Spectrochimica Acta. Vol. 44B, No I I, pp 106740841989 Printed in Great Britarn. 0584-8547/89 $03.W+ .M) Pergamon Press pk TUTORIAL REVIEW On the atomic state djs~ibution fun~io~ in ~uetively coupled plasmas- I. ~~~ynarni~ ~~lib~urn considered on the elementary level J. A, M. VAN DER MULLEN Bindhoven University of Technology, Physics Department, P.O. Box 513,560O MB Eindhoven, The Netherlands (Received 25 July 1988; in revised form 8 May 1989) Abstract-This is the first part of a series of articles dealing in a tutorial way with the atomic state distribution function in plasmas, especially inductively coupled plasmas. Attention is paid to the influence of the various types of equilibrium departures on this function. The deviation from its equilibrium value, i.e. the Saha-Boltzmann distribution function is traced back to the imbalance on the elementary level. The various stages of equilibrium departure are related to the dominance of specific kind of processes ordered in a hierarchy. This first paper of the series treats the state of thermodyn~c ~~lib~um (TE). Serving as a starting position for the study of~u~b~~ departure, TE is presented as a collection of elementary balances, all in eq~lib~um. Contents 1. 2. 3. 4. 5. 6. 7. Introduction The principle of microscopic reversibility Elastic collisions; the Maxwell balance and distribution (De)excitation processes; the Boltzmann balance and distribution Ionization and three particle r~ombination; the Saha balance and ~st~bution Matter-radiation interaction; Plan&s radiation law Discussion References Appendix 1. The difference between microscopic reversibility and detailed balancing and between a process and a reaction. Appendix 2. The number of states per unit of volume and energy range Appendix 3. Summary of the four types of balances 1. I~RoDu~I~N THE ATOMIC state distribution function (ASDF) which describes how atoms and ions are distributed over their various excited states contains a substantial amount of information about the plasma. For a plasma in thermodynamic equilibrium (TE) this ASDF is well known and given by the Saha-Boltzmann relation. In that case we only need the density of one excited state of each element to calculate important plasma parameters such as the chemical composition, the electron density and temperature. The velocity distribution of the particles is according to Maxwell, the ratio between the population densities of excited states of the same atomic system is given by the Boltzmann relation; the densities of subsequent ionic states are interrelated according to Saha, while the photon gas obeys Planck’s radiation law. Deviations from TE will affect these statistical laws and the relation between the atomic state population and the basic plasma parameters become much more complex. This 1067
Transcript
Spectrochimica Acta. Vol. 44B, No I I, pp 106740841989 Printed in Great Britarn.
0584-8547/89 $03.W+ .M) Pergamon Press pk
TUTORIAL REVIEW
On the atomic state djs~ibution fun~io~ in ~uetively coupled plasmas- I. ~~~ynarni~ ~~lib~urn considered on the elementary level
J. A, M. VAN DER MULLEN
Bindhoven University of Technology, Physics Department, P.O. Box 513,560O MB Eindhoven, The Netherlands
(Received 25 July 1988; in revised form 8 May 1989)
Abstract-This is the first part of a series of articles dealing in a tutorial way with the atomic state distribution function in plasmas, especially inductively coupled plasmas. Attention is paid to the influence of the various types of equilibrium departures on this function. The deviation from its equilibrium value, i.e. the Saha-Boltzmann distribution function is traced back to the imbalance on the elementary level. The various stages of equilibrium departure are related to the dominance of specific kind of processes ordered in a hierarchy. This first paper of the series treats the state of thermodyn~c ~~lib~um (TE). Serving as a starting position for the study of~u~b~~ departure, TE is presented as a collection of elementary balances, all in eq~lib~um.
Contents
1. 2. 3.
4.
5.
6.
7.
Introduction The principle of microscopic reversibility Elastic collisions; the Maxwell balance and distribution (De)excitation processes; the Boltzmann balance and distribution Ionization and three particle r~ombination; the Saha balance and ~st~bution Matter-radiation interaction; Plan&s radiation law Discussion
References
Appendix 1. The difference between microscopic reversibility and detailed balancing and between a process and a reaction.
Appendix 2. The number of states per unit of volume and energy range Appendix 3. Summary of the four types of balances
1. I~RoDu~I~N
THE ATOMIC state distribution function (ASDF) which describes how atoms and ions are distributed over their various excited states contains a substantial amount of information about the plasma. For a plasma in thermodynamic equilibrium (TE) this ASDF is well known and given by the Saha-Boltzmann relation. In that case we only need the density of one excited state of each element to calculate important plasma parameters such as the chemical composition, the electron density and temperature. The velocity distribution of the particles is according to Maxwell, the ratio between the population densities of excited states of the same atomic system is given by the Boltzmann relation; the densities of subsequent ionic states are interrelated according to Saha, while the photon gas obeys Planck’s radiation law.
Deviations from TE will affect these statistical laws and the relation between the atomic state population and the basic plasma parameters become much more complex. This
1067
1068 J. A. M. VAN DER MULLEN
relation is studied in the framework of collisional radiative (CR) models, which in the numerical approach normally consist of a large number of equations. Each equation describes how the population of one particular atomic state or level is produced and destroyed [l]. Since the destruction of the population of one level will contribute to the population production of another level, the levels and thus the equations are coupled to each other and the study of the atomic state distribution function in non-equilibrium is not simple. That is the reason why numerical CR models often look like blackboxes. A lot of parameters, rate coefficients and transition probabilities are inserted leading to the output of an enormous amount of data. The actual physics which leads from input to output is often totally obscured.
Another approach will be presented in this series of papers which deals with the effect of TE deviations on the ASDF. The aim of this series is to elucidate the relation between the cause, i.e. deviations from equilibrium, and the effect, i.e. the deviation of the ASDF from the Saha-Boltzmann value. It will be shown that the departure from TE moves along stages and this feature often affects the distribution function in a way that can be described analytically C2-43. The main reason for this analytical path along stages is that in each stage one particular type of elementary process is dominant over all others. So the structure in the TE deviations is based on the hierarchy in elementary processes.
To give an example. The fact that laboratory plasmas are created in a limited space has three consequences:
(1) different kind of material particles are subjected to different forces, (2) radiation will escape from the plasma without being reabsorbed, which violates
Planck’s radiation law, (3) gradients of densities and temperatures will be present, which causes transport of
particles and energy. If, however, on each location the interaction between material particles occurs frequently
#enough, then the distribution functions of Maxwell, Boltzmann and Saha will still be obeyed and the atomic state distribution will be the same as in the case of TE.
This stage of TE departure in which only Plan&s radiation law is violated is denoted by local thermal equilibrium (LTE) and is closely approached in various parts of the inductively coupled plasma (ICP). In this stage of LTE, moderate gradients of temperature and densities can be permitted but the plasma parameters have to be specified locally. If gradients become larger and/or the leak of radiation becomes much more severe, then the atomic state distribution function will be affected. First, the population density of the lower levels will deviate from the Saha value while the higher levels remain in equilibrium with the continuum. This difference between the behaviour of the lower and higher levels is due to the state dependent hierarchy in elementary processes. Electron collisions are the more frequent the higher the state of excitation, whereas the opposite applies to the radiative processes.
To study departures from TE we first need a proper understanding of the state of TE. This is why the first paper is dedicated to the situation of full equilibrium. In the second paper we will discuss the global aspects of TE departure in the relation with the causes, i.e. the existence of different forces for different types of particles, the leak of radiation and the presence of gradients. The dominance of processes will be shown to be dependent on the concentration and rate coefficients. Paper 3 is devoted to the rate coefficients in relation with atomic properties. Paper 4 deals with the classification of stages of equilibrium departures. Finally an example of a simplified CRM for the atomic argon system in the ICP will be given in paper 5.
As stated before, this first part of the series is devoted to the state of TE. One could object that this treatment is overdone since there are many textbooks on thermodynamics and statistical physics dealing with this subject. However, in almost any case the first acquaint- ance of students with thermodynamics is one in which a static impression is given of the state of thermodynamic equilibrium. It is presented as a state in which macroscopic properties like temperature and pressure are constant due to the fact that the system occupies the state of maximum entropy. The derivation of the statistical laws are subsequently based on that principle of maximum entropy.
On the atomic state distribution function in ICP-I. 1069
The derivation given in this paper is not based on the static behaviour on the macroscopic scale but on the dynamic properties on the microscopic or elementary level. This activity on the elementary level is ordered by the principle of microscopic reversibility (MR). This principle states that on the elementary level each forward process has a corresponding backward process, and that on that level the transition probability of the forward process equals that of the corresponding backward process. Apart from the ordering of reactions in balances of forward and corresponding backward processes, it will be shown that this principle of MR can be used to derive the statistical laws valid in TE. Moreover, since this derivation is based on the activity of elementary processes ordered in balances, it offers a good starting point for understanding the implications of deviations from equilibrium, since departure from equilibrium can be traced back to the imbalance of certain elementary balances.
In this series of articles we shall disregard molecules and molecular processes.
2. THE PRINCIPLE OF MICROSCOPIC REVERSIBILITY
Consider the reaction balance X+Yctx’+y’. (1)
In forward direction (left to right) the reaction between the particles X and Y leads to the creation of the particles X’ and Y’. From right to left we have the corresponding backward reaction. According to the law of mass action we can obtain the number of forward reactions per unit time and volume by multiplying the number density n, of particle X with that of the particle Y and a rate coefficient k +. For the backward reaction a similar expression can be obtained. If the balance of forward and backward reactions is in equilibrium (cf: Note l), i.e. if the number of forward reactions equals that of the backward reactions, the following relation between the densities will be applicable
k+ n’ n’ or K=-=S
kt 4 ny
n,n,k+ = n&k t
in which K is the temperature dependent equilibrium constant. The principle of detailed balancing (DB) states that in the presence of TE all the different balances are in equilibrium.
This applies to any reaction between any combination of particles and is valid on the macroscopic and elementary level. To clarify the difference between these two levels (cf: Appendix 1) we consider as an example the macroscopic balance of the electron induced excitation and deexcitation represented by
e+A,t,e+A,. (3)
In the forward (excitation) reaction the internal energy of the atom increases with the amount E,, at the cost of the kinetic energy of the colliding particles. This can be done by a large class of elementary processes. The members of this class can differ from each other with respect to the kinetic energy of the colliding particles, with respect to the orientation of the relative velocity, or with respect to the orientation of the intrinsic angular momenta. Giving the rate of the forward reaction in the macroscopic form as that of the lhs of eqn (2a), we sum up the effect of this whole class of elementary processes (cf: Appendix 1).
Note 1 In this study we distinguish between “balance” and “equilibrium”. We apply the meaning
based on the original concepts (cf Websters dictionary): balance = bi- + -1anc = two + scalepans, whereas equilibrium = equi- + -1ibris = same + weight. Thus a balance is a device which compares the weight of two objects and it can be in or out equilibrium. In our meaning the balance compares the “weight” of a forward with that of the corresponding backward process.
1070 J. A. M. VAN DER MULLEN
It is interesting to study the features of DB on processes and particles on the most elementary level, i.e. the level on which particles are classified according to the quantum state they occupy and on which a balance of processes describes how each particle changes from one particular initial to one particular final quantum state and vice versa. It can be proved that the presence of equilibrium of the elementary balance
X+YtrX+Y, (4) u B a’ p
leads to the simple relation for the elementary concentration
liAW&o) = fA(44 (B’) (5a)
or
(W
in which &(a) is what we call the elementary occupation (or elementary concentration) i.e. the number of X particles in state u. This state of a particle has different aspects. So, for instance, it has three translational (for each degree of freedom) and various internalfeatures (cf: Fig. 1). Comparing eqn (5) (elementary) with eqn (2) (macroscopic) we see that the concentration in the macroscopic case given in eqn (2) as the number of particles per unit volume is replaced (in eqn 5) by the number of particles per quantum state and, most
Process 1
e + a Ap~eo; +
ST 0 ST
“e- ‘a’ v; +
Erz EP Eai’ Ea - E1u -u
Process 2
e + (1. A,----,03 +
ST J7 sl
V- e “&I+- v; +
E a EP "ai= E" - E1u
-u
-1
JbJ
v;4-
=E E4 p
Vu
-1
Fig. 1. Two forward processes which differ from each other with respect to the electron spin state s and the angular momentum J of the atom. The initial electron energy E and the transfer AE= E,, are the same. The two processes contribute to the same reaction: the excitation of the atom from a lower state 1 to an upper state u (c$ the energy scheme). Still other members of the same excitation reaction can be obtained by varying the translational energies, the velocities, and the orientations of the
intrinsic angular momenta of the colliding particles.
On the atomic state distribution function in ICP-I. 1071
interesting, that the latter equation does not contain a reaction rate coefficient. This is related to the principle of microscopic reversibility (MR) which states that the probability that a particle leaves a quantum state via a certain route, equals that of the process at which a particle arrives at that quantum state via the reversed route. So the elementary pendant of k+ can be eliminated against that of kc yielding eqn (5). In all cases the elementary equilibrium constant Kel equals unity. This simple but powerful relation between elementary occupations is a direct consequence of the principle of microscopic reversibility and can be used to derive the distribution laws of Maxwell, Boltzmann, Saha and, after some adjustment, Planck.
The principle of MR is valid on the quantum state level and is the basis for the principle of DB applied to the macroscopic level (cf: Appendix 1). In contrast to DB, microscopic reversibility remains valid in the absence of TE.
In the next section this principle of MR will be used in combination with the conservation law of energy to derive the distribution laws of Maxwell, Boltzmann, Saha and Planck. In this derivation, valid in TE, we follow OS~ER [S]. In the next papers we will use the principle of MR to study the behaviour of the ASDF in the departure from TE.
Summary The principle of microscopic reversibility states that the probability that a particle leaves a
quantum state via a certain route Vorward process) equals that of the process at which the particle arrives at the quantum state via the reversed route.
This principle is valid irrespective of the presence of TE and is the basis for the principle of detailed balancing. The validity of DB, applied to the macroscopic level, generally depends on the state of departure from equilibrium.
If the balance between an elementary forward and corresponding backward process is in equilibrium, then the principle of MR induces a simple relation between the elementary occupation of quantum states. This relation is the elementary pendant of the mass action law applied to the equilibrium state of the balances between elementary processes.
3. ELASTIC COLLISIONS; THE MAXWELL BALANCE AND DISTRIBUTION
In order to derive the distribution laws we shall apply the principle of MR to various processes. We start with the study of elastic collisions, i.e. interactions in which material particles change only in translational state.
Consider the elastic balance given by
M Xi+Yj c* Xi+Yj (6)
in which i and j refer to the non-changing internal states of the particles X and Y while E denotes the kinetic energy. Such a balance will be denoted by a Maxwell balance and be marked with the symbol M.
In the following we assume the presence of TE and that all the balances of type M are in equilibrium so that the results of MR as given in eqn (5) may be applied and written in the form
4,(i, ~%)fi,ci, E,) = &(i, Eh)fi,(j, Eb). (7)
In this equation the states of the particles are specified by (k, E,), i.e. the combination of the internal state and the kinetic energy.
In order to derive the dependency of the occupation 3 on the translational energy E we apply the energy conservation law which in this particular situation states that kinetic energy is conserved
AE=E;--E,= -(I+Ep) (8)
i.e. the kinetic energy (AE) gained by particle X in the forward process equals the amount of
1072 J. A. ha. VAN DER MULLEN
kinetic energy lost by Y. Combining eqn (8) with (7) we find that
The third member H(AE) is introduced to express that the first and second member are independent of ES and EB. This can be understood by comparing balance (6) with the balance
na XiSYj ++ Xi -I- ‘, (IO)
E, E, E,fAE E,-AE
which only differs from balance (6) with respect to the kinetic energy of particle X, the initial energy equals E, instead of E,. By applying the principle of MR on this new balance (eqn 10) a variant of eqn (9) is obtained in which the first member changes but the second remains the same, proving that
%(i, EJ %(i, E,) M, 4 + AE) = &(i, E,+ AE)
= H(AE*) (11)
is independent of the initial kinetic energy value of the particle X. The same proof can be applied to show the E, independency of the second member of eqn (9). It can be proved (for instance by substitution) that the general solution of eqn (11) is given by
&(i, E) = it& 0) exp f - &Et (12)
in which 0, is some positive quantity determined by the boundary conditions. Substitution of this result in eqn (9) shows that for the particle Y an analogous relation holds and that
8, = e,( = e). (13)
This important result, known as the principle of thermal equilibrium, states that the different constituents of a gas mixture in TE do have a property in common. This property B is often replaced by its reciprocal l/e = kT in which T is the tem~rat~re of the system and k the Boltzmann constant.
The preceding discussion shows that the derivation based on the principle of MR produces the two main statistical features of the tr~slational behaviour of gases, i.e. -the constituents of a gas mixture in TE have the same tem~rature
and T,=T, (141
-the translational states are populated according to the Boltzmann exponent, i.e.
&(i, E)-exp(- E/kT). (15)
With these two results we arrived via different route (the principle of MR) at the same point as the derivation found in the textbooks which are normally based on the principle of maximum entropy.
The relations between the mean translational kinetic energy and tem~rature (l/2 mv2 = 312 kT) and between the value of &(i, 0) and the total number of N,(i) of particles X with internal state i, can easily be derived (see textbooks on statistical mechanic). An important quantity used in the derivation is the translational partition function given by
Q:“=N&i)/fi,(i, O)=Cexp(-EJkT) (161 a
in which eqn (12), the relation 0- - l/kT and the definition N,(i) = 4(i, 0)x exp (- EJkT) are
used. The summation extends over all the translational states and, ls expressed by the German name ~‘~~tandssumme”, it can be considered as the sum of states in which the states are weighted with their occupation probability, i.e. the Boltzrnann exponent. The formula of Q:’ found in the textbooks [6]
On the atomic state distribution function in ICP-I. 1073
shows that this sum of weighted states can be found by dividing the total volume V by the volume ux(mx, T) = h3/(2zm,kT)3/2 which apparently can be regarded as the mean volume of one translational state of the particle of kind X.
An interesting interpretation of this translational quantum volume can be given by applying the concept of the de Broglie wavelength. It is well known that a particle with momentum p = mv = (2 mE)‘/’ can be associated with a probability wavelength 1= h/p = h/(2 mE)“‘. This de Broglie wavelength puts a boundary on the size of the particle in the direction along the p vector. The particles in a gas have momenta dist~buted over ah possible magnitudes and orientations in three degrees of freedom. This dist~bution is characterized by the temperature and in each degree of freedom the average magnitude of p scales with (mkT) w This dependency makes the theoretical result for the thermal de . Broglie wavelength 2 = h/(2nmkT)“2 plausible. Since there are three translational degrees of freedom, the average particle will be bound in all directions within the distance 2 which implies that the average particle X has a “volume” v, = h3/(2rrm,kT)3/2.
As an example we can consider the electrons for which it is known that the Pauli exclusion principle applies. One quantum state may not contain more than one electron which means that electrons with the same spin (same internal state i) are not allowed to have the same translation states. So the average electron will not accept another electron in its quantum volume ti,(m,, T). And the total number of tr~slation~ electron states in volume Y will be Vlu(m,, Z”), as expressed in eqn (17).
With the use of eqns (16) and (17) we find that
K(i) N,(i) V Q&i, O)= ~ =L----- h3
Q:’ V Q: = ‘lx(i)(2,+k T)3/2 (18)
in which o,(i) (without ^!!) is the number density of particles X with internal state i. This implies that the elementary occupation can be expressed as
40, E) = q&&m,, T) exp ( - E/k T)
h3 =~~(i)(2~m~k~~,~ exp(-ElkT). (19)
This equation expresses that the number of occupied translational quantum states of kinetic energy E can be found by multiplying the number of particles per volume a,(i) with the average volume of a quantum state vx(mx, I) weighted with the Boltzmann factor exp
(-E/k[T3. Equation (19) gives the occupation of the translational states and can be regarded as the
basic form of Maxwell’s equation which is normally given as the number density of particles per kinetic energy range, i.e. dn/dE (cf: Appendix 2). This familiar form can be obtained by multiplying the number density of kinetic states per energy range dG/dE with the occupation, i.e.
dtlx (4 dE =$,(i, E)g.
It should be noted that the classical form of the Maxwell equation loses validity if the number of particles approaches the number of states. Therefore the condition 4 < 1 should hold. It can be seen from eqn (19) that the situation whereby q-, 1 occurs first for the lightest particle, i.e. the electron. However, in the ICP where the temperature is about 1 eV we find that vc(me, 1 eV)z 3.3 x 1O-28 m3, so that an electron density of 3 x 102’ me3 is needed to fill up all the translational electron states. Since this is about six orders of magnitude above the maximum electron density in the ICP, the degeneration of the electron gas is far from actual and the classical Maxwell equation may be used.
Summary The prineip~e of MR toget~r with the energy conservation law append to elastic processes
gives the ~oxwe~l d~stri~t~on law and the principle of t~rmu~ e4u~~ibrium which states that the constituents of a gas mixture have the same kinetic temperature.
1074 J. A. Ail. VAN DER MULLEN
4. (DE)EXCITATION PROCESSES; THE BOLTZMANN BALANCE AND DETRIBUTION
In this section we will study the implications of MR on the relation between the densities of different atomic levels belonging to the same atomic system. The term “atom” refers to atoms as well as ions. It will be shown that the principle of MR leads to the Boltzmann relation.
We consider a balance of inelastic processes given by
X,+A, t Xi+A, (21)
The forward reaction describes how a particle Xi excites an atom from a lower state 1 to an upper state u; E refers to the kinetic energy of the corresponding particles. We assume that only the internal state of the atom changes; the internal state of particle Xi remains constant. The role of particle Xi can be played by an electron, an ion or an atom. The corresponding backward process is the deexcitation process. This type of balance will be denoted by a Boit~ann balance and is labelled by the symbol B. If this balance is in equilib~um we may apply the principle of MR as given in eqn (5) and find that
il,& 4)9& &J = 44, ~~~(~ E6).
By using eqn (19) this can be simplifi~ to the so-called Bol~ann relation
(22)
(23)
The state density, i.e. the number density of atoms with internal state u, is denoted by q,,(u). The superscript b is used to indicate that these densities obey the Boltzmann relation. The increase of the internal energy of the atom is denoted by E,,. The simplification leading to eqn (23) is realized by using the energy conservation law
EXN = E, + E, -(E: + E;;t WI
and by dividing both sides of eqn (22) by the de BrogIie volume 4rn, T) of the particle Xi and the atom. The algebra of this s~plifi~tion shows that there are four ~nditions for the derivation of the Boltzmann relation:
(1) The particles X, and A must have a kinetic distribution according to Maxwell. (2) The temperatures of the particles have to be the same. (3) The mass of the particles is unchanged by the reaction. (4) No creation (ionization) nor destruction of particles is permitted. Thus the Boltzmann
relation only applies for levels of the same atomic (or ionic) system. The derivation shows how the internal energy is related to the change in translational energy and how the translation t~~rature of the particles is imposed on the internal state distribution. Violations of the Boltzmann relation can be expected if the temperatures of the particles Xi and A are different or, if the kinetic energy distribution of the particles is not according to Maxwell. We come back to this in paper 2.
Summary lf two atomic states are coupled with each other by ~~~tz~a~n balances in ~~ail~~~~~~ then
the level densities are related to each other by the ~o~t~mann relation. This is generally true if the interacting particles do have a Maxweliian distribution uf translational energy with the same temperature.
5. IONIZATION AND THREE PARTICLE RECOMBINATION; THE SAHA BALANCS AND ~~sTRIB~I~~
Next we consider the so-called Saha balance of ionization and 3 particle r~ombination marked with the symbol S
Xi+A,AXi+Al +e,
On the atomic state distribution function in ICP-I. 1075
The forward process describes how an atom in state p is ionized by the collision with the particle Xi. The ion thus created is in its ground state indicated by the subscript “l”, the created electron has a spin state “s”. Again we assume that the particle Xi does not change internally. Using the principle of MR the equilibrium state of this balance leads to the relation
ii.& QL(p, &)=9&i, Eh)&+(l, &)%(s, F;) (26)
which can be simplified by using the energy conservation law and dividing by the volume o(m, T) of the particle Xi and A. This leads to the Saha relation
V(p) = rl +(l)fi,(s, 0) exp (J&T)
or
V(P)=V+(l)% h3 2 (2nm,kT)3/2 exp (1,/k T) (27)
in which I, is the ionization potential of the atom in state p and q + (1) is the state density of the ion ground state. The factor 2 in the denominator of eqn (27) stems from the two different spin states of the electron making the density per spin state q,(s) = nJ2.
It is instructive to compare the Saha relation with the Boltzmann eqn (23). First we compare the derivations and find that the algebra leading from eqn (26) to (27) shows that the first three physical conditions needed to derive the Boltzmann relation should be valid for the derivation of the Saha law as well. That is, (1) the particles should have Maxwellian velocity distributions, (2) with the same temperature and (3) the mass of the atom and corresponding ion should be the same (we may neglect the electron mass). However, condition (4) conservation of the number of particles is not fulfilled since an extra particle, an electron, is created in the ionization process. This leads to the extra Q&s, 0) factor in the Saha equation. In Fig. 2 this factor fi,, which is a short notation for if&, 0), is depicted by the so- called “Saha jump”. It should be realized that this quantity depends only on translational properties of the electron gas [7,8] and that each succession of ion stages reflects the same Saha jump, irrespective of the element.
It is easy to show that the quotient qb(p)/qb(q) equals @(p)/@(q), provided that p and q belong to the same system. This means that two levels which both stand (apart from each other) in Saha relation with the ion state have densities between which the Boltzmann relation applies: Thus in TE the Saha and Boltzmann balances are cooperating processes. The Saha relation as given in eqn (27) gives more information than the Boltzmann relation
I
---ElP %O” Fig. 2. The atomic state distribution function in a plasma in TE of the element X (not hydrogen) presented in a plot of In t) vs the excitation energy El, The succession of the two subsequent systems X I and X II is effected by the Saha jump 4, = ?I~(s, 0) (cf: eqn 27), a quantity related to the electron gas. The slopes in both systems are the same and related to the temperature. Moving from X II to X III (possible for all elements except H and He) is accompanied by the same jump. And for a different species Y # X in the same plasma (in TE) the same Saha jump will be presented for any ion stage
succession. The graph of In t7 vs E,, in one system is known as a Boltzmann plot. Including levels of successive
systems we call the graph a Boltzmann-Saha plot.
1076 J. A. M. VAN DER MULLEN
since it relates the atomic state densities not only to each other but also to the properties of the electron gas (the jump in Fig. 2). This special feature of the Saha relation stems from the fact that the corresponding balance describes how three particles are created from two.
Summary The application of the principle of microscopic reversibility to the balance of ionization and
three particle recombination, the so-called “Saha balance”, gives the Saha equation, provided that the Saha balance is in equilibrium. The Saha and Boltzmann balances are cooperative. The information included in the Boltzmann relation between the densities of two atomic states is also included in the Saha equation. The Saha equation contains more information. The Saha jump is present for any element between each succession of ion stages.
6. MATTER-RADIATION INTERACTION; PLANCK’S RADIATION LAW
Until now we have only regarded the interaction between material particles and it was shown that the classical distribution laws of Maxwell, Boltzmann and Saha can be derived, provided that there are many more states than particles, i.e. 3 4 1. In the same way we can study the interaction of matter with a radiation field hoping to find Plan&s radiation law. The processes in question can be represented by the balance
A,c*hv+A, (28)
of spontaneous emission and absorption. In the forward process the decay of an atom from an upper to a lower state leads to the creation of a photon with energy hv. After the usual simplification we get the following relation ’
c=%rt, (29)
in which (in a short notation) +j, E ij(i, hv) is the number of photons per photon state. Note that the (internal state) index i refers to one of the two different states of polarization. Inserting the Boltzmann relation (eqn 23) we find that Qy = vu/q, = exp (- hv/kT) which does not reflect the same dependency as that given by Plan&s radiation law. The basic reason for this defect is that the radiation intensity such that &> 1, ice. that the photon states are occupied, is easily reached for low v values so that the classical treatment is no longer justified. In a proper treatment one should apply quantum statistics. But the same results can also be obtained with the help of soine intelligent guessing [S]. In this guessihg we are guided by a process not included in the balance (eqn 28): the process of stimulated emission. An essential feature of stimulated emission is that it always leads to a photo state already occupied (the created photon is coherent with the stimulating photon). And since the rate of this stimulation process is proportional to the intensity, it is also proportional to the number of occupied photon states, i.e. 4,. Moreover, it is known to be proportional to the spontaneous decay rate since transitions sensitive to spontaneous emission are sensitive to stimulated emission as well. Therefore it is plausible that the number of (stimulated) processes to photon states already occupied, equals 4” times that of the spontaneous type. The result is that the total number of forward reactions is enhanced in such a way that eqn (29) becomes
9”+4”?u=4V)1,. (30)
If stimulated emission is taken into account, the balance given in eqn (28) can be completed with an extra part so that the Planck balance marked with the symbol P reads in completed form
A, f-) hv+A, + Pw hv -. hv.
(31)
The second line indicates how the presence of a photon stimulates the process to the right. The double wavy arrow reflects that the created photon is coherent with the stimulating one. By inserting the Boltzmann relation as given in eqn (23) into (30) we find that in equilibrium
On the atomic state distribution function in ICP-I.
the elementary photon concentration should obey the relation
1077
1 1
4y=(~J~U)-l =exp(hv/lcT)-1 * (32)
This expression gives the occupation of the photon states and is the basic form of Plan&s radiation law which in its familiar form expresses how the energy is distributed over the frequency range. Since eqn (32) gives the number of photons per state we only need to know how the states are distributed ovef the frequency range. For this derivation, which is analogous to that of the Maxwell equation, we refer to Appendix 2 and the textbooks on statistical physics.
In this derivation we have sketched how the photon distribution follows from the Boltzmann relation between two atomic levels. This description suggests that the Planck radiation finds its origin in the photons created and destroyed by bound-bound transitions between atomic states that are populated according to Boltzmann. Thus, in this interpre- tation the Planck balances are placed below the Boltzmann balances in the hierarchy of processes. Of course there are also other processes such as free-free or free-bound transitions by which photons are created or destroyed, and we may regard the atoms as being imbedded in this radiation field so that the Boltzmann relation follows from Plan&s law [inserting eqn (32) into (30) gives the Boltzmann relation for rl(u)/q(l)]. This implies that the balances of Boltzmann and Planck are cooperative balances, or to state it differently: Equilibrium in the matter-radiation interaction results in the same distribution as that effected by the matter-matter interaction.
A special feature of radiation which should be mentioned is that radiation created in one spatial part of the plasma can irradiate another part and impose in that way the Boltzmann relation to that other part. This effect should be borne in mind when the transport of radiation in a non-equilibrium plasma is studied. This difficult subject will be discussed globally in Part 2.
Summary Plan&s radiation law can be derived by applying the principle of MR on the
matter-radiation interaction zf the stimulated emission is taken into account. It is found that Boltzmann and Planck are cooperative balances in the presence of TE.
7. DISCUSSION
In this paper we have classified reactions in balances of forward and corresponding backward processes. Next we distinguished four different types of balances which we called the balances of Maxwell, Boltzmann, Saha and Planck. They are reviewed in Appendix 3. By applying the principle of microscopic reversibility on these four balances we derived the distribution laws of Maxwell, Boltzmann, Saha and Planck. There is more, however. It appears that these balances are coupled and cooperative in the presence of TE, which implies that if one balance gets out of equilibrium then this can affect the equilibrium state of the other balances as well. For instance, if radiation will escape out of the plasma avoiding reabsorption, this will primarily affect Planck’s radiation law but will also disturb the balances and distributions of Boltzmann, Saha and Maxwell. The degree to which the balances are coupled and to which the disturbance of one balance will affect the other, depends on the rate of the various processes and the densities of the various species. This will be studied in the next papers.
A remark on the hierarchy of balances and processes should be made. We started with the Maxwell balance and distribution law from which we deduced the balances and distribution laws of Boltzmann and Saha. Via the Boltzmann relation we derived Planck’s radiation law. One could propose to do it in opposite direction, i.e. by starting with Plan&s radiation law and end with Maxwell’s distribution law. However, this is not along the line of decreasing hierarchy of processes. After all it is the Maxwell balances which are the most effective and in many cases they are needed for the (rekhstribution of energy. This will be elucidated by two examples.
1078 J. A. M. VAN DER MULLEN
Material particles can exchange their kinetic energy very effectively by elastic collisions and all kinds of energy combinations can result, provided that the energy conservation law is not violated. If, for instance, two electrons, with the kinetic energy of 1 and 2 eV respectively collide, this can result in a final situation in which both electrons have an equal kinetic energy of 1.5 eV, but all other distributions of the total energy of 3 eV among the particles are possible. The same applies to elastic collisions between ions or atoms, etc. However, such an interaction of energy redistribution cannot be found between photons since they do not interact with each other. Whereas different Maxwell balances are mutually coupled and self- supporting in the (re)distribution of kinetic energy between material particles, it appears that the (re)distribution of energy over the photons needs the material particles as intermedium. Thus the Maxwell balances create a “bath” in which the Planck balances are imbedded.
As a second example we try an alternative for the Boltzmann balance. We look for a reaction in which the internal energy distribution of atoms does not originate from the kinetic energy of the colliding particles but from a redistribution of the internal energy. However, the rate of the reaction
A,+A, + A,+A, (33)
by which the atoms initially in the states p and q end up with the states r and s is normally very low. Such a reaction can never be purely excitational since any combination of levels has a unique energy difference so that some translational energy will be needed or has to be created to fit the mismatch. Moreover, and that is very essential, the particles need translation energy to meet each other!
Of course the processes in eqn (33) will be very effective if p = r and q = s but this is not a (re)distribution of energy over the various atomic states but an exchange of the same energy between similar particles.
Much alike is the phenomenon of the excitation or change transfer between different kinds of atoms represented by the reaction equation
A*+X+A+X*. (34)
Although the rates of these reactions can be very high [7,9], they are only important for a narrow class of processes for which a good matching of internal energies exists. Again these are not effective channels of excitational energy (re)distribution but rather processes of exchange of energy of some particular values and the Maxwell balances are still needed as an intermedium for effective energy redistribution.
We stress the fact that the principle of MR can also be applied to the type of reactions represented by the eqns (33) and (34) and that the result of the application will match the Bol’tzmann-Saha equation. In this way these processes are cooperative with the Boltzmann and/or Saha balances which are connected with the Maxwell distribution.
Summary We may state that the group of Maxwell balances stands on the highest stage of the hierarchy
between process balances and can be regarded as a “bath” of energy (re)distribution in which the balances of Boltzmann, Saha and Planck are imbedded. This classijication in hierarchy will be worked out in the next papers and will be used to study the atomic state distribution in non- equilibrium situations.
REFERENCES
[l] D. R. Bates, A. E. Kingston and R. W. P. McWhirter, Proc. Roy. Sot. 267, 297 (1962). [2] T. Fujimoto, J. Phys. Sot. Japan 47, 265 (1979); 47, 273 (1979); 49, 1561 (1980); 49, 1569 (1980). [3] J. A. M. van der Mullen, Thesis, Eindhoven University of Technology (1986). [4] L. M. Biberman, V. S. Vorobev and I. T. Yakubov, Sou. Phys. Uspekhi 22,411(1979); Kinetics ofNonequilibrium
Low-temperature Plasmas. Consultant Bureau, New York (1987). [S] L. Oster, Am. J. Phys. 38, 754 (1970). [6] E. A. Guggenheim, Boltzmann’s Distribution Law. North-Holland, Amsterdam (1963). [7] J. A. M. van der Mullen, I. J. M. M. Raaijmakers, A. C. A. P. van Lammeren, D. C. Schram, B. van der Sijde and
H. J. W. Schenkelaars, Spectrochim. Acta 42B, 1039 (1987).
On the atomic state distribution fun&ion in XC%-L 1079
[8] J. A. M. van der Mullen, S. Now& A. C, A. P, van Lammeren, D. C. Schram und B. van der Sijde, Spectrochint. Actza 43B, 317 (1988).
[9] A. Goldwasser and J. M. Mermet, SpLrcrrochim. Acta 41B, 725 (1986).
There is a~~~~~ in the literature on the concept of detailed balanting (DB)~ Sometimes it is used to refer to microscopic realty (MR). In this s&es of papers we ~~g~ betueen MR and DR. The printiple of MR applies ta the el~~~y level and to proeessea A process describes how, at the elementary (q~nt~~ level, each involved particle changes &om one specisc initial to one speei8e quantum sta& Such a state has intermu and translational features. In our notation a state is designated by a lower ease Greek symbol. Thus, e.g. x = ii, El is the notation for a elementary state with internal property i and transIational energy E.
The principle of DB applies to a reuctlon, i.e. the total effect of the collection of processes with the same macroscopic result. To elucidate the diflerence we compare the two levels with each other for the case of an electron induced (dejsxcitation balance. So the meeroscapic result of the forward reaction is ebtron excitation which can be realized by several elementary processes labeled below by 1,2,3 . . .
MR elemerltary teyel a pracass belane.e e+A,oe+A, @ff Cr’Ir initial final
The mass action law in equilibrium: elementary macroscopic
~~(~~~~=~~tff’~~(~~ n,n,(l)k, = n,n&)k see also Pip. 1.
Starting with the laws of Maxwell and Planek in the pentagon of the number of particles per state (eqns 19 and 32) we can obtain the familiar form of these distriiution functions. All we need is the expression for the number of states per unit of vohune and energy range.
The determination can be realixed using the six dimensional phase space (3 spatial amI 3 momentum ~ensions~ where each particle occupies a volume h”. In this view the phase space has a cellular structure. The volume of each celI is ha. So in the phase space volume dl?=d3td3p the number of translational states is given by
For the material particles the ~sfo~~t~~~ from phase space to~o~u~tio~ space CBIx be e&ctuated applying d3p=&pzdp and p2=2mE which gives the number of states per unit of volume and energy range
The relative number of part&s per unit af volume and energy range can now be obtained eml$oy& eqn (20) giving the Maxwell energy distribution
For the photon gas we may also apply the phase space concept.Transfarmation to configuration space can be obtained using the relation p = E/c = hv/e which results in
so that the number of photons per unit volume and energy range equals
dG --d-$Q&.(i,hv))x2=
8x&-sh-”
exp(hvjkTf- t=
1080 J. A. M. VAN DER MULLEN
The factor 2 stems from the two different polarization states. Realizing that each photon has the energy hv we get for the energy density per unit of volume and frequency range
dE pV= ;(E)hv- =
8rtv3c-3h
dv exp(hv/kT)-1 (2.6)
which is the familiar form of the Plan&s radiation law.
APPENDIX 3. 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 ‘energy; M.A. amass action; SE. z stimulated emission.
