5.1. Resonance of an ion with a diocotron mode

The resonance condition between the transverse motion of an ion and a wave running along the azimuth in non-neutral plasma has the following form in the laboratory frame of reference (Davidson Reference Davidson1974):

(5.1)\begin{equation}{\omega _{\textrm{res}}} \approx m\omega _{\textrm{rot}}^i + n{\varOmega _i}\quad (m = 0, + 1, + 2, \ldots ,\;n = 0, \pm 1, \pm 2, \ldots ).\end{equation}

Here

(5.2)\begin{equation}\omega _{\textrm{rot}}^i = ( - {\omega _{\textrm{ci}}} + {\varOmega _i})/2 > 0\end{equation}

is the ‘slow’ ion rotation frequency, ${\omega _{\textrm{ci}}} = {e_i}B/({m_i}c) > 0$ is the ion cyclotron frequency and

(5.3)\begin{equation}{\varOmega _i} = {\omega _{\textrm{ci}}}{[1 + ({m_i}/{m_e})q(1 - f)]^{1/2}} = |{\omega _{\textrm{ce}}}|{[{({m_e}/{m_i})^2} + ({m_e}/{m_i})q(1 - f)]^{1/2}}\end{equation}

is the ‘modified’ cyclotron frequency of ions.

As is shown in § 3.5, diocotron mode (‘1’) (3.2) reaches the area of low frequencies and the zero frequency only when $R_p^2/R_w^2 > f$ (see figure 2a). We consider this condition to be fulfilled. The dependencies on q of the diocotron mode (‘1’) (3.2) and of the resonance frequencies of ions (5.1) for $m = 1$ and multiplicities of resonances $n ={-} 1,0, + 1$ are shown in figure 3.

Figure 3. Resonance frequencies of ions (5.1) (denoted by black solid lines) for $m ={+} 1$ and $n ={+} 1,\;0,\; - 1$ and the frequency of the diocotron mode ${\omega _1}$ (3.2) (denoted by a dotted line and the number ‘1’). The intersections of the frequencies are indicated by squares. The red square shows the Cherenkov resonance studied in the present article $(n = 0)$. The dependence of $\Delta \omega$ (5.15) is also shown (blue solid line). All frequencies are normalized to ${\omega _{\textrm{ci}}}$. Parameters of calculation: ${m_i} = 40\textrm{ a}\textrm{.u}\textrm{.}$ (argon); ${k_z}{R_p} = 0.03$; ${R_p}/{R_w} = 0.5$; $f = 0.1$.

In experiments (Vlasov et al. Reference Vlasov, Dobrokhotov and Zharinov1966; Nezlin Reference Nezlin1982; Sakawa & Joshi Reference Sakawa and Joshi2000; Jaeger Reference Jaeger2010; Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011; David Reference David2017), unstable low-frequency oscillations rotating along the azimuth in the positive direction $(\omega /m > 0)$ are observed. So, we study positive resonance frequencies. For $m = 1$ the lowest positive resonance ion frequency (5.1) is realized at $n = 0$. This is Cherenkov resonance between the diocotron mode and the ion rotation. The instability near this resonance was studied by Levy et al. (Reference Levy, Daugherty and Buneman1969) in the hydrodynamic description of electrons and ions and ${k_z} = 0$. We determine the position of the resonance $({\omega _1} = {\omega _{\textrm{res}}})$ for ${k_z} \ne 0$. Substituting the expression for the frequency of the diocotron mode (3.2) into the left-hand side of (5.1) and putting $m = 1$ and $n = 0$ in the right-hand side, we obtain the following equation for $q$:

(5.4)\begin{equation}\frac{{{m_i}}}{{2{m_e}}}\left[ {q\left( {\frac{{R_p^2}}{{R_w^2}} - f} \right) - {{({k_z}{R_p})}^2}} \right] ={-} 1 + {\left[ {1 + \frac{{{m_i}}}{{{m_e}}}q(1 - f)} \right]^{1/2}}.\end{equation}

By introducing a variable

(5.5)\begin{equation}y \equiv \frac{{{\varOmega _i}}}{{{\omega _{\textrm{ci}}}}} = {\left[ {1 + \frac{{{m_i}}}{{{m_e}}}q(1 - f)} \right]^{1/2}} \ge 1,\end{equation}

we obtain a simpler equation for $y$:

(5.6)\begin{equation}\eta {y^2} - 2y - \eta - k_z^2R_p^2({m_i}/{m_e}) + 2 = 0.\end{equation}

In (5.6), the following notation is introduced:

(5.7)\begin{equation}\eta \equiv (R_p^2/R_w^2 - f)/(1 - f),\quad R_p^2/R_w^2 > f.\end{equation}

We are interested in the area $R_p^2/R_w^2 > f$ where $\eta$ is positive. As follows from (5.7), the inequality $\eta < 1$ is always satisfied in this area.

From two roots of (5.6),

(5.8)\begin{equation}y = {\eta ^{ - 1}}[ + 1 \pm \sqrt {{{(\eta - 1)}^2} + \eta {{({k_z}{R_p})}^2}{m_i}/{m_e}} ],\end{equation}

only one root with the ‘+’ sign satisfies the inequality (5.5). We study only this root. It determines the resonance frequency of the ion (5.1) and the value of q at the point of intersection with the diocotron mode (3.2):

(5.9a,b)\begin{equation}{\omega _{\textrm{res}}}/{\omega _{\textrm{ci}}} = (y - 1)/2,\quad q \equiv {q_{\textrm{res}}} = ({m_e}/{m_i})({y^2} - 1){(1 - f)^{ - 1}}.\end{equation}

The resonance values (5.9) give an idea of the frequency and value of the parameter q near which the resonance electron–ion instability can arise.

Expressions (5.8) and (5.9) are simplified in limiting cases. When

(5.10)\begin{equation}[\eta /{(\eta - 1)^2}]{({k_z}{R_p})^2} \ll {m_e}/{m_i},\end{equation}

solutions (5.9) take the form

(5.11a,b)\begin{equation}{\omega _{\textrm{res}}}/{\omega _{\textrm{ci}}} \approx \frac{{1 - R_p^2/R_w^2}}{{R_p^2/R_w^2 - f}},\quad {q_{\textrm{res}}} \approx 4\frac{{{m_e}}}{{{m_i}}}\frac{{1 - R_p^2/R_w^2}}{{{{(R_p^2/R_w^2 - f)}^2}}}.\end{equation}

The resonance frequency ${\omega _{\textrm{res}}}$ (5.11) is proportional to the ion cyclotron frequency $({\omega _{\textrm{res}}} \propto {\omega _{\textrm{ci}}})$. The coefficient of proportionality is determined by the degree of filling of the waveguide with plasma $(R_p^2/R_w^2)$ and by the degree of plasma charge neutralization $(f)$. Expressions (5.11) do not depend on ${k_z}$. They are valid for sufficiently small values of ${k_z}{R_p}$ (5.10) and for the case ${k_z}{R_p} = 0$ studied by Levy et al. (Reference Levy, Daugherty and Buneman1969).

The expression for ${q_{\textrm{res}}}$ (5.11) gives numerical values that exactly correspond to the coordinates of the maxima of the growth rate obtained by Levy et al. (Reference Levy, Daugherty and Buneman1969). These coordinates are presented by dotted lines in figure 2 of their article in variables f and $\lambda = 4({m_e}/{m_i})/[q(1 - f)]$. The analytical expression for resonance values of $\lambda$ has the form

(5.12)\begin{equation}{\lambda _{\textrm{res}}} = 4\frac{{{m_e}}}{{{m_i}}}\frac{1}{{{q_{\textrm{res}}}(1 - f)}} = \frac{{{{(R_p^2/R_w^2 - f)}^2}}}{{[(1 - R_p^2/R_w^2)(1 - f)]}}.\end{equation}

It is valid within the entire acceptable range of variations of f and $R_p^2/R_w^2$. Expression (5.12) was written by Peurrung et al. (Reference Peurrung, Notte and Fajans1993) for the case $f = 0$. The value of ${\lambda _{\textrm{res}}}$ (5.12) does not depend on the mass of the ion $({m_e}/{m_i})$.

The resonance frequency ${\omega _{\textrm{res}}}$ (5.11) is lower than the ion cyclotron frequency $({\omega _{\textrm{res}}} < {\omega _{\textrm{ci}}})$ in the area

(5.13)\begin{equation}R_p^2/R_w^2 > (1 + f)/2.\end{equation}

This area is shown in figure 4. It takes the form of a triangle. On the lower boundary of the triangle, the equality ${\omega _{\textrm{res}}} = {\omega _{\textrm{ci}}}$ is fulfilled. On the upper boundary $(R_p^2/R_w^2 \to 1)$, we have ${\omega _{\textrm{res}}} \to 0$.

Figure 4. The area of low resonance frequencies $({\omega _{\textrm{res}}} \le {\omega _{\textrm{ci}}})$, with calculation using formula (5.11). The grey-shaded area in which $R_p^2/R_w^2 < f$ is not considered. The inequality (5.7) is not satisfied inside it.

If the inequality opposite to (5.10) is satisfied, the solution of the first equation (5.9) takes the following form:

(5.14)\begin{equation}{\omega _{\textrm{res}}}/{\omega _{\textrm{ci}}} \approx \frac{1}{2}\left\{ {\frac{{1 - R_p^2/R_w^2}}{{R_p^2/R_w^2 - f}} + {{\left[ {\frac{{(1 - f)}}{{(R_p^2/R_w^2 - f)}}k_z^2R_p^2\frac{{{m_i}}}{{{m_e}}}} \right]}^{1/2}}} \right\}.\end{equation}

The value of ${q_{\textrm{res}}}$ is determined by equation (5.9b). The second term in (5.14) is large compared with the first term. The resonance frequency (5.14) is always high compared with the ion cyclotron frequency $({\omega _{\textrm{res}}} \gg {\omega _{\textrm{ci}}})$. Expression (5.14) gives a larger value of ${\omega _{\textrm{res}}}$ than expression (5.11). With a decrease in the parameter $({k_z}{R_p})$, the frequency ${\omega _{\textrm{res}}}$ decreases to the level (5.11).

When $f \to R_p^2/R_w^2$ the value of ${q_{\textrm{res}}}$ increases. Note that expressions (5.4)–(5.14), as well as (3.2), are valid when ${k_z}{R_p} \ll 1$, ${q_{\textrm{res}}} \ll {q_{\max }}$.

5.2. Numerical estimations of resonance frequencies for typical values of experimental parameters

We obtain numerical values of the resonance frequencies from formulas (5.7)–(5.14) for experiments in which non-neutral plasma was produced as secondary plasma in an electron beam channel (Vlasov et al. Reference Vlasov, Dobrokhotov and Zharinov1966; Nezlin Reference Nezlin1982; Sakawa & Joshi Reference Sakawa and Joshi2000; Jaeger Reference Jaeger2010; Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011; David Reference David2017).

We choose the following typical values of the experimental parameters. Geometrical parameters of the plasma cylinder: ${R_p} \sim 1\;\textrm{cm}$; length, $L \sim 100\;\textrm{cm}$. The degree of plasma charge neutralization $f = 0.1$, and the working gas is argon (${m_i} = 40\;\textrm{a}\textrm{.u}\textrm{.}$, ${m_e}/{m_i} \approx 1.36 \times {10^{ - 5}}$), which is often used in experiments. At first we estimate the minimum value of ${k_z}$ for a plasma cylinder of finite length in the ‘usual’ way: ${k_z} \sim {\rm \pi}/L$. We give estimations for different degrees of filling of the waveguide with plasma: ${R_p}/{R_w} = 0.5;\textrm{ }0.75;\textrm{ }0.9$. These values were used in calculations by Levy et al. (Reference Levy, Daugherty and Buneman1969).

The inequality opposite to (5.10) is satisfied for chosen values of the parameters. In this case the resonance frequency ${\omega _{\textrm{res}}}$ and ${q_{\textrm{res}}}$ are determined by formula (5.14). It gives the numerical values shown in table 1. Note that the values of ${\omega _{\textrm{res}}}$ and ${q_{\textrm{res}}}$ for ${R_p}/{R_w} = 0.5$ are consistent with the position of the resonance in figure 3 (the red square).

Table 1. The resonance frequency ${\omega _{\textrm{res}}}$ and ${q_{\textrm{res}}}$ for different values of parameter ${R_p}/{R_w}$ (${k_z}{R_p} \simeq 0.03$, $f = 0.1$, ${m_e}/{m_i} \approx 1.36 \times {10^{ - 5}}$ (argon)). Calculations are according to formula (5.14).

The resonance frequencies in table 1 are high compared with the ion cyclotron frequency ${\omega _{\textrm{ci}}}$. For lighter ions (for example, hydrogen, helium), a thinner or a longer plasma cylinder, the left-hand side of inequality (5.10) can be of the same order as the right-hand side and even smaller. As a result, the resonance frequency will be determined by expression (5.11), which gives a smaller value of ${\omega _{\textrm{res}}}$.

In such estimations, it is important to correctly take into account the finite value of ${k_z}$ in experiments because of the sharp dependence of the diocotron frequency ${\omega _1}$ (3.2) on ${k_z}$. The factor of finite plasma length, which is present in every experiment, greatly complicates the theoretical consideration of the stability problem. In this regard, we note the article by Prasad & O'Neil (Reference Prasad and O'neil1983), in which the eigenfrequencies and eigenmodes of a long $({R_p}/L \ll 1)$ plasma cylinder of finite length, located in an infinite metal cylindrical waveguide, are determined. They showed that in such a plasma model the diocotron mode has a longitudinal wavelength that is much longer than the plasma length: $\textrm{|}{k_z}\mathrm{|\sim [}(1/|{\varepsilon _3}|)(L/{R_p}){\textrm{]}^{1/2}}(1/L) \ll (1/L)$. Evaluating $\textrm{|}{\varepsilon _3}\mathrm{|\sim 8/[}q{(1 - f)^2}\textrm{]}$ according to (2.8), we find the following value of the parameter $k_z^2R_p^2$: $k_z^2R_p^2\mathrm{\ \sim (}1 - f{\textrm{)}^2}(q/8)({R_p}/L)$. When $q \ll 8(L/{R_p})({m_e}/{m_i})$ the value of $k_z^2R_p^2$ satisfies inequality (5.10), and the resonance values ${\omega _{\textrm{res}}}$ and ${q_{\textrm{res}}}$ are determined by expressions (5.11). The numerical values for this case are shown in table 2. As is seen from table 2 and figure 4, in the long-wavelength limit, when inequality (5.10) is satisfied, the resonance frequency ${\omega _{\textrm{res}}}$ can be both low and high compared with the ion cyclotron frequency ${\omega _{\textrm{ci}}}$. When the waveguide is sufficiently fully filled with plasma (see formula (5.13)), the resonance frequency is small compared with the cyclotron frequency or of the same order of magnitude.

Table 2. The numerical values of resonance frequencies ${\omega _{\textrm{res}}}$ and ${q_{\textrm{res}}}$ for different values of parameter ${R_p}/{R_w}$. The value of parameter ${k_z}{R_p}$ satisfies the inequality (5.10). Calculations are according to formula (5.11) ($f = 0.1$, ${m_e}/{m_i} \approx 1.36 \times {10^{ - 5}}$ (argon)).

5.3. Expected frequencies of electron–ion instability

What frequency $\omega$ of the discussed non-neutral plasma instability due to the relative motion of electrons and ions along the azimuth can be expected? Generally speaking, its frequency is not equal to the resonance frequency (5.1).

This instability was studied by Levy et al. (Reference Levy, Daugherty and Buneman1969) (see also Davidson Reference Davidson1974). They described thresholds and maximum growth rates of instability in detail. However, the expression for the real part of the unstable frequency has not been calculated and written out explicitly. It seems that this is one of the reasons why this powerful instability, the conditions of the origin of which always exist in non-neutral plasma, was not used to explain the nature of low-frequency oscillations in linear devices, in which non-neutral plasma was produced in the electron beam channel (Vlasov et al. Reference Vlasov, Dobrokhotov and Zharinov1966; Nezlin Reference Nezlin1982; Pierre et al. Reference Pierre, Leclert and Braun1987; Sakawa & Joshi Reference Sakawa and Joshi2000; Jaeger Reference Jaeger2010; Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011; David Reference David2017).

There is also another reason. In the article by Levy et al. (Reference Levy, Daugherty and Buneman1969) it is stated that, for $f \ll 1$, the instability occurs only near the resonance of the diocotron mode with the ion rotation. The term ‘resonance’ was even included in the title of the paper. However, in experiments, the instability is observed in a wide range of field variations, i.e. outside the resonance too. This difference, noted by Peurrung et al. (Reference Peurrung, Notte and Fajans1993), makes the instability of Levy et al. (Reference Levy, Daugherty and Buneman1969) unlike the instability observed in experiments. This shortcoming of the theory was overcome by Fajans (Reference Fajans1993) in the framework of the nonlinear consideration.

This shortcoming can also be overcome within the framework of the linear theory of stability, taking into account kinetics of ions in the plasma model considered by Levy et al. (Reference Levy, Daugherty and Buneman1969).

The hydrodynamic description of ions used by Levy et al. (Reference Levy, Daugherty and Buneman1969) is not applicable when ions are unmagnetized $(q \ge {m_e}/{m_i})$. Ions are produced in experiments by electron impact of atoms (molecules) of the working (residual) gas. In crossed fields they oscillate along the radius with an amplitude comparable to the radius of the plasma cylinder itself. A detailed analysis of ion kinetics in crossed fields was also carried out by Levy et al. (Reference Levy, Daugherty and Buneman1969). A solution of the kinetic equation for ions was even obtained in that paper. But these results were not used in the study of instability. For such ions, a kinetic description is absolutely necessary.

The equilibrium distribution function of ions, which adequately takes into account the peculiarities of their production in crossed fields, was determined by Dem'yanov et al. (Reference Dem'yanov, Yeliseyev, Kirochkin, Luchaninov, Panchenko and Stepanov1988). It turned out to be anisotropic and non-Maxwellian in transverse energies and, therefore, very unstable. Due to anisotropy of ions such plasma is unstable even without the resonance with the electron mode (Mikhailovskii Reference Mikhailovskii1974; Yeliseyev Reference Yeliseyev2006). The stability of non-neutral plasma with such an ion distribution function was investigated by Yeliseyev (Reference Yeliseyev2010) for a waveguide completely filled with plasma, $m = 1$, ${k_z} \ne 0$ and $f \ll 1$. It is shown that near every harmonic of the ion resonance frequency (5.1) there exist not one, but two ion modes. The frequency of one of them is lower than the ion resonance frequency, the frequency of the other is higher. Both modes exist within the entire acceptable range of variation of the parameter q. Their frequencies repeat the dependence on q of ion resonance frequency (5.1). They are called ‘modified’ ion cyclotron modes. In the resonance area the electron mode is unstable with a fast growth rate. In a wide region outside the resonance, where $q < {q_{\textrm{res}}}$, the lower modified ion cyclotron mode is unstable, but with a slower growth rate $({\mathop{\rm Im}\nolimits}\, \omega \equiv \gamma \sim {\omega _{\textrm{pi}}})$. The real part of the frequency $({\rm Re} \omega )$ of the unstable mode in the area $q < {q_{\textrm{res}}}$ remains lower than the resonance frequency (5.1) by the value $\Delta \omega$ (Yeliseyev Reference Yeliseyev2010):

(5.15)\begin{align} \mathrm{\Delta }\omega & \equiv |{\rm Re} \omega - \omega _{\textrm{rot}}^i|\sim \gamma \sim {\omega _{\textrm{pi}}} = {\omega _{\textrm{ce}}}{[q(f/2)({m_e}/{m_i})]^{1/2}}\nonumber\\ & = {\omega _{\textrm{ci}}}{[q(f/2)({m_i}/{m_e})]^{1/2}} \simeq {\varOmega _i}{[f/(1 - f)]^{1/2}}. \end{align}

The dependence of $\Delta \omega$ (5.15) on q is shown in figure 3.

The resonance frequency (5.1) does not depend on the degree of filling of the waveguide $(R_p^2/R_w^2)$. So, it can be supposed that the value of $\Delta \omega$ (5.15) has the same order of magnitude when the waveguide is partially filled with plasma $(R_p^2/R_w^2 < 1)$. Because of the decrease in the value of $\Delta \omega$ (5.15), the real part of the frequency ${\rm Re} \omega$ can become comparable to the cyclotron frequency ${\omega _{\textrm{ci}}}$. The real part of the frequency ${\rm Re} \omega$ will behave depending on q in a manner similar to how $\Delta \omega$ behaves in figure 3.

In addition, the resonance frequency ${\omega _{\textrm{res}}}$ (5.1), which in the considered case of Cherenkov resonance equals $\omega _{\textrm{rot}}^i$ (5.2), itself decreases with a decrease in q. With it the frequency of unstable oscillations ${\rm Re} \omega$ decreases (see figure 3). Taking into account these two circumstances, the triangular area in figure 4, where ${\omega _{\textrm{res}}}/{\omega _{\textrm{ci}}} \le 1$, must be larger than is shown in the figure.

The equalities (5.15) contain also an estimation of the growth rate of instability ${\mathop{\rm Im}\nolimits}\, \omega = \gamma$. This is consistent with the frequency correction $\Delta \omega$. So the dependence of the growth rate $\gamma$ on q looks like the dependence of $\Delta \omega$ in figure 3. From (5.15) it follows that the growth rate exceeds the ion cyclotron frequency $(\gamma > {\omega _{\textrm{ci}}})$ in the area where $qf > 2{m_e}/{m_i}$. This inequality is satisfied in a wide range of acceptable parameter values.

Summarizing the above, we conclude that the diocotron mode interacting with ions having an anisotropic distribution function can lead to electron–ion instability at frequencies ${\rm Re} \omega$, which can be comparable to the ion cyclotron frequency ${\omega _{\textrm{ci}}}$, and have fast growth rates $({\mathop{\rm Im}\nolimits}\, \omega = \gamma > {\omega _{\textrm{ci}}})$ within a wide range of variations of parameter q, as is observed in experiments.

There is also an area of parameters in which the instability frequencies are large compared with the cyclotron frequency. Note that in different experiments with non-neutral plasma, produced in an electron beam channel, unstable oscillations are observed both at low frequencies $\omega \mathrm{\ < }{\omega _{\textrm{ci}}}$ (Jaeger Reference Jaeger2010; Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011; David Reference David2017) and at much higher frequencies (5–10) ${\omega _{\textrm{ci}}}$ (Sakawa & Joshi Reference Sakawa and Joshi2000).

5.4. About the instability of SLH modes

The family of SLH modes also passes through zero frequency and through the low-frequency area and can also be the cause of electron–ion instability. In the long-wavelength limit $({k_z}{R_p} \ll 1)$, SLH modes pass zero frequency at small values of q, while the diocotron mode crosses the value of zero of frequency at larger values of q (see figure 1a). Correspondingly, the growth rate of the instability associated with the diocotron mode is faster than the growth rates associated with the SLH modes. Note that the instability associated with the diocotron mode must develop at higher frequency than the instabilities associated with the SLH modes.

In the long-wavelength limit $({k_z}{R_p} \ll 1)$, the SLH modes are closely spaced around the value of $\omega = \omega _{\textrm{rot}}^e$. Different radial modes of this family can be excited with a small variation of the parameter q. However, in experiments, within a wide range of variation of fields, only the lowest radial mode is excited. This circumstance does not allow one to associate the observed low-frequency oscillations with SLH modes. The diocotron mode interacting with ions seems to be more preferable for explaining the nature of low-frequency oscillations observed in experiments with non-neutral plasma in linear devices (Vlasov et al. Reference Vlasov, Dobrokhotov and Zharinov1966; Nezlin Reference Nezlin1982; Sakawa & Joshi Reference Sakawa and Joshi2000; Jaeger Reference Jaeger2010; Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011; David Reference David2017) and other similar experiments.