Abstract
The common description of kinetic effects in low-pressure plasmas is based on the Boltzmann equation. This applies especially to the description of Ohmic (collisional) and non-local (stochastic/collisionless) electron heating, where the Boltzmann equation is the starting point for the derivation of the corresponding heating operator. Here, it is shown, that an alternative and fully equivalent approach for describing the interaction between electrons and electric fields can be based on the Fokker–Planck equation in combination with the corresponding Langevin equation. Although, ultimately the final expressions are the same in both cases, the procedures are entirely different. While the Fokker–Planck/Langevin approach provides physical insights in a very natural way, the linearized Boltzmann equation allows straightforward calculation but requires some effort to interpret the mathematical structure in terms of physics. The Fokker–Planck equation for the present problem is derived, with particular emphasis on the consistent treatment of velocity-dependent elastic collision frequencies. The concept is tested for a simple case by comparing it with results from an ergodic Monte-Carlo simulation. Finally, the concept is applied to the problem of combined Ohmic and stochastic heating in inductively coupled plasmas. The heating operator is first analyzed for an exponential model field profile. Self-consistent field profiles are determined subsequently. In this context, a generalization of the plasma dispersion function is introduced, which allows for arbitrary forms of the distribution function and velocity dependence of the elastic collision frequency. Combined with the Fokker–Planck heating operator, a fully self-consistent description of the plasma and the fields is realized. Finally, a concept for integrating the operator in a standard local Boltzmann solver and using the local solver for determination of the global electron velocity distribution function in a low-pressure plasma is provided.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Plasmas are many particle systems and, arguably, some of the most interesting physics in plasmas is associated with kinetic effects. There, only those charged particles in the ensemble, which have a proper energy or velocity, are taking part in certain interactions, either with fields or in collisions with neutral particles, due to resonances or thresholds, respectively. In turn, the interaction has an influence on the energy gain or loss of these particles and thereby on the distribution function of the ensemble. The standard way for describing kinetic effects of electrons and ions in plasmas is by solving the Boltzmann equation, including the Boltzmann collision operator (integral) for the interaction of electrons with neutrals. This concept, especially in the low-pressure range, where only binary collisions are important, is in principle highly precise and addresses all the relevant physics.
An important aspect of the Boltzmann equation is that it can conveniently be written as the sum of individual operators describing the various interactions of the charged particles with their environment. This includes, naturally, all the different elastic and inelastic collision processes, the latter including both excitation and ionization. An operator of key importance is describing the gain and loss of energy due to the interaction of electrons with external fields. This operator is usually derived from a polynomial expansion of the velocity distribution function, which in the case of small anisotropies is often limited to the first two terms, representing an isotropic and an anisotropic part (Holstein 1946; Allis 1956; Shkarofsky et al. 1966; Hazeltine and Waelbroeck 1998; Bittencourt 2004; Verboncoeur 2005; Makabe and Petrovic 2015; Loureiro and Amorim 2016). However, numerical solutions have also be obtained by extending the conventional two-term approximation to higher orders (Pitchford et al. 1981; Pitchford and Phelps 1982; Braglia et al. 1984; Leyh et al 1998; Stephens 2018).
Analytical solutions are rare and are feasible only in very limited and simplified cases. However, while deriving the various operators out of the Boltzmann equation, already some important physics insights can be obtained. Unfortunately, this is often less straightforward as it may seem since the particular structure of the Boltzmann equation requires a quite extensive use of mathematical transformations in the course of the derivation. For example, when describing local and non-local electron heating, the mathematical structure becomes very complex and an extra effort must be undertaken to extract the physical meaning.
An alternative concept to the Boltzmann equation is the Fokker–Planck equation combined with an appropriate Langevin equation (Keilson and Storer 1952; Risken 1989; Paul and Baschnagel 2013). While the Fokker–Planck equation provides two well-defined operators, representing drift and diffusion (either in configuration, velocity or energy space), the Langevin equation allows identifying the drift and diffusion coefficients required in the Fokker–Planck equation. In plasma physics, the Fokker–Planck/Langevin concept is used almost exclusively for describing Coulomb interactions between charged particles (Bittencort 2004; Hagelaar 2016) or collisions in general (Kolobov 2003). The main reason for the use in this context is that it allows addressing consistently the random interaction of an electron with multiple other charges within the Debye sphere, whereas the Boltzmann collision integral is binary by nature. Nevertheless, some attempts have been made to apply the Fokker–Planck concept also to the heating of charged particles, as for example to plasma oscillations (Lenard and Bernstein 1958), the multiple passing of ions through a local ion cyclotron resonance (ICR) in a Tokamak (Fish 1987) or the passage of electrons through the resonance region of an electron cyclotron resonance (ECR) discharge (Lieberman and Lichtenberg 1973). As will be shown here, the concept can be substantially extended and is well suited for describing the elastic collisional interactions (both, energy transfer and momentum scattering) together with the electron interaction with an external spatially inhomogeneous and temporally oscillating electric field. In particular, it allows describing in a consistent way the combined action of local Ohmic heating, due to electron-neutral collisions, and non-local or stochastic heating, due to spatial inhomogeneities of an oscillating electric field. The final expressions are identical to those obtained by the Boltzmann equation. However, the initial expressions as well as the way of progressing from there to the final result are very different. Although some mathematical transformations are still necessary, the particular structure of the Fokker–Planck/Langevin equations forces a more physics-oriented approach. Last but not least, it is certainly an important gain in its own right to have an alternative tool available for the kinetic description of electron-field interactions, which is not limited to the example(s) treated in detail here.
An essential point for the application of the heating operator is the fact that at sufficiently low neutral gas densities, the energy relaxation length of the electrons exceeds the system size. In this case, the particular spatial distributions of energy gain and loss in the system become irrelevant and the electrons can be described by a single global distribution function. This is the basic idea of the so-called non-local kinetic description of plasmas (Tsendin 1995, 2010). The pressure range for the non-local description and the onset of non-local electron heating are about the same and the two concepts can be combined. Since only a single global distribution function needs to be calculated, a local Boltzmann solver can be used after proper spatial averaging of the collisional operators in the Boltzmann equation and after replacing the local (Ohmic) heating operator by the extended operator allowing for both, local and non-local heating.
This concept of a global kinetic model was applied first by (Kortshagen et al. 1995 and Godyak and Kolobov 1998) and recently also by Yang et al. (Yang and Wang 2021; Yang et al. 2022). Here we present an extended concept that includes also surface losses and distinguishes between mean and surface densities of the electrons. Further, we show that the general heating operator has the same form as the local heating operator but with a different coupling function. This makes the adaptation of a local Boltzmann solver to the global kinetic model particularly simple. Compared to a full Particle-In-Cell Monte-Carlo-Collision (PIC-MCC) model, this global kinetic model for electrons has the advantage of yielding significantly faster numerical simulations, although certainly the wealth of information provided by a PIC-MCC simulation is much richer (e.g. Mattei et al. 2017). Nevertheless, from the distribution many other macroscopic quantities (e.g. transport parameters and rate coefficients) can be easily derived by building the corresponding moments. Indeed, the present most common use of local Boltzmann solvers is in numerical fluid models of low-temperature plasmas, where in the limit of local electron heating, at very short energy relaxation lengths, the aim is to use the calculated electron distribution functions for deriving macroscopic quantities (e.g. transport parameters and rate coefficients) as a function of the local reduced electric field. By this combination of local kinetics with fluid dynamics, a self-consistent simulation of ions and electrons is achieved (Sommerer and Kushner 1992; Salabas et al. 2002; Sakiyama and Graves 2006; Alves 2007; Niemi et al. 2009; van Dijk 2009; Brezmes and Breitkopf 2015; Alves et al 2018). Some publically available local Boltzmann solvers are LoKI-B (Tejero-del-Caz et al. 2019), BOLSIG + (Haagelar and Pitchford 2005), and MultiBolt (Stephens 2018). Such codes are usually very fast but naturally fail to address non-local electron heating at long energy relaxation lengths, i.e. low pressures.
Using a (general) homogenous Boltzmann solver that bridges the description of non-local and local cases allows a substantial extension of the concept to lower pressures and related different physics phenomena. As will be shown, in order to close the system of equations self-consistently for a global kinetic electron model, a fluid simulation for the ion-transport is required. This means that in the non-local case, the Boltzmann solver can no longer be operated independently of the ion model. The necessity to include the ions in the model reflects the fact that the treatment involves a full plasma instead of an isolated electron gas. In the simplest case, this can be an analytical model but in general, especially for non-trivial geometries, a numerical fluid code for the ions might be used.
The necessity to develop a non-local description for the interaction of electrons with an external electromagnetic field was first noticed in metals (Reuter and Sondheimer 1948) and later adopted to plasmas by Weibel (Weibel 1967). Up to the present day, the effect has been the subject of great interest in fundamental studies with focus on electron heating as well as the related field structure (anomalous skin effect) (e.g. Ichimaru 1973; Rauf and Kushner 1997; Kolobov and Economou 1997; Kolobov et al. 1997; Vasenkov and Kushner 2002; Kaganovich et al. 2004; Godyak 2005; Turner 2009; Yang and Wang 2021; Yang et al. 2022). It has also important consequences for applications e.g. for materials processing in the semi-conductor industry, where low-pressure discharges are commonly used (Hopwood 1992; Yu and Shaw 1995; Takagi et al. 2001; Lieberman and Lichtenberg 2005; Lallement et al. 2009; Donelly and Kornblit 2013; Makabe and Petrovic 2015; Yang et al. 2016). In brief, collisionless electrons in a homogeneous field would just experience some quiver motion without an average net energy gain. If, however, the field has a spatial structure and the electrons possess some thermal motion by which they move along the field gradient, then the electrons can gain more energy in a strong field region than loosing in a low field region. The argument can also be reversed if the electron moves thermally the other way but it turns out that on average there is a net energy gain. The conversion of the directional energy gain into heat in an irreversible scenario is realized by the collisional isotropization of the velocity vector within a large plasma volume beyond the region of interaction with the field. The effect depends naturally on the velocity distribution of the electrons and is strongly related to the electric field structure, which in turn is affected by the interaction of the field with the electrons, thus depending on their distribution function. This non-linearity evinces the need for a self-consistent formulation and solution of the problem, as addressed in this work, although model field profiles can help in understanding the basic effects of the energy gain. The problem of the field profile calculation is commonly treated by assuming a Maxwell distribution. Here, we will derive a more general self-consistent solution, which allows for arbitrary distribution functions and even includes velocity-dependent collision frequencies. Combining this result with the Fokker–Planck heating operator, the collisional operators from the Boltzmann solver, and an additional operator for surface losses, a full global kinetic model is presented, which is closed by an ion-fluid model.
The paper presents on the one hand a review of the present knowledge on describing local and non-local electron heating, and on the other hand it introduces a new approach based on the Fokker–Planck equation in combination with the Langevin equation to describe these heating effects, using a tutorial style and starting from first principles. Further, a couple of fresh ideas are introduced to allow a fully self-consistent description. This includes the generalized dispersion integral, which allows the use of arbitrary distribution functions and a velocity-dependent elastic collision frequency, the introduction of a kinetic surface loss operator, and the proposal of an iterative procedure to solve the kinetic description of electrons coupled to the fluid description of ions.
The paper is organized as follows: In Sect. 2 we first summarize some basic aspects of the Boltzmann equation but without detailed proofs, since those can be found in the rich literature on this topic. The main purpose of this section is to prepare the ground for the subsequent sections by providing concepts and expressions, which are needed later for further calculations or comparison. The Fokker–Planck equation is derived in Sect. 3. Here we go beyond the standard form of the equation by allowing also for velocity-dependent collision frequencies, which are essential for a realistic treatment in the present context. The diffusion coefficient in velocity space appearing in the Fokker–Planck equation is not specified by the equation but has to be determined externally. This is done in Sect. 4 by the Langevin equation. The equation itself is not solved. Instead, it is sufficient to derive the particular form for a certain interaction, since there is a one-to-one relation between the coefficients in the Langevin and the Fokker–Planck equation. Further, two important concepts are discussed in this section. Firstly, on a timescale much longer than the free-flight-time of an electron, velocity changes by the acceleration in an electric field and the subsequent random reorientation of the velocity vector in an elastic collision can be viewed as a single event. In this picture, the combined field-collision action is similar to the Brownian motion of a particle. The velocity changes entering the diffusion coefficient in the Fokker–Planck equation are then defined by simple integration of the equation of motion over a free acceleration period. Subsequently, stochastic averaging of the velocity changes resulting from the integration is performed by using suitable probability distributions. Secondly, the final form of the Fokker–Planck equation is obtained under the explicit assumption of the existence of a process that guarantees randomization of the velocity vector on the coarse timescale used in the present description. The process itself does not need to be specified. In a bounded plasmas, the necessary randomization is provided by elastic collisions, either locally within the heating zone, or in the larger volume beyond, as well as in reflections from walls, that usually have some random structure at least on a mesoscopic scale. Indeed, multiple interactions of an electron with the electric field in the heating zone are only possible in the presence of collisions to reverse the flight direction. Clearly, this return timescale is not resolved. In Sect. 5, the resulting Fokker–Planck heating operator combined with a simple dissipation term is compared to an ergodic Monte-Carlo collision model. This comparison verifies the proposed formulation and demonstrates the necessity to modify the standard Fokker–Planck equation in case of velocity-dependent collision frequencies.
In Sect. 6, we apply the general Fokker–Planck/Langevin concept to the particular problem of combined local Ohmic and non-local collisionless heating by a transversal electromagnetic wave entering the plasma from the boundary. This special scenario describes the typical situation in inductively coupled plasmas (ICPs), which represent one of the most common type of low-pressure plasma sources (Chabert et al. 2021). A general form of the heating operator for arbitrary spatial field profiles is derived. In addition, the operator and its properties are discussed in detail for a simple exponential model profile. The self-consistent profile is derived in Sect. 7. A dispersion integral is defined, that allows for arbitrary distribution functions and velocity dependence of the elastic collision frequency. The results are discussed for the limiting cases of local collisional Ohmic heating and non-local collisionless heating and conclusions are drawn on the conditions for pure stochastic heating. Notably, the combined action of local and non-local heating does not possess the simple form of a sum, where the terms term are describing the limiting cases separately, but has a complex structure, where both contributions are completely mixed.
Finally, Sect. 8 outlines the use of a local (homogeneous) Boltzmann solver as a mean for numerically calculating the global electron velocity distribution function. The presentation also includes the derivation of a surface-loss operator and the use of a fluid model for ion transport. The external input parameters are the same as in an experiment (gas type and pressure, chamber size, power etc.) and a concept is developed on how the internal parameters of the model can be determined by an iteration procedure. In Sect. 9, the results are briefly summarized and an outlook on further applications of the Fokker–Planck/Langevin concept is presented. Some of the more involving mathematical derivations of this work are moved to several appendices to enhance the readability of the main text.
2 Brief summary of the Boltzmann equation and classical kinetics
Most of the contents of this section can be found in textbooks and seminal publications on the topic (Allis 1956; Hazeltine 1998; Bittencort 2004; Lieberman and Lichtenberg 2005; Alves 2018; Makabe 2018). Therefore, results are mostly summarized without detailed proofs or derivations and the reader is referred to the rich literature for further details. Instead, the focus is mainly on results and aspects required in the main part of the paper, either for comparison or for highlighting certain aspects.
The Boltzmann equation describes the temporal and spatial development of volume elements in phase space containing a large number of particles. Nevertheless, compared to the entire system under description, these elements can still be considered as microscopic. According to this concept, a certain velocity distribution function \(f\left( {\vec{v},\,\vec{r},\,t} \right)\) is associated with each local point in space–time \(\left( {\vec{r},t} \right)\). Then \(f\,d^{3} v/n\) represents the probability to find a particle in the velocity-space volume element \(d^{3} v\) with \(n = \int {f\,d^{3} v}\) representing the local particle density. This is the normalization used for the isotropic velocity distribution function \(f_{0} \left( v \right)\) throughout this work. Alternatively, normalization can also be made to unity. In this case, the density becomes a factor in front of the distribution function. This option will be used in Sect. 8 in connection with the energy distribution function.
The formal development of the distribution function is provided by the total temporal variation and appropriate collision operators:
The indices at the nabla operators indicate derivatives with respect to space \(\left( r \right)\) and velocity \(\left( v \right)\). Here \(\vec{F}\left( {\vec{v},\vec{r},\,t} \right)\) is some force acting on the particles of mass \(m\). In plasmas, the force is the Lorentz force \(\vec{F} = q\,\left( {\vec{E} + \vec{v} \times \vec{B}} \right)\), with the charge \(q = \pm e\) (singly ionized atoms/molecules assumed) and the mass \(m = m_{j},j = e,i\) for electrons and ions, respectively. The fields follow self-consistently from Maxwell’s equations, where the moments of the distribution functions are required:
Integrals are carried out over the respective distribution functions of the ions and electrons in order to obtain the corresponding particle densities \(n_{j}\) and flow velocities \(\vec{u}_{j}\). Although formally easy to formulate, this system of partial integro-differential equations is not well suited for any practical purposes without further simplifications.
Throughout this paper, magnetic fields are neglected, which is justified in cases without external magnetic fields and for sufficiently high oscillation frequencies of the field and correspondingly low amplitudes. Collisions are considered as binary since the gas is usually quite dilute. A special case are Coulomb collisions in sufficiently dense plasmas at low neutral gas background. These collisions are usually treated by a Fokker–Planck approach. However, we will not go into any detail about these collisions within this work. Instead, the Fokker–Planck concept will be shown to be useful also for the description of the interaction of electrons with the electric field, which is so far not common in plasma science.
There is a subtle convention on the signs of the operators. Probably the most consequent formulation would be writing the various contributions to the variation of the distribution function (here indexed by \(j\)) as a sum, i.e. \(\partial f = \sum\limits_{j} {\left. {\partial f} \right|_{j} }\), with for instance \(\partial \left. f \right|_{1} = \partial_{t}\, f\) etc., and then demanding for a stable equilibrium \(\partial f = 0\). However, standardly the total temporal variation is kept on the lhs of the equation and all terms expressing the variation of \(f\) due to collisional processes are moved to the rhs of the equation. A positive sign is attributed to the formal temporal derivative denoting the collision operator, as is the case in Eq. (1). We will return to this sign convention when introducing the Fokker–Planck heating operator, since there the balance structure and thus the resulting sign conventions are slightly different.
Generally, binary collisions between particles of type a belonging to a distribution \(f_{a} \left( {\vec{v}_{a} } \right)\) and particles of type b belonging to a distribution \(f_{b} \left( {\vec{v}_{b} } \right)\) are described by the Boltzmann collision integral (Allis 1956; Bittencourt 2004):
For simplicity, any spatial and temporal dependences are dropped here with the convention that the collision integral applies at any point in space–time. The dash denotes the state after the collision and \(d\sigma /d\Omega\) is the differential cross section, which depends only on the absolute value of the relative velocity between the two particles and the corresponding angles in spherical coordinates.
A brief note might be added on the terminology, which sometimes can be confusing. In some communities, the above binary collision operator (3) is called the Boltzmann equation. Further, the Boltzmann Eq. (1) in the absence of collisions is often called the Vlasov equation or even Vlasov-Landau equation. In this work, Eq. (1) is called the Boltzmann equation and without collisions but with given external fields, it is called the collisionless Boltzmann equation. Only in cases where the fields are indeed calculated self-consistently from Maxwell’s equations using the currents and charge densities in the plasma as given by Eq. (2), the term Vlasov equation is used. Equation (3) is called the Boltzmann collision integral.
A simple but yet quite important collision operator results from the general expression (3) if the small electron to neutral mass ratio \(\mu = m_{e} /m_{N}\) is approximated as zero. In this case, the scattering is purely elastic with the direction of the electron momentum changed by the collision without any energy being transferred. In fact, the real interaction with finite-mass particles can be separated into two terms describing the small but finite energy transfer due to the finite mass ratio and the momentum scattering due to ideal elastic scattering. In the latter case, an initially anisotropic distribution \(f\) is converted into an isotropic distribution \(f_{0}\):
The anisotropic distribution function \(f\) is expressed as the sum of an isotropic part \(f_{0} = f_{0} \left( v \right)\) and a small anisotropic part \(f_{1}\), with \(\left| {f_{1} } \right| < < f_{0}\) assumed. The elastic collision frequency for momentum transfer is \(\nu_{m} = n_{g} \,\sigma_{m} \,v\), where \(n_{g}\) is the neutral gas density and \(\sigma_{m} \left( v \right)\) the corresponding collisional velocity-dependent cross section for momentum transfer. Neutrals are approximated to be at rest in comparison to the high speed of the electrons (typically 3 to 4 orders of magnitude difference). If the differential cross section is proportional to the inverse relative velocity, which is the case for Langevin scattering (induced dipole) between electrons and neutral atoms or molecules, \(\nu_{m} \, = {\text{const}}{.}\) results. Although this convenient approximation is often applied, the operator is generally correct even for a velocity-dependent collision frequency \(\nu_{m} \left( v \right)\). The above collision operator (4), usually called the Krook or relaxation operator, is the key interaction term for Ohmic heating. Alternatively, the operator together with the total derivate on the lhs of Eq. (1) might be derived from a path integral formulation with an exponential free flight probability distribution (Reif 1965):
where the mean free flight time is \(\tau_{m} = 1/\nu_{m}\). This probability concept will be used more intensely in connection with the Fokker–Planck and the Langevin equation.
Operators describing the collisional transfer of energy in inelastic collisions have in general a more complicate form. An important example is the operator accounting for the energy loss of an electron by exciting an atom or molecule from a discrete lower state to an excited state separated by an energy difference \(\varepsilon_{ex}\):
where \(\nu_{ex} \left( \varepsilon \right) = n_{g} \,\sigma_{ex} \,\sqrt {2\,\varepsilon \,/\,m_{e} }\) is the corresponding collision frequency with \(\sigma_{ex} \left( {\varepsilon < \varepsilon_{ex} } \right) = 0\). The operator is usually formulated for the electron energy distribution function \(F\left( \varepsilon \right)\) normalized to unity (the symbol F should not be confused with the general force introduced above), which is related to the isotropic part of the velocity distribution function \(\left( {\varepsilon = m_{e} \,v^{2} /2} \right)\):
conserving the particle number in the volume elements in phase space \(n_{e} \,F\left( \varepsilon \right)\,d\varepsilon = f_{0} \left( v \right)\,4\,\pi \,v^{2} \,dv\). It will be shown later that the surface loss of particles can be described by an operator of similar form. The inelastic operator (6) would converge to a differential expression in the limit \(\varepsilon_{ex} \to 0\). Usually, this transition cannot be made since for most relevant states \(\varepsilon_{ex}\) is larger or of the same order as the characteristic electron energy. Only for rotational transitions, the differential approximation can be used as a more convenient approximation (Makabe and Petrovic 2015; Ridenti et al. 2015).
Another operator of relevance to the present work is the operator describing the finite energy transfer in elastic collisions:
This operator follows from the general collision integral but will later be derived in an alternative way within the frame of the Fokker–Planck equation. Note that here too the elastic momentum transfer collision frequency \(\nu_{m} \left( v \right)\) can be velocity-dependent. Naturally, the operator is proportional to the small electron-neutral mass ratio \(\mu\). Equation (8) represents the only continuous dissipative operator in the electron Boltzmann equation, i.e. which can be expressed by a differential expression. All other dissipative operators contain energy differences rather than differentials, as in the inelastic excitation operator (6). Therefore, this operator often serves as a simple model dissipative term, allowing analytical solutions to the Boltzmann equation. We will make use of these analytic solutions later on as a simple test case for the Fokker–Planck heating operator.
Three special cases are of interest within this work: (a) non-local distribution functions in a static plasma potential, (b) local (Ohmic) electron heating in an oscillating field and, (c) local and non-local (stochastic) electron heating.
In unmagnetized plasmas of finite size without external fields, a self-consistent electrostatic plasma potential \(\Phi \left( {\vec{r}} \right)\) builds up in order to confine electrons and ensure equal fluxes of the light electrons and the far heavier (and often much colder) ions to the confining walls. Of course, the losses to the walls must be compensated by ionization within the volume. The (global) flux balance is a basic requirement for a static system. Neglecting the electron heating by external fields, which is usually of relevance only within a small part of the volume, and assuming that the energy relaxation length is larger than the characteristic system size \(L\), the collisonless and static Boltzmann equation for electrons reads:
Solutions to this equation are all functions of the total energy: \(f = f\left( {\varepsilon - e\,\Phi } \right)\). Consequently, the entire ensemble of electrons can be described by a single distribution function, where the form is independently determined by the balance between electron heating in external fields within a finite part of the volume and losses by inelastic collisions in the volume and transport to the walls. This is exactly the situation we want to address in this work. The spatial dependence of the distribution function and the density is introduced only by the spatial dependence of the plasma potential: \(f\left( {\varepsilon,\vec{r}} \right) = f\left( {\varepsilon - e\,\Phi \left( {\vec{r}} \right)} \right)\). For a Maxwellian velocity distribution function \(f = f_{M}\) at an electron temperature \(T_{e}\), this leads directly to the so called Boltzmann factor:\(f_{M} \left( {\varepsilon - e\,\Phi \left( {\vec{r}} \right)} \right) = \exp \left( {e\,\Phi \left( {\vec{r}} \right)/\left( {k_{B} T_{e} } \right)} \right)\,f_{M} \left( \varepsilon \right)\), where \(k_{B}\) is the Boltzmann constant. This implies that the spatial density profile is directly related to the plasma potential \(n\left( {\vec{r}} \right) = n_{0} \,\exp \left( {e\,\Phi \left( {\vec{r}} \right)/\left( {k_{B} T_{e} } \right)} \right)\). An experimental example of a non-local distribution function in a low-pressure inductively coupled RF plasma is shown in Fig. 1.
Experimentally obtained non-local distribution functions in a low pressure inductively coupled plasma (ICP) in argon. The distribution functions are measured at various radial positions in a cylindrical plasma by a Langmuir probe and the EEPFs are shifted by the local plasma potential. The insets show the radial plasma density profile, which can be well fitted with a Bessel function. Left figure: 0.1 Pa, Right figure: 1.0 Pa (From Zhu et al. 2015)
The requirement for this non-local behavior, where collisions are negligible for the global behavior, can be formulated as follows (Tsendin 1995, 2010; Tsankov and Czarnetzki 2017):
Assuming a Maxwellian distribution function and approximating the characteristic electric field caused by the plasma potential as \(k_{B} T_{e} \,/\,\left( {e\,L} \right)\), the average condition becomes:
Here, \(\left\langle {\nu_{col} } \right\rangle < \nu_{m}\) is the characteristic average inelastic collision frequency and \(v_{th}\) the thermal electron velocity. Since the thermal velocity of electrons in any low-temperature plasma is close to \(10^{6} \,{\text{m/s}}\) and the system size is typically of the order of \(L \approx 0.1\,{\text{m}}\), the inelastic collision frequency should be lower than typically \(10^{7} \,{\text{s}}^{{ - {1}}}\). The elastic collision frequency takes this value at a pressure of approximately 1 Pa. Therefore, one can expect non-local behavior at pressures below 1 Pa. Of course, for much smaller systems, the limiting pressure can be higher, as is the case for instance in (Kortshagen 1994).
The above discussion on neglecting collisions when forming a non-local distribution function should not lead to the misconception that collisions do not play a role at all. In fact, still collisional interactions and the characteristics of electron heating by external electric fields play a crucial role for determining the form of the distribution function. The non-local principle simply means that electrons confined in the plasma by the plasma potential cross with high probability the finite volume many times before they eventually experience a dissipating collision with an atom or are finally lost to the wall by overcoming the confining potential due to sufficient energy gain in the external electric field.
In the opposite case of very short energy relaxation lengths, the distribution function follows from a local balance with the external field. This is the classical case addressed by the so called ‘two-term approximation’ (Lorentz 1905; Hollstein 1946; Allis 1956) and solved numerically by many publically available codes, so called Boltzmann-solvers, like e.g. LoKI-B (Tejero-del-Caz et al. 2019, 2021) or BOLSIG + (Hagelaar and Pitchford 2005). Formally, the distribution function is expanded in an infinite series of Legendre polynomials, where in the two-term approximation the series is already terminated after the first order:
Differences to the two-term approximation are discussed for instance in (Pitchford et al. 1981; Pitchford and Phelps 1982; Braglia et al. 1984; Leyh et al 1998; Stephens 2018). A Boltzmann solver including higher order terms is provided by (Stephens 2018). The resulting equation for the isotropic part of the distribution function \(f_{0}\) can be organized so that the lhs represents a heating operator describing energy exchange with an external electric field and the rhs contains the sum of all dissipative operators:
The operator describing local (Ohmic) heating in a homogeneous harmonic electric field at an angular frequency \(\omega_{0}\) is (Lieberman and Lichtenberg 2005):
The field amplitude \(E_{0}\) is contained in the effective velocity \(v_{E}\), representing the velocity amplitude of the free oscillation of an electron in the field:
A requirement for justifying the truncation of the infinite Legendre polynomial series is that \(v_{E} < < v_{th}\). The dimensionless Ohmic coupling function is defined as:
with \(\beta = \nu_{m} /\omega_{0}\). It is important to note that apparently local Ohmic heating depends critically on the elastic collision frequency \(\nu_{m} \propto n_{g} \propto p\). At low gas pressures \(p\), where \(\beta \to 0\), the coupling function becomes very small and local Ohmic heating becomes inefficient. In this low collisionality regime, spatial inhomogeneities of the oscillating field can cause a non-local (often also called stochastic) heating. It will be shown later that a more general heating operator, describing both local Ohmic (high pressures) and non-local stochastic heating (low pressures) can be expressed using the same form as given by Eq. (14), but with a different and more complicate coupling function.
The action of the heating operator is visualized in Fig. 2, where a Maxwellian distribution function \(f_{M}\) at a temperature \(T_{e}\) is assumed. The figure shows that the operator redistributes cold electrons of energies lower than the mean energy of \(3/2\,k_{B} T_{e}\) (negative values) to higher energies (positive values). For convenience, \(\left. {\partial F/\partial t} \right|_{Ohm}\) is shown instead of \(\left. {\partial f_{0} /\partial t} \right|_{Ohm}\). A similar behavior as in the local Ohmic case will be found in Sect. 6 also for the non-local case.
Ohmic heating operator for a homogeneous oscillating electric field at constant collision frequency \(\nu_{m} /\omega_{0} = 1\) evaluated for a Maxwellian distribution function with an electron temperature \(T_{e}\). The operator is presented here for the electron energy distribution function \(F\left( \varepsilon \right)\), which makes the effect more obvious. On the ordinate \(\left. {\partial F/\partial t} \right|_{Ohm}\) is normalized to \(- m_{e} \,v_{E}^{2} \,\omega_{0} \,n_{e} /\sqrt \pi\). Negative values of the operator remove particles from the distribution and positive values insert these particles back into the distribution (indicated by the curved arrow). The positive and negative areas are identical so that particle number conservation is ensured. Zero crossing is at the mean thermal energy of \(3/2\,k_{B} T_{e}\) as indicated by the vertical dashed line
Finally, yet importantly, it should be briefly noted that the operator (14) allows also obtaining the operator for a static and homogeneous field by (a) letting \(\omega_{0} \to 0\) and (b) considering that the mean quadratic field in the oscillating case \(E_{0}^{2} /2\) translates into \(E_{0}^{2}\) for the static field:
We will make use of this result in Sects. 4 and 5 as a test case.
The more general case combining local and non-local electron heating allows for spatial variations. Here the ansatz is again a two-term distribution function but of a different kind than (12). The distribution function is composed of a homogeneous, isotropic, and stationary term and an anisotropic and oscillating (temporally varying) term:
The approximation assumes \(f_{0}\) to be homogeneous in configuration space and time, which is motivated by the above non-locality argument. Small variations of the plasma potential across the interaction (heating) zone with the electric field can be neglected in this context. Further, the small temporal oscillation of the homogeneous distribution is of second order only and can be neglected for the heating aspect. Note, however, that advantage is taken of the residual small oscillation at \(2\,\omega_{0}\) for instance in the so-called RF modulation spectroscopy, where tiny oscillations of the excitation rate and the related optical emission of the order of 1% are detected (Crintea et al. 2008; Tsankov and Czarnetzki 2011; Ahr et al. 2018). This optical diagnostic technique is an alternative to the so-called B-dot probes, more commonly used for the measurement of the field structure in inductively coupled plasmas (e.g. Han et al. 2019).
The ansatz (18) implies that the Krook operator for the elastic collisions becomes \(- \nu_{m} \left( v \right)\,f_{1} \,\). Then the Boltzmann equation can be separated into a stationary, homogeneous and isotropic equation similar to Eq. (13), on the one hand, and a dynamic and anisotropic equation, on the other hand:
where the lhs of the Eq. (19) is the general heating operator following from the Boltzmann equation (Boltzmann heating operator: BHO):
The bracket indicates averaging over space, time, and the solid angle in velocity space. It should be emphasized that the heating operator never resolves the timescales of the oscillation of the field or the collision rate. Like in the Ohmic case, heating needs to be balanced by dissipation, i.e. collisional energy transfer from electrons to neutrals, which is represented by the operator on the rhs. Here too, the volume average applies. The equation defining the anisotropic distribution function \(f_{1}\) and relating it to the isotropic function \(f_{0}\) reads:
This equation can be solved for \(f_{1}\) by applying Fourier transformation (FT):
Note that throughout this work, the symmetric variant of the FT is used, i.e. a factor of \(1/\sqrt {2\pi }\) for each dimension is applied symmetrically in the forward and backward transformation. Further, the above sign convention in the exponent is used throughout for all Fourier representations. In order to \(f_{1}\) and \(\vec{E}\) being real, it is required that:
Fourier transformation converts the second equation into an algebraic equation, which yields for the Fourier transform of the anisotropic distribution function:
The symmetry properties of the electric field then guarantee also the required symmetry properties of the anisotropic distribution function. Before proceeding with the determination of the heating operator, it should be mentioned that the anisotropic function \(f_{\begin{subarray}{l} 1 \\ \end{subarray} }\) and its Fourier transform \(\hat{f}_{1}\) are directly related to the current density and its Fourier transform, respectively:
Note that a numerical factor \(\sqrt {2\,\pi }\) follows for each of the four dimensions (spatial and temporal) from the particular choice of the form of the Fourier transformation as discussed above. In (25) we are taking the current density as a convolution \(\left( * \right)\) of a conductivity term, which in general is a tensor \(\left( =\right)\), with the electric field. This convolution contains already all non-local effects in space and time by connecting the electron flow at a given point in space and time to all earlier times and the related remote positions in space:
The tilde indicates that \({\underline{\underline{\tilde\sigma }}}\) has the dimension of conductivity \(\sigma\) per volume per time. Collisions naturally reduce the ‘memory’ timescale to about the inverse collision frequency \(1/\nu_{m}\). In the ultimate limit of very high collisionality, the expression becomes scalar (omitting the unity matrix) and local, which leads to Ohms law:
With Eq. (24), the Fourier transform of the conductivity tensor can immediately be identified:
The tensor components represent the contribution of an electric field in direction \(i\) to the current in direction \(j\). Naturally, in an isotropic plasma without a defined axis of symmetry, e.g. by an external magnetic field, only diagonal elements can contribute. Moreover, throughout this work only transversal electric fields are considered \(\left( {\vec{k} \bot \vec{E}} \right)\). Without loss of generality, the wave vector \(\vec{k}\) can be assumed to point in z-direction. Then the only non-zero elements of the tensor are the diagonal elements \(\hat{\sigma }_{xx} = \hat{\sigma }_{yy} = \hat{\sigma }\). Assuming a velocity-independent collision frequency \(\nu_{m}\), further simplifies the expression. In the limit of high-collisionality, the non-local term \(\vec{k} \cdot \vec{v}\) may be neglected in comparison to \(\nu_{m}\), so that again only the diagonal terms remain and take the same values. In this case, the conductivity is both, local and fully isotropic. When calculating the self-consistent field profile in the plasma in Sect. 7, use will be made of the above kinetic form of the Fourier transform of the conductivity.
It is tempting to apply an integration by parts to Eq. (28) in order to remove the derivative of the distribution function, but this is only practically feasible if the velocity dependence of the collision frequency can be neglected. Note also that the term proportional to \(\nu_{m}\) represents the real part of the conductivity and this is the only part that matters for the Fourier transform of the power density:
For a monotonously decaying isotropic distribution function \(\partial f_{0} /\partial v < 0\), which is most often the case, the real part of the conductivity is positive, which implies that also the power density is positive, as expected. We will return to these expressions later in connection with the Fokker–Planck operator and for the calculation of the self-consistent spatial electric field profile.
The Boltzmann heating operator follows formally by inserting (24) in (22) and the resulting expression again in Eq. (20). However, the different nested integrals are complicate and quite challenging for practical evaluation:
Some of these problems will appear in some form or another also in the frame of the Fokker–Planck treatment below. Here, we want to emphasize only two important aspects. It is interesting to solve the nested integrals for the trivial case of a homogeneous field, oscillating harmonically at an angular frequency \(\omega_{0}\) with an amplitude \(E_{0}\) pointing in the z-direction. After carrying out the frequency and temporal integrals, the expression reduces to a meaningful form:
This expression has already the form of a Fokker–Planck operator, or more precisely of the diffusion term of a Fokker–Planck operator, as will be shown in the following sections. Carrying out the angular average leads to exactly the same Ohmic heating operator as resulting from the two-term approximation (Eq. (14)). Details of the averaging procedure can be found in appendix A. One may conclude that, at least for the derivation of the heating operator (31), the two-term approximation to the Legendre expansion is not required. The result obtained is exact within the general ansatz of separating the distribution function in an isotropic and an anisotropic part, which is always possible without approximations for a homogeneous field.
Finally, it should be noted that the general Fourier amplitude \(\hat{f}_{1}\) (Eq. (24)) clearly exhibits resonances in velocity space at \(\omega \, = \vec{k} \cdot \vec{v}\), i.e. when the particle velocity equals the phase velocity of the wave. These resonances are the essence of collisionless heating. Collisions cause dam** of these resonances, which can only be neglected if \(\omega > > \nu_{m}\). For high collisionality \(\nu_{m} > > \vec{k} \cdot \vec{v}\), the spatial structure of the field becomes negligible and local Ohmic heating is recovered. Of course, this applies also for a homogeneous field, where \(\hat{\vec{E}} \propto \delta \left( {\vec{k}} \right)\).
The pressure range for non-local heating at typical radio-frequencies (RF) of the order of \(10\,{\text{MHz}}\) is again limited to a few Pa at maximum (typically \(\nu_{m} = {\text{a}}\,{\text{few}}\,10^{7} \,{\text{s}}^{{ - {1}}}\) at 1 Pa), quite comparable to the range characteristic for non-local distribution functions. Therefore, only a single global non-local distribution function needs to be determined. However, for lower RF frequencies in the range \(0.1\, - \,1.0\,{\text{MHz}}\) much lower pressures are required for stochastic heating to contribute. Such low pressures might be difficult to realize in an experiment due to the related strong collisionless losses to the walls, since then not only the electron mean free path but also the ion mean free path becomes comparable or larger than the system size. In addition, for such low frequencies, often ponderomotive effects cannot be neglected, which strongly complicates the physics (Cohen and Rognlien 1996).
In this work, emphasis is on typical RF frequencies in the range of 10 MHz and pressures in the range 0.1 Pa to a few Pa. Within this range, transition is made between dominant non-local heating and dominant local Ohmic heating of the electrons. In most cases, both mechanisms will contribute to some extend and have to be considered in a consistent way. These conditions are found in many applications, e.g. in semi-conductor processing, which emphasizes the relevance (Lieberman and Lichtenberg 2005). It is the aim of the subsequent sections to show how a consistent description of the electron heating mechanism can be achieved by using the Fokker–Planck/Langevin concept. The description will be generally obtained, yielding a natural combination of non-local (collisionless) heating and local Ohmic heating. Before proceeding with this task, the correct form of the Fokker–Planck equation and the related Langevin equation for the present problem is derived in Sects. 4 and 5. A test case is investigated in Sect. 6.
3 The master and the Fokker–Planck equations
On a very fundamental level, statistical processes can be described by so-called master equations. There are various alternative ways to approach the master equation concept. A slightly uncommon but well suited way to achieve this goal for electron collisions in plasma physics is to start with rate equations for discrete atomic states, where the population numbers of the states \(n_{j}\) at energies \(\varepsilon_{j}\) are connected by electronic collisions with rates \(A_{ij}\) (Fig. 3):
Scheme of discrete states ordered by energy and connected by transition rates
It is important to note that a fundamental assumption is made, considering any collisional interaction at a given time to be independent from all previous interactions. These are so-called Markov processes, where no memory effect is present. In this work, we will be dealing exclusively with Markov processes. Now transformation can be made from a discrete to a continuous system, which converts the discrete sums of Eq. (32) into integrals in energy \(\varepsilon\) over continuous rates. By further generalizing, the scalar variable energy \(\varepsilon\) can be replaced by the velocity vector space \(\vec{v}\), the population of states by the velocity distribution function \(f\left( {\vec{v},t} \right)\), and the discrete rates by continuous functions \(R\left( {\vec{v},\vec{v}^{\prime}} \right)\) that cause a transition from a state at velocity \(\vec{v}^{\prime}\) to another state at velocity \(\vec{v}\):
This is the master equation for the velocity distribution function \(f\left( {\vec{v},t} \right)\) (Reif 1965; Paul and Baschnagel 2013; Risken 1989). Although exact, it is very poorly suited for any practical calculation, particularly due to the fact of being an integro-differential equation. However, the master equation can be converted into a differential equation system by using the Kramers-Moyal expansion. The details can be found in the literature and only the general idea is briefly outlined here (Keilson and Storer 1952; Risken 1989; Paul and Baschnagel 2013). One starts with averaging an arbitrary well behaving function \(g\left( {\vec{v}} \right)\) by the above master equation:
The function \(g\left( {\vec{v}} \right)\) is now expanded into a Taylor series around \(\vec{v}^{\prime}\) in the second integral and subsequently a series of integrations by part and rearrangements is performed:
The resulting expression is the Kramers-Moyal expansion of the master equation, an infinite sum of derivatives of all orders. In this form it is exact and fully equivalent to the master equation. Some new effective rate coefficients \(M_{k}\) appear in this formulation:
where \(\Delta \vec{v} = \vec{v}^{\prime} - \vec{v}\) is the velocity change in a collision. Assuming now that the rates have significant values only for small changes in the velocity, as is the case for Brownian motion, the infinite series might be truncated after the second order, which finally yields the well-known standard form of the Fokker–Planck equation (Keilson and Storer 1952; Risken 1989; Paul and Baschnagel 2013):
where the first term is called the drift term and the second term, the diffusion term.
Although the previous truncation might look slightly arbitrary, it is demonstrated by the Pawula theorem that the truncation after the second order is either exact or not possible at any order (Pawula 1967). In this sense, the Fokker–Planck equation is an exact equation. The critical point is the smallness of the velocity change, where the reference is the typical average speed of the ensemble, i.e. the thermal speed.
The rate coefficients can be expressed in a more meaningful way by replacing the velocity integral by an integral over a set of statistical parameters \(\chi = \chi_{1},\,\chi_{2},\,\chi_{3},...\) and related independent probabilities \(P\left( \chi \right) = P_{1} \left( {\chi_{1} } \right)\,P_{2} \left( {\chi_{2} } \right)\,P_{3} \left( {\chi_{3} } \right)...\) which cause velocity changes \(\Delta \vec{v}\) with a collision frequency \(\nu_{c}\):
The determination of the velocity changes in a collisional interaction \(\Delta v_{i}\) is outside of the realm of the Fokker–Planck equation. We will return to this important point in the following section on the Langevin equation.
Naturally, the representation of the Fokker–Planck equation in velocity space is only one particular choice, although very important for this work. Alternatively, one can also choose the configuration space, which then allows describing the usual drift and diffusion phenomena. This was indeed the original formulation and the names of the two Fokker–Planck terms originate from this choice. We will briefly address this formulation at the end of this section. Another alternative is the formulation in energy space, which is particularly useful for calculating the small dissipation due to the finite energy transfer in elastic collisions. Calculating the corresponding operator (Eq. (8)) from the Boltzmann collision integral (3) is a lengthy and painstaking undertaking but as will be shown below, it can be obtained in a rather simple and straight forward calculation by using the Fokker–Planck equation. The first step involves a transformation from the velocity to the energy distribution function and from the velocity to the energy variable, in which case a one-dimensional equation results:
where \(\nu_{c} \left( {\varepsilon,\,\vartheta } \right)\) is the energy and scattering angle dependent elastic collision frequency. The energy transfer in an elastic collision is:
\(\Delta \varepsilon /\varepsilon\) is a small value due to the small mass ratio \(\mu < < 1\). Consequently, the diffusion term, scaling with \(\mu^{2}\), can be neglected in comparison to the drift term, scaling with \(\mu\). Finally, integration over all angles \(\vartheta\) gives:
where \(\nu_{m} \left( \varepsilon \right)\) is the elastic momentum-transfer collision frequency. The Fokker–Planck operator for energy transfer in elastic collisions (FPEC) finally reads:
Transforming from the energy representation back to the velocity distribution function yields exactly the same expression as derived from the Boltzmann collision integral in the frame of the two-term approximation (Eq. (8)). Therefore, the continuous dissipation due to energy transfer in elastic collisions can be understood as a drift in energy space. It will be shown below, that electron heating can be understood as diffusion in velocity space. However, before proceeding it is necessary to re-inspect more closely the above general form of the Fokker–Planck Eq. (37).
The above expansion is ordered in powers of the velocity change \(\left( {\Delta \vec{v}} \right)^{k}\). However, in Eq. (38), the functions \(R\left( {\vec{v},\vec{v}^{\prime}} \right)\) also depend in general on the velocity change \(\Delta \vec{v} = \vec{v}^{\prime} - \vec{v}\) and, therefore, they should be expanded too. It is usually assumed, in fact mostly without explicit notice, that the collision frequency and the probability functions do not depend on the velocity change. Then, naturally, the above expansion is correct. However, in the more general case, there is a velocity dependence, particularly of the collision frequency, and the term \(\psi \left( v \right) = \nu_{m} \left( v \right)P\left( {\chi,v} \right)\) must be expanded too. Such an additional expansion changes of course the order of powers in the drift and diffusion terms. In the diffusion term, this would lead to powers higher than 2, which can be neglected according to the Paluwa theorem. Consequently, we are only concerned with the expansion of the drift term and there only up to first order. Since the drift term is already of first order in \(\Delta \vec{v}\), this leads to a new second order term. This term then adds to the diffusion term, which is also second order, while the drift term remains unchanged.
One might argue that it is necessary to calculate an average value of \(\overline{\psi }\) with an unknown probability function \(W\left( {\delta \vec{v}} \right)\) for the velocity change. Fortunately, the first order expansion is independent of such a probability distribution:
The calculation is straightforward by expanding both \(\psi\) and \(W\) in powers of \(\left| {\delta \vec{v}} \right| \le \left| {\Delta \vec{v}} \right|\), which allows immediate integration. Then the quotient is expanded in powers of \(\Delta \vec{v}\), which finally leads to the above first order result. Indeed, the first order term is simply the linear average. In the above equation and all subsequent equations, we apply the Einstein summation convention.
Inserting the result of Eq. (43) in the Fokker–Planck Eqs. (37), (38) yields:
where, after the initial equality, the first two terms result from inserting the expansion (43) in the drift term, and the latter two terms result from applying the product rule to the diffusion term. Here, the index FPO is introduced to indicate explicitly the form of the Fokker–Planck operator.
If one further assumes that the velocity changes do not depend on the velocity, which is certainly the case for any electric force and all collisions of relevance for plasmas, then the final form of the Fokker–Planck operator becomes:
Note that the difference between Eqs. (37) and (45) is in the position of the derivative \(\partial /\partial \,v_{j}\) in the diffusion term. In the special case of a velocity-independent collision frequency \(\nu_{m}\), the two forms are identical. However, in the general case of \(\nu_{m} = \nu_{m} \left( v \right)\), the two equations differ significantly.
In this work, the emphasis is on the determination of the electron heating operator. In addition to the above stochastic average, the operator needs to be averaged also over the solid angle in velocity space, since it should act only on the isotropic part of the velocity distribution function. This immediately causes the drift term to vanish, showing that heating can indeed be understood as diffusion in velocity space. Further, in the diffusion term, only the diagonal elements remain. Thus, the final form of the Fokker–Planck heating operator (FPHO) is (the term \(\left\langle {\nu_{m} \,\Delta v_{j}^{2} } \right\rangle_{\chi }\) still needs to be specified, which is done in the next section):
where we have changed the sign of the operator, with respect to that of the corresponding term in the Fokker–Planck Eq. (45), for coherency with the writing of the Boltzmann Eq. (1) that features the temporal-variation and the heating terms on the same side. Note that no change of sign was needed in the Fokker–Planck operator for elastic collisions (FPEC) of Eq. (42), since the temporal-variation and the collisional terms are on opposite sides of the Boltzmann Eq. (1).
The necessity of the modification of the original Fokker–Planck equation will be investigated in more detail in the following section. However, a simple argument can be developed immediately by temporally switching to configuration space and focusing on the diffusion of particles in some environment. In this case, the velocity distribution function is replaced by the spatial density profile \(n\left( {\vec{x}} \right)\) with the independent variable changed by \(v_{i} \to x_{i}\). Neglecting the drift in external fields and additionally assuming a homogeneous temperature, the fluid equations describing the diffusion of particles are the continuity and the momentum balance equations (Lieberman and Lichtenberg 2005; Alves 2007). In the latter equation only the pressure gradient and the friction terms (momentum exchange with the environment) need to be considered, i.e. the temporal derivative of the drift velocity is neglected. This is equivalent to viewing the problem on a timescale longer than the inverse collision frequency, when the flow velocity \(\vec{u}\) equilibrates. Indeed, the Fokker–Planck equation never resolves the collision timescale. Further, the inertia term can be neglected since it is quadratic in the flow velocity, reasonably assumed to be much smaller than the thermal velocity. In general, the collision frequency might be a function of the spatial coordinates due to a potential inhomogeneity of the environment \(\left( {\nu \left( {\vec{x}} \right) = \,\left\langle {N\,\sigma \,v} \right\rangle } \right)\), i.e. either by variation of the density \(N\left( {\vec{x}} \right)\) and/or the composition of the environment (represented by the corresponding cross section \(\sigma \left( {\vec{x}} \right)\)). Under the previous assumptions, the relevant equations read:
where \(n\) is the density of the diffusing particles of mass \(m\). Combining both equations yields the diffusion equation with the diffusion constant \(D\), which can depend on the spatial coordinate via \(\nu\):
Note, that in plasmas the same form of the diffusion equation appears for the charged particles, particularly for ambipolar diffusion, where the temperature is the electron temperature, the mass is the ion mass, and the collision frequency is the ion-neutral elastic collision frequency (Lieberman and Lichtenberg 2005; Alves et al. 2007).
On the other hand, the modified Fokker–Planck Eq. (45) in configuration space reads:
where only the diagonal terms contribute since there is no mean directional displacement of the particles, i.e. \(\left\langle {\Delta x_{i} } \right\rangle = 0\). Equation (49) has the same form as Eq. (48) derived from the fluid dynamic picture, with the diffusion constant identified as:
In contrast, the standard form of the Fokker–Planck equation would lead to a diffusion equation of different form:
implying that the flux density is given by \(n\,u_{i} = - \left( {D\,\partial n/\partial x_{i} + \,n\,\partial D/\partial x_{i} } \right)\). Here, the second term is clearly artificial for a homogeneous temperature and not supported by the momentum balance Eq. (47). Therefore, Eq. (51) is an incorrect form of the diffusion equation, if the diffusion constant is spatially-dependent due to variations of the collision frequency. This simple example already provides a strong hint that the above modification of the standard form of the Fokker–Planck equation is essential. The differences between the modified and the standard forms of this equation will be highlighted in the following section for the specific example of the electron velocity distribution function in collisional plasmas.
4 The Langevin equation
The Langevin equation is another alternative to describe particle motion under the action of a stochastic force (Reif 1965; Paul and Baschnagel 2013). In the present context, the Langevin equation has particular relevance since it provides the link between the rate coefficients in the Fokker–Planck equation (i.e. the average quadratic variations of the velocity) and the equation of motion of particles in an external field.
The characteristic form of the Langevin equation, originally formulated to describe Brownian motion, consists of a friction term, proportional to the particle momentum \(\vec{p}\) via a collisional frequency \(\nu\) (acting as a friction coefficient), and a stochastic force term \(\vec{\Lambda }\left( t \right)\):
where the random force term can be described by summing the momentum changes \({\lambda }_{i}\) that occur in each particle collision i, at time \({t}_{i}\).
Here, we start with a slightly different equation, which is equivalent to the Langevin equation as will be shown later:
The expression (53) is similar to Newton’s equation of motion for a momentum \(\vec{p} = m\,\vec{v}\) under a force \(q\vec{E}\) (without loss of generality, here, an electric field \(\vec{E}\) acting on a charge \(q\) is chosen), but contains in addition the stochastic force term \(\vec{\eta }\left( t \right)\), defined by random delta function forces of strength \(\Delta \vec{p}_{ci}\), representing the momentum changes in a collision at time \(t_{i}\). As before, the introduction of the delta functions is motivated by the fact that the finite but very short timescale of the real interaction in a collision process is not resolved. The major aim of this section is to show that, for a sufficiently coarse timescale and ideal elastic collisions, the velocity changes in the interval between two collisions represent exactly those appearing in the Fokker–Planck equation.
Considering only ideal elastic collisions, where the small energy transfer is neglected, allows the momentum change to be expressed by a random rotation of the initial momentum vector (before the collision),
where \(\underline{\underline{R}}_{i}\) is a random rotation matrix and \(\underline{\underline{1}}\) is the unity matrix. The random rotations are a Markov process, so that rotations in subsequent collisions are uncorrelated. Further, the degree of rotation is statistically independent of the time between the collisions. Finally, it is assumed for simplicity that (a) the time between the collisions is independent of the electron velocity and (b) the applied force is constant, i.e. \(\vec{E} = \vec{E}_{0} = {\text{const}}.\), although it is rather straightforward to generalize the result also for oscillating force fields. The momentum gained in between two collisions, over a free flight interval \(\Delta t_{i} = t_{i} - t_{i - 1}\), by acceleration in the force field is:
Note that for a time dependent force, typically an oscillatory force, the corresponding momentum would read \(\Delta \vec{p}_{Ei} = q\int\limits_{{t_{i - 1} }}^{{t_{i} }} {\vec{E}\left( t \right)\,dt} = \,q\,\overline{\vec{E}}_{i} \,\Delta t_{i}\), where the bar indicates the average force over the interval \(\Delta t_{i}\).
Finally, the momentum direction after the action of the applied force is randomized immediately after the subsequent collision at \(t = t_{i}\), and using Eq. (54) its expression is:
This way, the action of the applied force is combined with the collision process into one effective process, which provides a certain variation in the momentum (and velocity) of the particle after each collision. The variation is random in both the direction, due to the random direction in the scattering process, and the strength, due to the randomness of the duration of the free-flight period. This is emphasized by Eqs. (55), (56) which yield:
Integrating the Langevin Eq. (52) over the same timescale \(\Delta t_{i}\) (not resolving the individual acceleration-scattering intervals) yields:
which shows that the first term in Eq. (57) can be associated with friction and the second term with a random force, i.e. \(\lambda_{i} = \underline{\underline{R}}_{i} \cdot q\vec{E}_{0} \,\Delta t_{i}\). For the present case, this later term is of major importance. These results also confirm that, under the present assumptions, Eq. (53) is indeed equivalent to the Langevin Eq. (52).
From Eq. (56) we also conclude that the momentum of a particle after \(N > > 1\) collisions is (\(\vec{p}_{0}\) is the initial momentum):
where \(\prod\limits_{i = j}^{N} {\underline{\underline{R}}_{i} = } \underline{\underline{R}}_{j}^{^{\prime}}\), because the multiplication of random rotation matrices is just another random rotation matrix. Apparently, the average value of the momentum per collision goes to zero, due to its isotropization caused by repeated random rotations in multiple elastic collisions \(\left( {N \to \infty } \right)\):
Taking the square, all mixed terms also average to zero and the direction of the momentum-vectors becomes irrelevant. Therefore, the average quadratic momentum per collision can be estimated from its change in any interval, due to the free acceleration in the field and the subsequent collision:
Since the contribution of the initial momentum vanishes for large numbers N, the final result reduces to the average over the squared momentum-change in between two collisions. This average can conveniently be carried out by using probability distributions for the two stochastic parameters, \(t_{i - 1}\) and \(\Delta t_{i}\), which enter the expression (55). The time \(t_{i - 1}\) is clearly homogenously distributed and the collision free interval \(\Delta t_{i}\) follows an exponential probability distribution (Eq. (5)) with the mean collision frequency for momentum change \(\nu_{m}\) given by:
This involves also a drift \(\left\langle {\Delta \vec{p}} \right\rangle = q\vec{E}_{0} \,\left\langle {\Delta t} \right\rangle = q\vec{E}_{0} /\nu_{m}\) consistent with the balance between the force and friction in the fluid dynamics picture.
In the average, as mentioned before, the indices may be dropped:
The average squared momentum change per time interval now becomes:
Finally, it is important to note that this quadratic momentum-change is distributed equally between the three Cartesian components since the random rotation matrices do not provide a preferred direction, i.e. each direction contributes 1/3 of the above expression:
Equation (65) allows to specify the stochastic average term in the FPHO (46), making the replacement \(\left\langle {\left( {\Delta \vec{p}_{N} } \right)^{2} } \right\rangle_{N} = \,\left\langle {m_{e}^{2} \,\Delta v_{j}^{2} } \right\rangle_{\chi } \,\) and noting that, for symmetry reasons, each of the three terms along a Cartesian direction makes an identical contribution to the solid angle average. This is equivalent to kee** the derivatives in only one direction, e.g. the direction of the field vector (here in z-direction), yet considering all three components of the quadratic momentum-change:
where \(q = - e\) and where we have used the result of the appendix A for the average over the solid angle in velocity space. Note that the FPHO (66) is identical to the result obtained from the Boltzmann equation for the Ohmic heating operator under the action of an AC field (Eq. (17)).
In summary, the above example, although explicitly formulated only for the simplest type of force, shows that the small velocity changes in the Fokker–Planck equation can be conveniently determined by integrating the equation of motion from a random initial moment over an arbitrary free-flight interval. In the case of a spatial-dependent force, also the initial positions have random coordinates, as will be discussed in detail in Sect. 6. Continuing along this path, the average quadratic variations of the velocity, needed to be specified in the FPHO, can be calculated by appropriate probability distributions. In this calculation, only one velocity component has to be considered in the Fokker–Planck equation since isotropization of the distribution function by elastic collisions is automatically included by this procedure. It should be emphasized that the isotropization process, as well as the random variations of the initial conditions (phase, position) in inhomogeneous and oscillating fields, are essential for creating irreversibility. Only in this way, a directional and deterministic energy gain from the field turns eventually into heat, randomizing the particle motion. However, randomization is not required within the interaction zone with the field but can happen in the larger plasma volume by collisions with neutrals or reflections from the walls, assuming they have at least some random mesoscopic structure.
Finally, the rate coefficient in the diffusion term of the Fokker–Planck equation is obtained from the average quadratic variations of the velocity, which relates to the average of the squared stochastic force (see Eqs. (57), (58) and (61)) (Paul and Baschnagel 2013), and therefore:
In the last step, a generalization from a constant force to a general time-dependent (periodic) force is made, as already outlined above. The result shows that from this perspective, the rate coefficient in the diffusion term of the Fokker–Planck equation can be calculated by simply integrating the equation of motion for a given force over a random free flight period \(\Delta t\), and then subsequently carrying out the average (over the square) with appropriate probability distributions.
An exact analytical integration of the equation of motion is not always possible. If the force causing the acceleration also depends on the spatial coordinate along the direction of force, approximations are necessary (Czarnetzki 2018). However, for fields where the spatial variation is perpendicular to the direction of force, the integration can always be performed.
For example, the particular case investigated in Sect. 6 considers a harmonically oscillating electric field that points in x-direction and varies in space along the z-direction. In this case the FPHO reads:
where the stochastic parameters for the averaging procedure are the initial coordinate \(\vec{r}\left( {t = t_{0} } \right) = \vec{r}_{0}\) (in fact, in the example only the z-coordinate matters so that \(\vec{r}_{0} \to z_{0}\)), the initial time \(t_{0}\), and the duration of the free (collisionless) flight period \(\tau\).
5 Test of the concept for a DC field by Monte-Carlo simulation
Before proceeding to study the case of an oscillating and spatially inhomogeneous field, we will first adopt the simplest possible situation of a homogeneous and constant electric field pointing in z-direction, to verify the modified and the standard FPHO (see Sect. 3) against an ergodic Monte-Carlo (MC) simulation. In order to further simplify the model conditions, we assume that heating is balanced only by the continuous energy losses due to elastic collisions, described by the differential operator (8), which allows exact analytical integration and provides explicit expressions for the velocity distribution function (Lieberman and Lichtenberg 2005).
In the MC simulation, the equation of motion is solved for a single particle interacting with the field and colliding with the surrounding atoms in a random manner. This is equivalent to solving the Langevin equation but including also dissipation. The ergodic principle implies that following a single particle for a long time is equivalent to observing a large number of particles in an ensemble at one time (Reif 1965). The distribution function follows by finally determining the probability to find the particle within a certain velocity interval. The simulation is described in more detail in appendix B. A similar concept has been applied in (Czarnetzki and Tarnev 2014; Tarnev et al. 2019), where the MC part includes even inelastic collisions.
As seen in Sect. 4, the heating operator for a DC field corresponds either to the Ohmic operator (Eq. (17)) obtained from the Boltzmann equation, or, equivalently, the corresponding Fokker–Planck operator (Eq. (66)):
The balance between heating and collisional dissipation then reads:
However, for the standard form of the Fokker–Planck Eq. (37), the heating operator on the left-hand side would read (where the subindex FPUO stands for Fokker–Planck unmodified (standard) operator):
The general solution of Eq. (70), adopting the modified Fokker–Planck equation, takes the well-known form:
where the constant \(C\) follows from normalization, i.e. \(4\pi \,\int\limits_{0}^{\infty } {f_{0} \,v^{2} \,dv} = \,1\,\,\left( {{\text{or}}\,n_{e} } \right)\). For the standard form of the operator (71), the solution reads:
The exponential function is identical in both cases but the preceding amplitude factors differ in general. Of course, for a constant (velocity-independent) collision frequency both results are identical to:
The result has the form of a Maxwell distribution, which can potentially cause misleading interpretations. Although Eq. (74) has the same form as a thermal equilibrium distribution, this special form is not at all the result of a thermal equilibrium. Indeed, thermal equilibrium requires interactions among the particles of the ensemble, a phenomenon not considered here, where all interactions are only with the atoms of the neutral gas, which is assumed as an independent ensemble not affected by the interaction with the electrons. Moreover, there is also no equilibrium between the electrons and the neutral gas, since the electron temperature is completely independent of the gas temperature, which in fact is approximated as zero in the above elastic dissipation operator.
The MC simulation for a constant collision frequency considers a velocity resolution \(\delta v = 1\) on a normalized scale, with normalization by \(v_{n} = r_{n} /t_{n}\), where \(r_{n} = e\,E_{0} \,/\,\left( {m_{e} \,\nu_{m}^{2} } \right)\) and \(t_{n} = \,\,1/\nu_{m}\), i.e. with the velocity normalized to the absolute value of the drift velocity resulting from fluid dynamics \(u_{z} = - e\,E_{0} /\left( {m_{e} \,\nu_{m} } \right)\). The velocity distribution function is normalized to 1 according to \(4\pi \,\int\limits_{0}^{\infty } {\left( {v/v_{n} } \right)^{2} f\left( {v/v_{n} } \right)} \,dv/v_{n} = 1\). In total \(10^{8}\) collisions are simulated for \(\mu = 10^{ - 4}\).
Not surprisingly, the analytical results for both operators agree well with the simulation result (Fig. 4). In the latter, some moderate statistical noise at low velocities is apparent, as an unavoidable consequence of the division of two small numbers in the statistical analysis (see appendix B). The deviation of the drift velocity in z-direction from its nominal value of -1 (normalized value) is only about \(6 \cdot 10^{ - 3}\).
Dimensionless isotropic velocity distribution function \(f_{0} \left( v \right)\) for a constant collision frequency \(\nu_{m}\). The velocity on the abscissa is normalized to the characteristic velocity of the system \(v_{0} = v_{n} = e\,E_{0} \,/\,\left( {m_{e} \,\nu_{m} } \right).\) The distribution function is normalized to 1. Solid line: analytical result, dots: ergodic MC simulation
While in the above example the results agree for both versions of the heating operator, they differ significantly for a constant mean free path \(\lambda_{m}\), where \(\nu_{m} \left( v \right) = v/\lambda_{m}\) depends on the velocity. In this case, the solution (72) based on the Boltzmann and the modified Fokker–Planck operator reads:
This is the so-called Druyvesteyn distribution. Quite in contrast, for the solution (73) based on the standard Fokker–Planck operator, the amplitude factor is not a constant but is proportional to the velocity:
Therefore, in the former case the distribution function is maximal at zero velocity and then decays monotonously, while in the latter case it is zero at zero velocity and has instead a maximum at \(v_{\max } = \,\,v_{\lambda } /\left( {3\,\mu \,} \right)^{1/4}\).
In this case the simulation is carried out for the same mass ratio \(\mu = 10^{ - 4}\) as before but for \(10^{7}\) collisions with a velocity resolution of \(\delta v = 0.1\), where the normalization is now to \(v_{n} = r_{n} /t_{n}\) with \(r_{n} = \lambda_{m}\) and \(t_{n} = \,\,\sqrt {m_{e} \,\lambda_{m} \,/\,\left( {e\,E_{0} } \right)}\). Very good agreement is obtained between the simulation and the analytical result (75) from the Boltzmann and the modified Fokker–Planck heating operators (Fig. 5). Again, the unavoidable increase of the statistical noise at low velocities is due to the division of two small numbers in the statistical analysis as outlined in the appendix B. In addition, an alternative distribution function is recorded, obtained from the simulation using a wrong statistical weight. Instead of assigning to each free-flight collision event \(i\) a statistical weight \(\tau_{i} /\sum\limits_{i} {\tau_{i} }\), where \(\tau_{i}\) is the free-flight time before the collision, the alternative distribution is computed by assigning the same statistical weight \(1/N\) to each event (where N is the total number of free-flights), regardless of the free-flight time before the collision. Of course, this alternative counting is not correct but interestingly it yields exactly the distribution function obtained by the standard Fokker–Planck operator. This may be interpreted as an indication that the effect of the incorrect position of the differential in the standard form of the Fokker–Planck Eq. (37) leading to (73) is equivalent to assigning a wrong statistical weight to the velocity intervals in between two collisions.
Dimensionless isotropic velocity distribution function \(f_{0} \left( v \right)\) for a constant mean free path \(\lambda_{m}\). The velocity on the abscissa is normalized to the characteristic velocity of the system \(v_{0} = v_{n} = \sqrt {e{\mkern 1mu} E_{0} {\mkern 1mu} \lambda_{m} {\mkern 1mu} /{\mkern 1mu} m_{e} }\). The distribution function is normalized to 1. Solid line: analytical result for the Boltzmann and the modified Fokker–Planck heating operator, solid dots: ergodic Monte-Carlo simulation, dashed line: analytical result for the standard form of the Fokker–Planck heating operator, open dots: simulation result for constant statistical weights independent of the free flight time
In conclusion, by comparing the results obtained with the modified Fokker–Planck equation to (a) the diffusion result in configuration space from fluid dynamics (see Sect. 3) and (b) the first principle ergodic Monte-Carlo simulation in velocity space, it is obvious that this modification is essential for obtaining physically correct results. Finally, the modification is also inevitable by the logic of the Kramers–Moyal expansion, which is based on sorting terms according to powers of the small changes in velocity. For consistency, this requires also an expansion of the rate coefficients, if these depend on the velocity changes, which then leads to the above modification, in perfect agreement with the examples.
Finally, a suggestion for interpreting the above distribution functions in term of Brownian motion is proposed. Brownian motion and the related diffusion mechanism are closely connected to the classical random walk process. In the simplest one-dimensional version, ‘a walker’ can take a step either to the left or to the right. The step size might be constant or randomly defined according to some given distribution (Pearson-Rayleigh Walk). After many steps, the probability distribution to find the walker at a given position is a Gaussian function (Reif 1965; Paul and Baschnagel 2013). The width of the Gaussian distribution is proportional to the root-mean-square (RMS) of the step size and the square root of the number of steps characterizes the diffusion mechanism. In the above two examples we have a kind of random walk too, although the distribution is stationary due to dissipation. For a constant collision frequency, it corresponds to a walk in velocity space (a constant free-flight time results in constant velocity steps \(e\,E_{0} /\left( {m\,\nu_{m} } \right)\), but for a constant mean free path it corresponds to a walk in energy space (constant energy steps \(e\,E_{0} \,\lambda_{m}\)). Therefore, it might not be accidental to find distribution functions in these two spaces corresponding to Gaussian functions. Particularly, for a constant collision frequency, the basic time-dependent Eq. (66) (before solid angle averaging and without balancing with dissipation) is the diffusion equation (Note the change of sign discussed when progressing from Eq. (45) to Eq. (46)), which has exactly the random walk solution with the diffusion constant in velocity space \(D_{v}\) (in units of m2 / s3):
Indeed, this equation yields a time-dependent Gaussian distribution function, well-know from diffusion theory (Reif 1965):
Comparing the exponents of Eq. (78) and the solution (74) obtained when balancing to dissipation, one may identify an effective time \(t_{m}\) for truncating the random walk and, from the diffusion constant, a mean velocity step-size \(\delta v_{z}\):
The truncation time is up to the factor 1/3 identical to the mean energy dissipation time \(\tau_{\varepsilon }\) and the velocity step-size of the ‘random walk’ is identical to the drift velocity in the external field, which seems quite reasonable in both cases. Nevertheless, one has to keep in mind that the whole argument should be understood only as an analogy since dissipation is a continuous process and no sudden termination of the random walk after a certain maximum number of steps is expected to take place. In any case, this tentative comparison is consistent with the common interpretation of heating as a diffusion process in velocity space (note that the interpretation for the case of a constant mean free path can be performed similarly), which is viewed here from a slightly different perspective.
6 Fokker–Planck heating operator for a transversal oscillating field
After derivation of the general concept of the FPHO in Sects. 3 and 4, and the demonstration of its application for some simple exemplary situations in Sects. 4 and 5, we are now finally in a position to apply the Fokker–Planck equation to a more challenging physical situation. We consider a harmonically oscillating electromagnetic field penetrating from outside into a confined plasma. The electric field is pointing in the x-direction, perpendicularly to the direction of propagation (z-direction), and exhibits a spatial profile along the direction of propagation, i.e. \(\vec{E} \bot \vec{k}\), where \(\vec{E}\) is the field vector and \(\vec{k}\) the wave vector. This is the typical situation for the inductive radio-frequency (RF) heating of a plasma, with typical angular frequencies of the order of \(\omega = 10^{8} \,{\text{s}}^{{ - 1}}\). The RF frequencies are ideal in the sense that they are located well above typical ion plasma frequencies (\(\omega_{pi} = 10^{5} - 10^{6} \,{\text{s}}^{{ - 1}}\)) but also well below the corresponding electron plasma frequencies (\(\omega_{pe} = 10^{9} - 10^{10} \,{\text{s}}^{{ - 1}}\)). This example contains already several aspects relevant also for other physical situations, hence with the concept laid out in this section generalization to other heating scenarios, e.g. microwave, ECR or INCA (Lieberman and Lichtenberg 1973, 2005; Czarnetzki and Tarnev 2014; Czarnetzki 2018), should be straightforward.
Naturally, any real radiation source will have a spatial profile also in the plane perpendicular to the propagation direction (x,y-plane). In fact, often planar spiral coils are used as antennas to launch the field. Geometry then allows a Cartesian coordinate system to be applied only locally. However, for sufficiently large antennas, the curvature effect is negligible and the contribution to stochastic (non-local) heating can be attributed only to the variation in the z-coordinate. The profile across the x,y-plane can then be accounted for by integrating over the local contributions. Therefore, the situation described here is assumed to be local in the x,y-plane but non-local along the z-direction. A counter example is the so called INCA concept, where a tailored vortex field structure for stochastic heating is created in the x,y-plane but non-local effects along the z-coordinate can be neglected (Czarnetzki 2018).
The spatial profile along the z-coordinate is essential for the non-local heating effect but does not need to be specified yet. It will be shown that the heating operator can be formulated in a very general way for arbitrary field profiles, to be determined in a self-consistent way. The determination of a self-consistent profile will be addressed in the next section, but in some cases results can be obtained also for an exponential model profile with a given decay length. This model profile describes exactly the propagation of an evanescent wave without dissipation. As will be shown in the next section, dissipation is always associated with spatial oscillations of the field structure. However, often these oscillations are small or become noticeable only when the field is already weak. Therefore, the exponential model profile captures the main aspects of a spatially inhomogeneous field in a rather simple and traceable way.
Under the above conditions, the FPHO operator is described by Eq. (68):
where the diffusion constant in velocity space \(D_{v} = \left\langle {\nu_{m} \,\,\Delta v_{x}^{2} } \right\rangle_{\chi } /2\) depends on the values of the absolute velocity and the velocity is z-direction. The elastic collision frequency introduces the dependence on the absolute velocity, as in the previous examples, and the spatial profile of the electric field along the z-coordinate contributes to the additional dependence on the velocity component in this direction. This is shown in detail by integrating the equation of motion with the electric field represented by its Fourier integral. However, for the solid angle average in velocity space it is sufficient to identify these general dependences. The details of the calculation are presented in the appendix A, and the result has essentially the same form as for the simpler cases discussed above:
The difference is only in the coupling function \(\Gamma \left( v \right)\). In the above local examples, the diffusion constant \(D_{v}\) does not depend on \(\xi = \cos \left( \vartheta \right)\), so that in those cases the integral simply reduces to \(\Gamma \left( v \right) = D_{v} \left( v \right)\). After averaging \(D_{v}\) over time and the free flight period, the local Ohmic operator is recovered. The averaged diffusion constant can then be separated into a preceding term proportional to \(v_{E}^{2} \sim E_{0}^{2}\) and the dimensionless Ohmic coupling function \(g_{O} \left( v \right)\) (Eq. (16)) in between the differentials. Generally, the same strategy is followed in the present case too but in addition, there is also a spatial average (The amplitude of the harmonic electric field can now have a spatial profile \(\vec{E}_{0} \left( {\vec{r}} \right)\).):
where \(\left\langle {v_{E}^{2} } \right\rangle_{V} = \left\langle {\left( {e\,\vec{E}\,/\left( {m_{e} \,\omega_{0} } \right)} \right)^{2} } \right\rangle_{V}\). The equation has then a form analogous to Eq. (14) or Eq. (17), respectively, where the change of the preceding factor 1/3 to 1/6 results from the temporal average of the squared harmonic field, which gives a factor ½. The spatial average in Eq. (82) results from the underlying assumption that we can deal with a global non-local distribution function as outlined in Sect. 2. Indeed, for energy relaxation lengths larger than the system size, heating contributes only as a volume averaged value. Further, it is assumed that over the limited range of the field penetration, the variations of the distribution function due to the local plasma potential profile can be neglected. At the low-collisionality conditions considered here, the major contribution to the plasma potential comes from the space-charge sheath region at the wall or the dielectric window in front of the antenna. Within this thin sheath, which has always a significantly smaller extension than the penetration depth, the electron density is negligibly low and does not contribute significantly to the heating or the field penetration.
In Eq. (82), the electric field average is kept inside the velocity-derivative operator since the field profile depends in general on the distribution function, as outlined in the next section. Thus, the formulation of Eq. (82) is convenient only for the special case of a predefined model profile independent of the distribution, as the one considered in this section, in which case one can move the quantity \(\left\langle {v_{E}^{2} } \right\rangle_{V}\) outside of the operator. Indeed, while the above form (82) nicely shows the analogy to the local Ohmic case, for practical purposes, particularly in the case of a self-consistent field, it is more convenient to keep the coupling function \(\Gamma \left( v \right)\) in the final operator, i.e. as done in Eq. (81). This avoids the construction of the dimensionless coupling function \(g_{OS} \left( v \right)\), which extends the dependence on the electric field to both, its nominator and denominator. In any case, one needs to evaluate the diffusion constant in velocity space (see Eq. (68)), where the stochastic parameters for the averaging are the spatial coordinates, the phase (initial times), and the free-flight time \(\tau \). In Eq. (67), the integration of the equation of motion within the free-flight time \(\tau\) can be carried out for arbitrary field profiles by introducing the Fourier representation of the field:
When integrating the equation of motion to calculate the velocity changes, the spatial coordinates enter as the initial value \(\vec{r}_{0}\) in addition to the initial time \(t_{0}\) (phase):
Naturally, this simple free-fall relation is correct only along directions perpendicular to the electric field. Since it is assumed here that \(\vec{k} \bot \vec{E}\), the term \(\vec{k} \cdot \vec{r}\) in the exponent of the Fourier representation singles out only these perpendicular terms. For longitudinal fields this simple relation would represent a first order approximation (Czarnetzki 2018).
The diffusion constant in velocity space has to be averaged over the stochastic parameters \(\chi = \,t_{0},\,\vec{r}_{0},\,\tau\). We will first workout the averages on the initial values \(t_{0}\) and \(\vec{r}_{0}\) applied to the squared velocity change:
The function \(\psi\) is an acronym for the expression:
Note that the sinc-functions converge to Dirac delta functions in the limit \(\mathop {{\text{lim}}}\limits_{a\, \to \,\infty } \, \left( {{\text{a}}\,{\text{sinc}}\left( {a\,x} \right)} \right) = \pi \,\,\delta \left( x \right).\) Note further that although the length scales of the plasma \(L_{i}\) are large, i.e. \(\left| {L_{i} \,k_{i} } \right| > > 1\), they remain finite and, therefore, the volume \(V = L_{x} \,L_{y} \,L_{z}\) is treated as finite and kept explicitly in the expression when taking the spatial limit:
Moreover, the sinc-function for the frequencies gives non-zero results only for pairs \(\omega + \omega ^{\prime} = 0\), in which case it equals 1, which is represented by the Kronecker delta-function \(\delta_{\omega, - \omega ^{\prime}}\). Essentially all inductively coupled discharges are operated only at discrete frequencies and in fact almost always at a single angular frequency \(\omega_{0}\), and therefore the Fourier representation of the field, which in general should have wave properties, takes the simple form:
Then, in order to meet condition (23) for the field being real, it is required that \(\hat{\vec{E}}_{0} \left( {\vec{k}} \right) = \hat{\vec{E}}_{0} \left( { - \vec{k}} \right)\). Further, it should be noted that in case of multiple discrete frequencies, the contributions of these frequencies would simply add linearly.
Introducing (87), (88) in (85) and simplifying gives:
The remaining average over the free-flight interval can be calculated without further difficulty using the exponential probability distribution (see Eq. (5)), and the final result reads:
where the integral kernel \(K\left( {\vec{k} \cdot \vec{v},\,\,\omega_{0},\,\,\nu_{m} } \right)\) is termed ‘conductivity kernel’ due to the similarity to the real part of the Fourier transform of the conductivity (Eq. (28)):
The dimensionless conductivity kernel contains two kinetic resonances (when the phase velocity equals the electron velocity) similarly to the general expression derived from the Boltzmann equation (Eq. (30)). These two resonances represent electrons moving along the direction of the spatial electric field variation, i.e. along \(\vec{k} = \pm \left| {k_{z} } \right|\,\vec{e}_{z}\), for the particular case under study in this section. Ohmic heating is recovered if at least one of the following two conditions applies (\(\ell\) is the characteristic length scale of the spatial variation):
The first condition represents dam** of temporal correlations due to collisions and the second condition represents locality, i.e. the lack of significant spatial inhomogeneity. Particularly, the latter condition is explicitly related to the thermal motion of the electrons, as expressed by the simple estimate on the rhs (thermal velocity:\(v_{th}\)). The meaning of this estimate is that stochastic heating requires that an electron must move thermally over the range of the spatial inhomogeneity in a time much shorter than the period of the harmonic oscillation, i.e. before the direction of the oscillating field reverses. Therefore, collisonless stochastic heating (energy gain) requires both, long time temporal correlation and non-locality.
In the local Ohmic case, by making use of Parseval’s theorem (Kaplan 1973) \(\left( {\int {\left| {\hat{\vec{E}}_{0} \left( {\vec{k}} \right)} \right|^{2} \,d^{3} k} = \int {\left| {\vec{E}_{0} \left( {\vec{r}} \right)} \right|}^{2} d^{3} r} \right)\), the standard Ohmic expression is indeed recovered for a spatially averaged (root mean square) field (cf. Eq. (31)):
where \(\beta = \nu_{m} /\omega_{0}\), like in Eq. (16). Since in this case \(\Gamma = D_{v}\), the local Ohmic heating operator for a harmonic field (14) is recovered. Since the real part of the Ohmic conductivity is \({\text{Re}} \left( {\sigma_{Ohm} } \right) = \varepsilon_{0} \,\omega_{0} \left( {\omega_{pe} /\omega_{0} } \right)^{2} \,g_{O}\) (Lieberman and Lichtenberg 2005), with the plasma frequency \(\omega_{pe} \sim n_{e}\) and \(\vec{j} = \sigma_{Ohm} \,\vec{E}\), one could, considering Eq. (29), alternatively also write:
This clearly emphasizes for this special case the relation between the diffusion constant in velocity space and the power density \(\partial P/\partial V\). Note that \(m_{e} \,D_{v}\) can be interpreted as the mean power contributed per electron. Indeed, the relation is more general, and can be derived by integrating the FPHO in the form of Eq. (81) or (82), respectively, to obtain the general average power density including also non-local heating effects:
The integration is straightforward and requires only one integration by parts. Note that the FPHO enters Eq. (95) with a minus sign to calculate the average power density, due to the convention of signs adopted when writing the operator in Eq. (46). Note further that the electron density is given by \(n_{e} = \int {f_{0} \,d^{3} v}\). The coupling function \(\Gamma \left( v \right)\) is directly related to the diffusion constant by its definition in Eq. (81) and in the isotropic Ohmic case, \(\Gamma = D_{v}\) applies. In case of \(\left\langle {v_{E}^{2} } \right\rangle_{V}\) and \(g_{OS}\) being constants, i.e. \(g_{OS} = g_{O}\) and \(\beta = {\text{const}}{.}\), these terms can be moved out of the integral so that after integration by parts the average power density becomes \(n_{e} \,m_{e} \left\langle {v_{E}^{2} } \right\rangle_{V} \,\omega_{0} \,g_{O} \,/2 = n_{e} \,m_{e} \,D_{v}\), which recovers Eq. (94). In the general case, but for a Maxwellian distribution function \(f_{0} \left( v \right) = f_{M} \left( v \right)\), the differentiation in the last line of (95) can be carried out directly, which then simplifies to:
In the case opposite to conditions (92), non-local (stochastic) heating effects become dominant. In this case, new variables \(\kappa_{ \pm } = \left( {\left| {k_{ \bot } } \right|\,v_{ \bot } \, \pm \omega_{0} } \right)/\nu_{m}\) can be introduced in the respective integrals resulting from the two terms of the conductivity kernel (91). The notation used here assigns “||” to vectors in the plane of the electromagnetic field and “\(\bot\)” to vectors in the direction of propagation of the wave (here, in z-direction), which is usually perpendicular to the boundary of the plasma.
With the new variables, the integrals can be given a similar form by switching to a general variable \(\kappa\) with differences only in the Fourier amplitudes of the field:
The two-dimensional wave vector \(\vec{k}_{\parallel }\) in the Fourier amplitude still allows averaging over the x,y-plane, while only the spatial variation along the z-direction of the wave penetration is assumed to contribute to the stochastic heating, as already outlined above, i.e. \(\left| {k_{\parallel } } \right|\, < < \,\left| {k_{ \bot } } \right|\) for significant values of the Fourier amplitudes. Due to the common factor \(1/\left( {1 + \kappa^{2} } \right)\), significant contributions to the integral can be expected only in the range \(\kappa^{2} < 1\). In this case, the Fourier amplitudes of the field become independent of \(\kappa\) if \(\nu_{m} < < \omega_{0}\). Of course, in order to be a meaningful argument, the Fourier amplitudes should indeed have significant values at \(\hat{\vec{E}}\left( { \pm \omega_{0} /v_{ \bot },\vec{k}_{\parallel } } \right)\), which requires \(\ell \,\omega_{0} < v_{ \bot } \approx v_{th}.\) This is the same condition as identified already above for non-local effects (Eq. (92)). The physical meaning is that the passage time of the electrons through the inhomogeneous region of the field must be shorter than the period of the field oscillation in order to avoid inefficient quiver motion of the electron. Under these conditions the Fourier amplitudes of the field can be moved out of the integral in \(\kappa\), which then simply yields a factor \(\pi\). Applying again Parseval’s theorem to the Fourier amplitudes, assumed as identical since in general \(\hat{E}^{*} \left( {\vec{k}} \right) = \hat{E}\left( { - \vec{k}} \right)\), yields:
where \(A_{\parallel }\) denotes the area of the field in the boundary plane, which is introduced by the two-dimensional spatial average of the field Fourier amplitude when applying Parseval’s theorem. Except for the factor \(A_{\parallel } /\left( {2\pi \,V} \right)\), the same relation was derived in (Aliev 1998). Equation (98) shows that only the Fourier amplitude at \(k_{ \bot } = \omega_{0} /v_{ \bot }\) contributes and, in general, stochastic heating scales with velocity like \(1/v_{ \bot }\). Further, it can be easily shown that \(\mathop {\lim }\limits_{{v_{\bot} \to \,0}} \left| {\hat{\vec{E}}_{0} \left( {\omega_{0} /v_{ \bot } } \right)} \right|^{2} /\,v_{ \bot } \to 0\) so that an apparent divergence due to this velocity dependence is avoided. Clearly, stochastic heating is a warm plasma effect that relies on the thermal motion of electrons through an inhomogeneous and oscillating field region.
The above expression applies when collisions can be neglected entirely. This requires some consideration on the underlying subtle assumptions. In the derivation of the basic Fokker–Planck heating operator, isotropization by elastic collisions was an essential point (see Sect. 4). Although here elastic collisions are neglected in the limited region of the heating zone, they can still play a role in the much larger volume of the plasma. Further, reflection of electrons by the space-charge sheaths in front of the usually irregular walls causes some isotropization. Therefore, the physical picture is that an electron experiences a certain velocity change by interaction with the field when passing through the heating zone. Before it returns to the heating zone a second time after some travel through the volume, the direction of its velocity vector has changed in a random way by either (elastic) collisions or reflections at the walls. This implies that the spatial position of the last collision before passing though the heating zone in free flight (acceleration) and the relative phase with respect to the oscillating field are totally random.
The calculation of an explicit general expression incorporating local Ohmic heating as well as non-local stochastic heating apparently requires the knowledge of the form of the field profile. We will handle the general situation of a self-consistent field in the next section. As discussed at the beginning of this section, an exponential profile with a decay length \(s\) can serve as a reasonable model for illustration purposes. In this case, it is convenient to adopt the formulation given in Eq. (82) with a dimensionless coupling function \(g_{OS}\) in analogy to the dimensionless Ohmic coupling function \(g_{O} \left( \beta \right)\) (Eq. (16)). In the following, \(x,y\) coordinates are used for the “\(\parallel\)”-plane, with the electric field pointing in x-direction, and the z-coordiante for the “\(\bot\)”-direction of the wave propagation. The Fourier transform of the exponential profile along the z-direction yields a simple expression:
Note that \(E_{0}\) can have an inhomogeneous profile over the \(x,y\)-plane, which does not contribute to stochastic heating but has to be considered in the integral (90) for the diffusion constant. In order to account for reflection of the electrons at the space charge sheath in front of the antenna (or more precisely in front of the dielectric window separating usually the antenna from the plasma), the field is mirrored at the reflection point (placed at the origin), by introducing the absolute value \(\left| z \right|\) and extending the z-space from minus to plus infinity in the Fourier-integral. However, this artificially doubles the plasma volume so that a factor ½ has to be introduced in the expression for the diffusion constant, which is proportional to \(1/V\). The integral in Eq. (90) can now be evaluated with the result:
where, \(\,w = v/v_{s}\) with \(\,v_{s} = s\,\omega_{0}\) and \(\xi = \cos \left( \vartheta \right)\). In separating the expression into the above two factors, again use has been made of Parseval’s theorem and the particular form of the electric field profile (Eq. (99)), which leads to the following relation:
Therefore, the preceding factor contains the spatially averaged quadratic electric field amplitude:
A dimensionless function \(\tilde{g}_{OS} \left( {w\xi,\beta } \right)\) for the combined action of Ohmic and stochastic heating is introduced, which still depends on the angle \(\xi = \cos \left( \vartheta \right)\):
The angle averaging by the integral in Eq. (81), turns \(\tilde{g}_{OS} (w\,\xi,\beta )\) into the dimensionless coupling function \(g_{OS} (w\,,\beta )\), which effectively is only a function of the absolute velocity, i.e. \(g_{OS} \left( v \right)\) since \(\beta = \beta \left( v \right)\). Only for the Ohmic case, the integral leaves the coupling function unchanged as was shown above. Naturally, in the pure stochastic case, the function does not depend on the collisionality, i.e. on \(\beta = \nu_{m} /\omega_{0}\).
Limiting cases of the general expression (103) are (\(w_{th} = v_{th} /v_{s},\)):
where the function describing pure stochastic heating reads:
Clearly, the local Ohmic case is recovered for high collisionality and in the collisonless limit, the result is in perfect agreement with the literature, where the respective expression of the diffusion constant was calculated in order to derive the mean power delivered by stochastic heating, although following a different conceptional and mathematical approach (Vahedi et al. 1995; Aliev et al. 1998).
Figure 6 shows a graphical presentation of the diffusion constant (6.20–6.22) for the exponential field, assuming a velocity-independent collision frequency. Naturally, the well-known Ohmic behavior is found along the \(\beta = v_{m} /\omega_{0}\) axis for a cold plasma with \(v_{z} /v_{s} = 0\). Interestingly, quite the same shape with even about the same amplitude is also found along the \(w\xi = v_{z} /v_{s}\) axis for vanishing collisionality \(v_{m} /\omega_{0} = 0\). While the maximum for pure Ohmic heating (on this normalized scale) is ½ at \(\nu_{m} /\omega_{0} = 1\), for pure stochastic heating the maximum is \(3^{3/2} /8 = 0.65\) at \(\left| {w\,\xi } \right| = \left| {v_{z} /v_{s} } \right| = \sqrt 3\). This indicates that stochastic heating can be as efficient as Ohmic heating. Further, comparison of the analytical expressions in the limiting cases above suggests the definition of an effective ‘stochastic collision frequency’ \(\nu_{sto} = \left| {v_{z} } \right|/s \approx v_{th} /s\), where again \(v_{z}\) is replaced by the characteristic thermal velocity for a simple estimate. Indeed, this is the same quantity appearing in the condition discussed above to identify non-local effects, and it corresponds to the mean rate at which electrons from the plasma volume ‘collide’ with the heating zone, if the heating zone is viewed as a macro particle.
Normalized diffusion constant in velocity space for an exponentially decaying electric field along the z-direction with decay length \(s\) as a function of the normalized oriented velocity \(w\,\xi = v_{z} /v_{s}\) and elastic collision frequency \(\beta = \nu_{m} /\omega_{0}\) (assumed as velocity-independent), where \(\,v_{s} = s\,\omega_{0}\) and \(D_{v0} \, = \omega_{0} \,\left\langle {v_{E}^{2} } \right\rangle_{V} /2\)
By inserting the diffusion constant (100–103) in the \(\Gamma\) integral (81), the final expression for the dimensionless coupling function \(g_{OS}\) is derived. For the model case of the exponential field, the integral can be evaluated exactly, yielding an expression that depends only on the absolute velocity via \(w = v/v_{s},\,\,\,\beta = \nu_{m} \left( v \right)/\omega_{0}\):
The limiting cases are:
Comparing the limiting case with the full expression clearly reveals that in the general intermediate case the combined action of local Ohmic and non-local stochastic heating is not simply the sum of both contributions. The structure of the dimensionless coupling function is complicate and in all terms, collisional \(\left( \beta \right)\) and non-local \(\left( w \right)\) contributions are mixed.
Figure 7 presents the dimensionless coupling function \(g_{OS}\) (106) for the exponential field, assuming a velocity-independent collision frequency \(\beta = {\text{const}}{.}\). Despite the latter assumption, the collision frequency is treated here as an independent variable since it depends also on the gas density. However, in general, the collision frequency can vary also with the velocity so that some effective trajectory along the surface of Fig. 7 is realized for a given velocity dependence of the elastic cross section.
Dimensionless coupling function \(g_{OS} \left( {w,\beta } \right)\) for an exponentially decaying electric field along the z-direction with decay length \(s\), as a function of the normalized absolute velocity \(w = v/v_{s}\) and collision frequency \(\beta = \nu_{m} /\omega_{0}\) (assumed as velocity-independent), where \(\,v_{s} = s\,\omega_{0}\)
On a first glance, Fig. 7 depicts a similar picture to that shown in Fig. 6, before the angle integration. However, there are some striking differences. Firstly, pure stochastic heating along the velocity axis rises at low velocities rather slowly with a positive curvature. Secondly, already for a moderate contribution of collisions (\(\beta = \nu_{m} /\omega_{0} > 1\)) the entire surface is rather flat along the velocity axis. This means that in this range stochastic heating does not contribute at all, since the same collisional contribution is found for all electron velocities. Only at much lower collisionality stochastic heating becomes effective, but then only for rather high velocities \(w = v/v_{s} > 2\), although there is no strict threshold. Beyond this value, the velocity dependence is again rather flat with a slow decay.
Using the above results allows now presenting the FPHO (81) for the combined Ohmic and stochastic heating in the desired form similar to the local Eq. (14):
The preceding factor of the operator has exactly the same meaning as for the homogeneous field solution but contains now the volume averaged field (see Eq. (14) for comparison).
The explicit analytical result for the exponential model profile allows also a visualization of the operator for a Maxwellian distribution function, similar to the local Ohmic case shown in Fig. 2. In addition to the collisionality, monitored by the relative frequency \(\beta = \nu_{m} /\omega_{0}\), now the stochasticity is also analyzed using the normalized thermal energy as a parameter that is varied in each of the three sub-figures in Fig. 8: \(\alpha = k_{B} T_{e} /\varepsilon_{s} = \,0,1,\,2,\,4,\,8,\,16,\,32\), where \(\varepsilon_{s} = \,m\,v_{s}^{2} /2,\,\,\,v_{s} = s\,\omega_{0}\). Throughout the chosen range of \(\alpha\), the amplitudes of the operator increase, although, the variation range becomes negligibly small already at a moderate collision rate of \(\nu_{m} /\omega_{0} = 0.5\). Only at very low collisionality, stochastic heating is really effective and can indeed compensate (and even slightly surpass) the loss of Ohmic heating. However, this requires a sufficiently high thermal velocity or energy, such that \(\alpha > > 1\), which reiterates the earlier statement that the thermal speed needs to be high enough for electrons to cross the heating zone on a timescale shorter than the oscillation period of the field. In any case, zero crossing of all curves is close to the thermal energy, like in the pure Ohmic heating case. It should be noted, that for even higher values of \(\alpha > > 30\) and at low collisionality, a very slow decrease of the amplitude of the heating operator is observed, which is related to the shorter interaction times with the field at higher thermal velocities. However, even at exceptionally high values of the order of \(\alpha = 100\), the decrease is not more than a few percent.
Fokker–Planck heating operator for an exponential decaying electric field along the z-direction at constant \(\beta = \nu_{m} /\omega_{0} = 0.1,\,\,0.5,\,\,1.0\) (subfigures (a–c), respectively), evaluated for a Maxwellian distribution function with an electron temperature \(T_{e}\). In each subfigure, the curves are for different values of \(\alpha = k_{B} T_{e} /\varepsilon_{s} = \,0,1,\,2,\,4,\,8,\,16,\,32\) with \(\varepsilon_{s} = \,m\,v_{s}^{2} /2,\,\,\,v_{s} = s\,\omega_{0}\). For all three collisional cases, the amplitudes of the various curves increase monotonously with \(\alpha\). Similar to Fig. 2 also here the heating operator for the normalized energy distribution function \(F\) is shown. The normalization is the same with the generalization \(v_{E}^{2} \to \left\langle {v_{E}^{2} } \right\rangle_{V}\), as discussed in the main text
This aspect can be better highlighted by calculating the power coupled to the electrons. The calculation can be performed by multiplying the heating operator by the kinetic energy and averaging over velocity or energy space, depending on the respective presentation, i.e. using Eq. (96). The integration is carried out numerically and the result shown in Fig. 9. In order to make the slight decay in the collisionless case more visible, the range of \(\alpha\) is extended up to 100. More importantly, the figure exhibits that stochastic heating basically keeps the power close to the level reached at optimum collisionality (\(\beta = 1\)). Without non-local effects (\(\alpha \le 1\)) heating would vanish with decreasing collisionality. However, with non-local effects, it stays about constant and becomes even effectively independent of \(\alpha\), provided that \(\alpha > > 1\).
Volume averaged power density for an exponential decaying electric field along the z-direction as a function of electron temperature and collisionality (assumed as velocity independent). The power density on the ordinate is normalized to \(m_{e} \,\left\langle {v_{E}^{2} } \right\rangle_{V} \,\omega_{0} \,n_{e}\)
In the above example of an exponentially decaying field profile, the calculation of \(g_{OS}\) followed the natural order of first carrying out the integration over the wavenumber in order to derive the diffusion constant, and subsequently integrating over the angle in velocity space in order to arrive at Eq. (106). However, this order can be reversed, which is technically not only slightly simpler in the integrations, but has significant advantage for an unknown profile. Since the Fourier transform of the field does not depend on the angle, the integration applies only to the conductivity kernel and can be carried out exactly for any field profile. In this case, the coupling function \(\Gamma\) reads (see Eq. (81) and (90)):
with
where the new dimensionless variable is \(\kappa = \left| {k_{z} } \right|\,v/\omega_{0}\). Expression (110) is, except for the factor 3/4 due to definition, identical to the one derived in (Godyak and Kolobov 1998) using the Boltzmann equation, although no details of the calculation are revealed in the letter. Now the general dimensionless coupling function according to Eq. (82) can be identified as:
The expression can be interpreted as a weighted average with an effective probability function given by the squared Fourier transform of the electric field. The characteristic coupling function \(\gamma_{OS}\) (Fig. 10) fully describes the response of the plasma while the electric field profile provides the weights of each region’s contribution to this response. In general, the function has a similar shape as the diffusion constant shown if Fig. 6. However, along the velocity axis for vanishing collisionality, there is a remarkable gap between zero and \(\kappa = \left| {k_{z} } \right|\,v/\omega_{0} = 1\). This gap is a consequence of the fact that stochastic heating requires a resonance between the velocity in the z-direction and the phase velocity of the wave. If the absolute velocity is lower than the phase velocity, this condition cannot be met—thus the gap. At higher absolute velocities, there is always the option for a change in the orientation of the velocity vector to meet the resonance condition by the projection on the z-axis. For the exponential model potential, the square of the electric field profile contributes to ~ 82% of the Fourier spectrum for \(k_{z} \,s\, < 1\) (Eq. (99)). For efficient stochastic heating it is then required that \(v > > v_{s} = s\,\,\omega_{0}\), which consistently repeats the above condition for stochastic heating from a different perspective.
Dimensionless function \(\gamma_{OS}\) as a function of the normalized wave number \(\kappa\) and collision frequency \(\beta\) (assumed as velocity-independent)
Finally, by using Eq. (110) into (109) and applying Parseval’s theorem to the integral over the squared field in the denominator of Eq. (111), one retrieves the expression (82) of the heating operator:
A mentioned already at the beginning of this section, the subtle difference to Eq. (108) is that now the factor containing the volume average of the squared effective velocity has to be kept within the differential. In general, the self-consistent spatial profile of the electric field depends on the distribution function itself, and so the volume-average term cannot be moved outside the operator. For comparisons to the simpler operators discussed above, it might be beneficial to introduce the dimensionless coupling functions (82) and (111), involving a ratio to the integral over the squared electric field. However, for practical purposes, it is easier to directly keep the \(\Gamma\)-integral in the FPHO, as shown in Eqs. (109) and (81). The problem of calculating the self-consistent Fourier profile of the field is addressed in the following section.
7 Self-consistent field distribution and the dispersion integral
Non-local stochastic heating depends critically on the spatial structure of the electric field, which should be calculated in a self-consistent way. This is a general problem, independent of the particular form and derivation of the heating operator, but is certainly of key importance to the implementation of the FPHO with self-consistent coupling functions (109–111) in a general situation. We will again consider a transversal electromagnetic field propagating in the z-direction with a frequency between the ion and the electron plasma frequency, assuming without loss of generality that the electric field vector points in the x-direction. In this case, the propagation of the wave within the plasma is described by the Helmholtz equation, which follows directly from Maxwell’s equations. As seen in the previous section, the heating operator depends only on the absolute square of the Fourier transform of the electric field. Therefore, we directly proceed to the Fourier transform of the Helmholtz equation:
It should be noted that we will make the implicit assumption of an infinitely extended plasma in the positive z-direction. Certainly, this is a strong simplifying approximation, which avoids standing wave effects. In real confined plasmas this can be an important issue and there is quite some work in the literature on the wave structure in small cylindrical plasmas with a coaxial solenoid as an antenna (Blevin et al. 1970; Alexandrov et al. 1984; Aliev et al. 1998; Tarnev et al. 2013) but in all cases a Maxwell distribution function is assumed. Such geometry effects are beyond the scope of the present paper and would only complicate the details of the general idea outlined in the following, which remains valid.
The current density is split into the internal currents in the plasma and external currents in the antenna launching the wave at the origin \(\left( {z = 0} \right)\): \(\hat{\vec{j}} = \hat{\vec{j}}_{pl} + \hat{\vec{j}}_{ex}\), where \(\,\hat{\vec{j}}_{pl} = \left( {2\pi } \right)^{2} \,\hat{\sigma }\,\hat{\vec{E}}\) (see Eq. (25)). Solving Eq. (113) for the field yields:
The conductivity is proportional to the electron density which in turn can be expressed conveniently by the electron plasma frequency, \(n_{e} \propto \omega_{pe}^{2}\):
Equation (115) defines the dimensionless dispersion integral \(I_{Z}\). By inserting in the above definition the general expression for the Fourier transform of the conductivity (28) and applying the specific directions of the field and the wavenumber defined above, the dispersion integral is immediately identified:
Note that the dispersion integral depends on the distribution function, which implies that the same dependence is found also for the squared Fourier transform of the electric field. This is the reason why, in Eq. (112), the volume average of the squared effective velocity cannot generally be moved outside the operator. An important symmetry property following directly from the structure of Eq. (116) is \(I_{z} \left( { - \omega } \right) = I_{z}^{*} \left( \omega \right)\). We will make use of this property later on when determining the absolute square of the Fourier transform of the field.
Combining the relation (115) between the dispersion integral and the conductivity with the formula for the power dissipated in the plasma (29), one immediately concludes that the latter depends only the imaginary part \(I_{ZI}\) of the complex dispersion integral \(I_{z} = I_{ZR} + i\,I_{ZI}\):
With the above definition of the dispersion integral, the Fourier transform of the electric field becomes:
where the term − 1 can be conveniently dropped within the plasma without loss of accuracy since by definition \(\omega_{pe} > > \omega\) therein. The denominator represents the dispersion relation of the wave in the plasma, which explains the choice of name for the integral (116):
Note that without a plasma \(\left( {\omega_{pe} = 0} \right)\), which requires to keep the term − 1, the classical dispersion relation in vacuum \(\omega = c\,k\) results. Moreover, only for pure real and positive values of \(I_{z}\) (if such a solution exist), the resulting wavenumber is real, which would represent a propagating wave without dam**. Otherwise complex values of \(k = k^{\prime} + i\,k^{\prime\prime}\) result and in this case, only the k-solution in the upper half of the complex plane describes a damped wave for a positive sign in the exponent in the Fourier transform, i.e. \(\exp \left( {i\,k\,z} \right) = \exp \left( {i\,k^{\prime}z - k^{\prime\prime}z} \right)\).
For a purely imaginary conductivity \(\hat{\sigma }\), i.e. in the absence of any dissipation (see Eq. (29)), \(I_{Z}\) is real and negative (Eq. (115)). In this case, by Eq. (118) the Fourier amplitude of the field is purely imaginary (assuming without loss of generality that \(\hat{j}\) is real). Then in Eq. (88) \(\hat{\vec{E}}\left( k \right)/i\), which is real, can be moved outside of the parenthesis. This common factor \(\hat{\vec{E}}\left( k \right)/i\) has the same structure as the Fourier transform of the model profile (99) used in the previous section. The back transform of the field then separates into the product of a spatially dependent part, which has the form \(\exp \left( { - \left| {z/s} \right|} \right)\), and a temporally oscillating part \(\sin \left( {\omega_{0} \,t} \right)\). Note that there is a \(\pi /2\) phase shift relative to the current in the antenna. The monotonous exponential spatial decay and the in-phase temporal oscillation at all positions are the characteristics of an evanescent wave. Consistently, the dispersion relation (119) yields only a pure imaginary wavenumber in this case, which indeed corresponds to a damped evanescent wave without any spatial oscillation, which recovers again the model profile. These aspects are discussed in more detail below in connection with the solutions in the limits of very high and negligible collisionality.
Because the solutions of the dispersion relation are poles of the Fourier transform of the field, the Fourier integral can be expanded to the upper half of the complex plane, where it is closed at infinity, so that it can be solved by the residuum theorem. Clearly only solutions of the dispersion relation with positive imaginary parts contribute. We will return to this aspect when discussing the field structure in the latter half of this section.
Before evaluating the dispersion integral (116) further, the connection to the well know plasma dispersion function \(Z\left( u \right)\) should be made (Fried and Conte 1961). The plasma dispersion function appears in the description of the Landau dam** (Stix 1992), in the related dispersion relation of longitudinal electrostatic waves in plasmas (Ichimaru 1973), and in other kinetic wave phenomena in plasmas. In the appendix C, we revisit the standard integration strategy for the plasma dispersion function and derive an alternative route for a more general treatment. The formal analogy between the plasma dispersion function (a24) and the dispersion integral (116), allows a similar mathematical treatment for both expressions.
In the derivation of the plasma dispersion function \(Z\left( u \right)\), it is assumed that (a) the velocity distribution function \(f_{0} \left( v \right)\) is Maxwellian with a corresponding thermal velocity \(v_{th} = \sqrt {2\,k_{B} T_{e} /m_{e} }\) and (b) that the elastic collision frequency \(\nu_{m}\) is velocity-independent (see appendix C). Under these assumptions, the dispersion integral \(I_{Z} (u)\) has a simple relation to the plasma dispersion function \(Z\left( u \right)\), where \(u = \left( {\omega + i\,\nu_{m} } \right)/\left( {\left| {k_{z} } \right|\,v_{th} } \right)\) is the complex dimensionless phase velocity:
where \(w_{z} = v_{z} /v_{th}\). Up to the dimensionless real phase velocity \(\omega /\left( {\left| {k_{z} } \right|\,v_{th} } \right)\), the dispersion integral and the plasma dispersion function are in fact identical under the above assumptions. In the collisionless case, where \(u = \omega /\left( {\left| {k_{z} } \right|\,v_{th} } \right)\) is real, the relation simplifies to \(I_{z} \left( u \right) = u\,Z\left( u \right)\). This connection to the plasma dispersion function is highlighted by the index ‘Z’ in the notation of the dispersion integral \(I_{Z}\).
Evaluating the dispersion integral for a general distribution function and an arbitrary velocity dependence of the collision frequency does not allow any of the convenient steps above. The remaining alternative for progressing with the evaluation is to carry out the angle integration, since neither the collision frequency nor the distribution function depend on the solid angle. For convenience, a dimensionless distribution function \(\tilde{f}_{0} \left( w \right) = \pi^{3/2} \,v_{th}^{3} \,\,\,f_{0} \left( v \right)/n_{e}\) with \(w = v/v_{th}\) is defined, which for a Maxwellian distribution becomes \(\tilde{f}_{0} = \exp \left( { - w^{2} } \right)\). Note that \(2/3\,\left\langle \varepsilon\right\rangle ={{k}_{B}}T_{e}^{\left( eff \right)}={{m}_{e}}\,v_{th}^{2}/2\). Further, \(\beta = \nu_{m} /\omega\) (same as in Eq. (16)) and \(\tilde{k} = k_{z} \,v_{th} /\omega\) are introduced as dimensionless variables. The \(\varphi\) integration of Eq. (116) yields a factor of \(\pi\):
where the \(\xi\) integration is contained in \(R\left( {\left| {\tilde{k}} \right|w,\beta } \right)\):
This integral can be solved exactly, allowing separating the real and the imaginary parts of the dispersion integral \(I_{z} = I_{ZR} + i\,I_{ZI}\):
This result for the plasma dispersion integral \(I_{Z} \left( {\left| {\tilde{k}} \right|,\beta } \right)\) applies for arbitrary distribution functions and velocity dependencies of the collision frequency. Further, the real and imaginary parts are both real for any combination of the two dimensionless parameters \(\left| {\tilde{k}} \right|\,\) and \(\beta\). Note that the same integration strategy is applied in appendix C to the plasma dispersion function, resulting in a generalized plasma dispersion function \(Z_{g}\), which is not limited to a Mawellian velocity distribution and a constant collision frequency.
The general result (123) and (124) will now be further evaluated and discussed for various limits. In the local limit \(\left| {\tilde{k}} \right|\,w \to 0\), these complicated expressions reduce to a simple and well-known form for any distribution function, if additionally the assumption of a constant collision frequency is made (second line in Eq. (125)):
By substituting Eq. (125) in Eq. (115), one obtains the well-known Ohmic conductivity. Indeed, also in this special case only the imaginary part of the dispersion integral \(I_{ZI} = \beta /\left( {1 + \beta^{2} } \right)\) is related to the power dissipation, as generally indicated by Eq. (117).
The collisionless case \(\beta \to 0\) allows integration by parts in (123) and (124) and the dispersion integral becomes:
In obtaining the limit of the imaginary part, use has been made of the relation (\(\left| {\tilde{k}} \right|w \ge 0\)):
where \(\theta \left( x \right)\) is the Heaviside step function. If in addition a Maxwellian distribution is assumed, the relation to the plasma dispersion function \(Z\left( u \right)\) is recovered here for the collisionless case, where \(u\, = 1/\,\left| {\tilde{k}} \right|\), so that then \(I_{z} \left( u \right)\, = u\,Z\left( u \right)\) as shown above (see appendix C for details).
We now return to the calculation of the self-consistent electric field distribution. The external current density \(\vec{j}_{ex} \left( {\vec{r},t} \right) = \vec{j}_{ex}^{\left( 0 \right)} \left( {\vec{r}_{\parallel } } \right)\,\delta \left( z \right)\,\cos \left( {\omega_{0} \,t} \right)\) is assumed to flow on the surface of the plasma volume and to oscillate harmonically at \(\omega_{0}\):
where \(\hat{\vec{j}}_{ex}^{\left( 0 \right)} \left( {\vec{k}_{\parallel } } \right)\) is defined here as the two-dimensional Fourier transform of the surface current density, which depends only on the two-dimensional wave vector in the plane of the antenna \(\vec{k}_{\parallel }\) at \(z = 0\). By the symmetry argument (23) it follows \(\hat{\vec{j}}_{ex}^{\left( 0 \right)} \left( {\vec{k}_{\parallel } } \right) = \hat{\vec{j}}_{ex}^{\left( 0 \right)} \left( { - \vec{k}_{\parallel } } \right)\), This definition introduces the factor \(1/\sqrt {2\,\pi }\) in the integration along \(z\). We assume that the spatial variation scale of the field across the x,y-plane of the antenna does not contribute significantly to stochastic heating as outlined above. Typically the antenna is a flat spiral coil of radius \(R\), with \(N\) windings carrying a current \(I\). At distances (in z-direction) larger than the spacing between the windings, the spiral can be well approximated to a thin disc carrying a homogeneous current density of the same total amount. In this case the surface current density becomes \(\vec{j}_{ext} = I\,N/R\,\,\theta \left( {R - r} \right)\,\vec{e}_{\varphi }\), where \(\theta\) is again the Heaviside function.
By using Eq. (115) in (114) in combination with the current density definition (128) and the symmetry property \(I_{z} \left( { - \omega } \right) = I_{z}^{*} \left( \omega \right)\) following from Eq. (116), the expression for the Fourier transform of the electric field reads:
Comparison with Eq. (88) allows identification of \(\hat{\vec{E}}_{0} \left( {\vec{k}} \right)\):
Noting the symmetry property of the Fourier transform of the surface current density, the field has indeed the required symmetry property. The phase factor \(i={{e}^{i\,\pi /2}}\) represents the fact that in vacuum the magnetic field near the coil (neglecting retardation effects on the short distances involved) is in phase with the current but the related electric field is 90 degrees phase shifted. Note that the plasma density (proportional to the square of the electron plasma frequency \(\omega_{pe}\)) is a scaling factor for the action of the dispersion integral. Note further, that neglect of the displacement current term in the above expression, requires \(\omega_{pe} > > \omega_{0}\).
We will now continue the analysis of the solutions of the dispersion relation, already addressed at the beginning of this section, by discussing the properties of Eq. (129). First, we introduce the following normalized quantities:
The dimensionless variable \(\rho\) is proportional to the electron pressure \(p_{e}\). Then the normalized square of the Fourier transform of the electric field and the related dispersion relation (neglecting also here transversal components of the wave vector) reads:
The zeros of the corresponding dispersion relation are poles of the Fourier transform of the field:
Applying the residuum theorem to the back transformation then leads a wave with a corresponding complex wavenumber \(\tilde{k} = \tilde{k}^{\prime} + i\,\tilde{k}^{\prime\prime}\). Although the dispersion relation (133) has a compact form, it has no general analytic solution due to the non-trivial dependence of the dispersion integral on the wavenumber. In the following we will discuss first the local Ohmic limit (\(\left| {\tilde{k}} \right| \to 0\)) and then the non-local collisionless limit (\(\beta \to 0\)).
Only for the Ohmic case, the dispersion integral is independent of the wavenumber \(\tilde{k}\). Then, the complex wavenumber is proportional to the square root of the plasma density, i.e. the normalized electron pressure \(\rho \propto n_{e} \,T_{e}^{\left( eff \right)}\). In this case (see Eq. (125)), \(I_{ZR} = - 1/\left( {1 + \beta^{2} } \right)\) and \(\left| {I_{ZI} } \right| = \beta /\sqrt {1 + \beta^{2} }\), which leads to an explicit and well-known solution of the dispersion relation (Lieberman and Lichtenberg 2005):
The two signs in the solution for the real wavenumber \(\tilde{k}^{\prime}\) are allowing a real field amplitude, while the imaginary part \(\tilde{k}^{\prime\prime}\) is positive in any case, since it represents dam**. The corresponding absolute square of the Fourier transform of the electric fields reads:
In case of vanishing collisionality \(\beta \to 0\), the real wavenumber \(\tilde{k}^{\prime}\), representing spatial oscillations, goes to zero, while the imaginary part converges towards \(\tilde{k}^{\prime\prime} = \sqrt \rho\). In this limit, the absolute square of the Fourier transform of the field also simplifies:
which has exactly the form of the model profile used in the previous section (Eq. (99)). The (normalized) skin depth is \(\tilde{s} = 1/\tilde{k}^{\prime\prime} = 1/\sqrt \rho\), i.e. the classical collisionless skin depth. Since the imaginary part of the dispersion integral is zero in this limit, so are the real part of the conductivity and the power coupled to the plasma (Eqs. (115) and (117)).
In case of high collisionality \(\beta > > 1\), the real and imaginary parts of the complex wavenumber become equal \(\left| {\tilde{k}^{\prime}} \right| \approx \tilde{k}^{\prime\prime}\). Since the (normalized) wavelength of the spatial oscillation is \(\tilde{\lambda } = 2\pi \,/\left| {\tilde{k}^{\prime}} \right|\), it follows that \(\lambda /s = \tilde{\lambda }/\tilde{s} \approx 2\pi > > 1\). Therefore, even with the oscillation present, the wave decays predominately exponential, which, in hindsight, justifies the use of the simple exponential model profile in the previous section.
In any case, dissipation leads to spatial oscillations and a finite phase velocity, which for the highly collisional case is much lower than the speed of light \(v_{ph} /c \approx \sqrt {2\,\nu_{m} /\omega_{0} } \left( {\omega_{0} /\omega_{pe} } \right) < < 1\). In contrast, without dissipation in the collisionless limit, the phase velocity goes to infinity, as is characteristic for an evanescent wave, i.e. all points in space oscillate in phase. The smallness of the phase velocity is also essential for the application of the so called ‘modulation spectroscopy’ in ICPs, where the slow propagation of the wave can be monitored (Crintea et al. 2008; Tsankov and Czarnetzki 2011).
The collisonless or stochastic case (\(\beta = 0\)) is discussed by assuming again a Maxwell distribution. In this case \(I_{Z} \left( {\left| {\tilde{k}} \right|,\beta = 0} \right) = u\,Z\left( u \right)\), where \(u = 1/\tilde{k}\) and \(Z\left( u \right)\) is the plasma dispersion function (see appendix C). Since the plasma dispersion function still contains an integral (the Dawson function), we discuss in the following separately the limits of small and large wavenumbers.
For small wavenumbers (equivalent to long wavelengths), the imaginary part of the dispersion integral converges to zero and the real part to \(I_{ZR} \left( {\left| {\tilde{k}} \right| < < 1,\beta = 0} \right) = - 1\). Consistent with the collisionless Ohmic case, also here no power is coupled to the plasma and the above result for the Fourier transform of the field is reproduced (Eq. (136)).
Stochastic non-local effects occur only for large wavenumbers (equivalent to short wavelengths). In this case, the dispersion integral (126) converges to a particularly simple form:
Note that for an arbitrary distribution function, the result would only differ by a factor \(\,\int\limits_{0}^{\infty } {2\,w\,\tilde{f}\left( w \right)} \,dw = O\left( 1 \right)\), which clearly is of the order of one. Equation (137) shows that the real part decays much faster than the imaginary part and can be neglected in comparison. In this case, the dispersion relation becomes:
The solution lying in the upper half of the complex plane is:
Like in the collisional Ohmic case, the wavenumber is complex with a real part \(\tilde{k}^{\prime} = \sqrt 3 /2\,\left( {\sqrt \pi \,\rho } \right)^{1/3}\), causing spatial oscillation, and an imaginary part, causing exponential dam** of the wave with a decay length \(s_{stoch} = 1/\tilde{k}^{\prime\prime} = 2/\left( {\sqrt \pi \,\rho } \right)^{1/3}\). Notably, \(\left| {\tilde{k}} \right| = \left( {\sqrt \pi \,\rho } \right)^{1/3} > > 1\). Therefore, the complex wavenumber scales by \(\rho^{1/3} \propto \,\left( {n_{e} \,T_{e} } \right)^{1/3}\), which is a little weaker than in the Ohmic case where \(\rho^{1/2}\) was found above (Eq. (134)).
The stochastic decay length is always larger than the classical collisonless skin depth \(s_{Ohm}\) resulting in the local case:
The phase velocity corresponding to the spatial oscillation would be \(v_{ph} /c = \left( {2/\sqrt {3\pi } } \right)\left( {\left( {v_{th} /c} \right)\left( {\omega_{0}^{2} /\omega_{pe}^{2} } \right)} \right)^{1/3} < < 1\), which is again much lower than the speed of light. Also here the wavelength of the oscillation is larger than the exponential decay length \(\lambda /s_{stoch} = 2\,\pi /\sqrt 3 > > 1\).
The complex wavenumber found above seems to indicate a similar field structure and behavior as in the Ohmic case. However, applying the residuum theorem to the integration of the Fourier integral over the complex plane requires a more complicate procedure. The absolute value of \(\left| {\tilde{k}} \right|\) in the dispersion relation causes a branch cut, which does not allow a simple integration along the real axis (Weibel 1967; Kolobov and Economou 1997; Aliev et al. 1998; Kaganovich et al. 2006). Further, so far only the leading order of the imaginary part was considered, while the real part as well as the contributions of smaller wavenumbers were neglected. Consequently, the resulting field structure is no longer a simple superposition of an oscillation and an exponential term. Nevertheless, due to the above scaling, neither the decay length nor the phase velocity can deviate much from the values in Eq. (139). Therefore, one can expect to account for the more complex situation by a simple scaling factor of the order of one. The so-called anomalous skin depth can then be expressed by introducing a dimensionless factor \(a \approx O\left( 1 \right)\) (normalized presentation, where the normalizing length scale is \(v_{th} /\omega_{0}\)):
Weibel proposes in his 1967 seminal publication in Physics of Fluids a definition of the effective skin depth s as twice the length of the \(1/e\) decay of the absolute square of the electric field (Weibel 1967). He concludes that the numerical proportionality factor becomes \(a = 8/9 \approx 0.89 \approx 1\). An alternative definition was proposed in (Kaganovich et al. 2006) as the inverse of the logarithmic derivative of the field at the origin, which leads to the same number.
The above discussion shows in particular that efficient stochastic heating requires \(\rho > > 1\). This means in practical terms \(\omega_{pe} /\omega_{0} > c/v_{th} \sim 200\) (using \(v_{th} \approx 1.5 \cdot 10^{6} {\text{m/s}}\), which is the characteristic velocity in the plasmas discussed here). Since \(\beta = \nu_{m} /\omega_{0} < < 1\) in the stochastic regime and \(\nu_{m} = n_{g} \left\langle {\sigma_{m} \,v} \right\rangle\), one can write for the ionization degree (\(n_{g}\): neutral gas density, \(\left\langle {\sigma_{m} \,v} \right\rangle \approx 1 \cdot 10^{ - 13} {\text{m}}^{{3}} {\text{s}}^{{ - 1}}\): rate constant for elastic collisions, \(\rho \approx 10\) for the estimate):
For an RF frequency of 1 MHz this requires \(n_{e} /n_{g} > > 0.6 \cdot 10^{ - 4} \approx 10^{ - 3}\), while at the standard frequency of 13.56 MHz, the requirement is \(n_{e} /n_{g} > > 9 \cdot 10^{ - 3} \approx 10^{ - 2}\). Although the conditions are more relaxed for the lower frequency, the one order of magnitude lower pressure, required by demanding \(\nu_{m} /\omega_{0} < < 1\), results in enhanced surface losses caused by a correspondingly reduced ion collisionality. These enhanced losses make it more difficult to meet the requirement on the ionization degree.
In the general case of arbitrary collisionality, wavenumber, and distribution function, the dispersion integral needs to be evaluated numerically. With the numerical solution for arbitrary wavenumbers, then the heating operator can be obtained by numerical integration of Eq. (109). Alternatively, one could solve numerically the dispersion relation and then use the residuum theorem. However, this seems to be more demanding. Firstly, evaluation of the dispersion integral for complex wavenumbers is a challenging task. Secondly, even with a known solution of the dispersion relation, application of the residuum theorem is not straight forward as is indicated by the difficulties already appearing for the large wavenumber limit in the collisionless case discussed above. In conclusion, direct integration of (109) using Eq. (132) is the more practical choice.
8 Concept for integrating the Fokker–Planck heating operator in a local Boltzmann solver
The previous sections were dedicated to deriving the FPHO for a general (self-consistently determined) spatial profile of the electric field, which has already provided a number of useful physical insights. However, the main aim is to make use of the operator in calculating the isotropic distribution function of the electrons, by using a local and stationary Boltzmann solver as will be outlined in this section. The general situation of a confined plasma of volume V, with electron heating occurring within a limited zone of area As and extension s, is sketched in Fig. 11. It should be emphasized that we assume an electron distribution function with non-local features, hence proposing its calculation by using a local Boltzmann solver (in a kind of global kinetic model), since there is no spatial resolution involved.
Simplified sketch of a confined plasma of volume \(V\) and surface area \(A\), which does not need to be of cylindrical form. The electron heating occurs within a limited zone of area \(A_{s}\) and extension \(s\) along the \(z\) coordinate. Heating is provided by the interaction of the electrons with an externally applied electromagnetic field, which propagates along the z-coordinate
A Boltzmann solver is usually applied for calculating the local distribution function under high-pressure conditions, i.e. when the energy relaxation length is much shorter than the spatial variation length of the electric field. Heating by the field is then balanced by energy losses due to elastic and inelastic collisions at any given point in space. Often, such a solver is combined with a fluid or a global model, where the electron transport coefficients and rate coefficients are computed by the Boltzmann solver. In this case, each point has a defined field, with a local distribution function and thereby local values of the transport and rate coefficients. In the low-pressure case of interest here, advantage can be taken of the non-local character of the distribution function, i.e. only one distribution function has to be computed for the entire plasma. This involves a volume average, which was performed already in the above derivation of the heating operator.
Therefore, the major approximation made in the present concept is the neglect of the variation of the form of the distribution function by the plasma potential \(\Phi \left( {\vec{r}} \right)\), of the order of the electron mean electron energy \(\left\langle \varepsilon \right\rangle\), within the quasi-neutral plasma. For a Maxwell distribution, this would be exact, but in general it is a reasonable approximation, as outlined in Sect. 2 when discussing the non-local concept. In any case, the electron density is more sensitive to the potential variation by the Boltzmann factor. This scaling results from the electron momentum balance equation under the assumption that the mean energy \(\left\langle \varepsilon \right\rangle\) is homogeneous, coherent with the assumption of a negligible variation of the distribution function across the bulk:
The related density variation cannot by neglected and is accounted for in the volume average of the different operators in the electron Boltzmann Eq. (13): the FPHO, the collisional loss operator, and the surface loss operator. Further, the value of the confining floating potential across the boundary sheath is calculated self-consistently. Note that the reference point for the monotonously varying potential is set to its maximum at the center of the plasma, so that \(\Phi \le 0\) and \(n\left( {\vec{r}} \right) \le n_{0}\) throughout.
Typically, the numerical solution of the electron Boltzmann equation is carried out in energy space (not velocity space), in which the FPHO (112) for the normalized energy distribution function reads (see also Eq. (7)):
Note that here the energy distribution function is normalized to unity, i.e. \(\int {F\,d\varepsilon = 1}\), which involves the scaling of the distribution function and the operator by the local density, as discussed in Sect. 2. Generally, the density profile would have to be included in the volume averaged performed in the derivation of the FPHO (81), i.e. it would have to be considered in the calculation of \(\left\langle {v_{E}^{2} } \right\rangle_{V}\). However, here the simplifying assumptions consider both, the penetration depth of the field and the radius of the antenna, much shorter than the plasma size in the corresponding directions. With these assumptions, the electron density can be approximated as homogeneous across the heating zone. Then the density at the edge of the quasi-neutral plasma \(n_{e}^{{\left( {edge} \right)}}\), the only region where the operator is meaningful, appears as a simple scaling factor for the energy-dependent operator (144).
In fact, the normalization to unity of the energy distribution function, in combination with the volume average introduced by formulating the heating operator, requires scaling all operators with a weighted volume averaged density. The volume average of the collisional loss operator reads (see also Sect. 2):
where \(n_{e}^{\left( V \right)}\) is the volume averaged electron density.
A heuristic form of a kinetic surface loss operator is introduced in (Kortshagen et al. 1995). Here we take a different approach, with the details of the derivation given in the appendix D. The operator describing the surface losses of electrons across the surface area \(A\) reads:
where \(\Phi_{s}\) is the floating potential across the sheath, i.e. between the edge of the quasi-neutral bulk (marked by the position where the ions reach their sound (Bohm) speed) and the surface of the confining wall. The surface loss operator introduces a particle loss frequency \(\nu_{loss}^{{\left( {electron} \right)}} \left( \varepsilon \right)\):
Similarly to (144), the loss operator (146) acts only at the edge of the plasma, thus it has to be scaled by the corresponding plasma density \(n_{e}^{{\left( {edge} \right)}}\). Strictly, both the floating potential and the edge density are functions of the particular surface coordinate (Mümken and Kortshangen 1996; Celik et al. 2012). Conducting walls correspond to an equipotential surface. The constant edge density, assumed in this work, corresponds to an equipotential surface too, as shown by Eq. (143). The potential difference between this surface and the wall is then a constant, given by the sum of the sheath potential, which may vary across the wall, and part of the bulk potential. This constant potential is estimated below (Eq. (151)) by assuming a local balance between the electron and the ion fluxes, which is strictly required only for dielectric walls. However, in the case of dielectric walls, neither the sheath potential nor the wall potential (hence the sheath edge) are constant since the bulk potential (potential difference to the maximum in the center) varies with the length of the connecting field lines as a consequence of ion friction (collisions with neutrals). The variation strongly depends on the particular geometry of the confining vessel and the surface material—usually a dielectric window in front of the antenna and conducting or isolating walls, depending on design and surface deposition conditions. In conclusion, considering these effects translates into a complicated average of the entire operator over the plasma surface. Accounting for these geometry effects would require a full 3-D plasma simulation, which goes far beyond the scope of the present work. In this sense, the sheath potential (Eq. (146)) and the constant edge density are only a compromise, representing effective average values for the electrons. Indeed, the ratio between the edge density and the volume averaged density is derived from a global balance below (Eqs. (152) and (153)). Further, in (Celik et al. 2012) it is shown that for the ionization rate, the difference between conducting and dielectric walls is negligibly small. This is consistent with the fact that the ionization rate derived below from an ion transport model (Eq. (150)) does not require any assumptions on the wall conductivity.
With the above approximations, the operators (144), (145), and (146) can be combined to write the electron Boltzmann Eq. (1) as a general balance equation (cf. the discussion about the sign convention, in Sects. 2 and 3) that, after division by \(n_{e}^{{\left( {edge} \right)}}\), reads:
The ratio \(n_{e}^{\left( V \right)} /n_{e}^{{\left( {edge} \right)}} > 1\) enhances the contribution of the collisional operator to the losses. The Eq. (148) now contains three unknown quantities to be self-consistently determined as eigenvalues to the problem:
The electric field at the surface \(E_{0}\) enters the heating operator via \(\left\langle {v_{E}^{2} } \right\rangle_{V} \,\), directly by Eqs. (100)–(102) in the case of the exponential decay model of Sect. 6, or through the intensity \(\left| {\vec{S}} \right|\) defined by Eq. (131), where it can be effectively replaced by \(j_{0}\), in the case of the self-consistent field model of Sect. 7. These three parameters are defined by the external (experimental) input parameters and the ion properties, as will be outlined below:
Ng: neutral gas density (and type).
V, A: plasma volume and surface area.
As: area of the heating zone.
P, \(\omega_{0}\): electric power coupled to the plasma at an angular frequency \(\omega_{0}\).
As will be shown below, the power defines only the absolute density but does not influence the form of the distribution function, at least when ignoring Coulomb collisions as will be discussed below.
Additional information needed for determining the three eigenvalues is provided by an ion model. This underlines very strongly the fact that, in the end, the description of a plasma requires the coupled treatment of the electron and the ion properties. For this purpose, it is usually sufficient to describe the ions by a fluid model using the continuity and the momentum balance equations. In general, the problem requires a numerical solution in 3-D, but in some special cases, analytical solutions can be found. The very simple case of a cubic geometry is solved in appendix F as an example, yielding for the ionic ionization frequency:
where \(\Lambda = V\,/\left( {A\,\lambda_{mi} } \right)\), \(\nu_{mi}\) is the elastic ion-neutral collision frequency and \(\lambda_{mi} = c_{i} /\nu_{mi}\) the ion mean-free-path at the ion sound (Bohm) speed \(c_{i} = \sqrt {2\left\langle \varepsilon \right\rangle \,/\left( {3\,m_{i} } \right)}\)(\(m_{i}\) is the ion mass and \(\left\langle \varepsilon \right\rangle\) the mean electron energy,). Further, for the cube \(V/A = L/6\), where \(L\) is the edge length of the cube.
The ion flux density to the wall is a boundary condition, which is independent of the particular ion transport model since in any case the edge of the quasi-neutral plasma is marked by ions reaching their sound speed (Lieberman and Lichtenberg 2005; Tsankov and Czarnetzki 2017). Then: \(n_{i}^{{\left( {edge} \right)}} \,c_{i} = n_{e}^{{\left( {edge} \right)}} \,c_{i}\). Stationarity requires balance of the ion and electron fluxes or the corresponding loss frequencies for the particles in the volume (see also Eqs. (146), (147)). The sheath potential is estimated using a local balance of the electron and ion fluxes:
Apparently, Eq. (151) defines the plasma potential, although the implicit nature of the equation is slightly inconvenient. Nevertheless, the special structure of the equation shows that \(\left| {e\,\Phi_{s} } \right| \propto \left\langle \varepsilon \right\rangle\) and otherwise the potential is only a function of the electron–ion mass ratio. In the special case of a Maxwell distribution, the equation has an analytical solution, which is indeed identical to the classical textbook formula for the floating potential (Raizer 1991; Lieberman and Lichtenberg 2005). One can use this explicit solution as a starting point for calculating analytically a first order correction to the potential in case of an arbitrary distribution function (see appendix E for details).
The volume average of the ion continuity equation yields (using quasi-neutrality):
For a known mean ionization frequency \(\left\langle {\nu_{iz} } \right\rangle^{{\left( {ion} \right)}}\), then the density ratio is also defined. For the above example of the cube, the latter is given by a simple linear relation obtained by combining Eqs. (150) and (152):
The above result shows for the particular geometry of a cube, that in Eq. (148) the density ratio can strongly enhance the volume collision term in comparison to the surface loss term.
Comparing the electron and the ion continuity equations shows that the corresponding ionization frequencies must be identical. This is also obvious by the fact that free ions and electrons are always generated by ionization in pairs:
Equation (154) defines the remaining open parameter, the electric field \(E_{0}\) or the current density \(j_{0}\), respectively. One possible strategy for finding the electric field amplitude is simply by iterating its value for different solutions of the Boltzmann equation, until Eq. (154) is satisfied. Alternatively, one can use the energy balance equation for the electrons and solve it for the electric field amplitude:
Of course, this equation would be trivially fulfilled, if all the terms result from the solution of the Boltzmann equation. The essential point to turn this into a meaningful expression for determining the field is to replace the average ionization rate by the corresponding rate following from the ion model. This is possible since the mean energy loss rate by ionization \(\left\langle {\varepsilon \,\nu_{iz} } \right\rangle\), entering in (155), is simply the ionization energy \(\varepsilon_{iz}\) times the average ionization rate \({{\left\langle {{\nu }_{iz}} \right\rangle }^{\left( electron \right)}}\):
Then the solution of equation (8.13) for the electric field reads (using also (152) for the density ratio):
The calculation of the three eigenvalues (8.7) is carried out iteratively, starting from some educated guess for the electron distribution function. The procedure can be applied for the exponential decay model as well as for the self-consistent field model. In the former case, also the decay length (skin depth) needs to be adjusted in the iteration using \(v_{th}^{2} = 4\,\left\langle \varepsilon \right\rangle /\left( {3\,m_{e} } \right)\) and Eq. (134) with \(s = 1/k^{\prime\prime}\). In the latter case, the heating operator contains integrals over the distribution function by the dispersion integral (see Eqs. (109)–(112), (123, 124) and (132)), which requires additional iterations for a full self-consistent solution. In addition, the non-linear effects can be further enhanced if including Coulomb collisions between the electrons, in which case also the collision operator contains integrals over the distribution function. Coulomb collisions can play a crucial role at low energies, due to the \(1/\varepsilon^{3/2}\) scaling on the one hand (Bittencourt 2004) and a reduced elastic collisionality due to the Ramsauer effect on the other hand (Raizer 1991), particularly in the case of noble gases (Alves et al. 2018). When including Coulomb collisions, the corresponding average rate depends on the volume averaged electron density, which should also be explicitly calculated.
The volume averaged electron density can be determined by the coupled power, via the integral over the heating operator:
The volume averaged density should represent a quantity, which is well suitable for comparison with experimental data, e.g. obtained from spatially resolved Langmuir probe (e.g. Zhu et al. 2015; Tsankov and Czarnetzki 2017) or microwave interferometry measurements (e.g. Tanskov et al. 2015). Obviously, the absolute densities scale linearly with the power coupled to the plasma as expected (Lieberman and Lichtenberg 2005; Ahr et al. 2018). The linearity follows from the simple argument that (a) for stationarity, power input and losses must be balanced and (b) all losses are proportional to the electron density. If Coulomb collisions are neglected, the calculation of the absolute electron density is not part of the iterative procedure, but can be performed subsequently. Note that Coulomb collisions between the electrons lead to thermalization (Maxwellization of the distribution function) but do not contribute to the energy dissipation. Under certain conditions, particularly high plasma densities and low pressure, gas heating and neutral gas depletion due to electron pressure can become important and deviations from the linear relation might be observed (O’Connell et al. 2008; Fruchtman 2017). However, this is an involving topic in itself, which is clearly beyond the scope of this work.
9 Summary and outlook
The Boltzmann equation and the Fokker–Planck equation both are obtained from the master equation but take quite different routes of approximations already at an early point (Oliveira 2019). When addressing energy exchange between electrons and external electric fields usually a so-called linearized version of the Boltzmann equation is used, where the distribution function is split into a homogeneous, stationary, and isotropic part and a spatially and temporarily varying anisotropic part. Implicitly, this linearization is based on the assumption of small average velocity changes in a single interaction between an electron and the field, although not stated explicitly in the derivation. The resulting equation is linear in terms of the isotropic distribution function but of second order in the velocity changes and represents an approximation with usually not well-studied convergence behaviour of the in principle infinite expansion. In contrast, the derivation of the Fokker–Planck equation is based on the smallness of the velocity changes from the start on. Further, by the Paluwa theorem it is shown that the second order equation is either exact or an expansion in the velocity changes is not possible at all.
In this work, we show that this linearized version of the Boltzmann equation and the Fokker–Planck equation, combined with the Langevin equation, are equivalent and lead to identical expressions for the heating operator, describing the energy exchange of the electrons with an external electric field. Nevertheless, the particular procedures for obtaining the final result are quite different. The Boltzmann equation offers the advantage of a formally straightforward calculation and the main challenge is in the detailed mathematics. However, the underlying physics is not revealed directly. It can only be extracted by subsequently interpreting the mathematical structure in terms of physics. On the other hand, the Fokker–Planck equation has already the form of a diffusion equation in velocity space. However, the mean quadratic velocity change appearing in the diffusion constant is not specified. This quantity is extracted from the Langevin equation and suitable probability distributions. Although the Langevin equation is not particularly useful here for determining the isotropic distribution function, its structure already allows identification of the velocity changes needed in the Fokker–Planck equation. This general procedure forces clear statements on the physics at every step. In particular, the role of isotropization and randomization in the large plasma volume beyond the heating zone, even in the so-called collisonless regime of heating, is clearly revealed. Comparing the two alternative approaches, the required overall mathematical effort is probably not very different, particularly for the solid angle and volume averages. Therefore, the Fokker–Planck/Langevin approach introduced here is a real and in fact very versatile alternative to the common use of the Boltzmann equation.
After having derived the general concept, this work addresses in detail the combined description of the Ohmic (collisional) and the stochastic (collisionless) electron heating, by using the Fokker–Planck equation to derive the corresponding operator. Stochastic heating in plasmas results from the interaction of the electrons with spatially inhomogeneous and temporally oscillating electric fields (typically in the RF range), during the thermal electron motion along the field gradients. The interaction region is limited to a thin layer defined by the penetration depth of the field, e.g. the sheath thickness for CCPs, the skin depth for ICPs or the gradient length of an inhomogeneous external magnetic field for ECRs.
The description of the phenomenon is not trivial due to both, conceptual and technical challenges. Stochastic heating is a non-local effect that is initiated by the interaction of the electrons with an inhomogeneous temporarily oscillating field within a localized heating region. The electrons enter and leave the heating region due to thermal motion, increasing on average their energy during the transition. Initially, this energy gain is anisotropic and depends on the particular structure of the field. However, the effect is subsequently isotropized within the plasma volume, due to electron collisions with the neutral gas (particularly elastic collisions), and with charged species (electrons and ions) at high plasma densities. Therefore, energy gain and irreversible randomization are taking place in two different regions of the plasma. Interestingly, recent detailed numerical investigations on collisionless heating in CCPs lead to similar conclusion (Lafleur and Chabert 2015). Note that the length scale of the interaction region with the field (skin depth) is assumed to be much shorter than the size of the plasma. Therefore, even when the probability for momentum changing collisions is low in the interaction region, it can still be significant in the larger plasma volume. Further, reflections from irregular walls (or more precisely from the sheath in front of the walls) can also provide randomization. In any case, collisions are necessary to reverse the direction of the electron flight so that they return multiple times to the heating zone. Since the velocity gain is a diffusion process, an electron reaching the ionization energy without any other intermediate collisions needs about \(N \approx (v_{iz} /\Delta v)^{2} \approx \varepsilon_{iz} /\Delta \varepsilon\) collisions, where \(\Delta \varepsilon < \,\left\langle \varepsilon \right\rangle\) is the average energy gain by a single interaction with the field in the heating zone. Typical numbers are of the order of 10. Accounting for inelastic collisions, leads to an even higher number, although quantitative estimates are more difficult and should be handled with numerical simulations.
In steady-state conditions, the electron velocity distribution function is essentially isotropic (yet kee** non-equilibrium features), becoming a function of the total (kinetic plus potential) energy and being globally defined by a non-local energy balance between the heating by the external field, and the losses by collisions in the volume and transport to the walls. The non-local regime requires electron energy relaxation lengths larger than the system size, which is satisfied at low gas pressure and introduces another conceptual challenge. Indeed, the low-pressure condition corresponds to a situation of low-collisionality between the electrons and the neutrals that should not be interpreted as a “total absence of collisions” (albeit the expression “collisionless case” is often used to identify stochastic-heating dominating conditions). As outlined above, a certain collisionality is also key to isotropize the electron distribution function. Thus, although at high-pressure one obtains a local regime associated with pure Ohmic (collisional) heating, in the opposite limit of low-pressure, the non-local regime that develops, corresponds to a combined stochastic plus Ohmic heating.
Efficient non-local (stochastic) heating thus requires:
-
(i)
A compact electric field, characterized by large wavenumbers, to ensure a fast passage of the electrons by thermal motion through the inhomogeneous field region on a timescale shorter than the oscillation period of the field:\(v_{th} > > s_{stoch} \omega_{0}\). Spatial oscillations (real part of the wavenumber) superimposed on the overall decay (imaginary part of the wavenumber) are inevitably related to dissipation, i.e. energy exchange between field and particles. The wavelength of these spatial oscillations is always much larger than the (exponential) decay length (skin depth). Sufficiently large wavenumbers require a high (normalized) electron pressure: \(n_{e} \,k_{B} T_{e}^{\left( eff \right)} \propto \rho = \left( {v_{th} /c} \right)^{2} \,\left( {\omega_{pe} /\omega_{0} } \right)^{2} \, > > 1\).
-
(ii)
Electric-field oscillation frequencies \(\omega_{pi} < < \omega_{0} < < \omega_{pe}\), i.e. well-below the electron plasma frequency (typically in the microwave range) and well-above the ion plasma frequency \(\omega_{pi} = \sqrt {m_{e} /m_{i} } \,\omega_{pe} = \sqrt \mu \,\omega_{pe}\) (assuming \(m_{i} = m_{N}\) and \(T_{i} < < T_{e}\)). For argon, for instance, the mass ratio is \(\mu = 1/7.5 \cdot 10^{4}\). Combined with the above condition in (i) on the normalized electron pressure \(\rho > > 1\) this leads to \(1 > \left( {\omega_{pi} /\omega_{0} } \right)^{2} > > \mu /\left( {v_{th} /c} \right)^{2} \, \approx \,1/2\). The latter number is for argon and for a typical ratio \(\left( {v_{th} /c} \right)^{2} \approx 1/4 \cdot 10^{4}\). Apparently, the result is not well compatible with the demand of static ions \(\left( {\omega_{pi} /\omega_{0} } \right)^{2} < 1\). Even by using a gas with higher mass, e.g. krypton, the number on the rhs would not change significantly, i.e. \(1/2 \to 1/6\). This indicates that pure non-local (stochastic) heating is probably difficult to realize. If the plasma density is sufficiently high to allow for a short skin depth, ions can no longer be considered as static and couple to the field too. Ion heating would channel energy away from the electrons into mostly heat (of the ions and the neutral gas), which in turn makes it difficult to realize high plasma densities. At lower plasma densities, ions are static but the skin depth is too large to allow efficient non-local energy gain. Therefore, a combined action of local and non-local heating is the likely scenario for most cases. Remarkably, this conclusion does not depend on the choice of the RF frequency. However, the condition \(\rho > > 1\)sets an upper limit to the frequency \(\omega_{0} < < \,\left( {v_{th} /c} \right)\,\omega_{pe} \approx 5 \cdot 10^{ - 3} \omega_{pe}\), which is easier realized at low RF frequencies.
-
(iii)
Low collisionality \(\nu_{m} /\omega_{0} < < 1\), which again combined with the demand of high plasma pressures from (i) limits the ionization degree to a minimum of about \(n_{e} /n_{g} > > \omega_{0} \,/10^{11} {\text{s}}^{{ - 1}}\) (see Eq. (142)), yielding estimates of \(n_{e} /n_{g} > > 10^{ - 3},\,10^{ - 2}\) at oscillating frequencies of 1 MHz and 13.56 MHz, respectively. In any case, the low collisionality condition requires a low-pressure operation, with the upper limit ranging typically from a few 0.1 Pa to a few Pa. The required low gas density and the related low ion collisionality enhances the surface losses and makes it difficult to comply with the requirement on the ionization degree. This again emphasizes from a different viewpoint the above conclusions that in most cases, both, local Ohmic and non-local stochastic heating are operational.
Technically, the formal description of the stochastic heating is also not without challenges. Here, we have adopted a deterministic approach, based on the solution of the differential electron Boltzmann equation, written as the balance between a heating operator and two loss operators, due to electron collisions (elastic and inelastic) with neutrals and the interaction with surfaces. The heating operator was deduced by combining the Fokker–Planck equation, to describe drift and diffusion phenomena in velocity space due to the continuous interaction with the external field, and the Langevin equation to determine the corresponding drift and diffusion coefficients. The new FPHO (i) accounts for both, non-local stochastic and local collisional heating; (ii) generalizes the applicability and/or corrects the expressions obtained heuristically in previous works (Vahedi et al. 1995; Aliev et al 1998; Godyak and Kolobov 1998; Kolobov and Godyak 2019) for the description of stochastic heating; (iii) gives dominant Ohmic heating and dominant stochastic heating, as limiting cases at high and low pressures, respectively; and (iv) it is well-suited to calculate global electron velocity distribution functions, by solving the electron Boltzmann equation with the same type of loss operators (collisional in the volume and surface) as in a local regime.
To describe global, stationary, and isotropic electron distribution functions, the FPHO is averaged over the plasma volume, the time and phase, and the full solid angle of the electron velocity. The latter averaging eliminates the drift component, and the whole treatment carefully places the velocity derivatives \(\partial /\partial v_{j}\) in the remaining diffusion component, which is relevant for a velocity-dependent momentum-transfer electron collision frequency \(\nu_{m}\). Moreover, the articulation of the Fokker–Planck and the Langevin equations allows identifying the stochastic parameters for the additional averages (the initial coordinate, the initial time, and the duration of the free (collisionless) flight period), thus formalizing the isotropization process by randomizing the particle motion (even in the presence of inhomogeneous and oscillating fields), to convert their internal energy into heat. Finally, this kinetic analysis has provided an expression for the diffusion coefficient of the FPHO, as a function of the external electric field, assumed as transversal, i.e. exhibiting a spatial variation perpendicular to the direction of the force.
The final expression of the FPHO deviates from the standard form of the Fokker–Planck equation by the position of one of the second order derivative \(\partial /\partial v_{j}\) in the diffusion term, that is moved directly to the distribution function. This change of position is a consequence of allowing velocity-dependent collision frequencies. The modified form of the FPHO is successfully verified against an ergodic MC simulation, assuming that the heating is provided by a homogeneous and constant electric field and the energy losses are due to elastic collisions. These conditions allow also an exact analytical integration of the Boltzmann equation for comparison. Further, the behaviour of the new operator is analysed for the particular case of a transversal electric field, pointing in the x-direction while exhibiting an exponential profile with a given decay length along the z-direction of propagation, similar to the field distribution in an ICP. For this case, we have obtained and discussed the Ohmic and the stochastic limits of the FPHO, as the function of the normalized transit velocity \(\omega_{0} \,s_{stoch} /v_{th}\) and the normalized collision frequency \(\nu_{m} /\omega_{0}\) (assumed as constant). The analysis is complemented with analytical calculations of the operator for a Maxwellian distribution function. The study is used also to prepare the FPHO for handling the more general situation of arbitrary space–time profiles of the electric field, namely by first integrating the conductivity kernel over the angle in velocity space, postponing the space–time integration of the electric-field profile. In this general expression, no restriction on the velocity dependence of the elastic collision frequency is made. The latter integration is one of the most challenging aspects in the general implementation of the FPHO.
Indeed, the electric-field spatial profile depends in general on the distribution function, meaning that it should be calculated in a self-consistent way. We have formulated the problem for arbitrary field profiles by introducing the Fourier representation of the field, thus converting the space–time analysis into a wavenumber-frequency analysis. The Fourier transform (FT) of a transversal electric field then becomes a function of the dispersion integral, which depends on the electron distribution, the wavenumber along the propagation direction, and the velocity-dependent elastic collision frequency. Ultimately, the FPHO is calculated for the coupling function \(\Gamma \left( v \right)\), which can be integrated using the squared FT of the field. An explicit solution for the squared FT of the field is provided in terms of the real and imaginary parts of the dispersion integral.
The new features brought by the FPHO proposed here, describing the combined action of Ohmic and stochastic heating, are essentially threefold. First, the study follows a deductive methodology based on a clear framework, which clarifies the approximations involved and provides a solid ground for using the operator and understanding its physical limits. Second, the operator is applicable to an arbitrary transversal field profile, which can be calculated in a self-consistent way. Third, the form of the FPHO is similar to that of other continuous operators included in the electron Boltzmann equation written under the classical two-term approximation, such as the Ohmic heating and the elastic collision operators. The latter feature facilitates the numerical implementation of the FPHO in typical local and stationary two-term Boltzmann solvers, to calculate the isotropic electron distribution function. In this case, besides the heating and the collisional loss operators, a third operator should be introduced describing the surface losses of electrons across the area of the plasma container. An additional Fokker–Planck operator, describing electron–electron Coulomb collisions, can also be introduced to better describe situations at high plasma densities (Sharkarofky et al. 1966; Hagelaar 2016; Alves et al.2018).
In the last section of this work, we present a possible procedure for the calculations when using a Boltzmann solver in an effective global kinetic model. This procedure highlights the need for the self-consistent determination of three eigenvalues to the problem:
-
(i)
The ratio of the volume-averaged electron density to the edge electron density, which appears when performing the volume average of all the operators in the Boltzmann equation.
-
(ii)
The floating potential across the space charge sheath at the wall, which intervenes in the surface-loss operator.
-
(iii)
The electric field at the excitation surface (or the corresponding external excitation current density), that enters the FPHO.
The full closure of the problem, with the iterative determination of the three eigenvalues, can only be achieved by further complementing the formulation with an ion (fluid) model. The ion transport model yields values for ionization frequency and the ionic flux towards the surface, which have to agree for consistency with the results from the global kinetic electron model. This highlights the fact that last not least a plasma consisting of electrons and ions and not only an isolated electron gas must be considered in the analysis. The numerical implementation and solution of the problem is beyond the scope of the present work. However, the workflow proposed here was already adopted as proof-of-concept in the LisbOn KInetics Boltzmann solver (LoKI) (Tejero et al. 2019), using a preliminary version of the FPHO (Alves 2019). Work is in progress to update and consolidate this implementation, using the formulation results presented in this work.
The procedures for deriving the heating operator and for implementing the operator into a global kinetic model using a local Boltzmann solver have been demonstrated for the case of an inductively coupled plasma. However, it should be rather straightforward to apply the same general procedure also to other heating scenarios, e.g. in the INCA discharge (Ahr et al. 2018) or in ECR discharges (Lieberman and Lichtenberg 2005). The detailed form of the coupling functions will likely be different but the general structure of the heating operator and the structure of the global kinetic model will remain the same. The preliminary results obtained in (Alves 2019) show that the computation time on an ordinary PC is of the order of 10 s. This is significantly shorter than PIC/MCC electrodynamic simulations (Alves et al. 2018; Matteia et al. 2017), which typically have runtimes on a timescale of weeks, which makes parameter variation and even code development very challenging.
Change history
13 May 2024
A Correction to this paper has been published: https://doi.org/10.1007/s41614-024-00159-2
References
P. Ahr, T.V. Tsankov, J. Kuhfeld, U. Czarnetzki, Inductively coupled array discharge. Plasma Sources Sci. Technol. 27, 105010 (2018)
A.F. Alexandrov, L.S. Bogdankevich, A.A. Rukhadze, Principles of Plasma Electrodynamics (Springer, Berlin, 1984)
Y.M. Aliev, I.D. Kaganovich, H. Schlüter, Quasilinear theory of collisionless electron heating in radio frequency gas discharges. Phys. Plasmas 4, 2413 (1998)
W. P. Allis, Handbuch der Physik 21, ed. S. Flügge, Heidelberg: Springer, 383–444,1956.
L.L. Alves, Fluid modelling of the positive column of direct current glow discharges. Plasma Sources Sci. Technol. 16, 557 (2007)
L.L. Alves, A. Bogaerts, V. Guerra, M.M. Turner, Foundations of modelling of nonequilibrium low-temperature plasmas. Plasma Sources Sci. Technol. 27, 023002 (2018)
L. L. Alves, On the influence of collisions on low-temperature plasmas, 72nd Annual Gaseous Electronics Conferenc, College Station / Texas, http://meetings.aps.org/Meeting/GEC19/Session/DT3.6 (2019).
J.A. Bittencort, Fundamentals of Plasma Physics (Springer, New York, 2004)
H.A. Blevin, J.A. Reynolds, P.C. Thonemanne, Penetration of an electromagnetic wave into a hot plasma slab. Phys. Fluids 13, 1259 (1970)
G.L. Braglia, J. Wilhelm, R. Winkler, Multiterm solution of Boltzmann’s equation for electrons in a gas in an electric field. Il Nuovo Cimento 80B, 21 (1984)
A.O. Brezmes, C. Breitkopf, Fast and reliable simulation of argon inductively coupled plasmas using COMSOL. Vacuum 116, 65 (2015)
Y. Celik, T.V. Tsankov, M. Aramaki, S. Yoshimura, D. Luggenhölscher, U. Czarnetzki, Electron cooling in decaying low-pressure plasmas. Phys. Rev. E 85, 046407 (2012)
P. Chabert, T.V. Tsankov, U. Czarnetzki, Foundations of capacitive and inductive radio-frequency discharges. Plasma Sources Sci. Technol. 30, 024001 (2021)
R.H. Cohen, T.D. Rognlien, Electron kinetics in radio-frequency magnetic fields of inductive plasma sources. Plasma Sources Sci. Technol. 5, 442 (1996)
D.L. Crintea, D. Luggenhölscher, V.A. Kadetov, Ch. Isenberg, U. Czarnetzki, Phase resolved measurement of anisotropic electron velocity distribution functions in a radio-frequency discharge. J. Phys. d: Appl. Phys. 41, 082003 (2008)
U. Czarnetzki, Kinetic model for stochastic heating in the INCA discharge. Plasma Sources Sci. Technol. 27, 105011 (2018)
U. Czarnetzki, K. Tarnev, Collisionless electron heating in periodic arrays of inductively coupled plasmas. Phys. Plasmas 21, 123508 (2014)
V.M. Donnelly, A. Kornblit, J. Vac. Sci. Technol. A 31, 050825 (2013)
G. Ekker, Theory of Fully Ionized Plasmas (Academic Press, New York, 1972)
N.J. Fisch, Theory of current drive in plasmas. Rev. Mod. Phys. 59, 175 (1987)
B.D. Fried, S.D. Conte, The Plasma Dispersion Funciton (Academic Press, New York, 1961)
A. Fruchtman, Neutral gas depletion in low temperature plasma. J. Phys. d: Appl. Phys. 50, 473002 (2017)
V. Godyak, Hot plasma effects in gas discharge plasmas. Phys. Plasmas 12, 055501 (2005)
V.A. Godyak, V.I. Kolobov, Effect of collisionless heating on electron energy distribution in an inductively coupled plasma. Phys. Rev. Lett. 81, 369 (1998)
G.J. Hagelaar, Coulomb collisions in the Boltzmann equation for electrons in low-temperature gas discharge plasmas. Plasma Sources Sci. Technol. 25, 015015 (2016)
G.J. Hagelaar, L.C. Pitchford, Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models. Plasma Sources Sci. Technol. 14, 722 (2005)
J. Han, P. Pribyl, W. Gekelman, A. Paterson, S.J. Lanham, C. Qu, M.J. Kushner, Three-dimensional measurements of plasma parameters in an inductively coupled plasma processing chamber. Phys. Plasmas 26, 103503 (2019)
R.D. Hazeltine, F. Waelbroeck, The Framework of Plasma Physics (Perseus Books, Reading, 1998)
T. Holstein, Energy distribution of electrons in high frequency gas discharges. Phys. Rev. 70, 367 (1946)
J. Hopwood, Review of inductively coupled plasmas for plasma processing. Plasma Sources Sci. Technol. 1, 109 (1992)
S. Ichimaru, Basic Principles of Plasma Physics: A Statistical Approach (Benjamin, Reading, 1973)
I. Kaganovich, O.V. Polomarov, C.E. Theodosiou, Landau dam** and anomalous skin effect in low-pressure gas discharges: Self-consistent treatment of collisionless heating. Phys. Plasmas 11, 2399 (2004)
I.D. Kaganovich, O.V. Polomarov, C.E. Theodosiou, Revisiting anomalous RF field penetration into a warm plasma. IEEE Trans. Plasma Sci. 34, 696 (2006)
W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, 1973)
J. Keilson, J.E. Storer, On Brownian motion, Boltzmann’s equation, and the Fokker-Planck Equation. Q. Appl. Math. 10, 233 (1952)
V.I. Kolobov, Fokker-Planck modelling of electron kinetics in plasmas and semiconductors. Comput. Mater. Sci. 28, 302 (2003)
V.I. Kolobov, R.R. Arslanbekov, Simulation of electron kinetics in gas discharges. IEEE Trans. Plasma Scie. 34, 895 (2006)
V.I. Kolobov, D.J. Economou, The anomalous skin effect in gas discharge plasmas. Plasma Sources Sci. Technol. 6, R1 (1997)
V.I. Kolobov, V. Godyak, Electron kinetics in low-temperature plasmas. Phys. Plasmas 26, 060601 (2019)
V.I. Kolobov, D.P. Lymberopoulos, D.J. Economou, Electron kinetics and non-Joule heating in near-collisionless inductively coupled plasmas. Phys. Rev. E 55, 3408 (1997)
U. Kortshagen, Experimental evidence on the nonlocality of the electron distribution function. Phys. Rev. E 49, 4369 (1994)
U. Kortshagen, I. Pukropski, L.D. Tsendin, Experimetnal investigation and fast two-dimensional self-consistent kinetic modeling of low-pressure inductively coupled RF discharge. Phys. Rev. E 51, 6063 (1995)
T. Lafleur, P. Chabert, Is collisionless heating in capacitively coupled plasmas really collisionless? Plasma Sources Sci. Technol. 24, 044002 (2015)
L. Lallement, A. Rhallabi, C. Cardinaud, M.C. Peignon-Fernandez, L.L. Alves, Global model and diagnostic of a low-pressure SF6/Ar inductively coupled plasma. Plasma Sources Sci. Technol. 18, 025001 (2009)
A. Lenard, I.B. Bernstein, Plasma oscillation with diffusion in velocity space. Phys. Rev. 112, 1456 (1958)
H. Leyh, D. Loffhagen, R. Winkler, A new multi-term solution technique for the electron Boltzmann equation of weakly ionized steady-state plasmas. Comput. Phys. Commun. 113, 33 (1998)
M.A. Lieberman, A.J. Lichtenberg, Theory of electron cyclotron resonance heating II: Long time and stochastic effects. Plasma Phys. 15, 125 (1973)
M.A. Lieberman, A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd edn. (Wiley, New York, 2005)
S.T. Lishev, A.P. Shivarova, K.T. Tarnev, On the inertia term in the momentum equation in the free-fall regime of discharge maintenance. J. Plasma Phys. 77, 469 (2010)
H.A. Lorentz, The motion of electrons in metallic bodies I. Proc. R. Acad. Arts Sci. 7, 438 (1905)
J. Loureiro, J. Amorim, Kinetics and Spectroscopy of Low Temperature Plasmas (Springer, Heidelberg, 2016)
T. Makabe, Velocity distribution of electrons in time-varying low-temperature plasmas: progress in theoretical procedures over the past 70 years. Plasma Sources Sci. Technol. 27, 033001 (2018)
T. Makabe, Z. Lj. Petrovic, Plasma Electronics (CRC Press Taylor and Francis, Boca Raton, 2015)
S. Mattei, K. Nishida, M. Onai, J. Lettry, M.Q. Tran, A. Hatayama, A fully-implicit particle-in-cell monte carlo Collision code for the simulation of inductively coupled plasmas. J. Comput. Phys. 350, 891 (2017)
G. Mümken, U. Kortshagen, On the radial distribution and nonambipolarity of charged particle fluxes in nonmagnetized planar inductively coupled plasmas. J. Appl. Phys. 80, 6639 (1996)
K. Niemi, S. Reuter, L.M. Graham, J. Waskoenig, T. Gans, Diagnostic based modeling for determining absolute atomic oxygen densities in atmospheric pressure helium-oxygen plasmas. Appl. Phys. Lett. 95, 151504 (2009)
W. Nolting, Theoretical Physics 1—Classical Mechanics (Springer, Heidelberg, 2016)
D. O’Connell, T. Gans, D.L. Crintea, U. Czarnetzki, N. Sadeghi, Neutral gas depletion mechanisms in dense low-temperature argon plasmas. J. Phys. d: Appl. Phys. 41, 035208 (2008)
M. J. de Oliveira, Boltzmann stochastic thermodynamics. Phys. Rev. E 99, 052138 (2019)
W. Paul, J. Baschnagel, Stochastic Processes—From Physics to Finance, 2nd edn. (Springer, Berlin, 2013)
R.F. Pawula, Approximation of the linear Boltzgnann equation by the Fokker–Planck equation. Phys. Rev. 162, 186 (1967)
L.C. Pitchford, A.V. Phelps, Comparative calculations of electron-swarm properties in N2 at moderate E/N values. Phys. Rev. A 25, 540 (1982)
L.C. Pitchford, S.V. O’Neil, J.R. Rumble Jr., Extended Boltzmann analysis of electron swarm experiment. Phys. Rev. A 23, 294 (1981)
Y. Raizer, Gas Discharge Physics (Springer, Berlin, 1991)
S. Rauf, M.J. Kushner, Model for noncollisional heating in inductively coupled plasma processing sources. J. Appl. Phys. 81, 5966 (1997)
F. Reif, Fundamentals of Statistical and Thermal Physics (Mc Graw-Hill, Boston, 1965)
G.E.H. Reuter, E.H. Sondheimer, The theory of the anomalous skin effect in metals. Proc. R. Society London Ser. A 195, 336 (1948)
M.A. Ridenti, L.L. Alves, V. Guerra, J. Amorim, Plasma sources science and technology, the role of rotational mechanisms in electron swarm parameters at low reduced electric field in N2, O2 and H2. Plasma Sources Sci. Technol. 24, 035002 (2015)
H. Risken, The Fokker–Planck Equation (Springer, Berlin, 1989)
Y. Sakiyama, D.B. Graves, Corona-glow transition in the atmospheric pressure RF-excited plasma needle. J. Phys. D: Appl. Phys. 39, 3644 (2006)
A. Salabas, G. Gousset, L.L. Alves, Two-dimensional fluid modelling of charged particle transport in radio-frequency capacitively coupled discharges. Plasma Sources Sci. Technol. 11, 448 (2002)
I. Shkarofsky, T. Johnston, M. Bachynski, The Particle Kinetics of Plasmas (Addison-Wesley Publishing Company, Reading, 1966)
T.J. Sommerer, M.J. Kushner, Numerical investigation of the kinetics and chemistry of RF glow discharge plasmas in He, N2, O2, He/N2/ using a Monte Carlo-fluid hybrid model. J. Appl. Phys. 71, 1654 (1992)
J. Stephens, A multi-term Boltzmann equation benchmark of electron-argon cross-sections for use in low temperature plasma models. J. Phys. D: Appl. Phys. 51, 125203 (2018)
T.H. Stix, Waves in Plasmas (American Insitute of Physics, New York, 1992)
K. Takagi, A. Ikeda, T. Fujimura, Y. Kuroki, Inductively coupled plasma application to the resist ashing. Thin Solid Films 386, 160 (2001)
K. Tarnev, D. Todorov, B. Gilev, On the spatial distribution of the electromagnetic field in small-radius planar coil inductive discharges. Plasma Sources Sci. Technol. 22, 055015 (2013)
Kh. Tarnev, E. Halova, U. Czarnetzki, AIP Conf. Proc. 2172, 070005 (2019)
A. Tejero-del-Caz, V. Guerra, D. Goncalves, M. Lino da Silva, L. Marques, N. Pinhão, C.D. Pintassilgo, L.L. Alves, The LisbOn Kinetics Boltzmann solver. Plasma Sources Sci. Technol. 28, 043001 (2019)
A. Tejero-del-Caz, V. Guerra, N. Pinhão, C.D. Pintassilgo, L.L. Alves, On the quasi-stationary approach to solve the electron Boltzmann equation in pulsed plasmas. Plasma Sources Sci. Technol. 30, 065008 (2021)
T.V. Tsankov, U. Czarnetzki, A discharge with a magnetic X-point as a negative hydrogen ion source. AIP Conf. Proc. 1390, 140–149 (2011)
T.V. Tsankov, U. Czarnetzki, Information hidden in the velocity distribution function of ions and the exact kinetic Bohm criterion. Plasma Sources Sci. Technol. 26, 055003 (2017)
T.V. Tsankov, R. Johnsen, U. Czarnetzki, Rydberg state, metastable, and electron dynamics in the low-pressure argon afterglow. Plasma Sources Sci. Technol. 24, 055001 (2015)
L. Tsendin, Electron kinetics in non-uniform glow discharge plasmas. Plasma Sources Sci. Technol. 4, 200 (1995)
L. Tsendin, Nonlocal electron kinetics in gas-discharge plasma. Physics Uspekhi 53, 133 (2010)
M.M. Turner, Collisionless heating in radio-frequency discharges: a review. J. Phys. d: Appl. Phys. 42, 194008 (2009)
V. Vahedi, M.A. Lieberman, G. DiPeso, T.D. Rognlien, D. Hewett, Analytic model of power deposition in inductively coupled plasma sources. J. Appl. Phys. 78, 1446 (1995)
J. van Dijk, K. Peerenboom, M. Jimenez-Diaz, D. Mihailova, J. Mullen, The plasma modelling toolkit Plasimo. J. Phys. D Appl. Phys. 42, 194012 (2009)
A.V. Vasenkov, M.J. Kushner, Electron energy distributions and anomalous skin depth effects in high-plasma-density inductively coupled discharges. Phys. Rev. E 66, 066411 (2002)
J.P. Verboncoeur, Particle simulation of plasmas: review and advances. Plasma Phys. Control. Fusion 47, A231 (2005)
E.S. Weibel, Anomalous skin effect in a plasma. Phys. Fluids 10, 741 (1967)
W. Yang, Y. Wang, Plasma. Phys. Control. Fusion 63, 035031 (2021)
W. Yang, S. Zhao, D. Wen, W. Liu, Y. Liu, X. Li, Y. Wang, J. Vacuum Scie. Technol. A 34, 031305 (2016)
W. Yang, F. Gao, Y. Wang, Plasma. Sci. Technol. 24, 055401 (2022)
Z. Yu, D. Shaw, Large area radio frequency plasma for microelectronics processing. J. Vacuum Sci. Technol. A 13, 871 (1995)
X.-M. Zhu, T.V. Tsankov, D. Luggenhölscher, U. Czarnetzki, 2D collisional-radiative model for non-uniform argon plasmas: With or without ’escape factor’, J. Phys. D 48, 085201 (2015)
Acknowledgements
The authors are grateful for numerous stimulating and inspiring discussion with Dr. Tsanko V. Tsankov. UC wants to thank Prof. Wolfgang Paul for an extensive exchange on the statistical properties of the Fokker-Planck equation during the early phase of this work. LLA wants to thank Dr. Antonio Tejero-del-Caz for useful insights about the numerical implementation of the FPHO in a local Boltzmann solver. LLA was partially supported by the Portuguese FCT-Fundação para a Ciência e a Tecnologia, under projects UIDB/50010/2020 and UIDP/50010/2020.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix
Appendix A: Solid angle averaging
The local Ohmic heating operator (31) for a harmonic and homogeneous electric field pointing in z-direction contains an average over the solid angle in velocity space. The operator is derived from the Boltzmann equation but the same problem arises for the Fokker–Planck heating operator (Eq. (66)). Further, the solid angle averaging is independent of the temporal variation and applies also for a homogeneous DC field. Equation (31) reads:
There:
The differential then becomes:
Therefore:
A similar but slightly more complicate averaging applies in the non-local case, described by Eq. (80), where \(D_{v} \left( {v,v_{z} } \right)\,\) is the anisotropic diffusion constant in velocity space:
There, differentiation is with respect to \(v_{x}\) but the diffusion constant also contains a \(v_{z}\) dependence so that both angles \(\vartheta\) and \(\varphi\) are involved:
The differential is transformed into spherical coordinates:
The \(\varphi\) average can be carried out straight away, since the first two terms yield \(\cos^{2} \left( \varphi \right)\) and the last term after differentiation \(- \sin^{2} \left( \varphi \right)\). Then averaging (\(\varphi\) integration and division by the full solid angle \(4\,\pi\)) yields a common factor 1/4 and change of sign in the last term. The \(\xi\)-differential can be removed by integration by parts, which allows combining the second and third terms:
The coupling function \(\Gamma \left( v \right)\) is defined as:
Note that for an isotropic diffusion constant in velocity space \(D_{v} \left( v \right)\), it follows that \(\Gamma \left( v \right) = D_{v} \left( v \right)\).
Appendix B: Brief outline of the ergodic Monte-Carlo simulation
The MC simulation consists of a predefined large number of N loops for a single particle, interacting with an electric field and colliding with the surrounding atoms in a random manner. Each loop corresponds to the free-flight of the particle, during which it is accelerated in the external static and homogeneous electric field of strength \(E_{0}\) and pointing in z-direction, and is followed by a collision at the end. The two cases of (i) constant collision frequency and (ii) constant mean free path are treated slightly differently.
For case (i), constant collision frequency, all variables are normalized to natural scales:
By this normalization the theoretical isotropic distribution function reads:
where \(v\) is the normalized velocity.
The (normalized) free flight time in each loop \(i\) is calculated by the probability distribution (5) and by drawing a random number \(P_{i}\):
By integrating the equation of motion over the free-flight time \(\tau\), considering the acceleration by the electric field along the z-direction and the initial velocities \(v_{z0}\) and \(v_{0}\), it follows that the particle travels a normalized distance \(dr = \sqrt {dx^{2} + dy^{2} + dz^{2} }\):
For the first loop the particle is assumed to be initially at rest. For the subsequent loops the initial velocities are updated at the end of the each loop, as follows. The mean velocity over the free-flight interval \(\tau\) is:
From this mean velocity, a velocity index number is calculated \(n = {\text{Int}}(\overline{v}/\delta v) + 1\), with \(\delta v\) denoting the (normalized) velocity resolution, and the corresponding value in a list \(\tilde{f}\left( n \right) = \tilde{f}\left( n \right) + \tau\) is increased by the free flight time as representing the statistical weight. The relation to the isotropic distribution function is \(\tilde{f}\left( v \right) = \,f\left( v \right)\,d^{3} v\,T_{j} = \,f\left( v \right)\,4\,\pi \,v^{2} dv\,T_{j}\), where \(T_{j}\) is the sum of all free flight periods up to loop \(j\). Further, the z-component and its absolute value at the end of the free flight period are calculated from the normalized relation:
Finally, to update the initial velocities, two new random numbers are drawn to define the scattering angles in spherical coordinates \(\cos \left( \vartheta \right) = 1 - 2\,P_{2},\,\,\,\varphi = 2\,\pi \,P_{3},\,\,\,P_{2,3} = [0,1]\). The two angles are treated differently due to the solid angle differential \(d\Omega = \cos \left( \vartheta \right)\,d\vartheta \,d\varphi\). The new initial values of the absolute velocity and the z-component after the scatting process (including finite momentum and energy transfer) are computed according to (Allis 1956; Nolting 2016):
After all loops have been computed, the final normalized distribution function (with an area of 1) is calculated according to:
The total time \(T\) is the sum of all individual free-flight periods. This realizes a dimensionless and normalized statistical weight of \(\tau /T\) for each count in a particular loop \(j\). Naturally, the total time \(T\) can be introduced only subsequently. The velocity interval \(d^{3} v = 4\pi \,v^{2} \,dv\) converges to zero as the velocity approaches zero. Therefore, also the counts in the distribution \(\tilde{f}\left( {v \to 0} \right)\) go to zero. As a consequence, the values of the final isotropic distribution \(f_{0} \left( {v \to 0} \right)\) are a result of the division of two small numbers, which causes inevitably strong statistical fluctuations, although the distribution \(f_{0}\) becomes maximal at \(v = 0\).
For case (ii), constant mean free path, the general procedure is identical but with some important differences. The normalization scales are now:
The normalized theoretical distribution function (75) following from the Boltzmann and the modified Fokker–Planck heating operator then reads:
In contrast, the normalized distribution function following from the standard form of the Fokker–Planck heating operator reads:
In this case, the free-flight distance \(dr\) (not the free-flight time \(\tau\)) is determined in each loop by drawing a random number:
The free flight time \(\tau\) follows from numerically solving Eq. (a13). This equation has two imaginary and two real roots, a positive and a negative one. Only the positive root is of relevance here. Solving this equation numerically enhances the calculation time significantly and so typically a lower number of collisions is calculated. It should be noted that in rare single events (probability in the range \(10^{ - 6} - 10^{ - 7}\)), the random flight distance \(dr\) can be so small that it falls within the numerical error radius. In these cases, the numerical value of the flight-time \(\tau\) is very close to zero and can even become accidentally negative, which is incorrect regardless of the smallness of the number. This case is checked in every loop and in the event of \(\left| \tau \right| < < 1\) and \(\tau < 0\), the approximate analytical solution to (a13) is applied instead: \(0 \le \tau = dr/v_{0} < < 1\).
The velocity distribution function is determined in the same way as in case (i). Additionally, we also compute an alternative distribution function, where each individual loop is weighted with 1 instead \(\tau\). In the subsequent final calculation (Eq. (a17)), the normalization is to the total number of collisions \(N\) instead of the total flight time \(T\). Therefore, the normalized statistical weight is constant and equals \(1/N.\)
Appendix C: Generalized plasma dispersion function
The classical plasma dispersion function appears frequently in kinetic wave phenomena and is central for the description of Landau dam** of electrostatic waves. For electrostatic waves, the dispersion relation follows from combining the Boltzmann equation with Poisson’s equation (Stix 1992; Bittencourt 2004):
The integral is simplified by assuming (a) a Maxwellian distribution function and (b) a velocity-independent collision frequency. The assumption (a) allows integration over the transversal velocity components and an explicit evaluation of the derivative. The subsequent evaluation involves integration over the complex plane (Ekker 1972). Only the final result is of interest here:
The plasma dispersion function \(Z\left( u \right)\) is defined as (\(F\left( u \right)\) is usually called the Dawson function) (Fried and Conte 1961):
By applying the same integration strategy as for the plasma dispersion integral in Sect. 7, the plasma dispersion function can be generalized so that it can be calculated for arbitrary distribution functions and velocity dependences of the collision frequency. After rearranging terms and using \(\partial /\partial v_{z} = \left( {v_{z} /v} \right)\,\partial /\partial v\), the above dispersion relation reads \(\left( {\xi = \cos \left( \vartheta \right)} \right)\):
Here, the definition of a generalized thermal velocity follows from comparison with the standard form of the dispersion relation for a Maxwell distribution (note the difference to the definition used in the main text):
Indeed, for a Maxwellian distribution function this definition is consistent with the expected thermal velocity \(v_{th}^{2} = 2\,k_{B} T_{e} /m_{e}\). With this general definition, the following normalized quantities can be defined:
Note that now the phase velocity \(u\) is defined above as real. A complex definition would include the factor \(\left( {1 + i\,\nu } \right)\), which depends on velocity, i.e. on the integration variable, which would not be useful in this context. The dispersion relation now reads:
The definition of the generalized plasma dispersion function \(Z_{g} \left( u \right)\) follows directly from a comparison to the above classical form (a23) of the dispersion relation, which suggests the right hand side to be defined as \(- u\,Z_{g} \left( u \right)\):
This result applies for any distribution function and any velocity dependence of the collision frequency. The functional dependence is noted only as \(Z_{g} \left( u \right)\), although the function also depends on the collision frequency \(\nu \left( w \right)\). However, in the general case this includes a complicate dependence on velocity, which may not exist in an analytical form. It is remarkable that in contrast to the standard derivation of the classical expression of the plasma dispersion function, no integration through the complex plane was required here. Further, the two terms of the classical plasma dispersion function represent the real and the imaginary parts only in the collisionless limit. In the collisional case, both parts become complex and separating the real and the imaginary parts is not straightforward. In the generalized plasma dispersion function, the real part and the imaginary part are easy to separate and both parts are conveniently given by real functions:
The simplest test for the integration concept is by comparing the collisionless limit for a Maxwellian distribution \(\tilde{f}_{0} \left( u \right) = \exp \left( { - w^{2} } \right)\). Then the sum of the two arctan-functions in (a29) converges in the range \(w \ge 0\) towards a step function \(\,\pi \,\theta \left( {w - u} \right)\), which allows explicit evaluation of the integral of the imaginary part. The real part can be evaluated by integrating by parts:
Both imaginary parts are clearly analytically identical and comparison of the real parts suggests equality of the Dawson function with the integral term:
Finding a suitable transformation in order to proof the analytical identity of the two quite different integrals is challenging. It may seem attempting to further evaluate the integral by performing an integration by parts, but the integrable divergence in the logarithm at \(w = u\) makes such a strategy questionable. At least the leading orders of the asymptotic behavior are easy to calculate explicitly and agree perfectly with the well-known behavior of the Dawson function (Fried and Conte 1961):
Further, the Dawson function and the above integral have the same symmetry property \(F\left( { - u} \right) = - F\left( u \right)\). In particular, the two expressions agree numerically perfectly well throughout as shown in Fig.
Negative of the real parts of the classical and the generalized plasma dispersion functions in the collisionless limit and for a Maxwellian distribution function (classical result using \(- {\text{Re}} (Z(u)) = 2\,F(u)\): thin solid line, present result using \(- Z_{g} (u)\): thick dashed line)
12. The relative deviation (Fig.
Relative numerical deviation between the two curves in Fig. 12.1. Positive values represent larger values of the Dawson function
13) is generally very small (\(< 10^{ - 3}\)) and clearly statistical, which can be attributed to the precision of the numerical integration techniques.
In the limit of\(u > > 1\), the generalized plasma dispersion function converges towards to the simple collisional limit, again in good agreement with the standard plasma dispersion function:
Finally, it should also be noted that here we arrived at the result (a32) by a significantly simpler and shorter way than by the classical derivation, which involves complicate integrations in the complex plane. The remaining integrals need of course to be evaluated numerically but this is in no way more complicate than the evaluation of the integral of the Dawson function.
Appendix D: Surface loss operator
Stochastic heating requires low pressures, i.e. large mean free paths. Then in addition to the (local) loss of energy by collisions also the (non-local) energy loss by electrons onto the walls is important. Plasma electrons are confined by the sheath potential in front of the wall and only electrons with kinetic energy larger than the confining potential can leave the plasma. Since the spatial extension of the space charge region generating the sheath potential is negligibly small compared to the typical plasma size (typically, two orders of magnitude below), no distinction is made between the area at the edge of the quasi-neutral plasma and the wall area.
Without loss of generality, one can choose a coordinate system with the z-axis perpendicular to the surface. Indeed, the calculation is similar to calculating the flux to a planar probe (Lieberman and Lichtenberg 2005). For an electron to be lost onto the wall, its velocity along the z-coordinate has to be higher than the threshold velocity introduced by the potential difference \(\Phi_{s} > 0\) between the edge of the quasi-neutral bulk and the wall:
The loss operator is then given by the integral over the entire wall surface \(A\), assumed to be an equipotential surface, followed by the division by the plasma volume \(V\), in order to obtain a volume average according to the divergence theorem. Further, the operator has to be averaged over the solid angle:
The integral yields:
and the corresponding operator for the energy distribution function reads:
In this form, the operator resembles very much an inelastic loss term for excitation collisions, where the plasma potential represents the threshold energy of the excited state. Note that the plasma potential in the operator is the sum of the floating potential over the sheath and the potential over about the first two ion mean-free-paths.
Appendix E: Floating potential calculation
The electron loss to the wall is controlled by the confining electrostatic floating potential \(\Phi_{s}\). The term “floating” refers to a local balance of the electron and the ion fluxes. This potential is defined by Eq. (151). After rearranging terms and taking the logarithm, the equation reads:
The dimensionless quantities are:
where \(y_{0}\) is the classical floating potential. Equation (a40) defines the normalized potential \(y = y\left( {y_{0} } \right)\) in a form that can usually be solved only numerically. However, for the special case of a Maxwell distribution \(f\left( x \right)\, = f_{M} \left( x \right) = e^{ - x} 2\,/\,\sqrt \pi\), the integral in (a40) can be solved analytically:
Indeed, in this case one obtains the expected classical floating potential (Lieberman 2005). Also, in general \(\Phi_{s}\) is only a function of the electron–ion mass ratio or \(y_{0}\), respectively. In general, one does not expect strong deviations from the classical value for a Maxwell distribution. Denoting \(y = y_{0} + \delta y\) and assuming \(\left| {\delta y} \right| < < y_{0}\), a first order expansion of \(f\left( {x + y} \right)\) in (a40) in the small correction \(\delta y\) yields for the integral:
where in the last step integration by parts has been applied. Then (a40) can be solved for \(\delta y\) by using a further expansion of the logarithm:
Naturally, \(\delta y = 0\) for a Maxwell distribution, since in this case the term in the brackets becomes zero. Depending on the particular form of the distribution function, \(\delta y\) can in general be either positive or negative. The first order correction now reads:
A valid guess on the accuracy of the above approximation might be provided by applying it to the Druyveystein distribution:
where \(\Gamma \left( a \right)\) is the Gamma function. A comparison of the numerical result from Eq. (a40) with the approximate analytical solution (a44) is shown in Fig.
Normalized sheath floating potential \(y\) as function of the normalized mass factor \(y_{0}\) (see (a41) for the definition of the classical floating potential) for a Druyvesteyn distribution function. Solid line: exact numerical solution, dashed line: approximate analytical solution
14. In the figure, a diagonal line would represent the linear relation \(y = y_{0}\) found for a Maxwell distribution. The results show that the potential is lower than \(y_{0}\), since the Druyvesteyn distribution decays faster with energy than the Maxwell distribution (\(- x^{2}\) vs. \(- x\) in the exponential function). Although the deviation from linearity is quite significant, the analytical approximation is still very close to the exact solution throughout. For realistic ion masses, the value of\(y_{0}\)does not exceed 6 (argon has \(y_{0} = 4.7\)). For this maximum value, the error introduced by the approximation is still below 8%. Therefore, one can conclude that the analytical approximation (a45) should be sufficient for all practical purposes.
Appendix F: Ion fluid transport model for a cubic geometry
The ion continuity and the ion momentum balance equation are solved for an illustration example in order to determine the ionic ionization rate \(\left\langle {\nu_{iz} } \right\rangle^{{\left( {ion} \right)}}\) as an eigenvalue of the equation set. The problem is solved for the particular case of a cube with edge length \(L\). A constant elastic ion-neutral collision frequency \(\nu_{mi}\) is assumed, i.e. the contribution of charge exchange collisions to the friction term is neglected. Further, the ions are assumed as cold \(T_{i} = 0\), so that the related pressure gradient term can be neglected, and the ionization rate \(\left\langle {\nu_{iz} } \right\rangle^{{\left( {ion} \right)}}\) is assumed to be homogeneous. The ambipolar electric field, causing the ion drift, is related to the density gradient by the Boltzmann factor (143). The normalized set of equations, consisting of the continuity and the momentum balance equation, reads in Cartesian coordinates:
where summation is over the three spatial coordinates \(x_{j},\,j = 1,2,3\). The dimensionless quantities are (\(n_{0}\) is the maximum plasma density at the center, \(c_{i}\) is the ion sound speed as defined in Sect. 8):
For the cube all three lengths are equal, so \(\Lambda_{j} = \Lambda = L/\lambda_{i}\). The transversal derivatives can in good approximation be neglected (Lishev et al. 2010):
A separation ansatz is made for the density \(n = \prod\limits_{j} {n_{j} \left( {\xi_{j} } \right)}\). Then:
Now the momentum balance equation reads:
The continuity equation can also be separated into a set of three equations using dimensionless factors \(0 \le \alpha_{j} \le 1\). For symmetry reasons it follows for the special case of a cube that \(\alpha_{j} = 1/3\):
Solving algebraically for the derivatives of the density and the velocity:
At \(u_{j} = 1\) all derivatives are diverging. This marks the edge of the quasi-neutral plasma and provides effectively a boundary condition at \(\xi_{j} = \pm \Lambda_{j} /2\) where \(u = 1\). Integration along the length using the monotonous variation of the velocity yields the eigenvalue equation for \(\nu\):
Generally, the 3 equations (a54) together with the sum rule for the \(\alpha_{j}\) (Eq. (a52)), determine all parameters. For the cube (\(\Lambda_{j} = \Lambda,\,\,\,\alpha_{j} = 1/3\)) there is only one equation:
Explicit inverted solutions can only be obtained for the limiting cases:
This motivates an empirical formula for the ionization rate over the entire variation range, which converges exactly to the limiting cases. The formula is constructed by setting \(1/\nu = 1/\nu_{l} + 1/\nu_{h}\), where \(\nu_{l}\) and \(\nu_{h}\) are the above (Eq. (a56)) limits for small and large values, respectively:
In fact, the empirical formula provides an excellent match for the variation of the ionization rate as a function of the normalized length \(\Lambda\) within the entire parameter range, as shown by Fig.
Normalized ionization frequency as a function of the normalized discharge length, following from the ion fluid model for the special case of a cube (solid line) as obtained from Eq. (a55). The dashed straight lines represent the solutions for the two limiting cases of low and high ionization rates or normalized length, respectively (Eq. (a56)). The curve represented by the dashed-dotted line, almost entirely hidden by the solid line, was obtained using the empirical matching formula (a57), which apparently shows very good accuracy throughout
15. Experimentally, the collisionless range \(\Lambda < < 1\) is usually difficult to realize since the power required for maintaining the discharge diverges.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Czarnetzki, U., Alves, L.L. Describing local and non-local electron heating by the Fokker–Planck equation. Rev. Mod. Plasma Phys. 6, 31 (2022). https://doi.org/10.1007/s41614-022-00086-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41614-022-00086-0