CC BY
11
UDC 533.9=111
A.P. Golovitskii
St. Petersburg State Polytechnical University, 29 Politekhnicheskaya St., St. Petersburg, 195251, Russia
ELECTRON ENERGY RELAxATION LENGTH IN CONNECTION WITH THE PROBLEM OF ELECTRON ENERGY DISTRIBUTION LOCALITY IN GLOW DISCHARGE PLASMA IN A xENON-CHLORINE MIxTURE
А.П. Головицкий
длина релаксации энергии электрона и проблема
ЛОКАЛЬНОСТИ РАСПРЕДЕЛЕНИЯ ЭЛЕКТРОНОВ ПО ЭНЕРГИЯМ В ПЛАЗМЕ ТЛЕЮЩЕГО РАЗРЯДА В СМЕСИ КСЕНОН-хЛОР
The analytic calculations of relaxation length of electron energy based on a probabilistic approach and numerical calculations of nonlocality effect on the plasma-chemical processes rates have been carried out for plasma of an electronegative glow discharge in a mixture of xenon and chlorine. It is shown, that for total pressure higher than 6 Torr and with amount of chlorine more than 1/25 the effect of electron energy distribution nonlocality is negligibly small and the local approach can be used for modeling such discharges.
GAS DISCHARGE, ENERGY RELAXATION, ELECTRONEGATIVE GASES, ENERGY DISTRIBUTION LOCALITY.
На основе вероятностного подхода выполнены аналитические расчеты длины релаксации энергии электрона, а также численные расчеты влияния нелокальности на скорости плазмохимических процессов в электроотрицательном разряде в смеси ксенона и хлора. Показано, что при общем давлении более 6 Торр и при доле хлора более 1/25 влияние нелокальности энергетического распределения электронов пренебрежимо мало, и для моделирования таких разрядов можно пользоваться локальным приближением.
ГАЗОВЫЙ РАЗРЯД, РЕЛАКСАЦИЯ ЭНЕРГИИ, ЭЛЕКТРООТРИЦАТЕЛЬНЫЕ ГАЗЫ, ЛОКАЛЬНОСТЬ ЭНЕРГЕТИЧЕСКОГО РАСПРЕДЕЛЕНИЯ.
In recent years, there has been significant interest in investigating and modeling the physical processes in middle pressure (5 — 40 Torr) electronegative (EN) discharges in the mixtures of inert gases and halogens because of their practical applicability as effective and powerful sources of ultraviolet radiation [1 — 5].
The physical processes in strongly EN plasmas are extremely complex, and therefore one obtains physical information about these processes mainly from numerical modeling [2, 6 — 8]. The vast majority of EN plasma models use so-called local approach, which means that the electron energy distribution function (EEDF)
and also all relevant plasma-chemical processes in the given place can be only expressed in terms of electric field E and neutrals density N being in the same place. We can accept that EEDF is local if
K (We ) <Xe/n VW , (1)
where XW (We) — relaxation length of electron energy We, XE/N — spatial scale of E/N ratio.
If (1) is not valid, then the kinetic energy of an electron cannot be expressed through the plasma conditions (at particular field E) measured in the place where this electron is located, i. e. EEDF is nonlocal [9, 10], and this fact
should be taken into account while modeling, especially if the field E is spatially nonuniform [10].
In the present paper we estimate the degree of EEDF nonlocality in EN plasmas of a gas mixture containing xenon and chlorine.
It is known that the nonisothermal plasma (at Te > T where Te, T are electron and ion temperatures) of a gas discharge containing both positive and negative ions stratifies across the current direction into two regions with different ion compositions and properties [6 — 8, 11]. In the central region («core»), the densities of positive np and negative nn ions are significantly more than the electron density ne, i. e. we can speak about an ionion (i-i) plasma where np « nn >> ne. In the case of strong electronegativity the diffusion of electrons in the «core» occurs to be almost free diffusion, the radial profile of ne is almost flat, and the radial space charge electric field Er is weak, close to zero. In the outer plasma region («edge»), the density of negative ions is very low np « ne >> nn, and we can speak about an electropositive (EP) electron-ion (e-i) plasma. In the «edge» area, there is an ambipo-lar diffusion regime, and the finite radial space charge electric field Er directed to the wall. The thickness of the «edge» can be estimated for cylindrical geometry as [12]:
8«n
R
2 Vwa
(2)
+ a
where a = vaeR2 / (|nT), va — attachment frequency, |n — negative ion mobility, Te — electron temperature, R0 — discharge tube radius.
In the narrow transition region between i-i-and e-i-plasma, there are strong variations of both positive and negative ion concentrations.
The estimation of XW and EEDF nonlocal-ity effects has got some difficulties because the value XW is not constant but it depends on the electron energy We. Therefore, at middle pressure it happens that inequation (1) can be valid for big electron energies (particulary, in pure inert gases where We is higher than the first excitation potential) while for lower energies (1) cannot be valid. In such a case, one should derive the kinetic Bolzmann equation taking into
account the electric field inhomogeneity which is a difficult task. According to [9, 13], EEDF for EP inert gases can be accepted as local if pR > 10 Torr-cm (p — gas pressure).
Three considerations prompted us to carry out the presented investigation.
Firstly, the value pR0 for the part of our discharges is just < 10 Torr-cm [1, 2], and an EEDF nonlocality can be suspected.
Secondly, for EP plasmas there is no stratification, radial field Er is distributed over the whole plasma cross section, and XE/N = R. But in case of EN plasma, XE / N = 8, and corresponding to (2), 8 can be ~ (0.1 - 0.2)R0 [8] i. e. XE/N is smaller than R0. Hence, a criterion (1) of EEDF locality for EP plasmas is weaker than for EN ones, and the problem of EEDF nonlocality in EN plasmas can become even more actual than for EP plasmas.
Thirdly, the gas mixtures considered here contain a noticeable amount of molecular EN gas where the electron energy losses can happen even for small initial energies - due to excitation of low lying vibrational and rotational molecular states and by dissociative attachment [14]. That can lead to a reduction of XW in comparison with XW in atomic EP gas. By numerical modeling an EN discharge in molecular oxygen [15], it has been shown that EEDF in the tube of 12 mm diameter is local at pressure p > 1 Torr, which corresponds to pR > 1.2 Torr-cm. Hence, for discharges in molecular EN gases, the EEDF nonlocality can occur at smaller pressure than in inert gases. This is an encouraging result but, unfortunately, there are no literature data either about A,W((e) or about EEDF nonlocality in the discharge plasma containing molecular chlorine.
Before estimating electron energy relaxation length XW (We) itself, let us obtain the expression for the electron free path length Xe ((e). Definitely,
^e (We ) = "
Vea (We )'
where ve is electron velocity, vea ((e) is the frequency of any electron-atomic (EA) collisions. since
Vea (We ) = (We )VgN, where ct fu" (We) — full EA collision cross-sec-
tion for electron energy We, we obtain
1
value down to 1/e, and from the equation
X (We ) =
N ct fuU (We )'
1
ctcI (Wei
(We )
<CT
full
If we consider the discharge in a gas mixture, for example in a mixture of xenon and chlorine, then
X (W) =_1_
^ NXeaXel1 (We ) + N^ < (We )'
In one elastic EA collision, an electron losses almost no energy. Therefore, we can accept that the probability of preserving electron initial energy We after one EA collision is equal to the probability of elastic collision:
ael (We) = ael (We)/ aful1 (We),
where ael (We) is the cross section for elastic EA collisions.
The same probability for a series of k successive collisions will be
a(k)(We ) = [«el (We )f .
we can calculate the value of k((e) — the number of collisions needed for the relaxation of electron energy We:
<• ; ^ (W. P A
k ( We ) = max
1, In
<CT
full
(We )
As electron motion in any gas at middle pressure is similar to the Brownian motion, the most probable electron displacement from an initial point after k (We) collisions corresponds to the desired electron energy relaxation length Xw (We):
V (We ) =4kkW) -Xe (We ) =
>IWë)
" NXe CT^ (We ) + Ncu CT»? (We)
(3)
In Fig. 1 the results of the calculations of If the gas mixture contains halogen mol- xw (We), according to Eq. (3), with the use of
ecules, then the denominator of ae/ (We) is always bigger than the numerator due to attachment and excitation of low lying molecular states, and the value of ael (We) is always smaller than 1 for all We. Hence, we can define the fact of relaxation of initial electron energy We after k collisions as diminution of a(f) (We)
the cross-section set taken from [14] are presented for a discharge in the mixture of Xe and Cl2. One can see that if We exceeds the first excitation potential of Xe, the value of XW((e) is small and we can state that the EEDF in this energy range is local. If NCl concentration is more than 1015 cm-3, then the EEDF is local
Fig. 1. Dependences on electron energy: a — number of collisions needed for energy relaxation; b: 1 — energy relaxation length, 2 — free path length; the horizontal line shows the tube radius R0 = 6 mm.
Gas mixture is: 6 Torr of Xe and 0.25 Torr of Cl2, discharge current I = 10 mA, XE/N » S » 0.1 cm
e
if We> 2.5 eV (slightly higher than excitation threshold of the Blnu state of Cl2 molecule). For the slowest electrons with We < 0.5 eV, the cross-section of dissociative attachment becomes large, and XW(We) becomes small. However, we can see that in the electron energy range 0.5 < We < 2 eV, V(We) « Xe/N = 8, and expression (1) for these electrons is no more valid. The cross-sections of EA collision processes in this energy range (dissociative attachment, molecular vibrational level excitation) are small, hence the probability of electron energy losses is small as well.
so, the EEDF in discharge plasmas containing inert gases and chlorine can turn out to be nonlocal only in the narrow electron energy range of 0.5 < We < 2 eV.
In discharges in EN gases and mixtures, the strong field inhomogeneity occurs in the «edge» (e-i plasma), where the radial field Er can be large (up to 100 — 400 V/cm at p « 6 Torr) within the distances of < 0.5 mm from the wall [6, 8]. Radial fields in the «edge» can even exceed the longitudinal field Ez (further in the «core» the radial field Er quickly reduces to zero). But the outer region contains relatively few electrons [8] and, due to the large total field, the fraction of «hot» electrons with big probability of energy-consuming inelastic collisions is significant. This means that XW (We) for
such a fraction is small (see Fig. 1), and «hot» electrons do not penetrate into the region of the «core». Slow electrons can penetrate from the «edge» into the «core» but due to the small concentration of electrons in the «edge», they cannot seriously effect EEDF in the «core».
The radial field in the «core» is small [6, 8], and the total electrical field practically is equal to the longitudinal field Ez which is uniform, and hence there is no difference between local und nonlocal EEDF.
There is one more aspect which should be considered. Together with electron energy relaxation in isotropic Brownian motion, there is a radial drift of electrons to the discharge center in radial field Er. If electrons during this drift have collisions with small energy losses only, they can receive an additional kinetic energy from the radial field.
But the slowest electrons with 0 < We < 0.5 eV cannot gain energy: XW((e) for such electrons is small (see Fig. 1, b, curve 1) because the attachment cross-section is large in this energy range [16], and in the attachment acts electrons disappear as free particles.
The said receive of kinetic energy from radial field is mostly relevant to the group of electrons with 0.5 < We < 2 eV where inelastic cross-sections are small and XW(We) is comparable with 8 . But let us notice that it is enough
Fig. 2. Radial field potential (a) and the value of Ar (b), limited above by XW = 0.2 cm (see Fig. 1); discharge conditions are the same as in Fig. 1; vertical line shows the boundary between i-i- and e-z'-plasmas
a) b)
*io16 8«e / 8f ion., cm 3s 1 -tgis 8ne / ht att., cm 3s 1
Fig. 3. Spatial distributions of ionization NXene {^¡v^ (a) and attachment Ncl ne (pave) (b) rates in local (1) and nonlocal (2) approach; the discharge conditions are the same as in Fig. 1
for these electrons to gain 2 eV, and their energy relaxation length becomes vanishingly small (see Fig. 1, b, curve 1).
Hence, the real energy relaxation length of electrons with initial energy 0.5 < We < 2 eV can be defined as the piece of radial coordinate Ar along which the potential of radial field changes by AU - 2 V (Fig. 2, a); more correctly, this length will be XW-(We) = min[Ar, XW (We)] where XW (We) can be estimated according to (3). In Fig. 2, there is an illustration to Ar computing and the result of such computing: Ar(r) - 2 / Er (r), where Er is in V/cm.
It is seen that in the «edge» (e-i plasma), where nearly all the radial potential fall is concentrated, the values of Ar are vanish-ingly small, and we can hence accept the EEDF in the «edge» as local. Only at the boundary between e-i- and i-i-plasmas and inside the «core» (where Er is close to zero), XwWe) - XW(We), and EEDF in the energy range 0.5 < We < 2 eV is nonlocal. But this group of electrons practically takes no part in plasma interaction processes due to smallness of all collision cross-sections for electrons of said energy range.
Assuming the afore-mentioned, we can suppose that the effect of EEDF nonlocality on properties of our discharges should be not significant. Model calculations [8] confirm this
assumption. As an illustration, the results of computing of rates of some plama-chemical processes both in local and nonlocal approach are presented in Fig. 3.
For the mixture of 6 Torr Xe and 0.25 Torr Cl2, the biggest difference among all collision rates was observed in the ionization rate NXene (ajVe), the residual for local and nonlocal approaches was of about 1.6 % with the main deviation taking place just near the tube wall (not more than 0.5 mm from the wall). The reason for such a weak reduction of ion-ization rate in the nonlocal approach lies in the small relaxation length (< 0.1 mm). As a result, the wall losses of «hot» electrons, which are not only able to produce ionization but also can penetrate through the wall potential barrier and die on the wall, are small.
For the attachment rate NCl n (a v) — the
Cl2 e \ a e /
most important process for EN discharges — where the process is maintained mainly by slow electrons [16], the residual turned out to be even smaller — about 1 %.
These differences have almost no effect on the spatial distributions of charged particles and excimer molecules densities (the concentration of residuals was less than 1%).
The residuals for the mixtures with higher gas pressure or with higher percentage of chlorine are smaller because XW (We) decreases
proportionally both to total pressure and to amount of molecular chlorine. For the mixture of 18 Torr Xe and 0.7 Torr Cl2, the residuals for rates of all the processes turned out to be no more than 0.4 %.
in a mixture of Xe and Cl2 with total pressure not less than 6 Torr and with amount of chlorine not less than 1/25, the effect of EEDF nonlocality on properties of discharges is insufficient, and the local approach can be
Hence, we can state that for gas discharges used in modeling such discharges.
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9. Tsendin L.D. Electron kinetics in non-uniform glow discharge plasmas. Plasma Sources Sci. Technol. 1995, V. 4, pp. 200—211.
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ГОЛОВИЦКИЙ Александр Петрович — доктор физико-математических наук, доцент кафедры физической электроники Санкт-Петербургского государственного политехнического университета. 195251, Россия, г. Санкт-Петербург, Политехническая ул., 29 [email protected]
© St. Petersburg State Polytechnical University, 2013
