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The Role of Electrostatic Instabilities in the Critical Ionization Velocity Mechanism

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The Role of Electrostatic Instabilities in the Critical Ionization Velocity Mechanism r ~! TRITA-EPP-77-18 THE ROLE OF ELECTROSTATIC INSTABILITIES IN THE CRITICAL IONIZATION VELOCITY MECHANISM M.A. Raadu July 1977 Department of Plasma Physics Royal Institute of Technology S-100 44 STOCKHOLM 70, Sweden THE ROLE OF ELECTROSTATIC INSTABILITIES IN THE CRITICAL IONIZATION VELOCITY MECHANISM M.A. Raadu Department of Plasma Physics, Royal Institute of Technology, Stockholm, Sweden Abstract The role of electrostatic instabilities in the critical ionization velocity mechanism is investigated. The analysis is based on the theory developed by Sherman which interprets Alfven's critical velocity in terms of a circular process. This process involves the acceleration of electrons by a two-stream instability modified by the presence of a mag- netic field. A general expression for the energy and momentum of ions and electrons associated with an electrostatic mode is derived in terms of the plasma dielectric constant. This is used in the case of the modified two-stream instability to determine the distribution of energy between ions and electrons. An extrapolation from the linear phase then gives an estimate of the energy delivered to the electrons which is compared to that required to ionize the neutral gas. 1. Introduction The critical ionization velocity mechanism involves the interaction of neutral gas with a plasma. In general ioni- zation of a neutral gas is possible if there is a source which can deliver the required energy. In a situation where plasma and neutral gas move relative to one another, energy is in principle available from the relative motion and may be used to produce ionization. Alfvén (1954, 1960) proposed that there should be a critical velocity for ionization to occur. He considered the situation where a continuous stream of neutral gas collides with a plasma in a magnetic field. The mechanism envisaged is a rapid increase in ionization when the velocity of the neutral gas u with respect to the plasma is such that the atoms have a corresponding kinetic energy equal to the ionization energy, eV.. Thus if the atomic mass is m_, Ck \ Vc - eV. (1) The required energy is released by the braking of the rela- tive motion between the neutral gas and the magnetized plasma. The original proposal for such a critical ionization velocity was made by Alfvén (1954) in connection with a theory of the origin of the solar system. There is now ample confirmation from a variety of laboratory experiments (see Danielsson, 1973, for an extensive review). In a collision between a neutral atom and an ion, where mo- mentum is conserved, only a fraction of the kinetic energy of the neutral atom given by the relative velocity can be released. Such collisions would imply a value of th»j critical velocity larger than originally proposed and also dependent on the mass of the ions involved. However the original pro- posal has been found to be in close agreement with experiment. This indicates that collective plasma processes may be in- volved. A particularly vivid demonstration of the critical velocity phenomenon is given by the plasma-gas impact experi- ment (Danielsson 1970, 1974; Danielsson and Brenning, 1975) in which a hydrogen plasma with super-critical velocity penetrates a small cloud o^ neutral gas, usually halium, in a region of transverse magnetic field. A strong inter- action occurs leading to rapid ionization and a braking of the motion to a velocity close to the critical value- Danielsson (1970) argues from the inefficiency of direct binary encounters that an efficient collective interaction between the plasma and the neutral gas must operate. High energy electrons are found to be produced by the interaction and these can account for the rapid ionization of neutral gas. Thus the critical velocity mechanism is a particularly interesting ionization process in which the collective plasma properties of the ionized gas play a vital role. From a theoretical point of view an efficient process for energizing electrons is needed. The initial energy is in the ion component of tha plasma. A similar problem is raised in the interpretation of experiments involving collisionless shock waves, .«'here a mechanism is required to preheat the electrons. In this connection Krall and Liewer (1971) con- sider the role of instabilities driven by the relative drift between ions and electrons across a magnetic field. They describe instabilities resulting from the coupling between a drift wave in a nonuniform plasma to either the ion plasma oscillation or a lower hybrid oscillation or alternatively a form of the two-stream instability. The growth rates are of the order of the ion plasma or lower hybrid frequency. Sherman (1969, 1972) argues that the critical ionization velocity mechanism is driven by a circular process involving the energization of electrons by the electrostatic two-stream instability modified by the presence of a magnetic field, such as described by Buneman (1962). There are, therefore, useful comparisons to be made between the critical velocity mechanism and processes in shock waves. If, in the critical velocity mechanism, electrons are being energized by the modified two-stream instability, it is important to know hov; the energy is shared between the ions and electrons. The situation considered is one in which the ions are inefficient- in producing ion ization. To account for ~l the observed critical velocity the bulk of the energy should go via the electrons to ionize the neutral gas. In this paper an estimate of the maximum electron energy is made extrapolating from the distribution of energy between the ions and the electrons in the linear phase and using general arguments about the saturation process. This analysis may be applied in a further development of the approach initiated by Sherman (1969) to estimate the critical velocity and is an extended version of a previous report ( Raadu, 1975 ). 2. Sherman's Critical Ionization Velocity Mechanism Sherman (1969, 1972) proposes a circular process for the critical velocity mechanism in connection with Danielsson*s (1970) experiment. In this experiment a beam of plasma is shot into a region of transverse magnetic field into which a cloud of neutral gas has been released. If ionization of the neutral gas occurs, newly formed ions with no initial drift velocity will be introduced within the plasma beam and thbre will be a relative drift between these ions and the electrons in the beam. The plasma beam enters the region of trahsverse magnetic field by setting up a polarization electric field and the plasma moves with the E/B drift velo- city. The newly formed ions and electrons are accelerated by this electric field and acquire a common drift velocity with the plasma beam. The polarization electric field is reduced by separation of the newly formed charges and the plasma beam is decelerated. Due to the relatively large inertia of the plasma beam ions, they will have a relative motion with rospect to the electrons during the deceleration of the beam. The motions of the beam particles and newly formed ions and electrons produced by ionization are indicated in Figure 1. Sherman argues that the relative drift motions between ions and electrons produced by these mechanisms can drive a two-stream instability modified by the presence of a magnetic field which accelerates electrons parallel to the magnetic field. These energetic electrons can then ionize the neutral gas. In this way the interaction is maintained by a circular process. The ionization energy is derived from the relative motion between ions and electrons accompany ing the introduction of newly formed charges within the plasma beam. An essential link in the process is the electron acceleration. From an analysis of the modified two-stream instability, Sherman argues that the unstab1e modes propagating nearly perpen- dicular to the magnetic field are the most significant and that their growth stops when the electron energy is of the order ^ n^u^r where nu and u, are the mass and relative velocity of the ions drifting with respect to the electrons. This energy must be greater than the ionization energy eV. for the circular process to work. If the newly formed ions are considered, then m. is equal to the mass of the neutral atoms, m , and sufficiently energetic electrons to ionize should be produced if the drift velocity u, exceeds the . critical velocity u defined by Equation 1. Since the newly formed ions have initially no mean velocity with respect to the neutral gas their drift velocity u, with respect to the plasma beam is equal to the relative velocity between the plasma and the neutral gas. Thus consideration of the saturation of instabilities driven by the drift motion of newly formed ions leads directly to a critical velocity for the plasma interaction with the neutral gas. This is the point of view adopted in the following analysis. Sherman (1969) develops the analysis of the modified two- stream instability given by Buneinan (1962) to describe the growth of the instability and the final saturation. The appropriate length and time scales for the development are such that the electron motion perpendicular to the magnetic field is restricted, but the ions may be regarded as moving freely. Under these circumstances the electrons are accele- rated primarily in the magnetic field direction. In the critical ionization velocity interaction newly formed elect- rons acquire the E/B drift velocity on the time scale of the electron gyroporiod. Moreover the adjustment of the electron drift velocity to changes in the polarization electric field associated with braking of the plasma motion takes place on a similar time scale. Thus the? electrons may be treated as having a common E/B drift velocity on longer time scales. The plasma ions and newly formed ions will initially have different mean velocities and velocity distributions, which develop under the combined action of the polarization ele ^tric field and the magnetic field.
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