Stability of very-high pressure arc discharges against perturbations of the electron temperature

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I. INTRODUCTION
Very high-pressure xenon arc lamps develop, under certain conditions, voltage oscillations accompanied by electromagnetic interference (EMI). 1-5The question arises in which region of the discharge the instability responsible for these oscillation develops and what its mechanism is.
This question was considered in Ref. 1.The near-cathode region was ruled out on the grounds that the measurements did not show any influence of the mode of operation of the cathode.The near-anode region was ruled out as well, one of the reasons being that all attempts of modeling an anodic sheath instability led to the conclusion that the anodic sheath is stable under the conditions considered.Accordingly, the conclusion was that the instability develops in the plasma column.This conclusion was supported by a local dispersion analysis, which revealed the potential presence of an instability of the energy balance of the electron gas in the plasma column.This instability has been described previously (e.g., Ref. 6, p. 61) and originates in the variations of heating of the electron gas by the electric field occurring faster than the variations of cooling by collisions with heavy particles.
However, subsequent experimental investigations of the voltage oscillations which can occur in xenon arc lamps operated at very high pressures have pinned down the instability to the near-anode region rather than to the plasma column. 3-5In particular, it was found that EMI is correlated with the temperature of the surface of the anode and the type of arc attachment to the anode.Therefore, the theoretical mechanism of instability leading to voltage oscillations under conditions of very high-pressure arc lamps must be revisited.A related topic of considerable interest is a potential relation between this mechanism and that of multiple attachments of high-pressure arcs to the anodes, e.g., Refs.7-13.These tasks are dealt with in the present work.The outline of the paper is as follows: A local dispersion analysis of the stability of the energy balance of the electron gas in a quasi-neutral plasma is briefly described in Sec.II and applied to a Xe plasma in Sec.III.The results are applied to investigate the stability of very high-pressure Xe lamps in Sec.IV.A relation between the mechanisms of the considered instability and multiple anodic attachments of high-pressure arcs is discussed in Sec.V. Conclusions are summarized in Sec.VI.

II. DISPERSION ANALYSIS
The instability of the energy balance of the electron gas in a quasi-neutral plasma and the procedure of its dispersion analysis have been described in the literature (page 61 of Ref. 6).Following Ref. 1, we start with the equations of conservation of number and energy of electrons in a collisional quasi-neutral plasma with a Maxwellian electron energy distribution.These equations can be written in the form (e.g., p. 428 of Ref. 14) @n e @t þ r Á n e vÀ j e e ¼ w; (1) þ r Á q e À E i j e e ¼ j e Á E À w elast À w rad : Here and further indices a, i, and e are attributed to the neutral particles, ions, and electrons, respectively; n a is the number density of a species a (a ¼ a, i, e); T e is the electron temperature; j e is the density of the electric current transported by the electrons; w is the net rate of production of pairs ion-electron in volume reactions; v is the mean mass velocity of the plasma; D=Dt ¼ @=@t þ v Á r is the material derivative; E i is the ionization energy; q e is the density of electron heat flux; E is the electric field; and w elast and w rad are the rates per unit volume at which energy is lost by the electron gas as a result of elastic collisions with heavy particles and, respectively, radiation.
A conventional expression describing ionization and recombination in monoatomic gases reads where the rate coefficients k i and k r are functions of the electron temperature: The electric field in the Joule heating term on the rhs of Eq. ( 2) will be eliminated with the use of Ohm's law, j e ¼ en e l e E, where l e is the mobility of electrons.
The rate of electron energy loss in elastic collisions with heavy particles is (e.g., page 428 in Ref. 14) Here, T h is the temperature of the heavy particles (atoms and ions) and v eh is the average frequency of momentum transfer in collisions of electrons with the heavy particles defined by the formula where C e ¼ ð8kT e =pm e Þ 1=2 is the electron mean thermal speed and Q ð1;1Þ ea and Q ð1;1Þ ei are average cross sections for momentum transfer in collisions between electrons and atoms and electrons and ions, respectively.
Making use of Eq. (1), Eq. ( 2) may be re-written as Here, the auxiliary quantity H abbreviates where l ½1 e ¼ e=m e v eh may be interpreted as an approximate value of the mobility of electrons.(In the accurate kinetic theory, l ½1 e represents the value given by the first approximation in the expansion in Sonine polynomials in the Chapman-Enskog formalism.) The linear stability theory is a conventional tool for investigation of the onset of instability.(Of course, a nonlinear analysis is required to study voltage spikes observed in Refs. 1, 5, which represent a result of development of the considered instability and cannot be described under the assumption of small perturbations.)Following the usual procedure (e.g., page 217 in Ref. 15), we represent the timedependent quantities as superpositions of the corresponding steady-state value and a small perturbation with an exponential time dependence, where k is the increment (rate of growth) of perturbations, which is the quantity to be found.Substituting the expressions in Eq. ( 8) into Eqs.( 1) and ( 6), expanding in n e1 and T e1 and retaining linear terms, one arrives at equations for n e1 and T e1 .Similarly to Ref. 1, this work is limited to a local dispersion analysis, which amounts to neglecting spatial gradients in the above-mentioned linear equations for n e1 and T e1 .It is well known that transversal perturbations, i.e., those which vary in a direction perpendicular to the discharge current and result in contraction of the discharge and appearance of filaments, perturb the discharge current, but the electric field remains unaffected, whereas longitudinal perturbations, i.e., those which vary in a direction along the discharge current and result in appearance of striations or domains, perturb the electric field, but the discharge current remains unaffected, e.g., pages 60 and 61 in Ref. 6 or pages 219 and 221 in Ref. 15.Since oscillations of the discharge voltage observed in Refs. 1, 5 mean that the electric field is perturbed, it is appropriate here to consider longitudinal perturbations.Therefore, the current density is treated as a given quantity not subject to perturbations.The contribution of the ion current is neglected, i.e., the (net) current density j is assumed to be equal to the electron current density j e .Variations of the number density of neutral atoms and of the temperature of atoms and ions are neglected as well.
In the above framework, the quantities w and H may be treated as functions of n e and T e : w ¼ w (n e , T e ), H ¼ H(n e , T e ), and the above-mentioned linear equations for perturbations assume the form (The derivatives are evaluated at n e ¼ n e0 , T e ¼ T e0 .)This system of linear algebraic equations for n e1 and T e1 admits nontrivial solutions, provided that its determinant vanishes.The latter condition represents a dispersion equation governing the increment k.A solution to this (quadratic) equation reads where The real part of the solution in Eq. ( 11) with the sign plus (k 1 ) is larger than or equal to the real part of the solution in Eq. ( 11) with the sign minus (k 2 ).Therefore, it is sufficient to evaluate the root k 1 : Re k 1 > 0 implies instability, with (Re k 1 ) À1 being a characteristic time of its development, and Re k 1 < 0 implies stability, with -(Re k 1 ) À1 being a characteristic time of decay of perturbations.

III. INCREMENT OF PERTURBATIONS IN A VERY HIGH-PRESSURE Xe PLASMA
From Sec.II, we now have a model giving the increment of perturbations of electron temperature in a uniform highpressure plasma.The input parameters include those characterizing the steady state in which stability is investigated: the electron and heavy-particle temperatures T e and T h , the electron density n e , and the neutral-atom density n a, or, alternatively, the plasma pressure p. (The index 0 referring to unperturbed plasma parameters is dropped for brevity here and further.)Another control parameter is the current density, j, which is needed for evaluation of derivatives of the function H(n e , T e ).
Results reported in this work refer to a very high-pressure Xe plasma.The transport and kinetic coefficients of this plasma and the radiation energy losses have been evaluated in the same way as in Ref. 16.The derivatives @H=@n e and @w=@T e ; @H=@T e have been found by means of numerical differentiation, the steps being 10 À3 n e and 10 À3 T e , respectively.An important particular case is that where the steady states being considered are are local thermodynamic equilibrium (LTE) ones, i.e., both the ionization and thermal equilibrium hold in these states: n e ¼ n S , T e ¼ T h (n S is the electron density evaluated by means of the Saha equation).It is sufficient to specify just three control parameters in this case: T e , p, and j.We assume a pressure of p ¼ 100 bar and a current density, j, in the range 10 6 -10 8 A/m 2 : parameters that are typical for very high-pressure xenon arc lamps, for example, for those studied experimentally in Ref. 5.This section is concerned with finding values of electron temperature at which the instability can develop; the question of whether these values do occur inside very high-pressure xenon arc lamps is treated in Sec.IV.
The real part of the increment of perturbations of the LTE high-pressure Xe plasma is shown in Fig. 1.One can see that there is a range of T e where Re k 1 is positive.We will refer to this range as the window of instability.Re k 1 is quite high inside this window, of the order of 10 ns À1 or higher.Outside this window, Re k 1 < 0; however, values of jRe k 1 j are substantially smaller than those inside the window (typically of the order of 1 ls À1 or smaller) and their sign cannot be seen from  I. The lower boundary is virtually independent of j and is close to 2400 K for all values of j.The upper boundary slowly increases with increasing j.The imaginary part of k 1 turns out to be negligible except in a narrow range of T e around the upper boundary of the instability window.
Also shown in Table I are the lower and upper boundaries of the instability window for a plasma in thermal non-equilibrium, T e > T h , but still in ionization equilibrium, n e ¼ n S , for two values of the temperature difference DT ¼ T e À T h (DT ¼ 1000 K, 3000 K), and for a plasma in ionization non-equilibrium, n e > n S , but still in thermal equilibrium, T e ¼ T h , for n e /n S ¼ 2. One can see that the instability window depends little on what is assumed for DT or n e /n S .
In order to elucidate the meaning of these results, let us consider the coefficients b and c defined by Eqs.(12).The coefficient c turned out positive in all the calculations performed in this work.Therefore, the sign of Re k 1 is determined by the sign of b, irrespective of k 1 being real or complex.It follows that the change of sign of Re k 1 seen in Fig. 1 is associated with the change of sign of b: b is positive inside the instability window and negative outside.At a state where the change of stability occurs, b ¼ 0 and the increment k 1 is imaginary: ; in other words, neutrally stable perturbations are oscillatory rather than stationary.Therefore, the fact that complex values of k 1 were detected near the upper boundary of the instability window, but not near the lower boundary, means that the vicinity of the lower boundary in which the perturbations are oscillatory is narrower than 1 K (this was the step in T e used in the calculations).
Furthermore, it was found that b 2 ) c in all the calculations except in the vicinity of the upper boundary of the instability window.In this case, one can simplify expression   (11) and find The physical sense is clear: one of the perturbation modes is fast and the other is slow; their increments equal b and c/b, respectively; if b > 0, both modes are growing in time and the rate of growth of perturbations is governed by the fast mode; and if b < 0, both modes are decaying and the rate of decay of perturbations is governed by the slow mode.This explains the difference in orders of magnitude of Re k 1 inside and outside the instability window, seen in Fig. 1, and also the fact that the vicinities of the boundaries of the instability window where k 1 is complex are so narrow.It was also found in the calculations that j@H=@T e j ) j@w=@n e j: the time of relaxation of the electron temperature is much shorter than the time of relaxation of the electron density, similarly to what happens in glow discharges (Ref.6, p. 60).Using this inequality and Eq. ( 12), one can express the increments of the fast and slow perturbation modes as, respectively, b % @H @T e ; c b % @w @n e À @w @T e @H @n e @H @T e À1 : The first expression in Eq. ( 13) represents a solution of Eq. ( 10), with the first term on the rhs of the latter equation being dropped.One can conclude that the instability seen in Fig. 1 indeed represents an instability of the energy balance of the electron gas and that the electron density remains frozen during the development of this instability, similarly to how it happens in glow discharges (Ref.6, p. 60).
In accordance with the above, the criterion of instability reads @H=@T e > 0 or, equivalently, where r 1 , r 2 , r 3 , and r 4 are derivatives with respect to T e of the corresponding terms on the right-hand side of Eq. ( 7), Obviously, these terms represent contributions to the increment or instability of, respectively, Joule heating of the electron gas, losses of electron energy in elastic collisions with heavy particles, losses of electron energy due to ionization, and losses of electron energy due to excitation of neutral atoms with subsequent emission of a photon.
Let us analyze the effect of these different terms r 1 , r 2 , r 3 , and r 4 on b (and thus on the sign of the increment Re k 1 ).If the steady state being considered is that of thermal equilibrium, T e ¼ T h , then Eq. ( 15) yields r 2 ¼ À2e=m a l ½1 e so that r 2 is negative.In fact, r 2 was found to be negative in all the calculations performed in this work and also in those with T e = T h .With increase of T e , k i (T e ) grows, while k r (T e ) decreases.Then it follows, from Eq. ( 3), that @w=@T e > 0, and it follows from the first equation in Eq. ( 16) that r 3 < 0. The radiation losses increase with increase of T e , hence r 4 < 0.
Thus, the terms r 2 , r 3 , and r 4 on the left-hand side of the inequality in Eq. ( 14) are negative, and the only term that may be positive is r 1 .This term is illustrated by Fig. 2. r 1 is positive inside the instability window, as it should have been expected, and also in a certain range of T e above the window.
The contributions of the different terms to the increment are illustrated by Fig. 3.For completeness, all the terms constituting the quantity b are shown, including the term r 5 ¼ @w=@n e from the (original) first equation in Eq. (12).Each term is normalized by jr 1 j þ… þ jr 5 j.For T e < 2400 K, r 1 is negative and much larger than all the other terms.For 2400 K< T e .6000 K, r 1 becomes positive and remains dominating.For T e exceeding approximately 6000 K, the terms r 2 and r 3 come into play and the contribution of r 1 starts decreasing.While the term r 2 reaches a value which does not exceed approximately 30% and then starts decreasing, the term r 3 continues to increase and, for T e & 7000 K, becomes dominating.The term r 4 is negligible for all T e .The term r 5 is negligible as well, in agreement to what was said above.
Thus, the most important mechanisms governing the (in)stability under the conditions of the very high-pressure arc discharges considered here are Joule heating of the electron gas (term r 1 ) and losses of electron energy due to ionization (r 3 ) and elastic collisions with heavy particles (r 2 ).Joule heating of the electron gas can produce both stabilizing (r 1 < 0) and destabilizing (r 1 > 0) effects; losses of electron energy can play only a stabilizing role (r 2 and r 3 < 0).With increasing T e , the instability appears when the effect of Joule heating switches from stabilizing to destabilizing and disappears when the destabilizing effect of Joule heating is overcome by the stabilizing effect of losses of electron energy, in the first place, due to ionization.
According to the first equation in Eq. ( 15), switching of the role played by the Joule heating from stabilization (r 1 < 0) to destabilization (r 1 > 0) is due to a non-monotonic dependence of electron mobility on T e .In order to illustrate this dependence, let us rewrite the first expression in Eq. ( 15) as J 2 e m e e 2 n 2 e @ @T e v eh f e ; where f e ¼ l e =l ½1 e is a kinetic coefficient of order unity and v eh is, as before, the average frequency of momentum transfer in collisions of electrons with the heavy particles.The dependence of v eh on T e can be derived from Eq. ( 5).The average electron-ion elastic-scattering cross section Q and the kinetic coefficient ea (here, l ea is the mobility of electrons in the gas of neutral atoms and l ½1 ea ¼ e=m e v ea ).The vertical dashed line in this figure represents the above-described lower boundary of the instability window, T e ¼ 2400 K.One can see that the instability window opens at the minimum of the function v ea =n a , in agreement with the above reasoning.The non-monotonic behavior of v ea is due to the nonmonotonic behavior of the cross section Q ð1;1Þ ea , which, in turn, is due to the Ramsauer effect.One can see that the lower boundary of the instability window is not far away from the Ramsauer minimum.(We remind that Q ð1;1Þ ea is the average cross section for momentum transfer in collisions between electrons and xenon atoms; the Ramsauer minimum in the energy-dependent momentum-transfer cross section is positioned at approximately 0.7 eV.)

IV. APPLICATION TO VERY HIGH-PRESSURE Xe LAMPS
In order to decide whether the instability can occur in a high-pressure gas discharge lamp, one needs to check whether values of the electron temperature within the instability window can occur inside the lamp.
Simple estimates for conditions of high-pressure discharge lamps 17 show that the plasma is close to LTE and the energy balance of the plasma is dominated by radiation (i.e., Joule heating in the plasma is locally compensated by radiation cooling) in the bulk plasma, which includes the arc column and near-electrode constriction zones.In such a case, the local temperature can be estimated in terms of the local current density and plasma pressure.Therefore, it is sufficient to specify just two control parameters in this case: p and j.
Results of evaluation of the increment for such conditions are shown in Fig. 5. Also shown is T eq , the local temperature evaluated assuming a radiation-dominated energy balance.One can see that T eq rises from about 7000 K at j ¼ 10 6 A/m 2 to about 13 000 K at 10 8 A/m 2 .Since even the lowest value (7000 K) is still above the instability window detected for these conditions in Sec.III, it is unsurprising that the increment k 1 , which is calculated for these conditions under the assumption T e ¼ T h ¼ T eq (and n e ¼ n s ), possesses a negative real part for all values of j, as seen in Fig. 5.It follows that the instability cannot develop in the radiation-dominated bulk of a very high-pressure Xe arc.
In the near-electrode layers of the arc (e.g., Ref. 17), the local energy balance of the plasma is perturbed by heat conduction and the plasma is no longer dominated by radiation; the local balance of charged particles is perturbed by ambipolar diffusion, i.e., the ionization equilibrium does not hold, and the heavy-particle temperature deviates from the electron temperature.Therefore, one needs to solve differential equations involving spatial derivatives in order to find an adequate description of the near-electrode layers, namely, equations of conservation of the charged particles, electron energy, and energy of the heavy particles.Such solutions under conditions of interest for this work have been reported in Refs.18 and 16 for the near-cathode and near-anode layers, respectively.An analysis of the modeling results 18 reveals that the electron temperature in the near-cathode layer is even higher than in the bulk plasma and thus also clearly above the instability window, so that the instability cannot develop in the near-cathode layer either.For the near-anode layer, on the other hand, modeling results 16 predict significantly lower values of T e , down to about 5000 K, which lies inside the instability window predicted here.This leaves the near-anode layer as the only region in a very high-pressure Xe lamp where the instability can occur.
This conclusion agrees with the experimental investigations 3-5 of voltage oscillations in very-high-pressure Xe arcs, which, as discussed in the introduction, pinned down the instability to the near-anode region.The theoretical time of development of the instability, (Re k 1 ) À1 , which is of the order of 0.1-1 ns, conforms to the rise time of a single pulse, which was experimentally determined to be about 800 ps or shorter. 1 Since the imaginary part of the increment is negligible in nearly all the cases, the perturbations grow monotonically on the linear stage of the development of the instability and the oscillations develop on the nonlinear stage.
When focusing the present stability analysis to the nearanode layer of very high-pressure Xe lamps, some of the early premises have to be revisited.The ionization degree corresponding to T ðstabÞ e under the conditions of Table I is below 0.1%, even in the case of a current density at the anode surface as high as 10 8 A m À2 .Perturbations of the density, velocity, and energy of charged particles and of the electric field do not appreciably affect the neutral atoms for ionization degrees that low, which justifies the neglect of variations of the number density and temperature of neutral atoms in the analysis of Sec.II.
The deviation of parameters of the xenon plasma at a pressure of 100 bar and temperatures exceeding 2000 K from the equation of state of ideal gas, estimated by means of the Van der Waals equation, is no more than 3%.The ideal-gas approximation is adequate under such conditions.
In order to get an idea of the time scale of relaxation of deviations from quasi-neutrality, let us estimate the time of diffusion of the electrons over a distance D equal to the scale of thickness of the near-anode space-charge sheath.Data in Fig. 5 of Ref. 16, which refers to a 100-bar xenon plasma as well, suggest D ¼ 0.3 lm.Setting the diffusion coefficient of electrons in the gas of neutral atoms D ea equal to 4 Â 10 À2 m 2 s À1 (a value for a 100-bar xenon plasma at T e ¼ T h ¼ 5 Â 10 3 K), one finds D 2 /D ea % 2 ps.This time is substantially shorter than the time of development of the instability (Re k 1 ) À1 .It follows that the plasma remains quasi-neutral while the instability develops, which justifies the corresponding assumption made in the analysis of Sec.II.
The treatment of Sec.II refers to spatially uniform plasmas, and its application to the near-anode layer, where gradients are considerable, is, strictly speaking, unjustified.However, this treatment still allows one to obtain a useful qualitative indication.Let us introduce the quantity The ratio j=en e represents the local mean speed with which the electrons drift toward the anode surface.Hence, s has the meaning of a characteristic distance over which the electrons drift while the instability is developing.If this distance is larger than the distance to the anode surface, then there will be no instability, because the time of drift of the electron gas to the anode surface is insufficient for the instability to develop.
Representative calculated values of s are shown in Fig. 6.One can see that s is quite high (of the order of millimeters and higher) near the upper boundary of the instability window.Given that the thickness of the near-anode layer is of the order of 0.1 mm, one should conclude that the instability cannot develop at T e near the upper boundary of the instability window.In other words, the T e window of instability at a given position in the near-anode layer is narrower than in a uniform plasma with the same parameters.
To treat the nonuniform near-anode plasma properly, the next step should be to perform a linear stability analysis of differential equations describing the current transfer through the near-anode layer, similarly to how stability of current transfer to arc cathodes has been investigated. 19,20 One can hope that this analysis will describe effects that remain unexplained at the present stage, in the first place, the surprising and very clean response of the instability to variations of the anode tip temperature. 5A further step will be to obtain a non-stationary numerical solution of the full (nonlinear) equations, which will provide information on the oscillations.
It must be considered why our main conclusion about the instability occurring in the near-anode layer, being in agreement with the experimental data, 3-5 contradicts Anders et al. 1 , who concluded that the instability occurs in the arc column.The dispersion analyses conducted in this work and Ref. 1 are not very different.In particular, our Fig. 1 is qualitatively similar to Fig. 6 of Ref. 1, except for one aspect: while jRe k 1 j in Fig. 6 of Ref. 1 exceeds 4 ns À1 for T e below the window of instability, it is much smaller, according to the calculations of this work.This difference becomes understandable if one assumes that the increment in Fig. 6 of Ref. 1 was evaluated neglecting perturbations of the electron density, i.e., equals @H=@T e in designations of the present work.Note that this assumption, while being inconsistent with n e1 appearing in Ref. 1 at the last phase of the derivation of Eq. ( 7), 1 is consistent with the structure of the rhs of the latter equation, although the last term on the rhs is difficult to understand anyway.
Worth noting are significant differences in values of transport and kinetic coefficients between Ref. 1 and the present work.In particular, the authors 1 seem to have taken into account only direct ionization of neutral atoms, but neglected stepwise ionization, which would amount to underestimating the ionization rate by several orders of magnitude: for example, the ratio of the coefficients k ½dir i =k ðstÞ i for T e ¼ 7000 K is about 2.8 Â 10 À3 .The fact that differences that significant have not produced a qualitative effect is remarkable and is likely to originate in the Arrhenius character of the processes involved.
Since the dispersion analyses conducted in this work and Ref. 1 are not very different, the difference in conclusions must originate in how results of the dispersion analysis are applied to conditions of very high-pressure Xe lamps.This is indeed the case: the authors 1 seem not to have realized that the electron temperature in the near-anode region may be significantly lower than in the arc column, as was subsequently indicated by the modeling. 16 It should be stressed that possibilities of measurements of the electron temperature near anodes of high-pressure arc discharges are limited.For example, experiments showed that T e in argon arcs at p ¼ 2 bar decreases toward the anode (Fig. 9 of Ref. 21), with the T e value at a point closest to the anode being around 8000 K; however, the spatial resolution was only of the order of 100 lm.This means that such measurements cannot detect a further decrease to values of the order of 5000 K within the last 100 lm from the anode, which is predicted by the modeling. 16In such a situation, pinning down the instability to the near-anode region 3-5 together with the dispersion analysis, indicating that the instability can occur only at T e values below those in the bulk plasma, represents an important, although inevitably indirect, experimental confirmation of the low T e values predicted by the modeling. 16

V. RELATION TO THE THEORY OF MULTIPLE ANODE ATTACHMENTS
Multiple attachments of a high-pressure arc to the anode are observed in the current range from tens of amperes to several hundred amperes, depending on the width of the electrode gap and the kind of the plasma-producing gas; see, e.g., Refs.7-13.A theoretical analysis of the instability leading to multiple attachments was given in Refs.7 and 11.The conclusion that is the most relevant to the present work is the following: if dl e /dT e > 0, which is the case if the charged particle density is high enough and electron-ion collisions prevail over electron-atom collisions, then the Joule heating produces a destabilizing effect.
This conclusion is just opposite to the conclusion of the present work: if dl e /dT e > 0, then it follows from Eq. ( 15) that r 1 < 0 and the Joule heating produces a stabilizing effect under the conditions of the present work, as discussed in Sect.III.Of course, this difference should have been expected: while the instability that causes multiple anode attachments consists in the development of transversal (perpendicular to the discharge current) perturbations of T e and, therefore, develops at constant electric field, the instability that causes voltage oscillations consists in the development of longitudinal (parallel to the discharge current) perturbations of T e and therefore develops at constant current density.The Joule heating of the electron gas in the case of a constant electric field is directly proportional to the electron mobility, while, in the case of a constant current density, it is inversely proportional to the electron mobility.Therefore, a growing dependence of the electron-atom collision frequency on the electron temperature that causes a negative derivative dl e /dT e provides a stabilizing mechanism for perturbations that develop at constant electric field and a positive feedback for perturbations that develop at a constant current density.

VI. CONCLUSIONS
A local dispersion analysis of very high-pressure Xe arc plasmas has confirmed the conclusion of Ref. 1 on a possible instability against perturbations of the electron temperature parallel to the arc current.The instability has been traced back to a growing dependence of the electron-neutral collision frequency on the electron temperature.This dependence ensures that the Joule heating of the electron gas (which, in the case of a constant current density, is inversely proportional to the electron mobility) is a growing function of T e , thus providing a positive feedback.As a manifestation of this mechanism, the value of T e that limits the instability window in Xe from below approximately corresponds to the Ramsauer minimum of the electron-atom cross section.In principle, the instability may occur not only in Xe, but also in other gases, for example, in Hg: there is no Ramsauer minimum in Hg and v ea for fixed n a monotonically grows for all T e of interest, including low values.
The conclusion of Ref. 1 that this instability develops in the arc column has not been confirmed: T e in a very highpressure, radiation-dominated Xe plasma is above the window of existence of the instability.The instability is not possible in the near-cathode layer either, where T e is still higher.But according to the modeling, 16 T e goes down to quite low values (of the order of 5000 K) in the near-anode layer; then this is the only region where the instability is possible.This conclusion agrees with the experimental observations of the last decade, 3-5 which pinned down the instability-caused voltage oscillations in very high-pressure Xe arcs to the near-anode region.The time of development of the instability (Re k 1 ) À1 conforms to the experimental rise time of a single pulse.Note that the above agreement represents an important, although inevitably indirect, confirmation of the theoretical conclusion 16 that T e in the near-anode layer of the very high-pressure arcs is quite low.
There is a similarity between the formalisms of the theory of the instability behind multiple anode attachments in high-pressure arcs and of the present theory of the instability leading to EMI.However, the mechanisms of the instabilities are different: the Joule heating effect that is stabilizing in one case is destabilizing in the other.(More precisely, it is stabilizing in the former case and destabilizing in the latter if dl e /dT e < 0 and the other way around if dl e /dT e > 0.) The treatment of this work, being based on the local dispersion analysis, represents just the first step in the development of the theory.The next step should be to perform the linear stability analysis of differential equations describing the current transfer through the near-anode layer.One can hope that this analysis will describe, in particular, the surprising and very clean response of the instability to variations of the anode tip temperature 5 and also will explain why this instability is not observed in very high-pressure Hg lamps.A further step will be to obtain a non-stationary numerical solution of the full (nonlinear) equations, which will provide information on the oscillations.

Fig. 1 .
The lower and upper boundaries of the instability window, T ðinstÞ e and T ðstabÞ e , are shown in Table

FIG. 1 .
FIG. 1. Real part of increment of perturbations of LTE high-pressure Xe plasma.p ¼ 100 bar.

FIG. 2 .
FIG. 2. Contribution of Joule heating to the increment of instability of electron energy balance in high-pressure LTE Xe plasma.p ¼ 100 bar, j ¼ 10 7 A/m 2 .Solid: negative values of r 1 .Dashed: positive values of r 1 .

.
À2e , apart from a weak dependence on T e through the Coulomb logarithm.Hence, at high T e , where the charged particle density is high enough and the Coulomb collisions prevail, n a Q It follows from Eq. (17) that r 1 < 0, i.e., Joule heating plays a stabilizing role.This means that the instability may occur only at low T e , where the charged particle density is low enough so that n a Q ð1;1Þ ea & n i Q ð1;1Þ ei , and only if the electron-atom collision frequency v ea ¼ 4 3 C e n a Q ð1;1Þ ea increases with T e .Note that, since variations of the number density of neutral atoms are neglected in the present analysis, the latter statement refers to the dependence of v ea on T e at fixed n a or, equivalently, to the dependence on T e of the ratio v ea =n a ¼ latter dependence evaluated for the case of Xe atoms is depicted in Fig. 4. Also shown are the average collision cross section Q ð1;1Þ ea

FIG. 4 .
FIG. 4. Kinetic coefficients characterizing transport of electrons in the gas of Xe atoms.

TABLE I .
The lower and upper boundaries of the instability window, T ðinstÞ