The influence of chemi-recombination and chemi-ionization processes on kinetics of non-equilibrium...

*Corresponding author. Tel.:#44-1865-273700; fax:#44-1865-273764.E-mail address: [email protected] (Z. DjuricH ).

Journal of Quantitative Spectroscopy &Radiative Transfer 70 (2001) 285}305

The in#uence of chemi-recombination and chemi-ionizationprocesses on kinetics of non-equilibrium helium plasma

Zoran DjuricH *, Anatolij A. MihajlovDepartment of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia

Received 15 May 2000

Abstract

The in#uence of He!HeH(n) chemi-ionization and their inverse dielectronic He!He!e and He!e

recombination processes on the populations of helium atoms in a weakly ionized helium plasma isestablished. On the basis of a collisional}radiative model of kinetics in such plasmas, the steady-state andtime-dependent values of the atomic levels' population densities are calculated. The main input parameters,such as atomic and electronic temperatures,

and

, respectively, and electron density, n

, have

been varied within the ranges: 10 000 K4420 000 K, 3000 K4

420 000 K and

4

, and

10 cm4n410 cm. The model for time-dependent calculations was extensively used but the main

contribution was made in the case of simulating the relaxation of the initially non-equilibrium plasma formedin an experiment. The in#uence of these ionization and recombination atomic processes was shown bycomparing results in cases when all the processes were included with those when the chemi-recombinationand ionization processes were excluded, either in the steady state or in the time-dependent regimes. 2001Elsevier Science Ltd. All rights reserved.

Keywords: Chemi-ionization and chemi-recombination processes; Weakly ionized helium plasma; Collisional}radiative kinetic model; Steady-state and time dependent atomic level population

1. Introduction

Knowledge of the atomic levels' populations in plasmas is an important theoretical problemwithmany practical implications: in calculations of optical, thermodynamical and transport propertiesof a given plasma [1}5]. The solving of these problems is simple if the plasma was in a local

0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 4 0 - 0

thermodynamic equilibrium (LTE). However, in the absence of LTE, the atomic levels' populationshave to be calculated by solving large systems of kinetic equations which describe all the importantcollisional}radiative processes in the plasma of interest. In these calculations, the radiative andcollisional processes of free electrons' scatterings on atoms and atomic ions, and radiative processesof interaction between atomic particles and a free electromagnetic "eld have mostly been con-sidered [6}8]. The processes of dissociative recombination of molecular ions in lower rovibrationalstates and some chemi-ionization processes (associative and penning ionization) involving atoms inground and lower excited states, were occasionaly included in calculations [9}12].However, in [13}15] we have shown that in weakly ionized gas plasmas, the dielectronic

recombination processes

e#A#APA#AH(n) and e#A(v,J)PA#AH(n)

and their inverse collisional chemi-ionization processes

A#AH(n)Pe#A#A and A#AH(n)Pe#A(v,J),

can be very important and sometimes dominant over other relevant ionization and recombinationprocesses. Later in the text the dielectronic recombination processes are denoted as chemi-recombination processes, in analogy with their inverse chemi-ionization processes. In the aboverelations, e denotes a free electron, A and A are an atom and an atomic positive ion in theirground states, and AH(n) denotes an atom in a highly excited (Rydberg) state with the principalquantum number n

processes must be treated together. Later it was con"rmed that the in#uence of the chemi-recombination process e#(A#A) on populations of AH(n) atoms is similar, and in some caseseven dominant in comparison with dissociative recombination process e#A

(v,J). Because of

that and the fact that both processes are caused by the same mechanism, they have to be treated aschannels of general dielectronic recombination processes. It is clear that because of the importanceof the e#A#A recombination channel, this general dielectronic process must not be identi"edwith its dissociative recombination channel.On the other hand, the dissociative recombination channel cannot be identi"ed with the

standard process of a dissociative recombination of molecular ions in lower rovibrational states(also in their ground electronic states). Namely, in weakly ionized hydrogen and helium plasmas,due to di!erent energies, this process does not in#uence populations of Rydberg atoms. However,until recently, the standard dissociation processes were usually considered. Only in some paperswere the dissociative recombination processes with molecular ions in highly excited rovibrationalstates treated [24,25], namely in the case of hydrogen and within a limited domain of application.For example, in [25] the recombination of H

ions in astrophysical plasmas was considered in

temperature domains up to 4000 K, while in the important case of astrophysical plasma of the Sunphotosphere, the temperature domain of interest is above that.This analysis makes clear that the importance and in#uence of chemi-recombination and their

inverse chemi-ionization processes on plasma kinetics is not yet well understood. Recently, it wasemphasized in [26] that any processes which can alter the calculations of degree of ionization ina given plasma have to be considered, especially when used in diagnostic methods based on theanalysis of atomic and ionic spectral line shapes. Chemi-recombination and chemi-ionization aresuch processes, and their in#uence on plasma kinetics still has to be thoroughly investigated.

2. Elementary processes and the collisional}radiative model

In this paper we will present a comprehensive collisional}radiative (CR) model which takes intoaccount all the processes relevant for a given non-equilibrium plasma, including the chemi-recombination and chemi-ionization ones. Using this CRmodel, the populations of atomic levels ina given plasma are calculated in steady-state and time-dependent regimes. Alternatively, it ispossible to perform all these calculations when the chemi-recombination and chemi-ionizationprocesses are excluded. Comparisons between the two calculations show the absolute in#uencethese processes have on the kinetics of a given plasma.Among the processes of particular interest in our CR model we will consider the chemi-

recombination processes due to the scattering of free electrons on molecular ions Heas well as on

the collisional quasi-molecular ion}atom complexes He#He, namelye#He

PHeH(n)#He, (1)

and

e#He#HePHeH(n)#He, (2)and the inverse chemi-ionization processes during HeH(n)!He atomic collisions, namely

HeH(n)#HePHe#e (3)

Z. Djuric& , A.A. Mihajlov / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 285}305 287

and

HeH(n)#HePHe#He#e, (4)where He"He(1s), He"He(1s), and HeH(n) is a helium atom in the Rydberg states with theprincipal quantum number n

Although the importance of these processes was indicated earlier, mostly at the level of kineticcoe$cients [13}15,18], we have carried out a full analysis of their in#uence within the completecollisional}radiative model.The model applied here concerns the case of helium plasma. All atomic levels with the principal

quantum number p44 are treated explicitly. The levels with p55 are considered to be hydrogen-like, and the populations of the sublevels are proportional to their statistical weights and thepopulation of the level as a whole. The number of levels treated in the model depend on the plasmaconditions. This number was found via a set of numerical calculations and is explained later in thetext.The model is used to calculate the steady-state values of populations when n

,

and

are the

given input parameters, and for the time-dependent calculations of these functions including thenumber densities of atomic and molecular ions, n and n

, and other derived functions, such as

the degree of ionization, energy losses etc. In the case of time-dependent calculations, the system ofdi!erential equations for particles' densities was completed with the appropriate di!erentialequations for n

,

and

. Initial data for time-dependent calculations are taken either from

experiments, or assumed to be steady-state, in which case the relaxation regime of plasma issimulated. The model can operate with all processes included and also if some of the processes areexcluded. This was the method to estimate the in#uence of the excluded processes by comparing theresults with those of the full model results.

3.1. Equations for the atomic-level populations

Additional assumptions made in the model are: (i) plasma is optically thin and photo-absorptionand photo-ionization processes are neglected; (ii) rate coe$cient for reaction (1) is expressed viaa rate coe$cient of dissociative recombination (2), as in our paper [14]. The standard di!erentialequation for the density n(p) of atoms in the state designed with p is:

n(p)t"

K(q, p)n(q)n!

K(p, q)n(p)n

#K(p)n

n!K

(p)n(p)n

#

A(q, p)n(q)!

A(p, q)n(p)#K(p)n

n

#K(p)n(1)n

n!K

(p)n(1)n(p), (5)

where K(p, q) is a rate coe$cient for pPq excitation (deexcitation) atom}free electron processes,K

(p) and K

(p) are rate coe$cients for atom}free electron ionization/recombination processes,

A(p, q) are Einstein coe$cients for pPq transitions, K(p) is a rate coe$cient for photo-

recombination and "nally K(p) and K

(p) are rate coe$cients for chemi-ionization and chemi-

recombination processes (1)}(4).A total number of Eqs. (5) is equal to the number of atomic levels treated in the model and in

principle is in"nite. In plasma, however, the "nite number of atomic states is realized because of thescreening e!ect. Assuming that all higher atomic states are hydrogen-like, the e!ective principle


quantum number NM H of the last realized state in plasma is given by the expression

NM H"I

I

, I"I

(n

,

)"e

r

, r"

k

4eZn

.(6)

Iis the ionization potential of the ground state of the hydrogen atom, I

is a reduction of the

ionization potential of the ground state of the helium atom, ris a Debye screening radius, and Ze

is a charge of the atomic ion.In the case of weakly ionized plasma NM H can be very large. On the other hand, starting from

a level denoted here with N

, practically all higher atomic states with N

4p4NM H are in LTE

with the continuum, and are populated according to the Saha}Boltzmann relation:

n

(p)"n

n

g(p)2g

2mk

e . (7)

The g(p) and g are the statistical weights of the atom in state p and the atomic ion in the groundstate respectively, I

is the ionization potential of the atom in the ground state and E

is the energy

of p-atom, while mrepresents the mass of an electron.

The system of Eqs. (5) has to be solved for all atomic levels which are not in the LTE, i.e.,for all levels p(N

, since the rest is populated accordingly to Eq. (7). Note that in all sums on

the RHS of Eqs. (5), contributions from all levels q5N

are known if n, n,

and

are

given/calculated. These sums are carried out up to the level NM H , which makes the system (5) "nitewith the "nite numbers of terms on the RHS of each equation. N

is not known in advance. This

number was chosen by a numerical procedure for every single calculation. Starting from the initialvalue ofN

, the series of calculations with the same set of input parameters, but with higher values

of N

, were performed until the di!erences in the calculated values of all n(p) considered are

smaller than the prescribed accuracy. In some cases, however, we found that N

"NM H and the

system (5) has to be solved for all atomic levels realized in a given plasma.The atomic level denoted by N

divides all states into two groups: (i) a group of lower levels

which are not in the LTE and whose populations must be calculated by solving the appropriatekinetic equations and (ii) a group of upper levels in the LTE which are populated according toEq. (7). This separation helps in establishing the system of equations to be solved.In order to complete the system of equations, one needs relations for

,

, n

and n. The

di!erential equations for n,

and

are derived from relations of particle and energy balances,

using the concept of two blocks of states. n can be found from the relation of quasi-neutrality inplasmas and within the approximation of the semi-classical model developed earlier.

3.2. Diwerential equations for ,

and n

All atomic levels from the upper block of states are in the LTE with the free electrons, and thetotal density of these particles,N

, is given by the expression

N"n

#

M H

n

(p). (8)

290 Z. Djuric& , A.A. Mihajlov / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 285}305

A formal di!erentiation of the last expression with respect to the time variable t, assuming thatn+n, gives

1n

dN

dt"a

1n

dn

dt #a1

d

dt , (9)where a

and a

are functions of n

,

and n

(p) of all states in the upper block, and can be

calculated if the values of these parameters are known or used from a previous iteration. Althoughthe upper block of states is in LTE with the continuum, there are unbalanced #uxes of particles forall processes involving the lower block of states. Considering all these processes, the kineticequation describing the population/depopulation of all states with the total densityN

is given in

a symbolic form

1n

N

t"

K(p, q)

n(q), (10)

where K(p, q) is a rate coe$cient for any of mentioned CR processes and n

are densities of

particles involved. Since the RHS of Eqs. (9) and (10) must be equal, we have the "rst di!erentialequation for

and n

in the form

a

1n

dn

dt #a1

d

dt "

K(p, q)

n(q). (11)

The second equation for nand

is obtained via a similar procedure, based on the energy

exchange between the lower and the upper blocks of states. The total energy of the upper block ofstates, Q

, is given by the expression

Q"

32k

#I

n#

M H

En

(p). (12)

Again, a formal di!erentiation gives

1n

dQ

dt"a

1n

dn

dt #a1

d

dt , (13)with similar functions a

and a

depending on variables n

,

and n

(p) from all levels in the

upper block. The equation of the energy balance for the upper block must contain the energy gains(losses) caused by all collisional and radiative processes between the two blocks of states. Thisequation can be symbolically written in the form

1n

Q

t" 1

n

P"

K(p, q)E

n(q), (14)

whereP

is an energy gain (loss) in unit time per unit volume caused by a particular CR process,and summation is over all CR processes involved. The quantityE

denotes a gain (loss) of energy

per single process.Apart from all non-elastic collisional}radiative processes, the electron}atomic particle elastic

scatterings have to be taken into consideration. Energy losses due to this mechanism are given by


the relation [34]

P"!n

2m

M 1!

Evn#

Ev(p)

n(p), (15)

where M is the mass of atomic particles and and

(p) are the cross-sections for the atomic

ion}electron and atom}electron elastic scatterings, Eis the free electron energy and 2 means

an averaging over the electron distribution function. In the second term in brackets, all cross-sections

in calculations are approximated with the value

(1), because the cross-sections for

the higher states are less known. In all calculations of interest we had n(1S)

The last equations in the system are those for solving atomic and molecular ion densities, n andn. The relation of electro-neutrality in plasmas is

n"n#n

. (22)

Although in many cases the assumption n;n is valid, n

has not been neglected whenever

processes (1)}(4) were included. A rigorous calculation of nis complicated: it is necessary to know

detailed information about the relevant energy levels, statistical weights etc., and then to performcomplex calculations for every set of input data. Simpler calculations suitable for our model arebased on a semi-classical model [29,30]. By applying the principle of detailed balance in the case ofprocesses (1)}(4), the "nal result can be written in a form similar to the Saha equation

n(1)nn

"2a

X(;/k

)

1!X(;/k

)e RdR

, (23)

X(R,)"

32;!; (R)

k/() if ;(R)(0,

0 if ;(R)50,

where ais the atomic unit length, R

is a solution of equation;

(R)"0 and;

is the adiabatic

energy of ground state (1

) of a quasi-molecular complex He#He; is an incomplete

function.

3.3. Rate coezcients

The choice of formulas for rate coe$cients is the most important factor which a!ects theaccuracy of calculations. Because the di!erential equations where the coe$cients appear arecomplicated and computationally expensive to solve, one also needs relatively simple expressionsfor rate coe$cients. Both accuracy and simplicity were the reasons for using mostly semi-empiricalformulas, as they are in good agreement with the more accurate quantum-mechanics calculationsand experimental results. Another factor which in#uences calculations is the representation of theinner structure of the helium atom in the model, as mentioned in a previous section. Our choice forthe rate coe$cients' formulas follows the subdivision of atomic levels. In all cases where the ratecoe$cients are not known, but the cross-sections for these processes are known, the relation

K(p)"v(p,E

)

(24)

is used, where (p,E) is the cross-section for a given process and v

is a free electron density. The

integration in (24) is either analytical or numerical, assuming that the distribution function isMaxwell's. With the known cross-section and this assumption, it is possible to calculate thequantities E

v(p,E

)

, needed in equations for energy balance.

Values of the Einstein coe$cientsA(p, q) for all sublevels with p44 were taken from tables givenin [35], while for levels with p'4 the quasi-classical expressions were used [36,37]. The ratecoe$cients for photo-recombination are calculated on the basis of known cross-sections forphoto-ionization [38,39] and the relation of detailed balance.


The rate coe$cients for atom}free electron excitation/deexcitation processes are calculated usingthe expressions for cross-sections and Eq. (24). Several expressions for cross-sections have beenapplied: one type for all levels with p44 [38,40}43], where all sublevels were explicitly treated, andanother for all levels with p'4, i.e., for the hydrogen-like states [5].The cross-sections and rate coe$cients for atom}free electron ionization and recombination

processes were calculated using expressions found in [40] for ionization of the ground state, and[5] for ionization of the higher states of a helium atom.Elastic scatterings of free electrons on atomic particles is a very important process in time-

dependent calculations, especially during the period when

in#uence the values of the coe$cients in the equations. The system of di!erential equations issolved using Adam's predictor}corrector method [51].In both cases (steady-state and time-dependent) calculations were performed for the full CR

model when all processes were included, as well as without processes (1)}(4). The di!erencesbetween these calculations show the in#uence of these processes on the kinetics of helium plasmaunder consideration.

4.1. Steady-state calculations

Input parameters ,

and n

have been varied in wide domains. We present only results

obtained for a limited range of input parameters corresponding to weakly ionized plasmas. Tables1}4 show the results of calculations for n

"10 cm and n

"10 cm, as the upper and

lower limits typical for tenuous and dense plasmas, respectively. A similar choice was made forelectron temperatures:

"10 000 K represents the upper limit for low-temperature plasmas and

"20 000 K is the lower limit for high-temperature plasmas. For every set of n

and

, several

characteristic values of varied, satisfying the condition

4

. For the full CR model, the

system of non-linear equations was solved. The iterative procedure based on Newton's method wascompleted when the relation

max

y(p)!y(p)(

was satis"ed. The y(p) is the value of function y(p) in nth iteration, and typically "10. Basedon the above-mentioned analysis of the LTE limit, our choice for N

was maximum 50, which

satis"es the condition that the upper block of states is in LTE with a continuum. However, thereare domains of

and n

where the number of the realized atomic states is smaller than N

. In

such cases all these states belong to the lower block and their populations were calculated bysolving the system of equations.In all Tables 1}4 the "rst column denotes the atomic levels, the second represents the appropriate

values of normalized populations, y(p), when processes (1)}(4) are excluded. The system of equationsin this case is linear and normalized populations are denoted as y

(p). The remaining columns

represent values of normalized populations for di!erent when all processes are included, reduced

to the corresponding y(p) values. The ratio y(p)/y

(p) in these columns is used to show the relative

importance of atomic processes on populations of atomic levels being considered.For the input parameters used here the relation n

(1S)

Table 1Steady-state values of normalized populations y

(p) when processes (1)}(4) are excluded (second column), for

n"10 cm and

"10 000 K; the remaining columns show the ratios y(p)/y

(p) for di!erent

4

, where y(p)

are normalized populations when all processes are included

p y(p)

(K)

10 000 8000 5000 4000 3000

1S 2.4910 3.14 3.97 3.56 2.22 0.9282S 10.2 5.1610 0.105 0.533 1.05 0.9282S 3.67 0.141 0.225 0.610 1.67 0.9812P 6.64 7.5610 0.109 0.209 0.465 0.8422P 0.128 3.66 4.73 4.31 2.66 1.093S 1.79 0.280 0.239 9.5510 6.3710 3.60103S 0.450 1.11 0.893 0.3 0.176 7.49103P 1.48 0.338 0.265 8.1810 4.5810 1.87103D 1.43 0.35 0.268 7.8310 4.2110 1.66103D 0.375 1.34 1.02 0.299 0.16 5.84103P 0.345 1.45 1.11 0.319 0.159 6.03104S 1.19 0.421 0.273 4.9010 1.9310 4.63104S 0.763 0.657 0.414 6.8810 2.5810 5.44104P 1.14 0.44 0.274 4.4010 1.6110 3.39104D 1.13 0.443 0.274 4.2810 1.5510 3.23104D 0.772 0.649 0.4 6.2610 2.2510 4.43104F 1.13 0.443 0.274 4.2710 1.5410 3.08104F 0.777 0.645 0.398 6.2210 2.2410 4.35104P 0.764 0.656 0.405 6.2810 2.2610 4.49105 1.00 0.501 0.277 3.1610 9.3210 1.51106 1.00 0.501 0.26 2.4610 6.5310 1.56107 1.00 0.501 0.249 2.1010 5.6010 3.70108 1.00 0.502 0.242 1.9410 6.4210 1.14109 1.00 0.503 0.238 1.9910 1.0910 3.251010 1.00 0.506 0.237 2.4710 2.4010 7.581011 1.00 0.513 0.242 3.8410 5.3110 0.14212 1.00 0.528 0.258 6.7510 0.103 0.22313 1.00 0.553 0.288 0.116 0.171 0.30714 1.00 0.586 0.333 0.18 0.249 0.38515 1.00 0.624 0.388 0.254 0.328 0.454

In the case when all processes are included, the normalized populations y(p) are functions of onemore parameter, the atomic temperature

. The values of ratios y(p)/y

(p) are shown in the

remaining columns in Tables 1}4. For n"10 cm and

"10 000 K (see Table 1) the LTE

limit is shifted toward those states with a much higher principal quantum number. This limit islowest for

"

and increases as

decreases. The value of y(1S) as a function of

has

a maximum for +8000 K. Other populations are mostly monotonically dependant on

. The

normalized populations y(p) of all states di!er signi"cantly from the corresponding y(p) values.

This means that the processes (1)}(4) are not only comparable with other relevant processes, butplay a dominant role in populating/depopulating the atomic levels of low-temperature non-ideal


Table 2Same as Table 1 but for

"20 000 K

p y(p)

(K)

20 000 15 000 10 000 5000 4000

1S 1.0810 1.00 1.01 1.01 1.05 1.092S 10.6 0.991 1.00 1.01 1.05 1.62S 3.64 1.00 1.00 1.01 1.04 1.102P 6.96 0.999 1.00 1.01 1.05 1.122P 0.188 1.00 1.00 1.01 1.04 1.113S 1.65 0.994 1.00 1.01 1.06 1.153S 0.66 1.00 1.01 1.01 1.07 1.153P 1.41 0.993 1.00 1.01 1.06 1.143D 1.36 1.00 1.01 1.02 1.07 1.143D 0.595 1.00 1.01 1.01 1.06 1.143P 0.542 1.00 1.01 1.02 1.06 1.144S 1.14 1.00 1.00 1.00 1.03 1.064S 0.879 1.00 1.00 1.00 1.02 1.044P 1.10 1.00 1.00 1.00 1.03 1.064D 1.09 1.00 1.01 1.01 1.03 1.064D 0.88 1.00 1.00 1.00 1.02 1.044F 1.09 1.00 1.00 1.01 1.03 1.064F 0.885 1.00 1.00 1.00 1.02 1.044P 0.871 1.00 1.00 1.01 1.02 1.045 1.01 0.99 1.00 1.00 1.00 1.016 1.00 1.00 1.00 1.00 1.00 1.017 1.00 1.00 1.00 1.00 1.00 1.008 1.00 1.00 1.00 1.00 1.00 1.009 1.00 1.00 1.00 1.00 1.00 1.0010 1.00 1.00 1.00 1.00 1.00 1.0011 1.00 1.00 1.00 1.00 1.00 1.0012 1.00 1.00 1.00 1.00 1.00 1.0013 1.00 1.00 1.00 1.00 1.00 1.0014 1.00 1.00 1.00 1.00 1.00 1.0015 1.00 1.00 1.00 1.00 1.00 1.00

plasmas. Recombination processes (1) and (2) are responsible for such behaviour at the smallervalues of

, and their inverse ionization processes (3) and (4) when

is higher.

For a detailed explanation we analysed rate coe$cients of chemi-ionization and chemi-recombi-nation processes as functions of

and

for all Rydberg states of interest, and compared them

with rate coe$cients for atom}electron ionization/recombination processes. Analysis showed thatthe in#uence of chemi-ionization processes (3) and (4) was pronounced for lower Rydberg states,especially with the principal quantum numbers n"3,4, but chemi-ionization becomes less impor-tant for higher Rydberg states, and can be neglected for n'6. In all cases the chemi-ionizatione$ciency increases if

increases with other parameters "xed. However, for given n and

, it

decreases with increase of .


Table 3Same as Table 1 but for n

"10 cm and

"10 000 K

p y(p)

(K)

10 000 8000 5000 4000 3000

1S 1.3510 8.5210 8.9610 9.6310 9.4810 5.98102S 1.19 5.5510 8.5710 0.174 0.148 7.12102S 0.914 7.2110 8.9510 0.131 0.124 6.74102P 1.14 5.7710 6.5410 8.8610 0.102 6.58102P 0.733 8.9410 9.3610 9.6010 9.6610 6.26103S 1.03 6.3910 4.2810 1.1510 4.7210 1.73103S 0.945 6.9610 4.3910 9.7710 3.5710 1.04103P 1.02 6.4510 3.9610 8.1510 2.8210 7.55103D 1.02 6.4510 3.8710 7.5210 2.5210 6.51103D 0.945 6.9610 4.1810 8.1110 2.7110 6.78103P 0.944 6.9710 4.1710 7.9910 2.6510 6.49104S 1.01 6.5210 3.3110 3.9510 9.6710 1.51104S 0.981 6.7110 3.3110 3.6010 8.5010 1.19104P 1.01 6.5210 3.1910 3.4010 7.7210 1.05104D 1.01 6.5210 3.1610 3.0710 7.3410 9.9104D 0.982 6.7010 3.2510 3.3710 7.5210 9.9104F 1.01 6.5210 3.1510 3.2710 7.2910 9.5104F 0.982 6.7010 3.2410 3.3610 7.4810 9.6104P 0.982 6.7010 3.2410 3.3410 7.4410 9.8105 1.00 6.5810 2.8510 2.1510 3.9010 3.6106 1.00 6.5810 2.6710 1.6710 2.6610 1.110

We noted a high e$ciency of chemi-recombination processes (1) and (2) for all n"3}10 indomains of the smallest

, but they are less e$cient for the highest n. Compared with the

ion}electron recombination, chemi-recombination processes are dominant in the whole domainof

for n46. For higher n this dominance is present but in the domain of

510 000. When

decreases, chemi-recombination processes are less important but cannot be neglected.

When increases, rate coe$cients for chemi-recombination processes decrease for all n410.

However, for every n there are domains of and

, where the in#uence of these processes is

signi"cant.The described characteristic behaviour of y(p)/y

(p) is much less pronounced in the case of

"20 000 K for the same value of n

. Table 2 shows that y(p) values are very close to y

(p)

values. Di!erences exist only for the smallest values of when

"4000 K. Therefore, when

increases, the in#uence of atomic processes (1)}(4) decreases until they become negligible. In

cases with a high , the atomic processes need be considered only if

Table 4Same as Table 3 but for

"20 000 K

p y(p)

(K)

20 000 15 000 10 000 5000 4000

1S 35.1 1.00 1.01 1.05 1.34 1.872S 1.15 1.00 1.02 1.05 1.42 2.132S 0.941 1.00 1.02 1.05 1.39 2.022P 1.10 1.00 1.02 1.06 1.41 2.082P 0.82 1.00 1.02 1.05 1.38 1.983S 1.01 1.00 1.02 1.06 1.39 1.913S 0.979 1.00 1.02 1.05 1.30 1.673P 1.01 1.00 1.01 1.05 1.32 1.713D 1.01 1.00 1.01 1.05 1.31 1.703D 0.979 1.00 1.01 1.04 1.29 1.613P 0.978 1.00 1.02 1.04 1.28 1.624S 1.00 1.00 1.01 1.02 1.12 1.204S 0.994 1.00 1.01 1.02 1.10 1.134P 1.00 1.00 1.01 1.02 1.10 1.144D 1.00 1.00 1.01 1.02 1.10 1.144D 0.995 1.00 1.00 1.02 1.08 1.114F 1.00 1.00 1.01 1.02 1.10 1.144F 0.995 1.00 1.00 1.02 1.08 1.114P 0.995 1.00 1.00 1.02 1.08 1.115 1.00 1.00 1.00 1.01 1.03 0.9496 1.00 1.00 1.00 1.00 1.00 0.8887 1.00 1.00 1.00 1.00 0.994 0.8718 1.00 1.00 1.00 1.00 0.991 0.8659 1.00 1.00 1.00 1.00 0.99 0.86310 1.00 1.00 1.00 1.00 0.99 0.86211 1.00 1.00 1.00 1.00 0.99 0.861

is an important result, especially if one compares it with the case when processes (1)}(4) areexcluded (second column in Table 3) when calculations showed that all levels with p54 are in theLTE. Therefore, the neglecting of atomic processes can lead to erroneous results. This is also true inthe case of time-dependent calculations, as shown in the next section.

4.2. Time-dependent calculations

The time-dependent model is applied in the case of non-equilibrium helium plasma in a relax-ation regime, after switching o! the electric "eld which supported a discharge, and is examined inpapers [9,20]. The initial values of electron and atom temperatures and electron density are

(0)"18 000 K,

(0)"4500 K and n

(0)"310 cm. The experiment was chosen be-

cause it represents a particular case between the non-equilibrium weakly ionized gases andnear-equilibrium gas plasmas. Also, in paper [20] all necessary data on populations of ground andsublevels with the principal quantum number p"2 were given. For all states with p'2 the initial


Fig. 1. Time evolution of and

functions when all processes are included in the model; the straight line de"ned by

equation (0)#d

(0)/dtt is used for calculating the characteristic relaxation time .

values are calculated using our steady-state model. In this calculation the number of explicitlytreated levels isN

"30. The "rst 300 iterations are completed with the time step t"10 s and

the next 2000 iterations with the time step t"2.510 s, giving a total calculation timet"5.310 s. This time is much smaller than the typical di!usion time for such plasmas and

validates our neglect of the di!usion processes. The choice of time steps satis"es the stability andconvergence requirements for our system of di!erential equations. The numerical error in calcu-lations is a combination of absolute and relative errors [51], and in all calculations it is less then10. Similarly to the steady-state calculations, two cases were considered: (a) all processes areincluded; (b) processes (1)}(4) are excluded.Fig. 1 shows functions

(t) and

(t) and a tangent line

(0)#d

(0)/dtt, when processes

(1)}(4) are included in the model. The tangent line's crossing with the (t) curve de"nes the

relaxation time . We calculated the relaxation time to be "1.77910 s while the measuredvalue is "2.010 s. Bearing in mind that after t"10 s the temperatures

and

are

practically equal, the calculated value for agrees very well with the measured one. Fig. 1 alsoshows a very fast drop of

, which is the consequence of the electron}atomic particles' elastic

collisions. Although the cross-sections for elastic collisions are very small, the total densities of


scattering particles are high enough to compensate for this and in terms of energy losses, to makethese processes very important. At the end of our calculations we "nd that

"4574 K and

"4502 K. When processes (1)}(4) are excluded, there are no signi"cant di!erences in the

behaviour of and

.

The behaviour of functions n(t), n(t) and n

(t) as well as n

(t)"nH

(t) when processes (1)}(4) are

excluded, was examined here. Functions n(t) and nH

(t) are decreasing and up to t&10 they are

practically equal. In the domains of t where (t)+

(t) we "nd that n

(t)(nH

(t), which is the

result of the in#uence of processes (1)}(4).One of the main objectives of these calculations was to examine the time evolution of popula-

tions n(p, t) for atomic levels with the principal quantum number p'1. For all atomic states withthe principal quantum number p"2, we established that signi"cant changes in the functions'values happen in the early stages of calculations, when the di!erences between

and

are

largest. These changes, however, are not as pronounced as in the case of (t). After approximately

t&0.510 the populations' functions are monotonic. Processes (1)}(4) have a signi"cantin#uence on the population of these atomic levels. When these processes are included, the densitiesof all these levels are up to 5 times greater than the corresponding densities when processes (1)}(4)are excluded. Also, a relatively good agreement between measured and calculated values for levels2S and 2Pwere obtained. A total agreement with experimental results was not an objective, sincea given plasma [20] contained a small amount of hydrogen atoms. Therefore, a fraction of excitedhelium atoms could take part in processes of Penning ionization. However, these processes havenot been taken into account, because in a given time interval (1 s) their in#uence can be neglected.The common feature for all populations n(p, t) with p'2 is their signi"cant change in the early

stages of the plasma relaxation processes, when all of them reach maximum values, which is inagreement with experimental results [20]. These changes are slower than changes in

(t), and

faster when processes (1)}(4) are included in the model. This is illustrated in Figs. 2 and 3, whichshow the n(p, t) dependency for all sublevels with the principal quantum number p"4. All densitiesare higher when processes (1)}(4) are included because chemi-recombination processes are verye$cient in populating these levels, as has been shown in our analysis of the results in thesteady-state case. Figs. 2 and 3 show that the results of calculations based on the complete CRmodel are qualitatively di!erent from those when processes (1)}(4) are excluded}in the latter caseall population functions show local minimums.Results shown in Figs. 1}3 were obtained under the assumption that a given plasma was

optically thick for the resonant emission of helium atoms, which is close to experimental conditions[20]. Bearing in mind very fast changes in the plasma, in order to make our model as close toexperiments as possible, we performed a series of calculations where the plasma's e!ective opticalthickness was varied. For instance, in the "rst case we had an optically thin plasma for all atomicspectral lines, while the last one had an optically thick plasma for all such lines. We founda signi"cant in#uence of processes (1)}(4) in all cases, with small di!erences between them.To conclude, the main results in these time-dependent calculations are: (i) the process of

decreasing towards

is extremely fast (a characteristic relaxation time in a given plasma is

"210 s) and is not sensitive to the inclusion of atomic processes, because the energy lossescaused by elastic electron}atomic particles' scatterings are the dominant mechanism responsiblefor electron temperature relaxation; (ii) the populations' functions are not monotonous during theperiod of

decreasing; the period of such behaviour of the populations' densities is longer than


Fig. 2. Time evolution of populations n(p, t) for all atomic levels with the principal quantum number p"4, all processesare included.

the electronic relaxation time ; during this period the in#uence of processes (1)}(4) on populations,especially levels with principal quantum numbers p"3,4, is very pronounced; (iii) when processes(1)}(4) are included, the calculated values of populations agree well with experimental values (forp"2 sublevels); without these processes the calculated values di!er very much from the measuredones; (iv) regarding the higher atomic levels it should be mentioned that during the calculating timefor all levels with 104p4N

, they are practically in the LTE.

5. Conclusions

The results presented here show the great in#uence of chemi-recombination and chemi-ioniz-ation processes (1)}(4) in populating the highly excited atoms in non-equilibrium helium plasmas,in broad ranges of electron densities and temperatures. Therefore, these processes must be includedin all collisional}radiative models dealing with such plasmas. Also, similar ionization and recombi-nation processes should be investigated in cases of non-equilibrium gas plasmas of di!erentchemical composition.


Fig. 3. Same as Fig. 2, but when processes (1)}(4) are excluded.

The important characteristic of the collisional}radiative model used here is the treatment of freeelectron density n

: it is assumed to be an unknown function which has to be calculated within the

model, and not a given constant which is the usual approximation. This approach is especiallyuseful when a high accuracy in n

calculation is required.

The developed model is #exible enough to include other processes, such as di!usion, photo-absorption and the in#uence of an external electric "eld etc. Although it is applied in the case ofhelium plasma, it is possible to make similar models for plasmas made in a mixture of di!erentgases. Since the programme is in a modular form, it is easy to change the routines describingdi!erent kinetic coe$cients, terms in equations, etc.

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