Ionization equilibrium model for the multiply charged ion formation

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Ionization equilibrium model for the multiply charged ion formation

S. Bliman, N. Chan-Tung

To cite this version:

S. Bliman, N. Chan-Tung. Ionization equilibrium model for the multiply charged ion formation.

Journal de Physique, 1981, 42 (9), pp.1247-1252. �10.1051/jphys:019810042090124700�. �jpa-00209315�

Ionization equilibrium model for the multiply charged ion formation

S. Bliman and N. Chan-Tung

Département de Recherches

la Fusion Contrôlée, ^Service IGn, Centre d’Etudes Nucléaires, ⁸⁵ X, 38041 Grenoble Cedex, ^France

(Reçu ^{le 10}

1981, ^{révisé le} 27 avril, accepté ^le 5 mai 1981)

Résumé.

Un modèle d’équilibre d’ionisation d’un plasma d’argon ^est proposé. ^La considération des collisions

élastiques ^entre électrons, ^{ions et} atomes neutres montre que les collisions ion-ion

^sont les processus dominants.

A partir de l’évaluation du coefficient de diffusion,

exprime

temps de vie des ions 03C4Zi qui ^varie

Zi2 (Zi ^{état de} charge ^de l’ion). ^{On résout les} équations ^donnant l’équilibre de la densité des ions dans les différents états de charge Zi

fonction de la densité électronique, ^des températures électronique ^et ionique; ^{les valeurs de}

ces grandeurs ^{sont celles} qui ^{sont observées dans les sources} à résonance cyclotron ^des électrons. La distribution des états de charge ^{des ions est calculée et} comparée ^{aux résultats de mesure} publiés. ^{L’accord est} satisfaisant.

L’accroissement de la densité électronique ^et la diminution de la pression de gaz neutre, tous autres paramètres

maintenus constants, permettraient ^{la création} d’ions argon hydrogénoïde.

Abstract.

A model is proposed ^to study

argon plasma ionization equilibrium. Considering elastic collisions between electrons, ^{ions and} neutrals, it appears that ion-ion collisions

dominant among these processes. From

an

evaluation of the diffusion coefficient, it is then possible ^{to express}

^{ion life time} 03C4zi which varies

Zi2 (Zi being ^{the ionic} charge state). ^Balance equations

then solved with values of electron number density, ^electron

and ion temperatures

^observed in E.C.R. ion sources (Electron Cyclotron ^{Resonance ion} sources). ^The charge

state distribution calculated from the model is compared ^with published experimental ^{results. The} agreement ^is

good. ^{It is} suggested ^that increasing electron number density ^and decreasing neutral gas pressure, all other para- meters being kept ^constant, would allow creation of hydrogen like argon ions.

Classification Physics ^Abstracts

52.80H - 52.80P - 52.75D - 29.25

1. Introduction.

_{In many} experiments performed

on highly charged ^{ion sources, it is difficult to} interpret

results considering ^the classical ionization equilibria

well described in the literature. Often in the source

plasmas, ions and electrons have different tempe- ratures ; ^{to reach} highly charged states, it is necessary to reduce the neutral _gas pressure ^to avoid charge exchange processes. Even the coronal equilibrium ^{is a}

too restrictive approach because electron number

density ^and temperature ^{are too low to fulfill the basic} assumptions.

The use of models previously proposed ^to interpret charge ^state distribution in the ^case of PIG sources

(G. ^Fuchs [1]), ^{EBIS sources} (A. ^Muller [2]) having

failed to give agreement with E.C.R. source charge

state distributions, ^it appeared quite interesting ^to

propose ^a model where, ^{as is shown} below, equilibrium

is reached taking ^{into account} step by step ionization, charge exchange and diffusion losses. Radiative and

dielectronic recombinations have both been neglected.

These _processes are associated with characteristic times too long compared ^with charge exchange ^and

diffusion times.

Elastic collisions between electrons, ^{neutrals and}

ions ^are reviewed, in order to assess which of them is associated to mean free paths comparable ^{to source}

characteristic dimensions. From this, it is then possible

to express ^a diffusion time which depends ^on Z 1 2 (Z; standing for the ion charge). ^Balance equations

are then written and solved introducing ^{electron num-}

ber density, electron and ion temperature ^{values and} neutral _gas number density ^as measured in different electron cyclotron ^{resonance ion source} experiments.

For a steady ^{state, the} charge ^state distribution is obtained and compared ^to experimental ^{ones. To} conclude, it is shown that increasing ne and decreasing

neutral _gas density ^allows hydrogenic ^{argon ions to be}

obtained.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042090124700

1248

2. Elastic collisions mean free paths.

^For any type of elastic collision, ^{the mean free} path ^may be expres- sed as the ratio of the mean velocity ^to collision fre- quency. Considering ^{an ion source} plasma, ^five types of collisions are recognized; taking ^{into account}

measured values of the physical parameters (N. ^Chan- Tung [3]) ^{of the} multiply charged E.C.R. ion source

(R. ^Geller [4] ; ^P. Briand et al. [5] ; P. Briand et al. [6]),

their associated mean free paths ^are estimated and

compared ^{to source} characteristic dimensions. It is

seen that for the values of characteristic parameters, the plasma may as ^a first approximation ^{be considered}

as completely ^ionized.

2.1 ELECTRON-ELECTRON COLLISIONS.

The elec- tron self-collision frequency (J. ^{L. Delcroix} [7]) ^is given by

where ne is the electron number density (cm- 3), Te their temperature (eV) and Aee

^{24 -} Ln (ne Te-1).

Introducing ^the parameter ^values gives the associated

mean free path

2.2 ELECTRON-NEUTRAL COLLISIONS.

In the elec-

tron temperature range, electron-neutral elastic col- lision cross-sections are considered as constant and of order (J. ^{L. Delcroix} [8]) ³ x10- 16 cm2. Then

In this case Âe-N - ^{3.1 x 105 cm.}

2.3 ELECTRON-ION COLLISIONS.

The electron- argon ion collision frequency ^{is to be written}

where nz; represents ^{the ionic} density ^{of ArZ t in cm- 3.}

From the plasma neutrality condition, ne = 1 nz, Zj,

Zi we define a mean value for nz, Z; consistent with observed charge ^state spectrum (P. Briand et al. [5]),

this mean value being equal ^{to 0.12} ne. Charge ^states greater ^{than 8} give ^a negligible contribution to the

mean value nz; Z; since their abundance is small with respect ^{to lower} charge ^states.

Introducing ^{it in} Ve-Arz; allows the mean free

path ^{to be} expressed ^{as :}

2.4 ION-NEUTRAL COLLISIONS.

For this process, the assumption is made that the cross-section is

independent of the ion charge ^and constant ; ^{in the} ionic temperature range (J N - Ar + is approximately (J. L. Delcroix [9]) 3 ^x 10-15 cm2. From this, the mean

free path ^is calculated :

2. 5 ION-ION COLLISIONS.

The elastic collision

frequency ^between charged particles ^{of same mass}

and of unlike charge ^states Z; ^and Z; writes (L. Spitzer [lo])

In this expression, the Coulomb logarithm ^is equal ^{to :}

Considering temperature ^and density ^values, AArzt ^-ArZi+ simplifies ^{and is} equal ^{to :}

A varies between two extreme values : a maximum when Z;

Z;

¹ (then A

12.6) ; ^for Zi

Z;

⁸ (then ^A

7.4). ^It is thus convenient to take a mean

value equal ^{to 10.}

The above-collision frequency ^is symmetric ⁱⁿ Z;

and Z;’. Introducing ^the experimental ^{mean ion} charge

and the plasma neutrality condition results in a

collision frequency of ions of charge Zi ^and charge Z; :

A summation has to be performed ^{on all} charges Zi’;

Z; is limited to 10 since experimentally, ^the highest charged ^state of argon ions observed is Ar1o+ [5].

In these conditions, ^{the mean free} path for elastic collisions of Ar’ t on any other charged argon ion is :

2.6 REMARK. - From the preceding calculations,

it _appears that _among all possible ^elastic collisions,

the only ^{one relevant to} the considered situation is the ion-ion collision. To this process is attached the

only ^{mean free} path much smaller than _any of the characteristic dimensions of the plasma in the ion

source

3. Plasma characteristic diffusion time.

In sources

designed ^{to obtain} highly charged ions, confining

configurations ^{are of} common use. Radial-diffusion

processes ^are thus inhibited (use is made of minimum B

magnetic configuration [6]). This leaves to be consi- dered diffusion losses in the direction of ion extraction

(axial direction).

In the transport equation ^{for the ion} density nzi in the Z;th ionization state, there need be introduced

a characteristic local diffusion time. For _argon ions A?- t it _may be written :

where L is the plasma length (the plasma length ^L

in the SUPERMAFIOS experiment ^{was 120} cm) ^and

Da ^{is taken to} be the axial ambipolar ^diffusion

coefficient. As a function of the free ion diffusion coefficient DZi, it writes :

The diffusion rate and as such the overall loss are thus determined by ion inertia (slowest species) [11].

Considering the ion self-collision frequency, Dz¡ ^is

calculated and thus the diffusion characteristic time is

Introducing ^the parameter ^{values as} ascribed above this finally simplifies ^{to :}

This diffusion time has in fact ^a value intermediate between classical and Bohm’s diffusion time.

4. Ionization equilibrium ^model.

^{The basic} assumptions ^are presented and the evolution equations

written.

4.1. ASSUMPTIONS.

1) The ionization process taken into account is a step by step ionization ; multiple

ionizations ^are neglected.

2) Calculated ionization cross-sections have been taken from W. Lotz [12,13]. Ionization rate coefficients obtained by folding ionization cross-sections with a

Maxwellian electron distribution at a temperature Te ^are introduced as generating ^terms [13].

3) The electron number density ^{is constant across}

the plasma region.

4) The ion diffusion time ’t Zi ^is expressed ^{as written}

.above.

5) In the ionization balance equations, ^{ion losses}

are due to :

-

diffusion,

-

ionization to the next higher charge state,

-

charge exchange ^{on neutrals to the two next}

lower charge ^{states : the} probability ^{that more than}

two electrons be captured being ^{one order of} magni-

tude smaller than for one or two electrons capture.

6) ^The charge exchange cross-section values are

those measured experimentally by C. Salzborn [14]

for argon ions up ^to Ar6+ and by other authors [15]

in the _energy range of 1.2 keV ^to 56 keV. For higher charges, ^the cross-sections ^are taken from [16] ^where

measured values are given ^for charges up ^{to +} 12, in the energy range 1 Z - 10 Z keV (where ^{Z is the}

incident ion charge). ^{For two electron} capture ^{in the}

argon ion charge exchange ^{collision on argon, mea-}

sured cross-section values are taken from S. Bliman

et al. [17]. It is assumed that the charge exchange ^cross-

sections at low energies ^{are constant and} equal ^those

measured in the keV _range. This assumption ^{seems to}

be supported by recently published ^{data obtained}

with Ne1o+ colliding ^neutral gas targets (energy

5-50 eV) [21] ^and by theoretical estimates [22] sup-

ported by experimental ^values [23].

7) Collisional radiative and dielectronic recombi- nations have been neglected. ^{The basis to this} assump- tion stems from a comparison of the characteristic times for the different collisional processes. In ^case

of Ar8 +, ^{the diffusion} time is of order 1 ms (section 3)

to be compared ^{with 14 ms for} single ^electron capture in charge exchange [16]. ^The scaling properties ^of

dielectronic recombination rate for the Ne-sequence [24] ^{allow an} estimate of the characteristic time for Ar8 + : it is of order 500 ms. As to the collisional radiative recombination, its associated time is much

larger (of order ^10-20 s).

4.2 BALANCE EQUATIONS.

Different models have been proposed ^{and solved} (Chan-Tung ^{[3] ; Fuchs [1] ;}

Müller et al. [2]). Taking ^{into account the above men-}

tioned basic assumptions, ^{a set of} differential _equa- tions is written, each of them describing ^{the evolution}

of one argon ion type.

4. 2.1 Set of equations.

^{The full set of} equations

is written for charge Z; ^{between 1} and 15 for _argon.

For charge ^{1 :}

1250

For charge ^{2 :}

For _any charge ^state ranging ^{from 3 to 15}

From inspection ^{of these} equations, ^three types ^of contributions are recognized ^{in the} right-hand ^{side :}

-

an ionization generating ^{term for the} charge

under consideration. In the multistage ^{E.C.R. source} (for ^Ar1+ ions), ^this term jo is in fact a flux of plasma injected from the first stage ^to the second where step by step ionization takes place;

-

a loss term where ionization to the next higher

state, charge exchange ^{to the next two lower states}

and diffusion are met;

-

a creation term where charge exchange processes feed the considered charge.

4.2.2 Solution.

The solution to this set of 15 equations is obtained assuming stationary ^state

and writing, ^{to close the} system, ^the plasma neutrality

condition. The system ^of algebraic equations ^{is then} easily ^solved.

4. 3 COMPARISON OF CALCULATED AND MEASURED CHARGE STATE DISTRIBUTIONS.

On figure ^{1 are} represented ^a calculated charge distribution for _argon

ions, corresponding ^to the above mentioned plasma parameter values and the experimental charge ^distri-

bution (Chan-Tung [3]). ^Good agreement ^{is observed} between the theoretical and the experimental ^C.S.D.

Figure ² represents ^a comparative ^{sketch of} experi-

Fig. ^1.

Comparison of calculated to measured charge ^{state distri-}

bution of argon ions (Te=1keV, Ti = 5 eV, no=1.06x 1010 cm-3,

ne = 5.5

1011

cm- 3) : ⁰ experimental points, A calculated points.

Fig. ^2.

Charge ^state distribution variation

function of neutral

gas pressure for argon ions of charge + 7, + 8, + 9, ⁺¹⁰ (Te ^{= 1} keV,

Ti

⁵ eV, ne = ^5.5

^10’ cm - 3).

mental and theoretical percentage values for Ar 7 + , Ar8 +, Ar9 +, ^{Ar10 + as a function} of neutral _gas pres- sure ; experimental ^values being ^{taken as above from} (P. ^{Briand et al.} [5]). The overall behaviour shows a

good agreement ^{between the} experimental ^{and the}

theoretical C.S.D. as to neutral _gas pressure effect.

This _may be interpreted ^{as follows : even} though ^the

neutral _gas number density decreases, ^{the term} representing ^the charge exchange contribution is still

important because of the increase of the cross-section which varies as azz - 1 ^aZ [16]. This counterbalances the ionization rapidly ^{since the} ionization cross-

section (J’ z,z + 1 ^is rapidly decreasing ^{when Z increases} [12].

It _may be underlined that a reconsideration of ioni- zation equilibrium models has been recently ^under-

taken in fusion plasma [18] ^{and in} astrophysics [19,20] :

it is shown that charge exchange collisions of ions on

neutrals are a factor influencing ionization equilibria

very seriously.

5. Conclusion.

The comparison ^{that has been}

made concerning ^the calculated and measured values of the charge ^{state of} argon ions in the E.C.R. source

has shown good agreement ^and gives support ^{to the} diffusion assumption according ^to which the ion diffusion time is mostly dependent ^on Z 2.

To improve ^the experimental results, ^{from this}

model’s approach, ^two possibilities ^{are obvious to}

reach higher charge ^states.

The first is to allow for a lower neutral gas pressure : this causes, all other parameters being kept ^constant,

a decrease in charge exchange collisions.

The second is, ^{in addition to} the pressure decrease,

an increase in electron number density.

To pinpoint the effect of these variations, figure ³ gives ^{the result of numerical} calculations showing

evolution of charge ^state distributions. The curve with open circles corresponds ^{to an} argon background

pressure of 10-10 torr (ne

^{5.5 x 1011} cm- 3 ; Te

¹ keV, T;

^{5 eV as} figure 1). ^{The curve with} triangles corresponds ^{to an} argon pressure of 10-10 torr and ne

^{2.2 x 1012 cm-3} ( Te

^{1 keV}

and Ti

^{5 eV as in} figure 1). ^{The mean ion} charge

is ^{slightly sensitive to plasma} conditions; lowering pressure displaces ^the barycentre ^{of the} charge ^state

Fig. ^3.

Calculated charge ^state distribution of argon ions for ^two different electron number densities. Argon pressure : 10-10 torr.

Te

¹ keV, Ti

^{5 eV.} Open ^circles ne

5.5

1011 cm-3 ; triangles ne

2.2

1012 cm- 3.

distribution : charge exchange is reduced allowing ^for higher charges ^and depopulating ^low charge ^states (their ^relative percentages ^thus becoming negligible).

It is seen in the last case where the set of 18 equations

has been solved (triangles ⁱⁿ figure 3) ^{that even at an}

electron temperature ^{as low as 1} keV, ^{Ar17 + is reached}

with an abundance of 0.1 %, ^the sharp decrease from Ar16 + being ^{due to the} ionization of a K shell electron.

Even though ^the assumptions of this model are

simple (excitation collisions and radiative recombi- nation have been neglected), they ^{allow however a}

reasonable description ^{of the} charge ^{state evolution}

as a function of the dominant experimental para- meters.

References

[1] FUCHS, G., IEEE Trans. Nucl. Sci. NS 19 (1972) ^160.

[2] MULLER, A., KLINGER, H. and SALZBORN, E., Nucl. Instrum.

Methods 140 (1977) ^181.

[3] CHAN-TUNG, N., Modèle d’équilibre d’ionisation pour

source

d’ions multiplement chargés ^du type R.C.E. ; appli-

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