Boltzmann Equation with Time-Varying Fields

Abstract

This chapter analyses the effects produced by a time-varying electric field in the electron kinetics. The behaviour exhibited by the electron velocity distribution function is controlled by two characteristic relaxation frequencies, one for energy and another for momentum transfer, when compared with the field frequency. The cases of high-frequency (HF) and radio-frequency (RF) fields are analysed separately, since they correspond to situations in which no time-modulation and large time-modulation exist, respectively, in the isotropic part of the electron velocity distribution. This chapter also analyses the electron kinetics under the simultaneous effects of a HF electric field and a stationary external magnetic field, with leads to electron cyclotron resonance (ECR) when the electron cyclotron frequency equals the field-frequency.

This chapter analyses the effects produced by a time-varying electric field in the electron kinetics. The behaviour exhibited by the electron velocity distribution function is controlled by two characteristic relaxation frequencies, one for energy and another for momentum transfer, when compared with the field frequency. The cases of high-frequency (HF) and radio-frequency (RF) fields are analysed separately, since they correspond to situations in which no time-modulation and large time-modulation exist, respectively, in the isotropic part of the electron velocity distribution. This chapter also analyses the electron kinetics under the simultaneous effects of a HF electric field and a stationary external magnetic field, with leads to electron cyclotron resonance (ECR) when the electron cyclotron frequency equals the field-frequency.

1 High-Frequency Electric Fields

1.1 Characteristic Frequencies

The situation to be analysed here is that of electron kinetics when a uniform time-varying electric field $E(t) = E_{0}\;\cos (\omega t)$ is applied to a plasma. A qualitative analysis of the time-dependence of the isotropic and the anisotropic components of the electron velocity distribution function under the effects of time-varying fields can be realised using the characteristic relaxation frequencies for energy and momentum transfer, ν _e and ν _m ^e respectively, previously introduced in Chap. 3 through equations (3.142) and (3.139)

$$\displaystyle\begin{array}{rcl} & & \nu _{e}(u)\; =\; \frac{2m} {M} \;\nu _{m}(u)\; +\;\sum _{j}\nu _{j}(u){}\end{array}$$

(4.1)

$$\displaystyle\begin{array}{rcl} & & \nu _{m}^{e}(u)\; =\;\nu _{ m}(u)\; +\;\sum _{j}\nu _{j}^{m}(u),{}\end{array}$$

(4.2)

where ν _m and $\sum _{j}\nu _{j}^{m}$ are the elastic and the inelastic electron collision frequencies for momentum transfer, the latter assumed here with transitions from the ground-state only, and in which the frequencies ν _j ^m equal the total frequencies ν _j in the case of isotropic scattering (see discussion in Sect. 3.1.2). Further, m∕M is the electron-molecule mass ratio and $u = \frac{1} {2}\;mv_{e}^{\;2}$ is the electron energy. The factor 2 in the first term of equation (4.1) holds as the Chapman and Cowling expression (3.84) is written in terms of the electron energy.

As we will show below, it is possible to distinguish the following situations in this analysis (see e.g. Loureiro 1993):

• (i) At low field frequencies such that ω ≪ ν _e , that is at radio-frequency (RF) fields, the isotropic component of the electron velocity distribution, i.e. the electron energy distribution function (EEDF), follows in a quasistationary way the RF field, presenting consequently a very large time-modulation in those parts of the relevant range of the electron energy where the above inequality is satisfied. Therefore, in this low-frequency limit, the EEDF can be obtained by solving the Boltzmann equation for a direct-current (DC) field for each time-varying value of the instantaneous RF field strength;

• (ii) When the field frequency increases up to $\nu _{e} \simeq \omega \ll \nu _{m}^{e}$, in most parts of the relevant electron-energy range, the time-modulation of the EEDF is significantly reduced and a time-resolved solution of the Boltzmann equation is required instead of a quasi-stationary one. However, in this range of ω values, since the inequality ω ≪ ν _m ^e holds in most of the significant electron-energy range, the anisotropic component of the electron velocity distribution is not significantly modified by the field frequency and, consequently, the magnitude of the EEDF is only slightly dependent on the field frequency;

• (iii) Finally, for higher values of ω, such that $\nu _{e}\ll \nu _{m}^{e} \simeq \omega$, the inequality ν _e  ≪ ω determines that no time-modulation of the EEDF can occur. However, the proximity of frequencies $\omega \simeq \nu _{m}^{e}$ produces a time-delay of the anisotropic component of the velocity distribution relatively to the field, which reaches $\phi = -\pi /2$ when ω ≫ ν _m ^e, producing as we will show below a strong reduction in the magnitude of the EEDF.

It follows from the present discussion that we can expect an important time-modulation of the EEDF for angular field frequencies ω < ν _e , in the whole significant electron-energy range. In an atomic gas, at field frequencies not too low (ω > 10⁸ s⁻¹) and at gas pressures typical of RF plasma processing (typically $p \sim 100$ Pa), the inequality ω < ν _e holds only in the high-energy tail of the EEDF, so that there is no time-modulation on the body of the distribution. We note that the characteristic frequency for energy transfer ν _e is a monotonously increasing function of the electron energy. Figure 4.1 shows the ratios ν _e ∕n _o and ν _m ^e∕n _o for argon as a function of the electron energy (Sá et al. 1994). The ratio ν _m ^e∕n _o largely exceeds ν _e ∕n _o in the energy range 0–40 eV, which is a consequence of the important contribution of elastic collisions to the total (elastic + inelastic) electron cross section for momentum transfer in Ar. On the other hand, $\nu _{e}(u) \simeq \nu _{m}^{e}(u)$ when $u \rightarrow \infty$, mainly due to the predominance of the ionization cross section in equations (4.1) and (4.2).

Fig. 4.1

Ratios of the characteristic relaxation frequencies to the gas number density for electron-neutral energy transfer, ν _e ∕n _o , and momentum transfer, ν _m ^e∕n _o , as a function of the electron energy, in argon (Sá et al. 1994)

However, this is no longer true if we deal with a molecular gas, where the characteristic frequency for energy transfer ν _e presents important values even at electron energies as low as a few eV (e.g. in N₂, at $u \simeq 2$ eV), as a result of the dissipation of electron energy in vibrational excitation. Here, the time-modulation of the body of the EEDF also occurs. Figure 4.2 shows the ratios of the characteristic relaxation frequencies to the gas number density ν _e ∕n _o and ν _m ^e∕n _o , as a function of the electron energy, in N₂ and H₂, in the case of absence of appreciable vibrational excitation, i.e. for T _v  = T _o , in order the effects of superelastic electron-vibration (e-V) collisions do not need to be taken into account (Loureiro 1993). The ratio ν _e ∕n _o presents a sharp maximum in N₂ at about 2 eV due to vibrational excitation and a monotonic growth at higher energies in accordance with the cross sections shown in Fig. 3.13 for this gas. On the contrary, in H₂ only a smooth maximum exists associated with vibrational excitation (see Fig. 3.11).

Fig. 4.2

Ratios of the characteristic relaxation frequencies to the gas number density for electron-neutral energy transfer, ν _e ∕n _o , and momentum transfer, ν _m ^e∕n _o , as a function of the electron energy, in N₂ (full curves) and in H₂ (broken curves) (Loureiro 1993)

In molecular gases the characteristic relaxation frequencies ν _e and ν _m ^e must include the contributions due to excitation of rotational and vibrational levels. Thus, even in the case of vanishingly small vibrational excitation, both frequencies (4.1) and (4.2) must be corrected to include inelastic and superelastic rotational exchanges (see Sect. 3.3.3) and inelastic vibrational exchanges, by writing (see also Makabe and Goto 1988 and Goto and Makabe 1990)

$$\displaystyle\begin{array}{rcl} & & \nu _{e}(u)\; =\; \frac{2m} {M} \;\nu _{m}(u)\; +\; 8B_{0}\;\frac{d\nu _{0}(u)} {du} \; +\;\sum _{v}\nu _{v}(u)\; +\;\sum _{j}\nu _{j}(u){}\end{array}$$

(4.3)

$$\displaystyle\begin{array}{rcl} & & \nu _{m}^{e}(u)\; =\;\nu _{ m}(u)\; +\; 2\;\nu _{0}(u)\; +\;\sum _{v}\nu _{v}(u)\; +\;\sum _{j}\nu _{j}(u)\;,{}\end{array}$$

(4.4)

with $\nu _{0}(u) = n_{o}\sigma _{0}\sqrt{2u/m}$, and where it is assumed $\nu _{J,J+2} +\nu _{J,J-2} \simeq 2\nu _{0}$ and $(u_{J,J+2} - u_{J,J-2})/2 = 4B_{0}$, with J denoting the rotational quantum number. On the other hand, ν _v is the collision frequency for vibrational excitation of v-th levels from v = 0, within the electronic ground state N₂(X $^{1}\Sigma _{g}^{+}$) or H₂(X $^{1}\Sigma _{g}^{+}$).

1.2 Power Absorbed at High-Frequency Fields

Let us consider now in equation (3.35) a time-varying electric field of frequency ω and amplitude E ₀, directed along the negative z axis: $\mathbf{E}(t) = -\;E(t)\;\mathbf{e_{z}}$. Assuming the only anisotropies present on the electron velocity distribution are those caused by the field and that they are sufficiently weak for a two-term expansion suffices (3.11), we may write from (3.41) and (3.42) the following equations for the evolution of the isotropic and the anisotropic components of the electron velocity distribution function

$$\displaystyle\begin{array}{rcl} & & \frac{\partial f_{e}^{0}} {\partial t} \; +\; \frac{eE(t)} {m} \; \frac{1} {3v_{e}^{\;2}}\; \frac{\partial } {\partial v_{e}}\left (v_{e}^{\;2}\;f_{ e}^{1}\right ) =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0}){}\end{array}$$

(4.5)

$$\displaystyle\begin{array}{rcl} & & \qquad \frac{\partial f_{e}^{1}} {\partial t} \; +\; \frac{eE(t)} {m} \;\frac{\partial f_{e}^{0}} {\partial v_{e}} \; =\; -\;\nu _{m}^{e}\;f_{ e}^{1},{}\end{array}$$

(4.6)

where the elastic and the inelastic collision terms in the equation for f _e ⁰ are given by equations (3.85) and (3.134), respectively, and the effective collision frequency for momentum transfer is given by (3.139).

In principle, f _e ⁰ and f _e ¹ are now functions of v _e and t, so that we may expand them in Fourier series in ω t

$$\displaystyle{ f_{e}^{l}(v_{ e},t)\; =\;\sum _{ k=0}^{\infty }\mbox{ Re}\{\overline{f_{ e}^{l}}_{,k}(v_{e})\;e^{jk\omega t}\}, }$$

(4.7)

where Re{ } means “the real part of”. The situation to be considered here firstly is that of a stationary plasma created by a high-frequency (HF) field. Thus, if the field frequency is significantly larger than the characteristic frequency for energy transfer, the isotropic component of the velocity distribution does not change appreciably during a cycle of the field oscillation and hence $\partial f_{e}^{0}/\partial t = 0$ in equation (4.5), while in equation (4.6) we have a dependence in ω t through the field. We may write then

$$\displaystyle{ f_{e}^{1}(v_{ e},t)\; =\; \mbox{ Re}\{\overline{f_{e}^{1}}\;e^{j\omega t}\} }$$

(4.8)

and to obtain from (4.6)

$$\displaystyle{ \overline{f_{e}^{1}}(v_{e})\; =\; -\; \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{eE_{0}} {m} \;\frac{df_{e}^{0}} {dv_{e}} \;. }$$

(4.9)

The drift or mean vector velocity v _ed = <​​ v _e ​​ > given by equation (3.18) also oscillates with the frequency ω and takes the form

$$\displaystyle\begin{array}{rcl} \mathbf{v_{ed}}(t)& =& \frac{1} {n_{e}}\int _{0}^{\infty }\frac{v_{e}} {3} \;\;\mathbf{f_{e}^{1}}(v_{ e},t)\;4\pi v_{e}^{\;2}\;dv_{ e} \\ & =& \mbox{ Re}\{\overline{V _{e}}_{d}\;e^{j\omega t}\}\;\mathbf{e_{ z}}\;, {}\end{array}$$

(4.10)

being the complex amplitude $\overline{V _{e}}_{d}$ after the substitution of (4.9) given by

$$\displaystyle{ \overline{V _{e}}_{d}\; =\; -\;\frac{eE_{0}} {n_{e}m}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}. }$$

(4.11)

Since v _ed = Re$\{\overline{\mu _{e}}\;E_{0}\;e^{j\omega t}\}\;\mathbf{e_{z}}$, we obtain the complex electron mobility

$$\displaystyle{ \overline{\mu _{e}}\; =\; -\; \frac{e} {n_{e}m}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;. }$$

(4.12)

On the other hand, the power absorbed by the electrons from the field, per volume unit, is

$$\displaystyle{ P_{E}(t)\; =\; (\mathbf{J_{e}}\;.\;\mathbf{E})\; =\; \mbox{ Re}\{\overline{\sigma _{c}}_{e}\;E_{0}\;e^{j\omega t}\}\;\mbox{ Re}\{E_{ 0}\;e^{j\omega t}\}\;, }$$

(4.13)

being $\overline{\sigma _{c}}_{e} = en_{e}\overline{\mu _{e}}$ the complex conductivity. Making the product of complex quantities, we obtain

$$\displaystyle{ P_{E}(t)\; =\; \frac{1} {2}\;\mbox{ Re}\{\overline{\sigma _{c}}_{e}\}\;E_{0}^{\;2}\; +\; \frac{1} {2}\;\mbox{ Re}\{\overline{\sigma _{c}}_{e}\;e^{j2\omega t}\}\;E_{ 0}^{\;2}\;, }$$

(4.14)

and because the second term vanishes as the time-average is taken, the time-averaged absorbed power is

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \mbox{ Re}\{\overline{\sigma _{c}}_{e}\}\;E_{rms}^{\;2}\;, }$$

(4.15)

with $E_{rms} = E_{0}/\sqrt{2}$ denoting the root mean square field and

$$\displaystyle{ \mbox{ Re}\{\overline{\sigma _{c}}_{e}\}\; =\; -\;\frac{e^{2}} {m}\int _{0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} }$$

(4.16)

the real part of the complex electron conductivity. The mean power absorbed per electron in eV s⁻¹, $\overline{P_{E}(t)}/n_{e}$, is usually represented in literature by $\Theta$ (see e.g. Ferreira and Loureiro 1984, 1989).

The time-averaged absorbed power rapidly decreases as ω increases beyond ν _m ^e, as a result of a progressive shift delay between the electron current density and the field which reaches $-\pi /2$ as ω ≫ ν _m ^e. In this limit of extremely high frequencies, we obtain from equations (4.10) and (4.11)

$$\displaystyle{ \mathbf{v_{ed}}(t)\; =\; \frac{eE_{0}} {m\omega } \;\cos (\omega t -\pi /2)\;\mathbf{e_{z}}\;, }$$

(4.17)

and consequently

$$\displaystyle{ P_{E}(t)\; =\; \frac{e^{2}n_{e}} {2m\omega } \;E_{0}^{\;2}\sin (2\omega t)\;, }$$

(4.18)

with $\overline{P_{E}(t)} = 0$. As a matter of fact, they are the collisions that are responsible for the energy transfer from the field to the electron motion so that as they exist the phase shift progressively departs from π∕2 and $\overline{P_{E}(t)}$ becomes non-null.

On the other hand, when the effective collision frequency for momentum transfer is independent of the electron energy, ν _m ^e(v _e ) = const, we obtain from equations (4.15) and (4.16) a similar expression to that of a DC field

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m\nu _{m}^{e}}\;E_{e}^{\;2}, }$$

(4.19)

having defined the effective electric field E _e given by

$$\displaystyle{ E_{e}\; =\; \frac{1} {\sqrt{1 + (\omega /\nu _{m }^{e })^{2}}}\;E_{rms}, }$$

(4.20)

which represents the equivalent field magnitude capable to produce the same energy absorption as a DC field. This effective field is E _e  = E _rms , for ω ≪ ν _m ^e, and $E_{e} = (\nu _{m}^{e}/\omega )\;E_{rms}$, for ω ≫ ν _m ^e, so that in this latter case we have $E_{e} \rightarrow 0$ as $\omega \rightarrow \infty$, due to the above mentioned phase shift delay of π∕2 between the electron current density and the HF field.

1.3 Stationary Electron Energy Distribution Functions

As the field frequency is sufficiently high so that the isotropic component of the electron velocity distribution remains time constant, we may assume $\partial f_{e}^{0}/\partial t = 0$ in equation (4.5) and write

$$\displaystyle{ -\;\frac{e} {m}\; \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{v_{e}^{\;2}} {2} \;\mbox{ Re}\{(\mathbf{f_{e}^{1}}\;.\;\mathbf{E^{{\ast}}})\}\right ) =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;, }$$

(4.21)

with E ^∗ denoting the complex conjugate of the electric field. Inserting now equation (4.9) in (4.21) and making the product of complexes quantities, we easily obtain

$$\displaystyle{ -\;\frac{1} {2}\left (\frac{eE_{0}} {m} \right )^{2} \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left ( \frac{v_{e}^{\;2}} {3\nu _{m}^{e}}\; \frac{1} {1 + (\omega /\nu _{m}^{e})^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \right ) =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;. }$$

(4.22)

From equations (4.19) and (4.20) we see that when ν _m ^e(v _e ) depends on the electron velocity, $\overline{P_{E}(v_{e},t)}/n_{e}$ may be seen as representing the time-averaged energy transferred from the field to an electron of velocity v _e , while $u_{c}(v_{e}) = \overline{P_{E}(v_{e},t)}/(n_{e}\nu _{m}^{e})$ is the energy transferred per collision. Comparing u _c with the term under brackets in (4.22), this equation may be written under the form

$$\displaystyle{ -\; \frac{1} {3m} \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (v_{e}^{\;2}\;\nu _{ m}^{e}\;u_{ c}\;\frac{df_{e}^{0}} {dv_{e}} \right ) =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;, }$$

(4.23)

or still under the following more explicit form using (3.85), (3.134), and (3.158) and multiplying both members by $\sqrt{u}$

$$\displaystyle\begin{array}{rcl} & -& \frac{d} {du}\left [\frac{2} {3}\;u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\;\frac{df_{e}^{0}} {du} \; +\; \frac{2m} {M} \;u^{3/2}\;\nu _{ m}\left (f_{e}^{0} + k_{ B}T_{o}\;\frac{df_{e}^{o}} {du} \right ) + 4B_{0}\;\nu _{0}\;\sqrt{u}\;f_{e}^{0}\right ] \\ & =& \sum _{i,j}\left \{\sqrt{u + u_{ij}}\;\;\nu _{ij}(u + u_{ij})\;f_{e}^{0}(u + u_{ ij}) -\;\sqrt{u}\;\;\nu _{ij}(u)\;f_{e}^{0}(u)\right \} \\ & & +\;\sum _{j,i}\left \{\sqrt{u - u_{ij}}\;\;\nu _{ji}(u - u_{ij})\;f_{e}^{0}(u - u_{ ij}) -\;\sqrt{u}\;\;\nu _{ji}(u)\;f_{e}^{0}(u)\right \}\;. {}\end{array}$$

(4.24)

This equation can now be written in terms of the electron energy distribution function (EEDF), f(u), normalized through condition (3.166). Both equations are identical since the two distributions are related one another through a constant factor.

Inspection of equation (4.24) reveals that the dependence of the EEDF on the parameters E ₀, ω, and n _o arises through the time-averaged energy gain per collision

$$\displaystyle{ u_{c}(u)\; =\; \frac{e^{2}E_{0}^{\;2}} {2m\;(\nu _{m}^{e}(u)^{2} +\omega ^{2})}\;, }$$

(4.25)

with $\nu _{m}^{e}(u)/n_{o} = \sqrt{2u/m}\;\sigma _{m}^{e}(u)$, so that we can express all results in terms of the two independent reduced parameters E ₀∕n _o and ω∕n _o as follows

$$\displaystyle{ u_{c}(u)\; =\; \frac{e^{2}} {2m}\; \frac{(E_{0}/n_{o})^{2}} {(\nu _{m}^{e}(u)/n_{o})^{2} + (\omega /n_{o})^{2}}\;. }$$

(4.26)

When ω ≫ ν _m ^e throughout the significant electron energy range, we have

$$\displaystyle{ u_{c}\; \sim \; \frac{e^{2}} {2m}\left (\frac{E_{0}} {\omega } \right )^{2} }$$

(4.27)

and the EEDF is function of E ₀∕ω only.

Since both members of equation (4.24) can be divided by the gas number density n _o , the other independent variables are the fractional population concentrations $\delta _{i} = n_{i}/n_{o}$ and $\delta _{j} = n_{j}/n_{o}$ in the inelastic and superelastic collision terms, respectively, and the gas temperature T _o due the small heating of electrons in collisions with non-frozen molecules (3.85). Generally, the populations of excited states are important only for the vibrational levels (the effects of rotational levels are already included through a continuous approximation, see Sect. 3.3.3) and the effects of v-th levels can be taken into consideration through a given vibrational distribution function (VDF) characterized by a vibrational temperature T _v . The determination of the VDFs is beyond the scope of this book, but the reader is invited to search this subject in Loureiro and Ferreira (1989, 1986) for H₂ and N₂ cases, respectively. In those papers it was shown that the coupling between the EEDF and VDF needs to be taken into consideration for a self-consistent determination of both distributions.

Figure 4.3a, b show the calculated EEDFs in argon obtained in Ferreira and Loureiro (1983) for various combinations of the reduced parameters E ₀∕n _o and ω∕n _o . The full curves in Fig. 4.3a are for constant $\omega /n_{o} = 4 \times 10^{-13}$ m³ s⁻¹ and various values of E _rms ∕n _o , with $E_{rms} = E_{0}/\sqrt{2}$, ranging from 15 to 100 Td ($1\,\mathrm{Td} = 1 \times 10^{-21}$ V m²). These distributions are remarkably different in shape from those obtained in a DC field at the same E∕n _o values, which are also shown in Fig. 4.3a for comparison (broken curves). The distributions shown in Fig. 4.3b are for constant $E_{rms}/n_{o} = 65$ Td and various values of ω∕n _o ranging from zero (DC case) to $\omega /n_{o} = 1 \times 10^{-11}$ m³ s⁻¹. From Fig. 4.3b it is seen that the EEDFs are strongly depleted for $\omega /n_{o} > 5 \times 10^{-12}$ m³ s⁻¹, because beyond this value we have ω ≫ ν _m ^e in the whole energy range (see Fig. 4.1) and, consequently, the absorption of energy is significantly reduced due to the electron current density becomes $\sim \pi /2$ out of phase to the field. It is also assumed here T _o  = 300 K for the gas temperature, so that the small heating of the electrons due to collisions with non-frozen atoms is negligible.

Fig. 4.3

Electron energy distribution functions in argon. (a ) Full curves: ω∕n _o = 4×10⁻¹³ m³ s⁻¹; Broken curves: DC case. The different curves are for the following values of E _rms ∕n _o (full) or E∕n _o (broken) in Td: (A) 100; (B) 65; (C) 30; (D) 15. (b ) $E_{rms}/n_{o} = 65$ Td and the following values of ω∕n _o in 10⁻¹² m³ s⁻¹: (A) DC case; (B) 0.4; (C) 3; (D) 5; (E) 7.5; (F) 10 (Ferreira and Loureiro 1983)

Moreover, it is also seen in Fig. 4.3b that the EEDFs are Maxwellian at the higher field frequencies $\omega /n_{o} > 5 \times 10^{-12}$ m³ s⁻¹ (Margenau 1946). This behaviour results from the fact that in Ar the EEDFs are strongly depleted at these ω∕n _o values and, consequently, the electrons do not gain enough energy to produce an inelastic collision. The collisions are of elastic type only and the effective collision frequency for momentum transfer is equal to the elastic collision frequency, ν _m ^e ≡ ν _m . Then equation (4.24) reduces to

$$\displaystyle{ \frac{d} {du}\left \{u^{3/2}\;\nu _{ m}\left [\left ( \frac{e^{2}} {3m}\left (\frac{E_{0}} {\omega } \right )^{2} + \frac{2m} {M} \;k_{B}T_{o}\right )\frac{df_{e}^{0}} {du} \; +\; \frac{2m} {M} \;f_{e}^{0}\right ]\right \}\; =\; 0, }$$

(4.28)

whose solution is a Maxwellian distribution at temperature

$$\displaystyle{ T_{e}\; = T_{o}\; +\; \frac{M} {6k_{B}}\left (\frac{eE_{0}} {m\omega } \right )^{2}. }$$

(4.29)

For $E_{0}/\omega = \sqrt{2} \times 10^{-8}$ V m⁻¹ s and T _o  = 300 K in argon, we obtain k _B T _e  = 0. 43 eV.

As seen in Fig. 4.3a the EEDFs in argon at high field frequencies sharply increase near the origin of low electron energies. They are also depleted relatively to DC case at intermediate energies and have larger slopes at high energies. This behaviour can be well understood through Fig. 4.4, in which the EEDFs having the same mean energy < ​​ u​​ > = 3. 5 eV are plotted for various values of ω∕n _o . We can see that the relative number of high-energy electrons increases with ω∕n _o , although < ​​ u​​ > is kept unchanged. This is a direct consequence of the increasing efficiency of the power transfer from the field to the electrons of energy well above the average. Indeed, the power transfer per electron u _c (u) ν _m ^e(u) (with u _c (u) given by equation (4.25)) is maximum when ω = ν _m ^e(u) and, in argon, the effective collision frequency for momentum transfer is an increasing function of energy, up to $u \sim 12$ eV (see Fig. 4.1). Therefore, as ω increases the electron energy u ^∗ for which the transfer of energy is maximum, ν _m ^e(u ^∗) = ω, also increases, which produces the strong enhancement of the high-energy tail of the EEDFs shown in Fig. 4.4 (Winkler et al. 1984; Ferreira et al. 1987; Karoulina and Lebedev 1988).

Fig. 4.4

Electron energy distribution functions in argon with the same mean energy of 3.5 eV for the following cases: (A) DC; (B) $\omega /n_{o} = 1 \times 10^{-13}$ m³ s⁻¹;(C) $\omega /n_{o} = 1.6 \times 10^{-13}$ m³ s⁻¹; (D) $\omega \rightarrow \infty$ (Ferreira et al. 1987)

The percentage electron energy losses by elastic, excitation, and ionization collisions are shown in Fig. 4.5 for argon, as a function of the time-averaged input power per electron at unit gas density, $\overline{P_{E}(t)}/(n_{e}n_{o})$, for the high-frequency (HF) limit (ω ≫ ν _m ^e(u) in the whole significant electron energy range) and for the stationary (DC) limit (Ferreira and Loureiro 1984). It may be concluded that, for similar $\overline{P_{E}(t)}/(n_{e}n_{o})$ values, the input energy transferred to ionization increases with increasing ω, i.e. when one goes from the DC case to HF. This is explained by an enhancement of the high-energy tail of the EEDF. The power transfer has a maximum at ω = ν _m ^e(u) and in argon ν _m ^e(u) increases with u in the range 0. 2 < u < 12 eV and becomes approximately constant at energies u > 12 eV, which corresponds to the high-energy tail of the distribution. Thus, as ω increases, the power transfer to the electrons in the tail of the distribution also increases. Besides elastic collisions and ionization, the excitation channels shown in Fig. 4.5 correspond to the cross sections represented in Fig. 3.10 as follows: conjoint excitation of the metastable states³P₂ +³P₀; resonant state³P₁; resonant state¹P₁; forbidden states 3p⁵4p with the threshold energy of 12.9 eV; and higher-lying optically allowed states with the threshold energy of 14.0 eV.

Fig. 4.5

Percentage electron energy losses in argon as a function of the average input power per electron at unit gas density, $\overline{P_{E}(t)}/(n_{e}n_{o})$, in HF (full curves) and DC (broken curves). The labels of the curves correspond to the following energy loss channels (see Fig. 3.9): (E) elastic; (M) metastable states³P₂ +³P₀; (P) resonant state³P₁; (R) resonant¹P₁; (F) forbidden states; (H) higher-lying allowed states; (I) ionization (Ferreira and Loureiro 1984)

Figure 4.6a, b show the EEDFs calculated in nitrogen for $E_{rms}/n_{o} = 100$ Td, various values of ω∕n _o , including the DC case, and for the cases of appreciable vibrational excitation of N₂(X $^{1}\Sigma _{g}^{+},v$) levels (figure a) and null vibrational excitation (figure b). As ω∕n _o increases up to 4 × 10⁻¹³ m³ s⁻¹ the relative number of high-energy electrons decreases due to small energy absorption, as a result of the increase of the phase shift between the electron current density and the electric field. However, the new effect shown here that did not exist in argon is the strong enhancement of the high-energy tail of the EEDF as the electron superelastic collisions are also taken into account. More precisely, the electron superelastic collisions with vibrationally excited molecules N₂(X $^{1}\Sigma _{g}^{+},v > 0$), i.e. the effect of the so-called electron-vibration (e−V) superelastic processes. Due to the strong peak at $u \sim 2$ eV in the electron cross section for vibrational excitation (see Fig. 3.13), the EEDFs exhibit a sharp decrease at this value of energy when the electron superelastic collisions are not taken into account, because even in DC case the E∕n _o values are not high enough in order an appreciable amount of electrons may cross over the energy barrier at 2 eV (figure b). The situation dramatically changes as the e-V superelastic collisions are included (figure a) being now the high-energy portion of the EEDF populated with electrons that receive energy from the de-excitation of v-th levels. Figure 4.6a is for T _v  = 4000 K and T _o  = 400 K, while Fig. 4.6b is for $T_{v} = T_{o} = 400$ K (see Loureiro and Ferreira 1986 for DC and Ferreira and Loureiro 1989 for HF cases).

Fig. 4.6

Electron energy distribution functions in N₂ for $E_{rms}/n_{o} = 100$ Td and the following values of ω∕n _o in 10⁻¹³ m³ s⁻¹: (A) DC case; (B) 1; (C) 2; (D) 4. Figure (a ): T _v  = 4000 K and T _o  = 400 K. Figure (b ): $T_{v} = T_{o} = 400$ K (Ferreira and Loureiro 1989)

Figure 4.7 shows the percentage electron energy losses in nitrogen through vibrational excitation and electronic excitation plus ionization, as a function of the time-averaged absorbed power per electron at unity gas density, $\overline{P_{E}(t)}/(n_{e}n_{o})$. An increase in ω∕n _o or in T _v results in a decrease of the power transferred to the vibrational mode and, consequently, in an increase of the power transferred to electronic excitation plus ionization. Both effects are consequence of the enhancement of the high-energy tail of the EEDF which occurs with increasing either ω∕n _o or T _v . The effects of the changes in ω∕n _o are qualitatively the same in nitrogen as those found before in Fig. 4.5 for argon.

Fig. 4.7

Fractional power transfer into vibrational excitation and electronic excitation plus ionization in N₂, as a function of the mean power absorbed per electron at unit gas density, for (A) DC and (B) $\omega /n_{o} = 4 \times 10^{-13}$ m³ s⁻¹, and for T _v  = 4000 K (full curves) and $T_{v} = T_{o} = 400$ K (broken curves) (Ferreira et al. 1987)

Finally, Fig. 4.8 shows the percentage electron energy losses into excitation of the most important triplet states of N₂ (A $^{3}\Sigma _{u}^{+}$, B $^{3}\Pi _{g}$, C $^{3}\Pi _{u}$) and ionization, for T _v  = 4000 K, and for the cases DC and $\omega /n_{o} = 4 \times 10^{-13}$ m³ s⁻¹. Due to the enhancement of the high-energy tail of the EEDF in HF, the excitation of the triplet states of N₂ is obtained at lower $\overline{P_{E}(t)}/n_{e}n_{o}$ values than in DC case (although as we have seen before the E ₀∕n _o values are much larger).

Fig. 4.8

Fractional power transfer into excitation of the triplet states in N₂: (A) A $^{3}\Sigma _{u}^{+}$; (B) B $^{3}\Pi _{g}$; (C) C $^{3}\Pi _{u}$; (D) ionization, as a function of the mean power absorbed per electron at unit gas density, for T _v  = 4000 K, and for DC (full curves) and $\omega /n_{o} = 4 \times 10^{-13}$ m³ s⁻¹ (broken curves) (Ferreira and Loureiro 1989)

2 Electron Cyclotron Resonance

2.1 Hydrodynamic Description

As seen in Sect. 4.1.2, when the electrons of a plasma are submitted to a HF electric field they oscillate at the same frequency of the field. However, in the collisionless limit ω ≫ ν _m ^e, they oscillate with a phase shift of π∕2 and the average energy acquired over a field period by the electrons is zero. They are the collisions that temporarily interrupt the movement of the electrons and make the phase shift deviates from π∕2. The electrons may absorb then energy from the field.

Due to this particular aspect of HF discharges, the electrons in HF fields may absorb significantly less energy than in DC fields. So, to avoid this inconvenience, the superimposing of a static magnetic field on the HF system is of great interest (Margot et al. 1992). Besides the fact that a discharge with this configuration significantly reduces the electron losses to the wall, the electrons when submitted to a static B field rotate circularly around its field lines with an angular electron cyclotron frequency $\omega _{ce} = eB/m$. If a HF wave with frequency ω also exists, with the electric field perpendicular to the direction of B field, rotating in the same direction, the electrons in their reference system see a DC field when both frequencies are adjusted, ω _ce  = ω. The electrons are then continuously accelerated until their movement is interrupted by a collision.

As the electron frequency for momentum transfer is independent of velocity, ν _m ^e(v _e ) = const, the electron movement is described by the hydrodynamic equation for local momentum conservation (see equation (3.60) in Sect. 3.1.5)

$$\displaystyle{ m\;\frac{\mathbf{dv_{ed}}} {dt} \; =\; -\;e\;\mathbf{E}\; -\; e\;[\mathbf{v_{ed}} \times \mathbf{B}]\; -\; m\;\nu _{m}^{e}\;\mathbf{v_{ ed}}\;. }$$

(4.30)

Here, v _ed = < ​​v _e​​ > is the electron drift or average vector velocity, E = E ₀ e ^jωt is the HF electric field with amplitude E ₀ and frequency ω, and B = B e _z is the static magnetic field oriented along the z axis. Since $\mathbf{v_{ed}} \propto e^{j\omega t}$, we may still write using complex quantities

$$\displaystyle{ (\nu _{m}^{e}\; +\; j\omega )\;\mathbf{v_{ ed}}\; =\; -\;\frac{e} {m}\;\mathbf{E}\; +\;\omega _{ce}\;[\mathbf{e_{z}} \times \mathbf{v_{ed}}]\;. }$$

(4.31)

This equation can be then decomposed through the three axis as follows

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e}\; +\; j\;\omega )\;\overline{v}_{ x}\; +\;\omega _{ce}\;\overline{v}_{y}\; =\; -\;\frac{e} {m}\;\overline{E}_{0x}{}\end{array}$$

(4.32)

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e}\; +\; j\;\omega )\;\overline{v}_{ y}\; -\;\omega _{ce}\;\overline{v}_{x}\; =\; -\;\frac{e} {m}\;\overline{E}_{0y}{}\end{array}$$

(4.33)

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e}\; +\; j\;\omega )\;\overline{v}_{ z}\; =\; -\;\frac{e} {m}\;\overline{E}_{0z}\;,{}\end{array}$$

(4.34)

where the velocity amplitudes are complex quantities to include origin phase shifts.

The vector equation is better written using tensor notation

$$\displaystyle{ \mathbf{v_{ed}}\; =\; -\widehat{\boldsymbol{\mu }_{\mathbf{e}}}\;\mathbf{E}\;. }$$

(4.35)

Introducing the quantities as in Allis (1956)

$$\displaystyle{ \overline{r}\; =\; \frac{1} {\nu _{m}^{e}\; +\; j\;(\omega -\omega _{ce})}\;,\;\;\;\overline{l}\; =\; \frac{1} {\nu _{m}^{e}\; +\; j\;(\omega +\omega _{ce})}\;,\;\;\;\overline{p}\; =\; \frac{1} {\nu _{m}^{e}\; +\; j\;\omega }\;, }$$

(4.36)

to correspond to right and left circular polarization of the E field normal to B, and E parallel to B, the mobility tensor writes as

$$\displaystyle{ \widehat{\boldsymbol{\mu }_{\mathbf{e}}}\; = \left (\begin{array}{ccc} \overline{\mu }_{xx} &\overline{\mu }_{xy}& 0 \\ -\overline{\mu }_{xy}&\overline{\mu }_{yy}& 0 \\ 0 & 0 &\overline{\mu }_{zz}\\ \end{array} \right )\;, }$$

(4.37)

in which the different components are

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{xx}\; =\; \overline{\mu }_{yy}\; =\; \frac{e} {m}\;\frac{(\overline{r} + \overline{l})} {2}{}\end{array}$$

(4.38)

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{xy}\; =\; j\; \frac{e} {m}\;\frac{(\overline{r} -\overline{l})} {2}{}\end{array}$$

(4.39)

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{zz}\; =\; \frac{e} {m}\;\overline{p}\;.{}\end{array}$$

(4.40)

The electron current density is then $\mathbf{J_{e}} = -en_{e}\mathbf{v_{ed}} =\widehat{\boldsymbol{\sigma } _{\mathbf{ce}}}\;\mathbf{E}$, being $\widehat{\boldsymbol{\sigma }_{\mathbf{ce}}} = en_{e}\widehat{\boldsymbol{\mu }_{\mathbf{e}}}$ the electron conductivity tensor.

As seen in equation (4.13), the power absorbed per volume unit is

$$\displaystyle{ P_{E}(t)\; =\; (\mathbf{J_{e}}\;.\;\mathbf{E})\; =\; (\widehat{\boldsymbol{\sigma }_{\mathbf{ce}}}\;\mathbf{E}\;.\;\mathbf{E})\;, }$$

(4.41)

being the time-averaged power given by

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{1} {2}\;\mbox{ Re}\{(\widehat{\boldsymbol{\sigma }_{\mathbf{ce}}}\;\mathbf{E}\;.\;\mathbf{E^{{\ast}}})\}\;, }$$

(4.42)

with E ^∗ denoting the complex conjugate. Defining now the components of the field as $\overline{E}_{x} = E_{0x}\;e^{j\omega t}$, $\overline{E}_{y} = E_{0y}\;e^{j(\omega t+\theta )}$ and $\overline{E}_{z} = E_{0z}\;e^{j\omega t}$, in order $\theta$ may represent the phase shift between $\overline{E}_{y}$ and $\overline{E}_{x}$, we obtain

$$\displaystyle\begin{array}{rcl} \overline{P_{E}(t)}& =& \frac{1} {2}\;\left [\mbox{ Re}\{\overline{\sigma }_{xx}\}\;E_{0x}^{\;2}\; + \mbox{ Re}\{\overline{\sigma }_{ yy}\}\;E_{0y}^{\;2}\; + \mbox{ Re}\{\overline{\sigma }_{ zz}\}\;E_{0z}^{\;2}\right ] \\ & & -\;\mbox{ Im}\{\overline{\sigma }_{xy}\}\;E_{0x}\;E_{0y}\;\sin \theta \;, {}\end{array}$$

(4.43)

in which

$$\displaystyle\begin{array}{rcl} & & \mbox{ Re}\{\overline{\sigma }_{xx}\}\; =\; \mbox{ Re}\{\overline{\sigma }_{yy}\}\; =\; \frac{e^{2}n_{e}} {m} \; \frac{\nu _{m}^{e}\;(\nu _{m}^{e\;2} +\omega ^{2} +\omega _{ ce}^{\;2})} {(\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2})\;(\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2})}{}\end{array}$$

(4.44)

$$\displaystyle\begin{array}{rcl} & & \mbox{ Im}\{\overline{\sigma }_{xy}\}\; =\; \frac{e^{2}n_{e}} {m} \; \frac{2\;\nu _{m}^{e}\;\omega \;\omega _{ce}} {(\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2})\;(\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2})}{}\end{array}$$

(4.45)

$$\displaystyle\begin{array}{rcl} & & \mbox{ Re}\{\overline{\sigma }_{zz}\}\; =\; \frac{e^{2}n_{e}} {m} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;.{}\end{array}$$

(4.46)

The angle $\theta$ is the polarization angle of the electric field in the direction perpendicular to the B field. For a right-hand circularly polarized wave, we have $E_{0x} = E_{0y} = E_{0}$, E _0z  = 0, and $\theta = -\pi /2$, so that equation (4.43) becomes

$$\displaystyle\begin{array}{rcl} \overline{P_{E}}_{(R)}& =& \left (\mbox{ Re}\{\overline{\sigma }_{xx}\} + \mbox{ Im}\{\overline{\sigma }_{xy}\}\right )\;E_{0}^{\;2} \\ & =& \frac{e^{2}n_{e}} {m} \;\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}}\;E_{0}^{\;2}\;,{}\end{array}$$

(4.47)

while for a left-hand circularly polarized wave, we have $\theta =\pi /2$ and we obtain

$$\displaystyle\begin{array}{rcl} \overline{P_{E}}_{(L)}& =& \left (\mbox{ Re}\{\overline{\sigma }_{xx}\} -\mbox{ Im}\{\overline{\sigma }_{xy}\}\right )\;E_{0}^{\;2} \\ & =& \frac{e^{2}n_{e}} {m} \;\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\;E_{0}^{\;2}\;.{}\end{array}$$

(4.48)

Equation (4.47) shows that for ω _ce  = ω, the power absorbed by the electrons from a right-hand circularly polarized wave has a maximum as a result of the exact match between the rotating field and the cyclotron motion of electrons. The power becomes equal to that absorbed from a DC field of amplitude E ₀. On the other hand, in the case of a left-hand wave the power absorbed is significantly reduced, being the ratio of the power absorbed by the two waves at ω _ce  = ω equal to $\nu _{m}^{e\;2}/(\nu _{m}^{e\;2} + 4\omega ^{2})$.

It is still worth noting that in the case of a linearly polarized wave in the plane perpendicular to the B field of amplitude E _0(l), which may be decomposed into two circularly polarized waves of amplitudes E _0(l)∕2, we obtain

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \;\left ( \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}} + \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\right )\left (\frac{E_{0(l)}} {2} \right )^{2}\;. }$$

(4.49)

Thus, when B = 0 this equation transforms into the power absorbed by a HF field of amplitude E _0(l) given by equation (4.19)

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{E_{0(l)}^{\;2}} {2} \;. }$$

(4.50)

2.2 Boltzmann Equation

In the presence of the E and B fields, and in the absence of space gradients, the electron Boltzmann equation takes the form given by equation (3.6)

$$\displaystyle{ \frac{\partial f_{e}} {\partial t} \; + \left (\left \{-\;\frac{e} {m}\;\mathbf{E}\; +\; [\vec{\boldsymbol{\omega }_{\mathbf{ce}}} \times \mathbf{v_{e}}]\right \}\;.\;\frac{\partial f_{e}} {\boldsymbol{\partial }\mathbf{v_{e}}}\right ) =\; \left (\frac{\partial f_{e}} {\partial t} \right )_{\!\!e-o}\;, }$$

(4.51)

with $\vec{\boldsymbol{\omega }_{\mathbf{ce}}} = eB/m\;\mathbf{e_{z}}$. In this case due to the magnetic field we need to consider the dependence on the azimuthal coordinate ϕ, so that the expansion of f _e (v _e, t) limited to the first order anisotropies takes the form (3.21)

$$\displaystyle\begin{array}{rcl} f(\mathbf{v_{e}},t)& =& f_{e}^{0}(v_{ e},t)\; +\; \left (\mathbf{f_{e}^{l}}(v_{ e},t)\;.\;\frac{\mathbf{v_{e}}} {v_{e}} \right ) \\ & =& f_{e}^{0}\; +\; p_{ 11}\;\frac{v_{ex}} {v_{e}} \; +\; q_{11}\;\frac{v_{ey}} {v_{e}} \; +\; p_{10}\;\frac{v_{ez}} {v_{e}} \;.{}\end{array}$$

(4.52)

Then, the term with the magnetic field in equation (4.51) is

$$\displaystyle{ \left ([\vec{\boldsymbol{\omega }_{\mathbf{ce}}} \times \mathbf{v_{e}}]\;.\;\frac{\partial f_{e}} {\boldsymbol{\partial }\mathbf{v_{e}}}\right )\; =\;\omega _{ce}\;q_{11}\;\frac{v_{ex}} {v_{e}} \; -\;\omega _{ce}\;p_{11}\;\frac{v_{ey}} {v_{e}} \;. }$$

(4.53)

Since $v_{ex}/v_{e} =\sin \theta \;\cos \phi$ and $v_{ey}/v_{e} =\sin \theta \;\sin \phi$, equation (4.53) leads to the appearing of two new terms in the equations for the anisotropic components p ₁₁ and q ₁₁, being equation (3.42) replaced with the following system of equations

$$\displaystyle\begin{array}{rcl} & & \frac{\partial p_{11}} {\partial t} \; -\;\frac{eE_{x}} {m} \;\frac{\partial f_{e}^{0}} {\partial v_{e}} \; +\;\omega _{ce}\;q_{11}\; =\; -\;\nu _{m}^{e}\;p_{ 11}{}\end{array}$$

(4.54)

$$\displaystyle\begin{array}{rcl} & & \frac{\partial q_{11}} {\partial t} \; -\;\frac{eE_{y}} {m} \;\frac{\partial f_{e}^{0}} {\partial v_{e}} \; -\;\omega _{ce}\;p_{11}\; =\; -\;\nu _{m}^{e}\;q_{ 11}{}\end{array}$$

(4.55)

$$\displaystyle\begin{array}{rcl} & & \frac{\partial p_{10}} {\partial t} \; -\;\frac{eE_{z}} {m} \;\frac{\partial f_{e}^{0}} {\partial v_{e}} \; =\; -\;\nu _{m}^{e}\;p_{ 10}\;,{}\end{array}$$

(4.56)

while for the isotropic component we have in place of (3.41)

$$\displaystyle{ \frac{\partial f_{e}^{0}} {\partial t} \; -\; \frac{e} {m}\; \frac{1} {3v_{e}^{\;2}}\; \frac{\partial } {\partial v_{e}}\left (v_{e}^{\;2}\;(p_{ 11}\;E_{x} + q_{11}\;E_{y} + p_{10}\;E_{z})\right ) =\; I^{0}(f_{ e}^{0})\; + J^{0}(f_{ e}^{0})\;, }$$

(4.57)

with the elastic and the inelastic collision terms given by equations (3.85) and (3.134).

Considering now the field frequency much larger than the characteristic relaxation frequency for energy transfer ν _e , in order we may assume the time invariance of f _e ⁰, the equations (4.54), (4.55) and (4.56) take the following form using complex quantities

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e} + j\omega )\;\overline{p}_{ 11}\; +\;\omega _{ce}\;\overline{q}_{11}\; =\; \frac{e} {m}\;\overline{E}_{x}\;\frac{df_{e}^{0}} {dv_{e}}{}\end{array}$$

(4.58)

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e} + j\omega )\;\overline{q}_{ 11}\; -\;\omega _{ce}\;\overline{p}_{11}\; =\; \frac{e} {m}\;\overline{E}_{y}\;\frac{df_{e}^{0}} {dv_{e}}{}\end{array}$$

(4.59)

$$\displaystyle\begin{array}{rcl} & & (\nu _{m}^{e} + j\omega )\;\overline{p}_{ 10}\; =\; \frac{e} {m}\;\overline{E}_{z}\;\frac{df_{e}^{0}} {dv_{e}}{}\end{array}$$

(4.60)

and as in (4.37) we may express under tensorial form

$$\displaystyle{ \left [\begin{array}{c} \overline{p}_{11}\\ \\ \overline{q}_{11}\\ \\ \overline{p}_{10}\\ \end{array} \right ]\; =\; \frac{e} {m}\left [\begin{array}{c@{\qquad }cc} \overline{E}_{x}\;(\overline{r} + \overline{l})/2\; \qquad &\;j\overline{E}_{y}\;(\overline{r} -\overline{l})/2& 0\\ \qquad \\ -\; j\overline{E}_{x}\;(\overline{r} -\overline{l})/2\;\qquad & \;\overline{E}_{y}\;(\overline{r} + \overline{l})/2 & 0\\ \qquad \\ 0 \qquad & 0 &\overline{E}_{x}\;\overline{p}\\ \qquad \end{array} \right ]\;\frac{df_{e}^{0}} {dv_{e}} \;. }$$

(4.61)

The drift or average vector velocity (4.10)

$$\displaystyle{ \mathbf{v_{ed}}\; =\; \frac{1} {n_{e}}\int _{0}^{\infty }(\overline{p}_{ 11}\;\mathbf{e_{x}} + \overline{q}_{11}\;\mathbf{e_{y}} + \overline{p}_{10}\;\mathbf{e_{z}})\;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} }$$

(4.62)

may be written in the form (4.35) being the components of the tensor mobility as follows

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{xx}\; =\; -\; \frac{e} {mn_{e}}\int _{0}^{\infty }\left (\frac{\overline{r} + \overline{l}} {2} \right )\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}{}\end{array}$$

(4.63)

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{xy}\; =\; -\;j\; \frac{e} {mn_{e}}\int _{0}^{\infty }\left (\frac{\overline{r} -\overline{l}} {2} \right )\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}{}\end{array}$$

(4.64)

$$\displaystyle\begin{array}{rcl} & & \overline{\mu }_{zz}\; =\; -\; \frac{e} {mn_{e}}\int _{0}^{\infty }\overline{p}\;\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}{}\end{array}$$

(4.65)

and $\overline{\mu }_{yy} = \overline{\mu }_{xx}$. Obviously when ν _m ^e(v _e ) = const and because of

$$\displaystyle{ \int _{0}^{\infty }\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\; =\; -\;n_{e}\;, }$$

(4.66)

we return back to equations (4.38), (4.39) and (4.40).

Let us consider now equation (4.57) for the isotropic component assuming ω ≫ ν _e , and hence $\partial f_{e}^{0}/\partial t = 0$. Substituting the equation (4.61) for the anisotropic component in equation (4.57), written as in equation (4.21), we obtain

$$\displaystyle\begin{array}{rcl} & -& \left ( \frac{e} {m}\right )^{2} \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\Bigg[\;\frac{v_{e}^{\;2}} {2} \;\Bigg(\;\mbox{ Re}\left \{\frac{\overline{r} + \overline{l}} {2} \right \}(E_{0x}^{\;2} + E_{ 0y}^{\;2})\; +\; \mbox{ Re}\{\overline{p}\}\;E_{ 0z}^{\;2} \\ & & -\;\mbox{ Im}\left \{j\left (\frac{\overline{r} -\overline{l}} {2} \right )\right \}\;2\;E_{0x}\;E_{0y}\;\sin \theta \;\Bigg)\;\frac{df_{e}^{0}} {dv_{e}} \;\Bigg]\; =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\!, {}\end{array}$$

(4.67)

in which

$$\displaystyle\begin{array}{rcl} & & \mbox{ Re}\left \{\frac{\overline{r} + \overline{l}} {2} \right \}\; =\; \frac{\nu _{m}^{e}\;(\nu _{m}^{e\;2} +\omega ^{2} +\omega _{ ce}^{\;2})} {(\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2})\;(\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2})}{}\end{array}$$

(4.68)

$$\displaystyle\begin{array}{rcl} & & \mbox{ Im}\left \{j\left (\frac{\overline{r} -\overline{l}} {2} \right )\right \}\; =\; \frac{2\;\nu _{m}^{e}\;\omega \;\omega _{ce}} {(\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2})\;(\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2})}{}\end{array}$$

(4.69)

$$\displaystyle\begin{array}{rcl} & & \mbox{ Re}\left \{\overline{p}\right \}\; =\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;.{}\end{array}$$

(4.70)

For a right-hand circularly polarized wave $E_{0x} = E_{0y} = E_{0}$, E _0z  = 0, and $\theta = -\pi /2$, we obtain

$$\displaystyle\begin{array}{rcl} & -& \left ( \frac{e} {m}\right )^{2} \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\;\left [v_{e}^{\;2}\;\left (\mbox{ Re}\left \{\frac{\overline{r} + \overline{l}} {2} \right \} + \mbox{ Im}\left \{j\left (\frac{\overline{r} -\overline{l}} {2} \right )\right \}\right )\;E_{0}^{\;2}\;\;\frac{df_{e}^{0}} {dv_{e}} \right ] {}\\ & & =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0}) {}\\ \end{array}$$

and therefore

$$\displaystyle{ -\;\left ( \frac{e} {m}\right )^{2} \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\;\left [v_{e}^{\;2}\;\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}}\;\;E_{0}^{\;2}\;\;\frac{df_{e}^{0}} {dv_{e}} \right ]\; =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;, }$$

(4.71)

while for a left-hand circularly polarized wave $\theta =\pi /2$, we find

$$\displaystyle{ -\;\left ( \frac{e} {m}\right )^{2} \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\;\left [v_{e}^{\;2}\;\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\;\;E_{0}^{\;2}\;\;\frac{df_{e}^{0}} {dv_{e}} \right ]\; =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;. }$$

(4.72)

We may check that in case of a linearly polarized wave in the plane perpendicular to the B field of amplitude E _0(l), decomposed into two circularly polarized waves of amplitudes E _0(l)∕2, we have

$$\displaystyle\begin{array}{rcl} & -& \left ( \frac{e} {m}\right )^{2} \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\;\left [v_{e}^{\;2}\;\left ( \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}}\; +\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\right )\;\frac{E_{0l}^{\;2}} {4} \;\;\frac{df_{e}^{0}} {dv_{e}} \right ] \\ & & =\; I^{0}(f_{ e}^{0})\; +\; J^{0}(f_{ e}^{0})\;, {}\end{array}$$

(4.73)

so that as ω _ce  = 0, equation (4.73) transforms into the previous expression (4.22) derived for a linearly polarized HF field.

Finally, let us consider the most general case of a right-hand circularly polarized wave of amplitude E _0(R) simultaneously with a left-hand circularly polarized wave of amplitude E _0(L). Multiplying both members of equations (4.71) and (4.72) by $\sqrt{u}$, we obtain the following expression for the Boltzmann equation in terms of the EEDF f _e ⁰(u) equivalent to equation (4.24) as B is discarded

$$\displaystyle\begin{array}{rcl} & -& \frac{d} {du}\left [\frac{2} {3}\;u^{3/2}\;\nu _{ m}^{e}\;\frac{e^{2}} {m}\left ( \frac{E_{0(R)}^{\;2}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}}\; +\; \frac{E_{0(L)}^{\;2}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\right )\;\frac{df_{e}^{0}} {du} \right ] \\ & & =\; \frac{d} {du}\left [\frac{2m} {M} \;u^{3/2}\;\nu _{ m}\left (f_{e}^{0} + k_{ B}T_{o}\;\frac{df_{e}^{o}} {du} \right ) + 4B_{0}\;\nu _{0}\;\sqrt{u}\;f_{e}^{0}\right ] \\ & & \qquad +\sum _{i,j}\left \{\sqrt{u + u_{ij}}\;\;\nu _{ij}(u + u_{ij})\;f_{e}^{0}(u + u_{ ij}) -\;\sqrt{u}\;\;\nu _{ij}(u)\;f_{e}^{0}(u)\right \} \\ & & \qquad +\;\sum _{j,i}\left \{\sqrt{u - u_{ij}}\;\;\nu _{ji}(u - u_{ij})\;f_{e}^{0}(u - u_{ ij}) -\;\sqrt{u}\;\;\nu _{ji}(u)\;f_{e}^{0}(u)\right \}.{}\end{array}$$

(4.74)

Equation (4.74) determines the EEDF in terms of E _0(R)∕n ₀ and E _0(L)∕n ₀ (or E _0(l)∕n ₀), ω∕n ₀ and ω _ce ∕ω. When $\omega _{ce}/\omega = 1$ the right-hand circularly polarized wave leads to electron cyclotron resonance.

2.3 Power Absorbed from the Field

Following a procedure identical to (4.43) we obtain the following expression for the mean power absorbed from the HF field, per volume unit, for the case of right-hand and left-hand circularly polarized waves

$$\displaystyle{ \overline{P_{E}(t)}\; =\; en_{e}\left (\mbox{ Re}\left \{\overline{\mu }_{xx}\right \} + \mbox{ Im}\left \{\overline{\mu }_{xy}\right \}\right )E_{0(R)}^{\;2} + en_{ e}\left (\mbox{ Re}\left \{\overline{\mu }_{xx}\right \} -\mbox{ Im}\left \{\overline{\mu }_{xy}\right \}\right )E_{0(L)}^{\;2}\;. }$$

(4.75)

Individually, the power absorbed from the right-hand and the left-hand circularly polarized waves are as follows using the EEDF with the normalization (3.166)

$$\displaystyle\begin{array}{rcl} & & \overline{P_{E}}_{(R)}\; =\; -\;\frac{2} {3}\;\frac{e^{2}n_{e}} {m} \int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;\; \frac{E_{0(R)}^{\;2}} {\nu _{m}^{e\;2} + (\omega -\omega _{ce})^{2}}\;\; \frac{df} {du}\;du{}\end{array}$$

(4.76)

$$\displaystyle\begin{array}{rcl} & & \overline{P_{E}}_{(L)}\; =\; -\;\frac{2} {3}\;\frac{e^{2}n_{e}} {m} \int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;\; \frac{E_{0(L)}^{\;2}} {\nu _{m}^{e\;2} + (\omega +\omega _{ce})^{2}}\;\; \frac{df} {du}\;du\;.{}\end{array}$$

(4.77)

Figure 4.9 presents the mean power per electron at unit gas density, $\overline{P_{E}(t)}/n_{e}n_{o}$, absorbed from the R and L waves in argon, as a function of ω _ce ∕ω, for the reduced rms field $E_{rms(l)}/n_{o} = 20$ Td of a linearly polarized wave decomposed into two circularly polarized waves, and for $\omega /n_{o} = 1.17 \times 10^{-13}$ m³ s⁻¹ and 4. 78 × 10⁻¹³ m³ s⁻¹ (Loureiro 1995). The full curves represent results obtained from equation (4.74) in the absence of electron-electron (e-e) collisions, while the broken curves show for comparison the mean power calculated as the EEDFs are assumed Maxwellian (3.168), with their temperatures obtained from the energy balance equation (3.174)

$$\displaystyle{ \overline{P_{E}(t)}\; =\; P_{el}\; +\; P_{inel}\;, }$$

(4.78)

in which the average power loss terms are given by (3.151) and (3.176). In the case of a gas pressure p = 133. 3 Pa (1 Torr) and temperature of neutrals T _o  = 300 K, the values of ω∕n _o used here correspond to $\omega /2\pi = 600$ MHz and 2.45 GHz. The equality ω _ce  = ω is achieved for B = 214 and 875 G, respectively. Figure 4.9 shows that maximum heating is obtained for a Maxwellian EEDF. Further, the power absorbed from the L wave decreases as ω∕n _o increases from 1. 17 × 10⁻¹³ to 4. 78 × 10⁻¹³ m³ s⁻¹, which signifies higher efficiency in ECR when a linearly polarized wave is launched. The small maximum in the power absorbed from the L wave as ω _ce  = ω results from the changes on the EEDFs for conditions close to ECR originated by the R wave.

Fig. 4.9

Mean power absorbed per electron at unit gas density against ω _ce ∕ω in argon, for $E_{rms(l)}/n_{o} = 20$ Td and $\omega /n_{o} = 1.17 \times 10^{-13}$ m³ s⁻¹ (R ₁; L ₁) and 4. 78 × 10⁻¹³ m³ s⁻¹ (R ₂; L ₂). R and L are for the R and L waves. Full curves: absence of e-e collisions; Broken curves: Maxwellian EEDFs (Loureiro 1995)

Figure 4.10 shows values of $\overline{P_{E}(t)}/n_{e}n_{o}$ calculated for $E_{rms(l)}/n_{o} = 120$ Td, keeping all other conditions as in Fig. 4.9. From inspection of Figs. 4.9 and 4.10 we conclude that the differences between the values of $\overline{P_{E}(t)}/n_{e}n_{o}$ calculated in the absence of e-e collisions (i.e. with non-Maxwellian EEDFs) and in the case of Maxwellian EEDFs significantly reduce as E _rms(l)∕n _o increases, in consequence of modifications on the shape of the EEDFs.

Fig. 4.10

As in Fig. 4.9 but for $E_{rms(l)}/n_{o} = 120$ Td (Loureiro 1995)

Figure 4.11 shows values of $\overline{P_{E}(t)}/n_{e}n_{o}$ in Ar, as a function of the reduced rms amplitude of a linearly polarized wave, E _rms(l)∕n _o , and for ω _ce  = ω. As in Fig. 4.9 we consider $\omega /n_{o} = 1.17 \times 10^{-13}$ and 4. 78 × 10⁻¹³ m³ s⁻¹ in the absence of e-e collisions (full curves) and assuming Maxwellian EEDFs (broken curves). We note that the results shown in Fig. 4.11 for the R wave are equivalent to those in a DC electric field, if the L wave was omitted and a R wave with E _0(R) = E _DC was directly launched instead of the linearly polarized wave. The values for R waves are indistinguishable for the two values of ω∕n _o .

Fig. 4.11

Mean power absorbed per electron at unit gas density against E _rms(l)∕n _o in argon, for ω _ce  = ω and $\omega /n_{o} = 1.17 \times 10^{-13}$ m³ s⁻¹ (R ₁; L ₁) and 4. 78 × 10⁻¹³ m³ s⁻¹ (R ₂; L ₂). R and L are for the right and left waves. Full curves: absence of e-e collisions; Broken curves: Maxwellian EEDFs (Loureiro 1995)

Finally, Fig. 4.12 shows 2/3 of the average electron energy, < ​​ u​​ > , as a function of the ratio of the rms amplitude of a linearly polarized wave to the gas number density, E _rms(l)∕n _o , in argon at ECR conditions (i.e. for ω _ce  = ω), and for $\omega /n_{o} = 1.17 \times 10^{-13}$ and 4. 78 × 10⁻¹³ m³ s⁻¹. As before the full curves are obtained in the absence of e-e collisions, while the broken ones are for Maxwellian EEDFs. In the case of these latter, 2∕3 < ​​ u​​ > corresponds to the electron kinetic temperature k _B T _e of the distributions.

Fig. 4.12

2/3 of the average electron energy, < ​​ u​​ > , against E _rms(l)∕n _o in argon, for ω _ce  = ω, and for (A) $\omega /n_{o} = 1.17 \times 10^{-13}$ m³ s⁻¹ and (B) 4. 78 × 10⁻¹³ m³ s⁻¹. Full curves: absence of e-e collisions; Broken curves: Maxwellian EEDFs (Loureiro 1995)

3 Radio-Frequency Electric Fields

3.1 Time-Dependent Boltzmann Equation

As the field frequency decreases up to values of the same order or smaller than the characteristic relaxation frequency for energy transfer, ω < ν _e , with ν _e given by equation (4.1), the collisional energy dissipation is faster than the radio-frequency (RF) field variation, ν _e ⁻¹ < T, with T denoting the field period, and a large time-modulation occurs in the isotropic component of the electron velocity distribution function, i.e. in the EEDF (Margenau and Hartman 1948; Delcroix 1963,1966; Winkler et al. 1987; Capitelli et al. 1988; Loureiro 1993). The energy range where the time-modulation takes place depends on the form as ν _e varies with the electron energy. In argon, for example, ν _e sharply increases with the energy u (see Fig. 4.1), so that at field frequencies in RF range and for gas pressures typically $p \sim 100$ Pa, the inequality ω < ν _e holds only in the high-energy tail of the EEDF (Sá et al. 1994). On the contrary, in a molecular gas such as H₂ or N₂ the above inequality holds at much lower electron energies, due to the contribution of vibrational excitation to the characteristic relaxation frequency ν _e , and a large time-modulation occurs also in the bulk of the distribution (Loureiro 1993). This is particularly evident in N₂ (see Fig. 4.2), where a sharp and pronounced maximum exists at $u \sim 2$ eV in the total cross section for vibrational excitation (see also Fig. 3.13).

The EEDF in the collision dominated bulk plasma created by a RF field can be obtained by solving the time-dependent, spatially homogeneous, Boltzmann equation under the action of a time-varying sinusoidal electric field

$$\displaystyle{ \mathbf{E}(t)\; =\; \mathbf{E_{0}}\;\cos (\omega t)\;, }$$

(4.79)

with $\mathbf{E_{0}} = -\;E_{0}\;\mathbf{e_{z}}$, and where ω is the angular field frequency. Then, the electron velocity distribution function f _e (v _e, t) can be obtained by solving the time-dependent Boltzmann equation

$$\displaystyle{ \frac{\partial f_{e}} {\partial t} \; -\left ( \frac{e} {m}\;\mathbf{E}\;.\;\frac{\partial f_{e}} {\boldsymbol{\partial }\mathbf{v_{e}}}\right )\; =\; I(f_{e})\; +\; J(f_{e})\;, }$$

(4.80)

where e and m are the electron absolute charge and mass, respectively, v _e is the electron velocity, and I and J denote the collision operator for elastic and for inelastic collisions, respectively. Here, as before, we will neglect processes which result in the production or loss of electrons, i.e., production of secondary electrons by ionization, electron-ion recombination, and electron attachment, so that J(f _e ) includes only the effects of energy-exchange processes, both for inelastic and superelastic collisions. The electron velocity distribution is normalized through the condition

$$\displaystyle{ n_{e}\; =\;\int _{\mathbf{v_{e}}}f_{e}(\mathbf{v_{e}},t)\;\mathbf{dv_{e}}\;, }$$

(4.81)

where n _e denotes the electron number density assumed here time-independent.

Equation (4.80) is solved by expanding f _e in Legendre polynomials in velocity space and Fourier series in time

$$\displaystyle{ f_{e}(\mathbf{v_{e}},t)\; =\;\sum _{l=0}\sum _{k=0}\mbox{ Re}\{\overline{f_{k}^{l}}(v_{e})\;e^{jk\omega t}\}\;P_{ l}(\cos \theta )\;, }$$

(4.82)

with $\theta$ denoting the angle between the instantaneous velocity v _e and the direction of the anisotropy along the z axis, Re{ } means “the real part of”, v _e  = |v _e |, and $\overline{f_{k}^{l}}(v_{e})$ is a complex function expressing the time delay of the electron transport with respect to the applied electric field (4.79). Here, we will assume that the anisotropies resulting from the field are sufficiently small, so that the first two terms in spherical functions suffice for the expansion

$$\displaystyle{ f_{e}(\mathbf{v_{e}},t)\; \simeq \; f_{e}^{0}(v_{ e},t)\; +\; f_{e}^{1}(v_{ e},t)\;\cos \theta \;. }$$

(4.83)

Under this assumption, the lowest-order approximation for the expansion in Fourier series, allowing a periodic time-variation in the isotropic component of the electron velocity distribution (that is, in the EEDF), is

$$\displaystyle{ f_{e}^{0}(v_{ e},t)\; \simeq \; f_{0}^{0}(v_{ e})\; +\; f_{2}^{0}(v_{ e})\;\cos (2\omega t +\phi _{ 2}^{0}(v_{ e}))\;, }$$

(4.84)

while for the anisotropic component we have

$$\displaystyle{ f_{e}^{1}(v_{ e},t)\; \simeq \; f_{1}^{1}(v_{ e})\;\cos (\omega t +\phi _{ 1}^{1}(v_{ e}))\;. }$$

(4.85)

As shown below, the fact that the isotropic and the anisotropic components of the electron velocity distribution are functions of E(t)² and E(t), respectively, originates that the isotropic velocity distribution, f _e ⁰(v _e , t), only has even harmonics in the Fourier expansion, whereas f _e ¹(v _e , t) only has odd ones (Margenau and Hartman 1948; Delcroix 1963,1966). Then, the normalization condition (4.81) appropriate to the present simplification should be written as follows

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }f_{ 0}^{0}(v_{ e})\;4\pi v_{e}^{\;2}\;dv_{ e}\; =\; n_{e}\;,{}\end{array}$$

(4.86)

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }f_{ 2R}^{0}(v_{ e})\;4\pi v_{e}^{\;2}\;dv_{ e}\; =\; 0\;,{}\end{array}$$

(4.87)

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }f_{ 2I}^{0}(v_{ e})\;4\pi v_{e}^{\;2}\;dv_{ e}\; =\; 0\;,{}\end{array}$$

(4.88)

where f _2R ⁰ and f _2I ⁰ denote, respectively, the real and imaginary part of the complex amplitude $\overline{f_{2}^{0}} = f_{2}^{0}\;\exp (j\phi _{2}^{0})$ of the time-varying isotropic component at frequency 2ω.

Introducing (4.82) into equation (4.80), one obtains the following system of nonlocal equations in velocity space for f ₀ ⁰, $\overline{f_{2}^{0}}$, and $\overline{f_{1}^{1}}$, respectively

$$\displaystyle\begin{array}{rcl} & & \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{eE_{0}} {2m} \;v_{e}^{\;2}\;\mbox{ Re}\{\overline{f_{ 1}^{1}}\}\right ) =\; I^{0}(f_{ 0}^{0})\; +\; J^{0}(f_{ 0}^{0})\;,{}\end{array}$$

(4.89)

$$\displaystyle\begin{array}{rcl} & & j2\omega \;\overline{f_{2}^{0}}\; +\; \frac{1} {3v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{eE_{0}} {2m} \;v_{e}^{\;2}\;\overline{f_{ 1}^{1}}\right ) =\; I^{0}(\overline{f_{ 2}^{0}})\; +\; J^{0}(\overline{f_{ 2}^{0}})\;,{}\end{array}$$

(4.90)

$$\displaystyle\begin{array}{rcl} & & j\omega \;\overline{f_{1}^{1}}\; +\; \frac{eE_{0}} {m} \;\frac{df_{0}^{0}} {dv_{e}} \; -\;\frac{eE_{0}} {2m} \;\frac{d\overline{f_{2}^{0}}} {dv_{e}} \; =\; I^{1}(\overline{f_{ 1}^{1}})\; +\; J^{1}(\overline{f_{ 1}^{1}})\;.{}\end{array}$$

(4.91)

The elastic and inelastic collision terms of the isotropic components, I ⁰ and J ⁰, are given by (3.85) and (3.133), whereas in equation (4.91) we have

$$\displaystyle{ I^{1}(\overline{f_{ 1}^{1}}) + J^{1}(\overline{f_{ 1}^{1}})\; =\; -\;\nu _{ m}^{e}\;\overline{f_{ 1}^{1}}\;, }$$

(4.92)

with ν _m ^e given by (4.2). As seen in equation (3.139) the use of an effective collision frequency in equation (4.92) is justifiable in gases for which the inelastic scattering is not negligible but is nearly isotropic. The evaluation of such an approximation in N₂ is treated in Phelps and Pitchford (1985).

Substituting (4.92) into equation (4.91), one obtains for $\overline{f_{1}^{1}}$

$$\displaystyle{ \overline{f_{1}^{1}}\; =\; -\; \frac{eE_{0}} {m\;(\nu _{m}^{e} + j\omega )}\;\frac{df_{0}^{0}} {dv_{e}} \; -\; \frac{eE_{0}} {2m\;(\nu _{m}^{e} + j\omega )}\;\frac{d\overline{f_{2}^{0}}} {dv_{e}} \,. }$$

(4.93)

This expression when inserted in equations (4.89) and (4.90) allows us to write the following system of equations for f ₀ ⁰, f _2R ⁰, and f _2I ⁰ (Loureiro 1993)

$$\displaystyle\begin{array}{rcl} & & -\; \frac{1} {6v_{e}^{\;2}}\;\left (\frac{eE_{0}} {m} \right )^{2} \frac{d} {dv_{e}}\left [v_{e}^{\;2}\left (A\;\frac{df_{0}^{0}} {dv_{e}} \; +\; \frac{A} {2} \;\frac{df_{2R}^{0}} {dv_{e}} \; +\; \frac{B} {2} \;\frac{df_{2I}^{0}} {dv_{e}} \right )\right ] \\ & & \;\;\;\;\;\;\;\;\;\;\;\;=\; I^{0}(f_{ 0}^{0})\; +\; J^{0}(f_{ 0}^{0})\;, {}\end{array}$$

(4.94)

$$\displaystyle\begin{array}{rcl} & & \qquad - 2\omega \;f_{2I}^{0} -\; \frac{1} {6v_{e}^{\;2}}\;\left (\frac{eE_{0}} {m} \right )^{2} \frac{d} {dv_{e}}\left [v_{e}^{\;2}\left (A\;\frac{df_{0}^{0}} {dv_{e}} \; +\; \frac{A} {2} \;\frac{df_{2R}^{0}} {dv_{e}} \; +\; \frac{B} {2} \;\frac{df_{2I}^{0}} {dv_{e}} \right )\right ] \\ & & \;\;\;\;\;\;\;\;\;\;\;\;=\; I^{0}(f_{ 2R}^{0})\; +\; J^{0}(f_{ 2R}^{0})\;, {}\end{array}$$

(4.95)

$$\displaystyle\begin{array}{rcl} & & \qquad 2\omega \;f_{2R}^{0} +\; \frac{1} {6v_{e}^{\;2}}\;\left (\frac{eE_{0}} {m} \right )^{2} \frac{d} {dv_{e}}\left [v_{e}^{\;2}\left (B\;\frac{df_{0}^{0}} {dv_{e}} \; +\; \frac{B} {2} \;\frac{df_{2R}^{0}} {dv_{e}} \; -\;\frac{A} {2} \;\frac{df_{2I}^{0}} {dv_{e}} \right )\right ] \\ & & \;\;\;\;\;\;\;\;\;\;\;\;=\; I^{0}(f_{ 2I}^{0})\; +\; J^{0}(f_{ 2I}^{0})\;, {}\end{array}$$

(4.96)

with

$$\displaystyle{ A\; = \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;;\;\;\;\;\;\;B\; = \frac{\omega } {\nu _{m}^{e\;2} +\omega ^{2}}\;. }$$

(4.97)

Once the functions f ₀ ⁰(v _e ), f _2R ⁰(v _e ), and f _2I ⁰(v _e ) are obtained, the anisotropic component $\overline{f_{1}^{1}}(v_{e})$ is calculated from equation (4.93). Then, the drift velocity v _ed(t) = v _ed (t) e _z, the energy-averaged electron energy < ​​ u​​ > (t), or, for example, the electron rate coefficient C _ij (t) for excitation of a j state from i sate can be readily obtained.

Transforming now as in (3.166) the isotropic component of the electron velocity distribution f _e ⁰(v _e , t) to the electron energy distribution function F ⁰(u, t) as follows

$$\displaystyle{ f_{e}^{0}(v_{ e},t)\;4\pi v_{e}^{\;2}\;dv_{ e}\; =\; n_{e}\;F^{0}(u,t)\;\sqrt{u}\;du\;, }$$

(4.98)

and f ₀ ⁰(v _e ), f _2R ⁰(v _e ) and f _2I ⁰(v _e ) transformed to F ₀ ⁰(u), F _2R ⁰(u) and F _2I ⁰(u), while the anisotropic component $\overline{f_{1}^{1}}(v_{e})$ is transformed to $\overline{F_{1}^{1}}(u)$

$$\displaystyle{ \overline{f_{1}^{1}}(v_{e})\;4\pi v_{e}^{\;2}\;dv_{ e}\; =\; n_{e}\;\overline{F_{1}^{1}}(u)\;\sqrt{u}\;du\;, }$$

(4.99)

with $u = \frac{1} {2}\;mv_{e}^{\;2}$ denoting the electron energy, we obtain successively for the different quantities:

• (i) Drift velocity

  $$\displaystyle{ v_{ed}(t)\; =\; \mbox{ Re}\{\overline{V _{d}}_{0}\;e^{j\omega t}\}\;, }$$

  (4.100)

  with $\overline{V _{d}}_{0}$ given by

  $$\displaystyle{ \overline{V _{d}}_{0}\; =\; \frac{1} {3}\sqrt{ \frac{2} {m}}\int _{0}^{\infty }u\;\overline{F_{ 1}^{1}}(u)\;du\;; }$$

  (4.101)

• (ii) Energy-averaged electron energy

  $$\displaystyle{ <\!\! u\!\! >\!\! (t)\; =\;\int _{ 0}^{\infty }u^{3/2}F^{0}(u,t)\;du\;, }$$

  (4.102)

  which can also be written under the form

  $$\displaystyle{ <\!\! u\!\! >\!\! (t)\; =\;<\!\! u\!\! > _{0}+ <\!\! u\!\! > _{2R}\;\cos (2\omega t)- <\!\! u\!\! > _{2I}\;\sin (2\omega t)\;, }$$

  (4.103)

  with < ​​ u​​ > ₀ given by

  $$\displaystyle{ <\!\! u\!\! > _{0}\; =\;\int _{ 0}^{\infty }u^{3/2}F_{ 0}^{0}(u)\;du\;, }$$

  (4.104)

  and where < ​​ u​​ > _2R and < ​​ u​​ > _2I are given by similar expressions but in which F ₀ ⁰(u) is replaced with F _2R ⁰(u) and F _2I ⁰(u), respectively;

• (iii) Electron rate coefficient for a $i \rightarrow j$ state transition

  $$\displaystyle{ C_{ij}(t)\; =\;<\!\! v_{e}\;\sigma _{ij}\!\! >\;= \sqrt{ \frac{2} {m}}\int _{0}^{\infty }u\;\sigma _{ ij}(u)\;F^{0}(u,t)\;du\;, }$$

  (4.105)

  or in the form

  $$\displaystyle{ C_{ij}(t)\; =\; (C_{ij})_{0} +\; (C_{ij})_{2R}\;\cos (2\omega t)\; -\; (C_{ij})_{2I}\;\sin (2\omega t)\;, }$$

  (4.106)

  with (C _ij )₀ given by

  $$\displaystyle{ (C_{ij})_{0}\; = \sqrt{ \frac{2} {m}}\int _{0}^{\infty }u\;\sigma _{ ij}(u)\;F_{0}^{0}(u)\;du\;, }$$

  (4.107)

  and where (C _ij )_2R and (C _ij )_2I are given by similar expressions using F _2R ⁰(u) and F _2I ⁰(u).

The energy-averaged power balance equation may also be decoupled into three equations as follows

$$\displaystyle\begin{array}{rcl} (P_{E})_{0}& =& (P_{el})_{0} +\; (P_{rot})_{0} +\; (P_{inel})_{0}{}\end{array}$$

(4.108)

$$\displaystyle\begin{array}{rcl} (P_{E})_{2R}& =& (P_{el})_{2R} +\; (P_{rot})_{2R} +\; (P_{inel})_{2R} +\; (P_{\omega })_{2R}{}\end{array}$$

(4.109)

$$\displaystyle\begin{array}{rcl} (P_{E})_{2I}& =& (P_{el})_{2I} +\; (P_{rot})_{2I} +\; (P_{inel})_{2I} +\; (P_{\omega })_{2I}\;,{}\end{array}$$

(4.110)

in which the terms in the left-hand side members are

$$\displaystyle\begin{array}{rcl} (P_{E})_{0}& =& \frac{en_{e}E_{0}} {2} \;\mbox{ Re}\{\overline{V _{d}}_{0}\}{}\end{array}$$

(4.111)

$$\displaystyle\begin{array}{rcl} (P_{E})_{2R}& =& (P_{E})_{0}{}\end{array}$$

(4.112)

$$\displaystyle\begin{array}{rcl} (P_{E})_{2I}& =& \frac{en_{e}E_{0}} {2} \;\mbox{ Im}\{\overline{V _{d}}_{0}\}\;.{}\end{array}$$

(4.113)

The terms for P _el , P _rot and P _inel are identical to equations (3.176), (3.177), and (3.151) replacing F ₀ ⁰, F _2R ⁰(u) and F _2I ⁰(u), whereas the terms (P _ω )_2R and (P _ω )_2I associated with the interchange in equations for F _2R ⁰(u) and F _2I ⁰(u) due to the first term in equation (4.90), taking into consideration the phase-shift ϕ ₂ ⁰ in equation (4.84), are

$$\displaystyle\begin{array}{rcl} (P_{\omega })_{2R}& =& -\;2n_{e}\omega <\!\! u\!\! > _{2I}{}\end{array}$$

(4.114)

$$\displaystyle\begin{array}{rcl} (P_{\omega })_{2I}& =& 2n_{e}\omega <\!\! u\!\! > _{2R}\;.{}\end{array}$$

(4.115)

Finally, it follows from this formulation that the instantaneous energy-averaged power absorbed from the field is given by

$$\displaystyle{ P_{E}(t)\; =\; (\mathbf{J_{e}}\;.\;\mathbf{E})\; =\; (P_{E})_{0}\;[1 +\cos (2\omega t)]\; -\; (P_{E})_{2I}\;\sin (2\omega t)\;, }$$

(4.116)

with $\mathbf{J_{e}} = -\;en_{e}\mathbf{v_{ed}}$ denoting the electron current density.

3.2 Time-Dependent Velocity Distributions

The time-dependent electron Boltzmann transport equation written in the form (4.94), (4.95) and (4.96) can be solved to yield the EEDF, F ⁰(u, t), as a function of the independent parameters: ratio of the electric field amplitude to the gas number density, E ₀∕n _o ; ratio of the angular field frequency to the gas density, ω∕n _o ; gas temperature, T _o ; and vibrational temperature, T _v . The latter used to take into account the effects produced by the inelastic e-V processes starting from a level v > 0, and the superelastic e-V processes (Loureiro 1993).

Figure 4.13 shows the EEDFs calculated in nitrogen for $E_{0}/n_{o} = 60\sqrt{2}$ Td ($1\,\mathrm{Td} = 1 \times 10^{-21}$ V m²), $\omega /n_{o} = 5 \times 10^{-16}$ m³ s⁻¹, and $T_{v} = T_{o} = 400$ K, i.e. in the absence of appreciable vibrational excitation, and for the different times during the half period of the RF electric field shown in the inset.

Fig. 4.13

EEDFs in N₂ for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $\omega /n_{o} = 5 \times 10^{-16}$ m³ s⁻¹, and $T_{v} = T_{o} = 400$ K. The various curves are the following instants with T denoting the field period: (A) 0; (B) T/6; (C) T/4; (D) T/3; (E) T/2 (Loureiro 1993)

For the considered RF field the EEDFs oscillate with twice the field frequency presenting a modulation of many orders of magnitude and a small phase delay, relative to the applied electric field, in those parts of the electron-energy range where ω is appreciably smaller than the relaxation frequency for energy transfer, ν _e , shown in Fig. 4.2. The sharp maximum of ν _e ∕n _o at $u \simeq 2$ eV, of about 10⁻¹³ m³ s⁻¹, due to vibrational excitation is clearly larger than the value chosen here for ω∕n _o , so that for electron energies around 2 eV the EEDFs follow the RF field in a quasi-stationary way excepting when the field goes through zero. We note that in this region of electron energies the EEDFs are maximum when the absolute value of the RF field passes through its maximum, decrease strongly as the absolute value of the field decreases, and reach a minimum when the field passes through zero. In particular, for zero field, the EEDFs are extremely reduced and most of the electrons have only very small energy.

The other parts of the EEDFs can be interpreted as well by looking at the dependence of the relaxation frequency ν _e on the electron energy shown in Fig. 4.2. There are two energy regions u < 1. 5 eV and 4 < u < 8 eV practically devoid of vibrational or electronic inelastic processes, which correspond to relatively small values of ν _e ∕n _o . In both regions the inequality ω ≪ ν _e is no longer valid so the EEDFs follow the RF field with a much smaller modulation and large phase delay. On the other hand, in the high-energy range u > 8 eV, dominated by the excitation of electronic states, the relaxation frequency ν _e has approximately the same amplitude as for $u \simeq 2$ eV, so that the EEDFs show a very large modulation as well. It is also interesting to note that, as a result of the nonequilibrium between the EEDFs and the applied RF field, the EEDFs are different at the instants t and $T/2 - t$, with T denoting the field period. The EEDFs are naturally larger when the RF is decreasing, 0 < t < T∕4, because of memory effect. Obviously, the differences between the EEDFs at both instants are decreasingly smaller as ν _e ∕n _o increases, which is the case as the electron energy increases from $\simeq$ 8 to $\simeq$ 14 eV (see curves B and D in Fig. 4.13).

Figure 4.14 shows the EEDFs calculated in nitrogen for the same values of $E_{0}/n_{o} = 60\sqrt{2}$ Td and $T_{v} = T_{o} = 400$ K as in Fig. 4.13, but for the higher value of $\omega /n_{o} = 1 \times 10^{-14}$ m³ s⁻¹. In this case we have ω > ν _e in large parts of the relevant range of the electron energies so that the amplitudes of the time modulation of the EEDFs are strongly diminished with respect to Fig. 4.13.

Fig. 4.14

EEDFs in N₂ for the same values of E ₀∕n _o and T _v  = T _o as in Fig. 4.13, but for $\omega /n_{o} = 1 \times 10^{-14}$ m³ s⁻¹. The various curves are for the following instants with T denoting the field period: (A) 0; (B) T/6; (C) T/4 (Loureiro 1993)

Figure 4.15 shows the EEDFs calculated in nitrogen, still for the same values of E ₀∕n _o and T _v  = T _o as in Figs. 4.13 and 4.14, but for the higher value of $\omega /n_{o} = 1 \times 10^{-13}$ m³ s⁻¹. In this case we have no time-modulation at all because the inequality ω > ν _e holds in the entire electron-energy range. On the other hand, as long as ω is smaller than the relaxation frequency for momentum transfer ν _m ^e, the time-averaged value of the EEDF, $\overline{F^{0}(u,t)} = F_{0}^{0}(u)$, is only slightly dependent on ω. In the limit ω ≪ ν _m ^e, we have $A \simeq 1$ and $B \simeq 0$ in equations (4.97), and equation (4.94) becomes identical to the Boltzmann equation for a DC field of magnitude equal to the effective field strength $E_{eff} = E_{0}/\sqrt{2}$, excepting that here we must also keep in equation (4.94) the term with F _2R ⁰. However, as the field frequency increases beyond ν _m ^e, the time-averaged value $\overline{F^{0}(u,t)} = F_{0}^{0}(u)$ is strongly reduced. As we have seen in Sects. 4.1.2 and 4.1.3, when ω ≫ ν _m ^e the time delay of the electron current density approaches T∕4 and the electrons cannot gain energy from the field on the average. In this limit the time average over one period of equation (4.116) yields only a vanishingly small value $\overline{P_{E}(t)} = \overline{(\mathbf{J_{e}}\;.\;\mathbf{E})} \simeq 0$. For frequencies of this order, which correspond to microwave fields at gas pressures typically $p \sim 100$ Pa, the EEDFs exhibit a strong peak at zero energy. See, for example, the EEDFs in argon for the highest values of ω∕n _o shown in Fig. 4.3.

Fig. 4.15

EEDFs in N₂ for the same values of E ₀∕n _o and T _v  = T _o as in Figs. 4.13 and 4.14, but for $\omega /n_{o} = 1 \times 10^{-13}$ m³ s⁻¹. The curves are for the instants: (A) 0; (B) T/4 (Loureiro 1993)

This latter aspect is well understood through the evolution of the drift velocity with the ω∕n _o values. Figure 4.16 shows the drift velocity v _ed = v _ed (t) e _z calculated for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $T_{v} = T_{o} = 400$ K, and various values of ω∕n _o between 5 × 10⁻¹⁶ and 1 × 10⁻¹² m³ s⁻¹. This figure shows that for $\omega /n_{o} < 1 \times 10^{-14}$ m³ s⁻¹ there is no phase delay with respect to the applied field $\mathbf{E}(t) = -\;E_{0}\;\cos (\omega t)\;\mathbf{e_{z}}$; for $\omega /n_{o} \simeq$ 1×10⁻¹³ m³ s⁻¹, the delay is $\simeq -\;\pi /4$ in agreement with the fact of $\omega \sim \nu _{m}^{e}$ over most of the relevant electron-energy range; and for the highest values of ω∕n _o the delay approaches $-\;\pi /2$. On the other hand, for the highest values of ω∕n _o the drift velocity strongly reduces in magnitude.

Fig. 4.16

Electron drift velocity in N₂ as a function of the reduced time t∕T, with T denoting the field period, for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $T_{v} = T_{o} = 400$ K, and the following values of ω∕n _o in m³ s⁻¹: (A) 5 × 10⁻¹⁶; (B) 1 × 10⁻¹⁴; (C) 1×10⁻¹³; (D) 5 × 10⁻¹³; (E) 1 × 10⁻¹² (Loureiro 1993)

The large modulation of the EEDFs in the electron-energy range 1. 5 < u < 4 eV is strongly reduced when we consider the effects produced by the e-V superelastic collisions. Figure 4.17 shows the EEDFs for the same values of E∕n _o , ω∕n _o , and T _o as in Fig. 4.13, but for the case of vibrational excitation corresponding to a characteristic vibrational temperature T _v  = 4000 K. The comparison between Figs. 4.13 and 4.17 allows to evaluate the effects caused by the e-V superelastic collisions in reducing the effectiveness of the characteristic relaxation frequency for energy transfer ν _e , shown in Fig. 4.2, in the electron-energy range under discussion. The e-V superelastic processes produce a decrease in the amplitude and an increase in phase delay of the EEDFs, for energies 1. 5 < u < 4 eV, as well as an enhancement of the high-energy tail of the EEDFs. This latter aspect has already been discussed for the EEDFs obtained in a DC field in Sect. 3.6 Finally, it is worth noting that the enhancement of the high-energy tail of the EEDFs from Figs. 4.13 to 4.17 is not very significant because we have chosen here a relatively high value of E ₀∕n _o . In the case of lower E ₀∕n _o values, e.g. as small as $30\sqrt{2}$ Td, the effects produced by the e-V superelastic collisions would become much larger (see Sect. 3.6).

Fig. 4.17

EEDF in N₂ as in Fig. 4.13 but for T _v  = 4000 K (Loureiro 1993)

The EEDFs in hydrogen are much lesser modulated than in nitrogen due to the relaxation frequency for energy transfer is about one order of magnitude smaller. Figure 4.18 shows the EEDFs in H₂ calculated for $E_{0}/n_{o} = 30\sqrt{2}$ Td, $\omega /n_{o} = 2 \times 10^{-15}$ m³ s⁻¹, $T_{v} = T_{o} = 400$ K, and for the different times shown in the inset. As seen from this figure, as compared with those presented before for N₂, the time modulation is clearly less pronounced here, which is a consequence of the relative magnitudes of the relaxation frequency for energy transfer in both gases plotted in Fig. 4.2. Besides the fact of ν _e ∕n _o is smaller by one order of magnitude in H₂, it does not present any pronounced maximum due to the dissipation of electron energy in vibrational excitation as it exists in N₂ at $\sim$ 2 eV. The absence of such a maximum in H₂ also signifies that the effects of e-V superelastic collisions produce only minor modifications in the EEDFs in H₂ as compared to those observed in N₂.

Fig. 4.18

EEDFs in H₂ for $E_{0}/n_{o} = 30\sqrt{2}$ Td, $\omega /n_{o} = 2 \times 10^{-15}$ m³ s⁻¹, $T_{v} = T_{o} = 400$ K, and for the following instants: (A) 0; (B) T/6; (C) T/4 (Loureiro 1993)

Once the EEDF are obtained, the various energy-averaged quantities with interest for plasma modelling can also be obtained as a function of time. Figure 4.19 shows the energy-averaged electron energy, < ​​ u​​ > ​​ (t), calculated in nitrogen for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $T_{v} = T_{o} = 400$ K, and various values of ω∕n _o between 5 × 10⁻¹⁶ and 1 × 10⁻¹² m³ s⁻¹. This figure shows that there is a marked reduction in the time modulation (at the frequency 2ω) as ω∕n _o increases together with a reduction in the amplitude of the time-averaged values. There is also an increasing phase delay which approaches − π, i.e. a time delay $-\;T/4$, as $\omega \rightarrow \infty$, but this latter aspect is not particularly visible in this figure.

Fig. 4.19

Energy-averaged electron energy in N₂ as a function of the reduced time t∕T, for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $T_{v} = T_{o} = 400$ K, and the following values of ω∕n _o in m³ s⁻¹: (A) 5 × 10⁻¹⁶; (B) 1 × 10⁻¹⁵; (C) 1 × 10⁻¹⁴; (D) 1×10⁻¹³; (E) 1 × 10⁻¹² (Loureiro 1993)

Finally, Fig. 4.20 shows the electron rate coefficient, C _X ^B(t), for excitation of the triplet state N₂(B $^{3}\Pi _{g}$) at the same conditions as the previous figure for < ​​ u​​ > ​​ (t). Both the time modulation and the reduction of amplitude for the highest values of ω are particularly visible now because the rate coefficient depends of the high-energy tail of the EEDF only, whereas < ​​ u​​ > ​​ (t) in Fig. 4.19 results from an integration over the entire energy range. Also the increasing phase delay approaching $-\;T/4$ is more clearly visible in this figure. It is worth noting here that instead of equation (4.106) the electron rate coefficient for excitation of a given $i \rightarrow j$ transition may also be written under the form

$$\displaystyle{ C_{ij}(t)\; =\; (C_{ij})_{0} +\; (C_{ij})_{2}\;\cos (2\omega t +\phi _{ 2}^{0})\;, }$$

(4.117)

with $(C_{ij})_{2} \simeq (C_{ij})_{0}$ and $\phi _{2}^{0} \simeq 0$ when ω ≪ ν _e , and (C _ij )₂ ≪ (C _ij )₀ and $\phi _{2}^{0} \simeq -\;\pi$ when ω ≫ ν _e . Then, the term (C _ij )₀ is strongly reduced when ω ≫ ν _m ^e.

Fig. 4.20

Electron rate coefficient for excitation of N₂(B $^{3}\Pi _{g}$) state as a function of the reduced time t∕T, for $E_{0}/n_{o} = 60\sqrt{2}$ Td, $T_{v} = T_{o} = 400$ K, and the following values of ω∕n _o in m³ s⁻¹: (A) 5 × 10⁻¹⁶; (B) 1 × 10⁻¹⁵; (C) 1 × 10⁻¹⁴; (D) 1 × 10⁻¹³ (Loureiro 1993)

It follows from the present analysis that the instantaneous energy-averaged power absorbed from the RF field and the instantaneous power lost in electron collisions are not in phase with each other. The time-modulation at frequency 2ω of the instantaneous power lost by electron collisions is strongly diminished and its phase-delay approaches −π (i.e. $\Delta t = T/4$) as ω∕n _o increases, such as it occurred with < ​​ u​​ > ​ (t) and C _X ^B(t). On the contrary, the instantaneous power absorbed from the field shows no reduction of its modulation and has an increasing phase-delay going to $-\pi /2$ (i.e. $\Delta t = T/8$, see equation (4.116) and Exercise 4.6). We note that $P_{E}(t) \simeq -\;(P_{E})_{2I}\sin (2\omega t)$ in equation (4.116) as $\omega \rightarrow \infty$. Obviously, the time-averaged values of the energy-loss and energy-gain terms exactly compensate each other and both go towards zero as ω∕n _o increases beyond ν _m ^e∕n _o .

Appendices

Appendices

1.1 A.4.1 Effective Collision Frequency and Electron Density in High-Frequency Discharges

When the effective collision frequency for momentum transfer ν _m ^e, given by equation (3.139), is independent of electron velocity, the equation for electron momentum conservation (3.60) in high-frequency (HF) fields reduces to

$$\displaystyle{ n_{e}\;m\;\frac{\mathbf{dv_{ed}}} {dt} \; +\; n_{e}\;e\;\mathbf{E}\; =\; -\;n_{e}\;m\;\nu _{m}^{e}\;\mathbf{v_{ ed}}, }$$

(4.118)

with v _ed denoting the electron drift velocity. Here, we are assuming the field frequency much larger than the characteristic relaxation frequency for energy transfer ω ≫ ν _e , in order the electron density does not change appreciably during a cycle of the field oscillation. Then, for an applied field of frequency ω, we have $\mathbf{dv_{ed}}/dt = j\omega \;\mathbf{v_{ed}}$, and the drift velocity is given by the well known formula

$$\displaystyle{ \mathbf{v_{ed}}\; =\; -\; \frac{e} {m\;(\nu _{m}^{e} + j\omega )}\;\mathbf{E}\;. }$$

(4.119)

The electron current density $\mathbf{J_{e}} = -\;en_{e}\;\mathbf{v_{ed}}$, allows to obtain the complex conductivity from $\mathbf{J_{e}} = \overline{\sigma _{c}}_{e}\;\mathbf{E}$, with the form

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}n_{e}} {m\;(\nu _{m}^{e} + j\omega )}\;, }$$

(4.120)

whereas the time-averaged power absorbed from the field (4.15) is

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{1} {2}\; \frac{e^{2}n_{e}\;\nu _{m}^{e}} {m\;(\nu _{m}^{e\;2} +\omega ^{2})}\;E_{0}^{\;2}\;. }$$

(4.121)

However, when the frequency ν _m ^e is velocity-dependent, we must consider the expression (4.9) for the complex anisotropic component

$$\displaystyle{ \mathbf{f_{e}^{1}}\; =\; \overline{f_{ e}^{1}}(v_{ e})\;e^{j\omega t}\;\mathbf{e_{ z}}\; =\; -\; \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{eE_{0}} {m} \,\frac{df_{e}^{0}} {dv_{e}} \;e^{j\omega t}\;\mathbf{e_{ z}}\;, }$$

(4.122)

in the collision term of equation for momentum conservation (3.55)

$$\displaystyle{ \mathbf{I_{1}}\; =\; -\int _{0}^{\infty }m\;\frac{v_{e}} {3} \;\nu _{m}^{e}\;\mathbf{f_{ e}^{1}}\;4\pi v_{ e}^{\;2}\;dv_{ e}\;, }$$

(4.123)

assuming here the interactions of collision type only and the applied field with the form $\mathbf{E} = -\;E_{0}\;e^{j\omega t}\;\mathbf{e_{z}}$. The complex drift velocity (4.10) and (4.11) is then

$$\displaystyle{ \mathbf{v_{ed}}\; =\; \overline{V _{e}}_{d}\;e^{j\omega t}\;\mathbf{e_{ z}}\; =\; -\;\frac{eE_{0}} {n_{e}m}\;\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \,dv_{e}\;e^{j\omega t}\;\mathbf{e_{ z}}\;, }$$

(4.124)

so that substituting equation (4.123) in equation (4.118), we find

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V _{e}}_{d}\; -\; n_{e}\;eE_{0}\; =\; eE_{0}\int _{0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;. }$$

(4.125)

In the case of a velocity-independent frequency ν _m ^e, we immediately obtain equation (4.119) from (4.125). In fact, integrating by parts

$$\displaystyle{ \int _{0}^{\infty }\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\; =\; -\int _{0}^{\infty }f_{ e}^{0}\;4\pi v_{ e}^{\;2}\;dv_{ e}\; =\; -\;n_{e}\;, }$$

(4.126)

we find

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V }_{ed}\; -\; n_{e}\;eE_{0}\; =\; -\;eE_{0}\; \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;n_{e} }$$

(4.127)

and therefore

$$\displaystyle{ \overline{V _{e}}_{d}\; =\; \frac{eE_{0}} {m\;(\nu _{m}^{e} + j\omega )}\;. }$$

(4.128)

A different situation occurs as ν _m ^e depends on the electron velocity. In this case in order to express equation (4.125) with the form (4.118) and to obtain an equivalent expression (4.120) for the complex conductivity, we should consider an effective complex collision frequency $\overline{\nu }_{eff}$ in equation (4.125), such as

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V _{e}}_{d}\; -\; n_{e}\;eE_{0}\; =\; -\;n_{e}m\;\overline{\nu }_{eff}\;\overline{V _{e}}_{d}\;. }$$

(4.129)

Then, using equation (4.124) for $\overline{V _{e}}_{d}$, we obtain

$$\displaystyle{ \overline{\nu }_{eff}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;v_{e}^{\;3}\;dv_{ e}\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;v_{e}^{\;3}\;dv_{ e}\right )^{\!-1}\;. }$$

(4.130)

Obviously $\overline{\nu }_{eff}$ allows to obtain ν _m ^e again as this latter frequency is time-independent. When such is not the case, equation (4.129) allows to obtain an equivalent expression to equation (4.128) but with a complex frequency in place of ν _m ^e

$$\displaystyle{ \overline{V _{e}}_{d}\; =\; \frac{eE_{0}} {m\;(\overline{\nu }_{eff} + j\omega )}\;, }$$

(4.131)

whereas the complex conductivity is

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}n_{e}} {m\;(\overline{\nu }_{eff} + j\omega )}\;. }$$

(4.132)

However, separating $\overline{\nu }_{eff}$ in real and imaginary parts, $\overline{\nu }_{eff} =\nu _{R} + j\;\nu _{I}$, we obtain an exact equivalent expression (4.120) as follows

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}\;n_{e}^{{\ast}}} {m\;(\nu ^{{\ast}} + j\omega )}\;, }$$

(4.133)

with n _e ^∗ and ν ^∗ denoting an effective electron density and a new effective collision frequency given by

$$\displaystyle\begin{array}{rcl} & & n_{e}^{{\ast}}\; =\; \frac{n_{e}} {1 +\nu _{I}/\omega }{}\end{array}$$

(4.134)

$$\displaystyle\begin{array}{rcl} & & \nu ^{{\ast}}\; =\; \frac{\nu _{R}} {1 +\nu _{I}/\omega }\,.{}\end{array}$$

(4.135)

Separating equation (4.130) in equations for the real and imaginary parts of complex quantities, we obtain from equation for the imaginary part

$$\displaystyle{ \nu _{R}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\; =\; \left (1 + \frac{\nu _{I}} {\omega } \right )\int _{0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\;, }$$

(4.136)

from which we may write

$$\displaystyle{ \nu ^{{\ast}}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\right )^{\!-1}\;. }$$

(4.137)

On the other hand, from the comparison between the imaginary part of the complex conductivity (4.12)

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; en_{e}\;\overline{\mu _{e}}\; =\; -\;\frac{e^{2}} {m}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} }$$

(4.138)

and the imaginary part of equation (4.133), we obtain

$$\displaystyle{ n_{e}^{{\ast}}\; =\; -\;(\nu ^{{\ast}\;2} +\omega ^{2})\int _{ 0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;. }$$

(4.139)

When ν _m ^e is velocity-independent, equations (4.137) and (4.139) reduce to ν ^∗ = ν _m ^e and n _e ^∗ = n _e . Finally, in terms of the EEDF normalized through equation (3.166), ν ^∗ and n _e ^∗ are written as follows

$$\displaystyle\begin{array}{rcl} & & \nu ^{{\ast}}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\right )^{\!-1}{}\end{array}$$

(4.140)

$$\displaystyle\begin{array}{rcl} & & n_{e}^{{\ast}}\; =\; -\;\frac{2} {3}\;n_{e}\;(\nu ^{{\ast}\;2} +\omega ^{2})\int _{ 0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\;.{}\end{array}$$

(4.141)

As firstly pointed out in Whitmer and Herrmann (1966), the two effective parameters ν ^∗ and n _e ^∗ have the advantage to permit the writing of a relation of the type (4.120) for the electron conductivity in HF fields, for gases in which the collision frequency for momentum transfer depends on the electron energy.

Exercises

Exercise 4.1:

Determine the expression of the velocity-averaged collision frequency for momentum transfer valid as ω ≫ ν _m ^e, that allows to keep the time-averaged power absorbed from the field $\overline{P_{E}(t)}$ with the same form as the effective collision frequency for momentum transfer is independent of energy.

Resolution:

When ω ≫ ν _m ^e and ν _m ^e(v _e ) = const, we obtain the following expression for the time-averaged power absorbed from the field from equations (4.15) and (4.16)

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \;\frac{\nu _{m}^{e}} {\omega ^{2}} \;E_{rms}^{\;2}\;,}$$

so that as ν _m ^e(v _e ) ≠ const, we must consider the energy-averaged frequency

$$\displaystyle{\nu ^{{\ast}}\; =\; -\; \frac{1} {n_{e}}\int _{0}^{\infty }\nu _{ m}^{e}\;\frac{df_{e}^{0}} {dv_{e}} \;\;\frac{4\pi v_{e}^{3}} {3} \;dv_{e}\;,}$$

in order an equivalent expression for $\overline{P_{E}(t)}$ may be used

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \;\frac{\nu ^{{\ast}}} {\omega ^{2}}\;E_{rms}^{\;2}\;.}$$

Integrating by parts the frequency ν ^∗ can also be written as

$$\displaystyle{\nu ^{{\ast}}\; =\;< \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{\nu _{m}^{e}\;v_{e}^{\;3}} {3} \right ) >\;.}$$

Using the normalization (3.166) the first expression for ν ^∗ writes as follows

$$\displaystyle{\nu ^{{\ast}}\; =\; -\;\frac{2} {3}\int _{0}^{\infty }\nu _{ m}^{e}\;u^{3/2}\; \frac{df} {du}\;du\;.}$$

This frequency corresponds to the effective collision frequency introduced in Appendix A.4.1, in the limit ω ≫ ν _m ^e, and it differs from the effective collision frequency $\nu _{m}^{{\prime}}$ used with a DC field to write the electron mobility as $\mu _{e} = e/(m\;\nu _{m}^{{\prime}})$ (see Exercise 3.6). Obviously, both result in ν _m ^e when the frequency ν _m ^e is velocity-independent.

Exercise 4.2:

Obtain the time-averaged power absorbed from the field by the electrons, for the case of a high-frequency field ω ≫ ν _m ^e, directly from equation (4.22) for the isotropic component of the electron velocity distribution f _e ⁰.

Resolution:

Multiplying the symmetric of the left-hand side member of equation (4.22) by the electron velocity $u = \frac{1} {2}mv_{e}^{\;2}$ and integrating in all velocity space, we obtain

$$\displaystyle{\overline{P_{E}(t)}\; =\;\int _{ 0}^{\infty }u\;\;\frac{1} {2}\left (\frac{eE_{0}} {m} \right )^{2} \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{v_{e}^{\;2}} {3} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \right )\;4\pi v_{e}^{\;2}\;dv_{ e}\;.}$$

Integrating now by parts, we find

$$\displaystyle\begin{array}{rcl} \overline{P_{E}(t)}& =& -\;\frac{1} {2}\left (\frac{eE_{0}} {m} \right )^{2}\int _{ 0}^{\infty } \frac{d} {dv_{e}}\left (\frac{1} {2}\;mv_{e}^{\;2}\right )\;\frac{v_{e}^{\;2}} {3} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\;\frac{df_{e}^{0}} {dv_{e}} \;4\pi \;dv_{e} {}\\ & =& -\;\frac{1} {2}\;\frac{(eE_{0})^{2}} {m} \int _{0}^{\infty }\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} {}\\ \end{array}$$

and identifying Re$\{\overline{\sigma _{c}}_{e}\}$ given by equation (4.16) in this expression, we can write $\overline{P_{E}(t)}$ under the form

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{1} {2}\;\mbox{ Re}\{\overline{\sigma _{c}}_{e}\}\;E_{0}^{\;2}\;,}$$

in accordance with equation (4.15).

Exercise 4.3:

Write the expression for the mean power absorbed from the field per electron, $\overline{P_{E}(t)}/n_{e}$, in terms of the mean energy absorbed per collision u _c (u) by an electron of energy u given by equation (4.25).

Resolution:

Replacing equation (4.25) into equations (4.15) and (4.16), we find

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; -\; \frac{1} {n_{e}}\int _{0}^{\infty }\nu _{ m}^{e}\;u_{ c}\;\;\frac{df_{e}^{0}} {dv_{e}} \;\;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;.}$$

Making the replacement of v _e with u and using the EEDF normalized such as

$$\displaystyle{\int _{0}^{\infty }f\;\sqrt{u}\;du\; =\; 1\;,}$$

with both distributions linked each other through equation (3.167), we obtain

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; -\;\frac{2} {3}\int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\;\; \frac{df} {du}\;\;du\;.}$$

On the other hand, starting from the first expression in the answer to Exercise 4.2, we also obtain

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; \frac{2} {3}\int _{0}^{\infty }u\; \frac{d} {du}\left (u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\;\; \frac{df} {du}\right )\;du}$$

and integrating by parts we find the same expression.

Exercise 4.4:

Write the expression for the complex mobility (4.12) and for the time-averaged absorbed power from the field per volume unit (4.15) and (4.16) in the case of an HF field, using the EEDF normalized through equation (3.166).

Resolution:

The two expressions asked are:

$$\displaystyle\begin{array}{rcl} \overline{\mu _{e}}& =& -\;\frac{2} {3}\; \frac{e} {m}\int _{0}^{\infty } \frac{u^{3/2}} {\nu _{m}^{e} + j\omega }\;\frac{df} {du}\;du {}\\ \overline{P_{E}(t)}& =& -\;\frac{2} {3}\;\frac{e^{2}n_{e}} {m} \;E_{rms}^{\;2}\int _{ 0}^{\infty }u^{3/2}\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\; \frac{df} {du}\;du\;. {}\\ \end{array}$$

Although the notation seems to be identical the first expression represents a complex quantity and the second a time-averaged value of a real quantity.

Exercise 4.5:

Write the expressions of the time-averaged gain produced by the field in velocity space, $\overline{G_{E}(t)}$, and of the power absorbed by the electrons in terms of this gain, for the case of a HF frequency electric field.

Resolution:

The gain produced by a HF field in velocity space is obtained by inserting (4.20) into equations (3.92) and (3.95)

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;\frac{1} {6}\left (\frac{eE_{0}} {m} \right )^{2} \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \;4\pi v_{e}^{\;2}\;,}$$

so that using equation (4.25), we still have

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;\frac{1} {3}\;\frac{\nu _{m}^{e}\;u_{c}} {m} \;\frac{df_{e}^{0}} {dv_{e}} \;4\pi v_{e}^{\;2}\; >\; 0\;.}$$

On the other hand using equation (3.119), the time-averaged power absorbed by the electrons is given by

$$\displaystyle\begin{array}{rcl} \overline{P_{E}(t)}& =& -\int _{0}^{\infty }u\;\frac{d\overline{G_{E}}} {dv_{e}} \;dv_{e}\; =\;\int _{ 0}^{\infty }\overline{G_{ E}}\;mv_{e}\;dv_{e} {}\\ & =& -\int _{0}^{\infty }\nu _{ m}^{e}\;u_{ c}\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{3}} {3} \;dv_{e}\;. {}\\ \end{array}$$

In terms of the EEDF normalized through equation (3.166), we find

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;n_{e}\;\frac{2} {3}\;u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\; \frac{df} {du}}$$

and

$$\displaystyle{\overline{P_{E}(t)}\; =\int _{ 0}^{\infty }\overline{G_{ E}}\;du\; =\; -\;n_{e}\;\frac{2} {3}\int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\; \frac{df} {du}\;du\;.}$$

Exercise 4.6:

By writing the drift velocity of electrons under a RF field as $v_{ed}(t) = V _{d0}\;\cos (\omega t + \Phi )$ show that the power absorbed from the field (4.116) can be expressed as

$$\displaystyle{P_{E}(t)\; =\; (P_{E})_{0} +\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\cos (2\omega t + \Phi )\;.}$$

Obtain the limit values of P _E (t) at low and high field frequencies.

Resolution:

The power absorbed is given by

$$\displaystyle\begin{array}{rcl} P_{E}(t)& =& (\mathbf{J_{e}}\;.\;\mathbf{E})\; =\; en_{e}\;V _{d0}\;\mbox{ Re}\{e^{j(\omega t+\Phi )}\}\;E_{ 0}\;\mbox{ Re}\{e^{j\omega t}\} {}\\ & =& \frac{en_{e}E_{0}} {2} \;V _{d0}\;[\cos (\Phi ) +\cos (2\omega t + \Phi )]\;, {}\\ \end{array}$$

so that using equation (4.111) we obtain

$$\displaystyle{P_{E}(t)\; =\; (P_{E})_{0} +\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\cos (2\omega t + \Phi )\;.}$$

When ω ≪ ν _m ^e, v _ed (t) and E(t) are in phase so that

$$\displaystyle{P_{E}(t)\; =\; en_{e}E_{0}V _{d0}\;\cos ^{2}(\omega t)\;,}$$

with $V _{d0} = eE_{0}/(m\nu _{m}^{e})$ and $\overline{P_{E}(t)} = en_{e}E_{0}V _{d0}/2$. On the other hand, when ω ≫ ν _m ^e, the phase shift of the drift velocity approaches $-\;\pi /2$ and we obtain

$$\displaystyle{P_{E}(t)\; =\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\sin (2\omega t)\;,}$$

with $V _{d0} = eE_{0}/(m\omega )$ in accordance with equation (4.18), and therefore $\overline{P_{E}(t)} = 0$.