1452 ISSN 1063-7842, Technical Physics, 2019, Vol. 64, No. 10, pp. 1452–1461. © Pleiades Publishing, Ltd., 2019. Russian Text © The Author(s), 2019, published in Zhurnal Tekhnicheskoi Fiziki, 2019, Vol. 89, No. 10, pp. 1535–1544. Stable and Unstable Regimes of Plasma Diodes in the Presence of Electron Collisions S. Pramanik a , V. I. Kuznetsov b, *, and N. Chakrabarti c a Department of Applied Mathematics, University of Calcutta, Kolkata, 700009 India b Ioffe Institute, St. Petersburg, 194021 Russia c Saha Institute of Nuclear Physics, Kolkata, 700064 India *e-mail: victor.kuznetsov@mail.ioffe.ru Received February 8, 2019; revised February 8, 2019; accepted April 22, 2019 Abstract—A thermionic energy converter as the Pierce diode of which the interelectrode space is filled with a uniform background of immobile ions is considered. The features of stability of its time-independent solu- tions are investigated under electron collisions with background gas. The solutions are obtained analytically by the Lagrange method. A dispersion equation is derived using a perturbative approach. The aperiodic solu- tions of this equation, as well as the oscillatory ones, are obtained and their properties are studied too. The regions of unstable aperiodic modes are determined. It is shown that all the oscillatory modes are stable. DOI: 10.1134/S1063784219100189 INTRODUCTION The study of the space-charge-limited current in a vacuum diode was initiated through the pioneering works of Child and Langmuir [1, 2]. Their theoretical investigation was further extended by Bursian and Pavlov for an electron beam with finite injection velocity [3]. Later, Pierce studied a planar diode in the presence of a uniform ion background [4]. They found the existence of a threshold value of diode current above which an aperiodic instability (Bursian–Pierce instability) is developed in the system and, as a result, the current across the electrodes drops substantially [5]. This threshold point is called “space charge limit” (SCL) [6]. In the past few decades, plasma diodes have been studied actively [7–14] as they offer a wide range of applicability in modern technology. For example, there are many scientific instruments such as thermi- onic energy converters (TIC) [15], gas-filled diodes in nuclear fusion [16], laser printers [17], and microwave oscillators [18] that work on the basis of plasma diode technology. The steady states of plasma diodes are often repre- sented in the {ε 0 , δ} parametric plane. Here ε 0 and δ are the emitter field strength and the diode gap, respectively [19–23]. For the Bursian diode (the elec- tron beam moves with no ions in the background), the branches of solutions plotted in this parametric plane are named the Bursian family. In case of the Pierce diode (a uniform background of ions occupies the interelectrode gap), steady states can be segregated into the Bursian and the non-Bursian family of solu- tions. For purely monoenergetic electron beams, the Bursian branches can be separated into two parts, namely, normal-C branch (below SCL) and C-over- lap branch (above SCL). C-overlap branches end at “zero-points” which refer to the moment when the electron velocity vanishes within the plasma region at the first time [20–23]. Stability properties of these branches were analyzed by means of a dispersion rela- tion [20–28]. In [15], it was proposed to use a thermionic energy converter as a generator of alternating current. Its operation is based on the phenomenon of Bursian– Pierce instability. However, observations show that occurrence of electron–atom collisions may kill this instability. Therefore, the presence of electron scatter- ing may hinder the AC generation process by a TIC. As shown in [12], a thermionic energy converter can be simply modeled by a Pierce-like plasma diode. In our recent work, we explored plasma stationary states in such a diode under the assumption that a monoenergetic flow of electrons was injected from the emitter plate and suffered collisions with atoms when moving within the interelectrode space [29]. In this article, we investigate the effect of collisions on the stability properties of plasma diodes for both Bursian and non-Bursian families. It is shown that the pres- ence of collisions has profound effects on the stability properties of the steady states. The paper is organized as follows. Section 1 con- tains the basic governing equations and the treatment using the Lagrangian formalism. Time independent solutions are derived in Section 2. In Section 3, a rele- PLASMA