equilibrium.

Since this stage of TE departure is approached in several parts of the TCP it deserves special attention and it is especially important to know the conditions under which its occurence may be assumed. The validity or violation of a balance equilibrium depends on the rate of

On the atamic state distribution function in ICP-XI. 3

the balances. Apart from the concentration of the interacting species this rate depends on the rate coefficient. It is remarkable that the dist~bution laws in equiiib~um can be derived without information on the rate ~oe~cients; they are eliminated due to the principle of MR, while the presence of ~quilib~um cannot be investigated without knowledge of the rate coefficients. Thus the study of TE departure should be accompanied by the study of rate coefficients. This will be done in Part III of this series; in Part IV we shall show that the hierarchy of rate coefficients induces a systematic trend in the further stages of equilibrium departures. In this paper we will find that global effects of TE deviations on the ASDF can already be understood. Special attention will be focused on the stage of LTE and its validity regime. It turns out to be impossible to define the boundaries of LTE. Local thermal equilib~um is a fiGon as well. Only a part of the balances of the Maxwell, Boltzmann and Saha types can be in equilibrium.

2. DIFFERENT FORCES APPLIED TO DIFFERENT PARTICLES; VIOLA~ON OF LIE

An essential feature of laboratory plasmas is that different particles are subjected to different forces. For example, in an TCP the electrons are heated by the RF field while the cooling of the plasma is realized by an outer flow of atoms. This will induce a temperature difference between the electrons and the heavy particles; the plasma is no longer isothermal. The temperature difference T, # T,, can be substantial due to the fact that the energy exchange from an electron to a heavy particle is not very effective. This can be understood if one considers an elastic collision between an electron with energy E and an atom initially at rest. The energy exchange from electron to atom will not exceed 4nr/M in which m and M are the mass of the electron and atom respectively. So the small mass ratio m/tM causes an isolation between the translational energy of the electrons at one side and the heavy particles at the other. For the same reason the energy exchange between similar particles will be very effective (mass ratio of unity), so that the Maxwell balances for the energy exchange between like particles are easy to maintain. This means that the plasma can be regarded as being composed of two groups of particles, the heavy particles and the electrons, each having their own temperature and Maxwell distribution.

Now we consider the effect of the temperature difference on the ASDF. Since electrons as well as heavy particles collide with excited atoms, their difference in temperature will induce a competition between the Boltzmann balances ruled by the heavy particles:

and those ruled by the electrons

e+A, +% e+A,. (lb)

The first balance, in which H represents the heavy particle, tends to impose the heavy particle temperature on the atomic excited state distribution in the way discussed in Part I (CJ: Appendix l), while the second balance favours the electron temperature. In order to judge which particle wins this competition we have to study the rates of the corresponding reactions. A general expression for this rate of the corresponding forward reaction is given for the first balance by

n H $) KH& u) PaI

and for the second balance by

n, n(l) K’(l, u). (2b)

In these equations n, and n, are the densities of the heavy particles and the electrons, respectively, while n(p) refers to the density of the atoms in state p. The rate coefficient K(1, u) = <IS~~U> is the average of the product of cross-section and velocity.

First we compare the electrons with the ions. Since the rates contain the velocity of the particles, the collisions by electrons are in favour with respect to those by the ions because

4 J.A.M.vAN DER MULLEN

the mean electron velocity is much larger at a comparable temperature while the densities are equal (charge neutrality).

The rate coefficient for the electron induced transitions is also larger than that for transitions induced by atoms, since, apart from the larger velocity, electrons are charged whereas atoms are neutral. This results in ~~(1, u) << ~~(1, u). However, care has to be taken since the low ionization degree can disfavour the electrons. A more detailed study [2,3] shows that electron collisions are dominant as long as the ionization degree n,/n, exceeds 10m5, which condition is amply fulfilled in the ICP.

Consequently we may state that the main processes for (re)distribution of atoms over their excited states are those induced by electron collisions (eqn la) and that only a small ~rturbation can be expected from the heavy particles if T, f Tti

To understand what this dominance of electron collisions means for the ASDF we study the equilibrium of the electron maintained elementary balance B, of the Boltzmann type

e,+A,Ae,+A, (3)

where E refers to the kinetic energy of the particles while s refers to one of the spin states of the electron. According to the principle of MR the equilibrium of this balance implies that

ti,@> &) 4A& &) = %h Eh) fiA@h Efl)* (4)

Substituting the Maxwell value of the elementary occupation given in eqn I (19) (cf: Appendix 1) we obtain

h3 Fi,(i, S) = vlY(i) (2nm,kTY)3,2 exp( -EIki”,)

= v,(i) u&m,, T,) exp( - E/k T,)

which applies for any particle y (in this case A or e) with its corresponding temperature. Dividing the left and right hand sides of eqn (4) by u&t,, r,) and uA(mA, Th), respectively, we find that

E’--E Eb-Es * + k~ .

e h

Since there is almost no exchange of translational energy between the electron and the atom, the kinetic energy of the atom will remain practically unchanged and the internal energy of the atom is “coupled” to the kinetic energy of the electron. Mathematically expressed this means that, owing to the fact that rn~~M~<< 1, we get Eb= E, and consequently E*-- E,= -Et”, which implies that

r(WG=exp (- U’kT,). (7)

This is the Boltzmann relation containing the electron temperature only. Therefore, as a result of the dominance of electron collisions, the electron temperature is imposed on the atomic state dist~bution.

The essence of the phenomenon described in the preceding discussion can be formulated by stating that the translational coupling between electrons and atoms is worse than the coupling between the translational distribution of the electrons and the excitation of the atoms. This is closely related to the fact that an electron induced (de)excitation process can be regarded as a collision between two electrons of which one is bound [3].

The same reasoning applies to the Saha balance of ionization and recombination

e,+A,Ae,+A:+ci (8)

E, E, Ea E;, E;.

On the atomic state distribution function in ICP-II. 5

If this balance is in equilibrium, we can write (cf: Appendix I)

%!(% GJ UP9 &7) = 4&Y Eh) 4*+ (1, Efl) ii,li, E’,). (9)

After elimination of the volumes u,(m., Z’,,) = v(m.+, Th) and using the (near) equality E,=Eb we find

h3 (2Km,k r,)3/2 exp(I,/kll”,) (10)

i.e. the Saha equation containing the electron temperature only. This is a good example of how the principle of MR, which holds irrespective of the stage of

equilib~um departure, offers the possibility to find the atomic state dist~bution function in non-TE situations. The elegance of this derivation avoids mistakes as are often found in literature [4,5], where the Saha equation of a two temperature plasma is derived by using in various detours complicated thermodynamic quantities which are not defined in non-TE situations.

Even in the case that the temperatures of electrons, ions and atoms are different (three tem~rature plasma T,# T,# TA+), the principle of MR can be used to derive the Saha equation. In that case it can be found that

r]“(p) = ( TAIT,t+ J3” Y+ 5 h3

2 (2rcm,k TJ3/* exp(I,/k TJ (11)

in which the “extra” factor (T,/TA+)312 at the r. h. s. originates from the ratio uA+ (Q+,

TA + )/V.&Q> I-A). However, the occurrence of a difference between the ion and atom temperatures is not to

be expected in the ICP. First, there are no mechanisms strong enough to pull the two species A and A+ away from of each other; second, the mass ratio is unity which makes the energy exchange very effective; third, the presence of equilibrium of Saha balance will constantly exchange atoms into ions, and vice versa, which necessarily means mixing of the two species and equalization of the ion and atom temperatures.

Summary

The electron is the most eflective material particle in the maintenance of the Bo~t~~nn and Saha balances. The small mass ratio rn~~M gives the electron gas thermal freedom but also eflectuates that the atomic state distribution is ruled by the electrons only provided that the ionization degree is large enough. In that case the temperature rejected in the internal state distribution of atoms is related to the translational temperature of the electrons and not to that of the heavy particles. This stems from the fact that the electron induced (de)excitation process is mainly an interaction between two electrons of which one is bound. ff a plasma is non isothermal and the electron ruled balances of Bo~t~mann and Saha are eflective enough, we may use the electron temperature determined Saha-Boltzmann equation as the atomic state distribution function (ASDF).

3. THE TRANSPORT OF RADIATION; VIOLATION OF PE AND DISTURBANCE OF LBE

Let us regard the competition between a Planck and an electron induced Boltzmann balance for the same atomic transition

A,+-+hv + A,

+- p+ = hv hv

(12a)

Be

e+A .+--+e+A,. (12b)

6 J. A. M. VAN DER &fULLEN

If radiation escapes from the plasma, the radiation field will lose intensity, a deviation from Plan&s law can be expected and the spontaneous decay will force the density of the lower levels to increase at the cost of that of the upper level, i.e. the leak of radiation pulls the P balance to the right. The electron maintained Boltzmann balance (eqn 12b) can restore this if the frequency of the electron interaction exceeds that of the leak of radiation. This leak of radiation can be described by the expression

n(u) Ju, 1) A(u, 1) (13)

in which A(u,l) is the transition probability and n(u,l) the escape factor. The plasma is transparant for the radiation if n(u, I)= 1 representing the case of maximum escape. The more &u, I) approaches 0, the more radiation will be trapped and the more the Planck intensity will be reached (for that specific wavelength). In fact /I contains information about the absorption process and can be regarded as a measure for the equilib~um deviation of the corresponding Planck balance.

Here we arrive at a stage in which we can discuss the competing properties of the ICP. On one hand it should be an optically thin source (i.e. JI= 1 for any transition) so that the intensity is directly related to the density of the excited level. On the other hand it would be nice if this radiant flux caused only a small leak; i.e. if the electrons were not substantially disturbed in maintaining the balances of Boltzmann and Saha, since then we would only need to know the density of one level of each element, The rest of the densities can simply be computed using the Saha-Boltzmann law. But this condition is reached at a high n, value, which, owing to the frequent free-free and free-bound transitions scaling with a:, favours the existence of a Planck background radiation.

The preceding dicussion makes clear that it is useful to know especially the lower boundary of the n, value, i.e. the critical density, for which LTE is still present. For that purpose we need a measure to compare the total rate of the electron induced transitions with the leak of radiation. To this end we define for the atom A in state p the number of electron collisions per (effective) radiative lifetime given by the formula

R,(P) = n,k( p)/ A( p) (14)

in which k(p) =c k(p, q) is the total rate coefficient for transitions induced by electron collisions and A(p)= cA(p, l)A(p, 1) the total probability for radiative decay. The sum- mations extend over ail (relevant) levels. Since k(p) and A(p) are functions of the atomic level p, this holds for R, (p) as well and at a given (intermediates n, value we can distinguish between collisional A, (p) 3 1 and radiative levels R,(p) < 1. On the other hand we can also define for each level p a critical value of the electron density n:(p) such that A,(p)= 1; i.e.

nXk(p) = A(P). (15)

The condition that a particular level is completely collisional, i.e. that the density is not affected by radiative transfer, can now be formulated as the condition

n,k(p) @ A(P) (16)

or h, g n:(P).

This is related to the well-known Griem criterion [6]. In the third paper of this series we will discuss this criterion in a more quantitative manner. A global result is that for a given atomic system the critical density is the lower the higher the excitational state of the level. Thus the demand that all the levels of one particular atomic system are collisional mostly implies that tz, exceeds the critical density for the first excited level, i.e.

s, (2) > I or n, > n:(2). (17)

However, moving to the next ion stage (for instance, from Fe I to Fe II), with the demand that those levels be collisional as well, requires a larger n, value since n:(p) scales (for hydrogen-like ions) with Z7 in which Z is the charge number of the core. The third ion stage needs an n, value which is even larger, etc. This makes clear that complete LTE, that is, for all ionic systems, cannot be established. In fact, LTE is a fiction.

On the atomic state distribution function in ICP-II. I

Apart from the radiation leak we have to study the LTE deviations caused by plasma irradiation. An example of an irradiation. increasingly found in the literature [7] is that of laser irradiation. It can be seen from eqn (30, Part I = 18, Part II) that in the limit of high intensity spontaneous emission will lose importance with respect to stimulated emission so that we are left with the balance of absorption and stimulated emission. Thus the elementary occupation equation

%I(1 +4,)=r1,4, (18)

can be simplified to L .

V”VY = ?IrlY

which for the atomic state distribution results in

(19)

1”=11. (20)

This relation is the same as that described by a Boltzmann factor (cf: eqn 23 in Part I) containing an infinite temperature. Therefore this situation can be interpreted as one in which the extreme high ‘temperature’ of the laser field is imposed on the atomic state distribution. Such a Planck equilibrium between a part of the atomic system and the laser radiation field is denoted by a partial local Planck equilibrium (pLPE). Partial, because it only applies to a part of the energy levels (in this example: two levels) and local because it only concerns a spatial part of the plasma.

Another example of pLPE is that in which a hotter plasma part irradiates a colder one. Since the absorption rate in the colder part may exceed the local rate of spontaneous decay, a negative value of the escape factor may be expected, i.e. the density of upper levels will increase at the cost of the lower level. This describes that the higher temperature tends to be imposed on the cold plasma. Again these effects are of minor importance if the n, value is high enough, and a key parameter of the LTE presence is the n, value as described by the Griem criterion (eqn. 16). However, the Griem criterion is not a sufficient condition for the presence of LTE. It is merely a demand for the levels to be collisional, i. e. independent of radiation transport, and depends on the ion stage in question.

There is another point which has to be considered. As seen in the introduction, there are three causes for distortion of the distribution function; the first (Cl), discussed in section 2, is the temperature difference T, # T,,, the second (C2) is the transport of radiation treated in this section and the third (C3) deals with the transport of material particles and demands a supplementary condition. The impact of diffusion on the distribution function noted before by DRAWIN and EMARD [S] will be discussed in the following section.

Summary The transport of radiation violating Plan&s equilibrium will not substantially disturb the

Boltzmann equilibrium and the densities of the levels involved, provided the number of electron induced transitions per radiative lifetime is much larger than unity. This puts a demand on the electron density, which depends on the atomic system and the level in question. This demand known as the Griem criterion is not a sujficient condition for the presence of LTE.

4. THE INFLUENCE OF TRANSPORT; DISTURBANCE OF LSE AND LBE

In this section we consider one atomic system, the first system of the main gas of the ICP, which in most cases is the argon system Ar I. We assume that n, >> nE(I, 2), i.e. that all levels in the main atomic system are collisional (cfi eqn. 16), so that the transport of radiation has no impact on the distribution function of that system.

Consider a part of the skin of the plasma where locally appreciable ionization takes place, and charged particles are created. This will create gradients in the densities of charged particles, which, in their turn, cause locally an outward transport (diffusion) of electrons and ions. In a steady state there must be locally a balance between the creation and outward

8 J. A. M. VAN DER MULLEN

transport of charged particles. Let us consider what this means for the Saha balance:

A,+esA++e+e. (21) n u 11

The outward transport of charged particles, indicated by the vertical arrows, will disturb the Saha balance; the presence of LSE will be violated. The drain of charged particles will prevent the recombination to be as large as the ionization. This “overionization” is only possible if the ground state density exceeds the corresponding Saha value. We say that the ground state is overpopulated and this situation will be characterized by the parameter

b(l)=n(l)/n”(l). (22)

A further demand for a steady state is that there will be an inward transport of ground state atoms indicated by the upward arrow at the Ihs of eqn (21).

In the previous discussion we only regarded the direct (Saha-) coupling between ground state and ion state. However, an important ionization channel in the ICP is that of stepwise excitation. This is effectuated as follows: owing to the overpopulation of the ground state the Boltzmann balance between the ground state and first excited state will be disturbed, i.e. the excitation exceeds the LTE value. This will result in an overpopulation of the first excited state (level 2), and, if level 2 were separated from the rest of the system and only connected with level 1 by electron collisions, its density would increase until it reached the value at which the Boltzmann balance 1~ 2 is in equilibrium again. This hypothetical situation can be characterized as b( 1) = b(2). However, level 2 is connected with level 3, which will result in disturbance of the B balance 1~2. Therefore we can expect that b(l)> b(2). The same reasoning applies for the balance 2-3, etc. The chain of disturbed Boltzmann balances can be depicted by

A, s A, 9 A,-+. . . . . . . . -+A,+e + + +

E(l) > eb(2) > i(3) . . . (23)

t 1 1

in which the bold horizontal arrows symbolize that the excitation is larger than the deexcitation (thin arrow). This chain reflects a stepwise flow over the system of excited states. By approaching the ion state the Saha balance of ionization and three-particle recombina- tion will increase in frequency, the influence of the excitation flow will diminish, at least relatively, and the Saha value of the excited state density will be established. We say that the highly excited states are in partial local Saha equilib~~ (pLSE) (cf: Fig. 1).

The above scenario makes clear that, although the electron density can be so high that radiative processes are unimportant, still deviations from LTE can be expected: the Griem criterion is not a sufficient condition for the presence of LTE. We should also put a restriction on the transport properties of the plasma part, i.e. the number of processes of outward transport of electrons should be much smaller than the equilibrium value of the number of excitation processes in the first step. Mathematically this reads

Vny, K n@(l) k,, (24)

in which w, is the drift velocity of the electrons. (The quantity V. ny, has the property that integration over a volume represents the number of electrons leaving that volume per unit time.) If the outward transport is realized by ambipolar diffusion we may equate

V.nw== -V.L),Vn,r D&A; (25)

in which Ae is the gradient length of the electron density (the distance over which the electron density reduces by a factor l/e ~0.37) and D, the ambipolar diffusion coefficient, so that eqn (24) can be written as

Da/A,2 << n”(1) k,, (26)

which puts a demand on the gradient length of the electron density.

On the atomic state distribution function in ICP-II. 9

t V-n.w+ outward

I \ E-O -E

E-O

Fig. 1. An ionizing system. (a) Inward transport ofground state atoms and outward transport ofions and electrons supported by an effective stepwise flow I’” over the atomic energy levels. This effective flow equals the number of excitations minus the number of deexcitation processes. (b) A sketch of the distribution function typical for ionizing systems. The number density per statistical weight is denoted by q(p)=n(p)/g(p). The Saha value is depicted by n’(p). (c) A sketch of the distribution function in the b(p)=q(p)/$(p) representation. Note that the lower levels are overpopulated; i.e.

b>l.

The impact of transport on the distribution function is depicted in Fig. 1. The correspond- ing b representation for which only the net transport over the levels needs to be given is depicted in Fig. l(c).

Of course the above description gives only the global trend of how the atomic state distribution is affected by outward transport of charged particles. In Part IV we will present a more quantitative description of the distribution function in ionizing plasma regions.

Opposite to the ionizing plasma we have the recombining plasma regions in which inward diffusion of charged particles is accompanied by outward diffusion of ground state particles supported with a deexcitation flow (cf: Fig. 2). Again the Boltzmann balances chain the levels in a stepwise manner. Figure 2 gives an example in which the ground state is under- populated, i.e. b(l)< 1. However, this is not condition for a plasma to be recombining. It should be realized that in most cases recombination is supported by radiative decay processes, which (in the optically thin case) always go in the downward direction. Thus it is in principle possible that a plasma is recombining whereas b( 1) > 1. For a collisional system for which the relation n, >> n:(2) holds (cf: eqn 16), the condition that the plasma is recombining implies that b(1) < 1. In that case the condition for b(1) approaching unity is the same as that given in eqn (26).

1 V njif1 outward

J. A.M. VAN DER MULLEN

a

b

I v.n+w, Inward

-- E.0 -E

Ob -E

E=O

Fig. 2. A recombining system. (a) Inward transport of charged particles and outward transport of ground state particles supported by an effective recombination flow Rcff over the system of energy levels. (b) Sketch of the distribution functron in the n(p) representation. (c) The ~rresponding b(p) representation. Note that lower lying bvels are Under~puIated, b< 1, which is a condition for a

collisional system [n, > nf(2) cc eqn. 161 to be recombining.

The above discussion applies to the main gas in the ICP. An analogous treatment can be given for the various anaiytes. Then it turns out that this will entail a condition for the gradient length of the ion density of the system in question. This matter will be discussed in a later study.

Summary The presence of gradients in the electron density will create in- or outward transport of

charged particles, which will afjkt the equilibrium state of the Saha balance between the ground and ion states of the main gas in the ICP. In an ionizing part of the ICP this results in an overpopulation of the ground state, b(l) > 1, while an underpopulation of the ground state, b(l)< 1, implies that the plasma is recombining. Thefact that the levels between ground and ion states are interconnected by Boltzmann balances in a stepwise manner implies that the Saha deviations of the ground state will propagate in the system to higher excited states. So the existence of LTE requires the ful~lment of two conditions by the electron gas

(1) the absolute concentration should be so large that all collisiona~ transitions dominate over radiative decay,

(2) the gradient length of the electron density should be so large that transport processes will not affect the local Saha equilibrium.

On the atomic state distribution function in ICP-II.

5. CONCLUDING REMARKS

11

In Part I it was shown how in TE the atomic state dist~bution function (ASDF) results from the equilib~um state of the balances of Boltzmann and Saha (cf: Appendix 1). In this part we have studied qualitatively how types of TE departures can affect the ASDF. Three types of causes were discussed:

(Cl) the temperature inequality ( Th # T,), (C2) the escape of radiation (violating Plan&s law), (C3) the presence of gradients (particle transport).

Under moderate conditions we reach the first stage of TE departure denoted by LTE being a combination of LIE, LME, LBE and LSE. We also discussed the transition region between LTE and a next stage of TE departure and we did so with the aim to understand globally the effect of the departure on the ASDF. It was found that cause (Cl) is not important provided that the degree of ionization is large enough. The (de~xcitation, ionization and recombina- tion processes are ruled by the electrons, and only their temperature is relevant in the Saha-Boltzmann relation. In fact, the atomic state distribution is the result of LS,E and LB,E in which the subscript “e” expresses that the balances are governed by the electrons. To avoid the influence of cause (C3) a limit should be put on the gradient lengths of the concentrations of the electrons and the different ionic species. However, difficulties are encountered in the formulation of the condition such that cause (C2) has no or only moderate effects. The only thing we can do is to postulate a minimum of the n, value for a specific atomic system. Clearly, the condition of L&E for the Ar I system, denoted by LS,E(Ar I), is not the same as that for L&E { Ar II >, L&E ( Fe 11 or LS,E( Mg II>. If we wanted LS,E for systems with all the various charge numbers of the core, we would have to increase n, scaling with Z7 with as consequence that we would reach the fictitious state of TE. Apparently, LTE is a fiction as well.

From this discussion it appears that the state of LTE is rather subtle and has many aspects. This is the cause of the confusion often found in the literature. Several authors use

F(E)

T

bulk tall

excttatlon of

non-pLMeE

between top levels

El2 ---WE

deexcltatlonc excltahon PLSeE l~>fi

pLBeE (p>p*) escape of radlatlon

Fig. 3. An example of violation of LM,E, the presence of pLMeE and the connection between the various elementary balances. The production of excited atoms must be realized by electrons with large energy having E > Et,. For a low degree of ionization and/or a substantial escape of radiation this can affect the Maxwell balances which supply the tail of the energy distribution function resulting in a depletion of the tail. A basic source of equilibrium departure can be the escape of radiation denoted by the curvy arrow. Thus the imbalance of Planck balances disturbs some Boitzmann, Saha and Maxwell balances. The effect of this tail disturbance on Maxwell balances in the bulk of the distribution is in many cases very limited. In such a case we deal with a situation of pLM,E (i.e. partial local Maxwell equilibrium maintained by the electrons). The presence of pLM,E is in many cases sufficient to maintain Saha equilibrium in the upper part of the atomic energy scheme denoted by

PLSNP’ P* 1.

12 J. A. M. VAN DER MULLEN

the expression local thermal equilibrium to indicate that all the different species have the same translational temperature. So they deal with the LIE aspect of LTE. Historically they may be correct since the principle of thermal equilibrium (cf eqns 13 and 14 in Part I) was defined as such. Most others use the same abbreviation “LTE” to indicate the equilibrium state of the Saha and Boltzmann balances. In fact they deal with the LSE aspect of LTE or in most cases even with LS,E, which may exist in the plasma whereas the LIE aspect of LTE is violated. But even for the authors for which LTEE LSE there is ambiguity in the LTE criterion. In most cases the only restriction is that on the n, value so that the disturbing influence of radiation transport is ruled out. This condition, known as the Griem criterion, is incomplete. One should also put a limit on the magnitude of cause (C3): the transport of material particles generated by the gradients. However, this is hardly ever found in the literature.

Another remark should be made on the concept of partial LTE (pLTE). This abbreviation pLTE is used to denote the equilibrium state of a part of the atomic system where for each level the ionization is balanced by recombination. A more proper name for this equilibrium state is pLS,E. In the future we will, unless stated differently, use the concepts LTE and pLTE such that LTE = LS,E and pLTE = pLS,E, both applied to the atomic system (or part of the atomic system) under study.

This paper is concluded with Fig. 3, which depicts how the various types of balances are coupled to each other and Appendix 2, in which the various abbreviations are found. One could object that this list is too long. However, the main reason for the misunderstanding often found in the literature is that one often does not recognize the wide variety in departures from TE.

REFERENCES

[l] J. A. M. van der Mullen, Spectrochim. Acta 44B, 1067 (1989). [2] L. M. Biberman, V. S. Vorobev and I. T. Yakubov, Sou. Phys. Uspekhi 22, 411 (1979); Kinetics of

Nonequilibrium Low-Temperature Plasmas. Consultants bureau, New York (1987). [3] J. A. M. van der Mullen, Thesis, Eindhoven University of Technology (1986). [4] I. Prigogine, Bull. C1. SC. Acad. R. Belg. 26, 53 (1940). [S] A. V. Potapov, High Temp. 5, 48 (1966). [6] H. &tern, Phys. Rev. 13, 1170 (1963). [7] P. E. Walters, G. L. Long and J. D. Winefordner, Spectrochim. Acta 39B, 69 (1984). [S] H. W. Drawin and F. Emard, 2. Naturforsch. Z&I, 1289 (1973).

APPENDIX 1. SUMMARY OF THE FOUR TYPES OF BALANCES

The four types of balances; the corresponding laws of mass action and the results for the elementary occupation (number of particles per quantum state). Abbr.: E. cenergy; M. A. amass action; S. E. =stimulated emission.

MAXWELL

kinetic E

Conservation

law of M.A.

result:

xi + Y,- M x, + Y,

E, E, Eh Eb

E, + E, = Eh + E#

fi.(i, -%)rj,(j, E,) = il.& Eh)ti$j, Q)

fi,(k. E)=r~,(k)u,(m, T) exp(--ElkT): with u,= h”/(2nm,kr)3’2

no changes m

internal

states i or j

BOLTZMANN Xi + A, AX, + A,

kinetic E E, E, Eh E; Conservation E, + E, = Eb + E; + E,,

law of M. A. %(i. E,)4, (P, %) = rj,(i, Eh)fi,(u, Eb) result: rl(u)/n(J) = eW---E,,lW

On the atomic state distribution function in ICP-II. 13

SAHA

Conservation

Law of M.A.

result

PLANCK

X, + A, AX, -I- A; e,

E, + Ep = E; + E; + E; + I,

ti.6, E&o, Es) = ri,(i, WA+ (1, E#).(s, Er)

tlYp) = rt+Wi,exp &.!kT); ri,=ti.(s,O)

A, +-+ hv + A, + P =a hv + hv

Law of M.A.: tt.(l + &I= = fi”rt, taking SE. into account

fi=rj(i,hv) = [exp(hv/kT)-11-l

APPENDIX 2. A LIST OF ABBREVIATIONS OF VARIOUS TYPES OF EQUILIBRIA

The abbreviation E refers to equilibrium. The third column depicts the relation with local Saha equilibrium LS,E which, related to the atomic system of interest, is the most important feature of LTE as far as the atomic state distribution function is concerned. The last column refers to parts in the text where the equilibrium in question is discussed in more detail.

LIE LPE pLPE LM,E

PLM,E

LB,E

PLB,E LS,E{X I}

PLS,{X I1

local isothermal E; T. = T,, local Planck E partial local Planck E local Maxwell E for the electrons. partial LM,E

local Boltxmann E ruled by the electrons. partial LB,E local Saha E for system X I ruled by the electrons partial L&E, i.e. L&E for a part of the system {X I}

not needed for LS,E. not needed for LS,E.

(sect. 2) (sect. 3) (eqns 19,20)

needed for L&E. not suff. for LS,E needed for pLS,E needed but not sufficient for LS,E.

not sufficient for LS,E{X II}

(Fig. 2)

(sect. 4)