Physics of Spotless Mode of Current Transfer to Cathodes of Metal Vapor Arcs

A fresh attempt is made to clarify the physics of the diffuse, or spotless, mode of current transfer that may occur on cathodes of vacuum arcs if the average cathode surface temperature is high enough, about 2000 K. It is shown that in the case of chromium cathode the usual mechanism of current transfer to arc cathodes cannot sustain current densities of the order of 105-106 A · m-2 observed in the experiment, the reason being that the electrical power deposited into electron gas in the nearcathode space-charge sheath is insufficient. It is hypothesized that the electrical power is supplied to the electron gas primarily in the bulk plasma, rather than in the sheath, and a high level of electron energy in the vicinity of the sheath edge is sustained by electron heat conduction from the bulk plasma. Estimates of the current of ions diffusing to the sheath edge from the quasineutral plasma gave values comparable with the experimental current density, which supports the above hypothesis. On the contrary, the spotless attachment of vacuum arcs to gadolinium cathodes may be interpreted as a manifestation of the usual mechanism of current transfer to arc cathodes. Results given for gadolinium cathodes by a model of near-cathode layers in vacuum arcs conform to available experimental information.

occurs in cases where the average temperature of the cathode surface is high enough, typically around 2000 K.The latter can be achieved by placing the (evaporating) cathode into a thermally insulated crucible made of a material for which a vacuum arc would burn at a higher voltage than for the cathode material.Characteristic features of this discharge are a relatively low current density at the cathode, of the order of 10 5 -10 6 A • m −2 , and the ability to generate steady highly ionized plasma containing no microdroplet fraction.The latter feature may be attractive for applications [6], [7].
Different spot and spotless modes on cathodes of ambientgas arcs are described by a unified theory [8], which is based on the concept of self-organization in bistable nonlinear dissipative systems and applies not only to arc cathodes but also to cathodes of dc glow discharges.Recently, the same approach was applied to simulation of spots on cathodes of vacuum arcs [9], [10].
However, understanding of the spotless mode on cathodes of vacuum arcs remains elusive: it was concluded in the pioneer work [1] that none of the known mechanisms of current transfer to the cathode surface and, in particular, of the electron emission is capable of producing the above-mentioned current densities, and this conclusion still stands (except for cathodes made of Gd, which is characterized by a very low work function).Given that the effort invested by different groups in the experiment and its theoretical analysis has been quite significant [1]- [7], [11]- [15] and that the spotless arc attachment to cathodes, being in essence a 1-D and stationary phenomenon, represents a much simpler object than cathode spots, this state of the art is rather surprising and detrimental not only to potential applications of spotless vacuum arc discharges but also to the vacuum arc physics in general.
A fresh attempt to elucidate basic mechanisms of the spotless vacuum arc attachment to cathodes is undertaken in this paper.

A. Nonemitting Cathodes
The plasma in the experiments [1]- [7] may be considered as a thermal plasma of saturated vapor of the cathode material.Ionization (Saha) equilibrium holds in the bulk of the plasma but is violated in a thin layer adjacent to the near-cathode space-charge sheath; the so-called ionization layer.If the ionization of neutral atoms occurs primarily in the quasineutral plasma, then the maximum ion current that can be extracted from the plasma (ion saturation current) is limited by ambipolar diffusion of the ions across the ionization layer.In the simplest case, where the ions in the ionization layer are predominantly singly charged and variations of the temperatures of the electrons and the heavy particles, T e and T h , are negligible in the ionization layer, the ion saturation current may be estimated analytically [16] Here, D ia is the coefficient of binary diffusion of the ions and the atoms; n i is the charged particle density at the edge of the ionization layer, which may be evaluated by means of the Saha equation in terms of T e , T h , and the plasma pressure p; C is a numerical coefficient which depends on the ratio T e /T h and the ionization degree at the edge of the ionization layer and varies between 1.49 and 1; and k i is the ionization rate constant (a function of T e ).Note that D a has the meaning of the ambipolar diffusion coefficient, d is usually called the ionization length and represents a scale of thickness of the ionization layer.
To evaluate the ion current by means of (1), one needs to specify T e , T h , and p.It is natural to assume that the heavy particle temperature T h equals the cathode surface temperature T w and to set p equal to the pressure of saturated vapor of the cathode material at the temperature T w .
The mechanism of energy exchange in the near-cathode region needs to be identified to understand how T e should be properly specified.The case considered in this section is the one where the temperature of the cathode surface is not sufficiently high to produce appreciable electron emission and the current is transported to the cathode surface by ions entering the sheath from the ionization layer.This regime is similar to the ion current regime of electrostatic (Langmuir) probes.The power deposited by the arc power supply into the near-cathode space-charge sheath is transported by the ions to the cathode.A high level of electron temperature in the ionization layer, which is necessary to sustain a sufficiently high charged-particle density and, consequently, a sufficiently high ion current to the cathode, is ensured by electron heat conduction from the bulk plasma.It is necessary to analyze the electron energy balance of the bulk plasma to self-consistently determine the electron temperature in the ionization layer and thus the ion current.

B. Emitting Cathodes
Under typical conditions of arc discharges, the temperature of the cathode surface is sufficiently high to produce an appreciable electron emission.The ion current in such conditions constitutes a few tenths (usually no more than 0.3) of the net current.Hence, a few tenths of the power deposited into the near-cathode space-charge sheath goes to the ions and is transported to the cathode surface.This power ensures heating of the cathode up to temperatures sufficient for electron emission.The rest of the deposited power goes to the electrons.This power is partly spent in the near-cathode region on ionization of neutral atoms, which should be sufficient for providing the necessary ion current, and partly is transported into the bulk plasma by the electron current.
The latter means that the flux of electron energy is directed not from the bulk plasma to the near-cathode layer, as in the case of nonemitting cathodes, but rather the other way round.Therefore, balance of electron energy in the near-cathode layer may be evaluated and the local electron temperature found independently of the bulk plasma.This is the approach employed in models of near-cathode layers in arc discharges both in ambient gases and vacuum (see [17] and references therein and [18]- [20], respectively).In addition to the ion current, such models compute also the electron emission current with account of Schottky correction or thermo-field emission mechanism and the current of plasma electrons reaching the cathode against the retarding electric field in the space-charge sheath.
One can conclude that there is a substantial difference between evaluation of current transfer to cathodes in thermal plasmas in the cases of emitting and nonemitting cathodes.A relevant model of near-cathode layer in arc discharges may be employed in the former case.In the latter case, the ion current to the cathode cannot be self-consistently evaluated from analysis of the near-cathode region only, without evaluation of the electron energy balance of the bulk plasma.This difference stems from the opposite directions of transport of electron energy: it is transported by the electron current in the direction from the near-cathode region into the bulk plasma in the case of emitting cathode, while in the case of nonemitting cathode the energy is transported by electron heat conduction in the direction from the bulk plasma into the near-cathode region.

III. CHROMIUM CATHODE
The cathode material on which many experiments have been performed is chromium [1], [5], [7].In this section, estimates of current transfer to chromium cathodes in conditions of these experiments are made on the basis of the qualitative considerations given in Section II.
Let us first assume that the electron emission from the cathode is significant.As discussed in Section II-B, one can perform a self-consistent evaluation by means of a relevant model of near-cathode layers in arc discharges in this case.In this paper, the model of near-cathode layers in vacuum arcs [20] is used.Note that a characteristic feature of nearcathode layers in vacuum arcs is the ionization of evaporated metal atoms occurring primarily in the outer section of the near-cathode space-charge sheath, rather than in the quasineutral plasma (ionization layer) as in the case of nonemitting cathodes discussed in Section II-A.This feature is reflected in the model [20], which comprises a self-consistent description of the space-charge sheath with ionization of evaporated atoms and with a maximum of potential occurring inside the sheath.Also considered is balance of the evaporated metal atoms with their eventual return to the cathode if ionized before the potential maximum, as well as balance of the electron energy.The results of calculations performed by means of the model [20] are shown in Fig. 1. (Here, j i is the density of ion current to the cathode surface, j em is the density of electron emission current, j is the net current density, j R is the thermionic emission current density evaluated by means of the Richardson formula, and T e is the temperature of electrons in the near-cathode layer.)The material constants for Cr cathodes were set as described in [20].The work function A f in these calculations was set equal to 4.58 eV; a value for polycrystalline chromium surface recommended in [21].(This value is given also in [22] and a similar value of 4.5 eV is given in [23].) One can see that the density of current of ions coming to the cathode surface from the space-charge sheath slightly exceeds the electron emission current density.The electric field at the cathode surface is not high enough for the thermofield electron emission mechanism to come into play and the electron emission is of thermionic nature.Therefore, the electron emission current density j em shown in Fig. 1 (which was evaluated by means of the Murphy and Good formalism) is very well represented by the Richardson-Dushmann formula.On the other hand, j em exceeds the Richardson value j R ; hence, the Schottky correction to the work function is appreciable (e.g., 0.13 eV for T w = 2850 K).The current of the fast plasma electrons capable of overcoming the retarding electric field of the sheath and reaching the cathode surface is negligible and not seen on the graph.
Although the density j of net electric current at the cathode surface exceeds the Richardson value j R by a factor of 3-4, this difference is insufficient to bring the estimates close to the experiment: j attains the value of 10 5 A • m −2 at T w 2700 K, which is significantly higher than the measured temperatures around 2000 K.
A natural question is if the value A f = 4.58 eV of the work function of chromium, used in calculations shown in Fig. 1, is adequate under conditions being considered.Reference [24] makes an explicit distinction between thermionic and photoelectric work functions; for chromium, the values of 3.90 eV are given for the former and of 4.4 eV for the latter.Without discussing the reliability of these data (while the thermionic and photoelectric work functions differ in semiconductors, they are usually considered to be the same in metals), we only note that the calculations similar to those shown in Fig. 1 but with A f = 3.90 eV [15] gave results that are closer to the experiment, however, the difference is still significant.
Thus, the estimates have indicated that the usual mechanism of current transfer to vacuum arc cathodes in the case of chromium cathode cannot sustain current densities of the order of 10 5 -10 6 A • m −2 observed in the spotless attachment [1], [5], [7].The reason may be understood from very low values of the electron temperature T e and the corresponding ionization degree in the near-cathode layer predicted by the model (see curve T e in Fig. 1): 3960 K and 2.3 × 10 −3 , respectively, for T w = 2700 K and still lower for smaller T w .Since the plasma is highly ionized under the experimental conditions [1], [5], [7], these values clearly are unrealistically low.One should conclude that work of the electric field over the emitted electrons inside the sheath, which is the main source of electron energy in the near-cathode layer in the model [20], is insufficient to maintain an appreciable ionization degree due to very low electron emission current.
It follows that the usual mechanism of current transfer to vacuum arc cathodes is irrelevant in the case of spotless attachment to chromium cathodes and one should consider the case of nonemitting cathode, discussed in Section II-A.A question arises if the density of the bulk plasma under the experimental conditions [1], [5], [7] is sufficiently high to ensure the density of ion current to the cathode of the order of 10 5 -10 6 A • m −2 .As discussed in Section II, the electron temperature T e in the near-cathode layer in the case of nonemitting cathode is governed by electron heat conduction from the bulk plasma (rather than by a local energy supply as assumed in computations shown in Fig. 1) and a self-consistent determination of T e would involve an evaluation of the electron energy balance of the bulk plasma.Leaving such evaluation beyond the scope of this paper, we will set T e equal to 8000 K: at this value the Cr plasma under the pressure of 1 mbar (which is the pressure of saturated Cr vapor at the cathode temperature of 2000 K) approaches full ionization.The ion current density computed by means of (1) with the use of this value is depicted by the curve marked j (d)   i in Fig. 2. (In these calculations the pressure of saturated vapor of Cr and the ionization rate constant have been estimated, as described in [20], and the ion-atom binary diffusion coefficient was estimated in terms of the resonant charge exchange cross section given in [22]).One can see that the ion current density is of the same order of magnitude as the experimental current density.
Also shown in Fig. 2 is the ionization length d, the Debye length λ D , and the mean free path for collisions between the ions and neutral atoms λ ia = [(n (S)  a  for momentum transfer in ion-atom collisions.Conditions of the validity of (1) read λ D , λ ia d L, where L is a characteristic dimension of the plasma.Given that L is of the order of 1 cm in the conditions of experiments [1]- [7], one can see that the inequalities λ D d L are satisfied but the inequality λ ia d is not; in fact, λ ia exceeds d by a factor of about 3.An approximate theory for such conditions was developed by means of the so-called multifluid approach [25].The ion current evaluated by means of [25, eq. ( 50)] with the coefficient C 1 replaced by C 2 given by [16, eq. ( 14)] is depicted by the curve marked j i , it is still not far away from the experiment: j Thus, estimates of the ion saturation current give values consistent with the assumption that the current in the spotless attachment of vacuum arc to chromium cathode is transported by ions diffusing to the sheath edge from the quasi-neutral plasma.

IV. GADOLINIUM CATHODE
A significant amount of experimental data on spotless vacuum arc attachment has been obtained for cathodes made of gadolinium [2]- [4], [6], [7].The value of work function recommended in [21] for a polycrystalline surface of gadolinium is 3.1 eV, i.e., quite low, and a still lower value of 2.9 eV is given in [23].Accordingly, it is well known that thermionic emission current is high enough to ensure the spotless attachment of vacuum arcs to a gadolinium cathode observed in the experiments [2]- [4], [6], [7].However, it is of interest to apply to gadolinium cathodes the same approach that was applied above to cathodes made of chromium, to once again validate the approach and demonstrate the possibility of self-consistent calculation of some parameters of the spotless vacuum arc on a gadolinium cathode.
The results of calculations for gadolinium cathodes are shown in Figs. 3 and 4. (In these calculations, work function  of Gd was set equal to 3.1 eV, the pressure of saturated vapor of Gd has been estimated according to [23], the ionization rate constant was evaluated as described in [25] with the use of the cross section of direct ionization of Gd atoms from [26], and the ion-atom diffusion coefficient and the average cross section Q(1,1) ia were evaluated in terms of the resonant charge exchange cross section given in [22]).One can see that the current density computed by means of the model of near-cathode layers in vacuum arcs [20] (curve j in Fig. 3) exceeds 10 5 A • m −2 for T w 2000 K, i.e., is in the right order of magnitude.On the other hand, the ion saturation current from the Gd-vapor plasma ( j ( f ) i in Fig. 4) does not exceed approximately 10 4 A • m −2 .It follows that the spotless attachment of vacuum arcs to gadolinium cathodes is a manifestation of the usual mechanism of current transfer to vacuum arc cathodes and cannot be explained by current of ions diffusing from the quasi-neutral plasma, in contrast to the case of chromium cathodes.Let us consider in some detail results given by the model described in [20] and shown in Fig. 3.The electron temperature in the near-cathode layer is around 3.1 × 10 4 K, i.e., rather high, and is still higher for higher values of the sheath voltage; e.g., T e ≈ 6.3 × 10 4 K for U = 20 V.This is a consequence of the significant electrical power supplied to the electron gas in the sheath, which, in turn, originates in the significant electron-emission current.
The net current density j is close to j em and slightly bigger than j R .Current transported by fast plasma electrons attaining the cathode (curve j e ) is about 12% of the net current density, while the ion current j i is about 4%.It should be stressed that the smallness of j i is not a consequence of the return of most part of evaporated atoms to the cathode in the form of ions; e.g., for T w = 2000 K about 43% of the ions are produced after the potential maximum in the sheath and escape into the plasma.Rather, the smallness of the contribution of the ion current originates in the well known [2]- [4], [6], [7] fact that the flux of emitted electrons from a gadolinium surface is significantly higher than the flux of evaporated atoms, which results from the low work function of gadolinium and relatively high evaporation energy.(The latter equals 3.73 eV per atom [22]; for comparison, values for Cr and Cu are 3.55 and 3.16 eV, respectively [24].) Values of the ratio j i /j em that low are unusual for arc cathodes and mean that the cooling of the cathode by electron emission exceeds heating by the ion current.Therefore, a possible source of heating of gadolinium cathodes in the experiments [2]- [4], [6], [7] is radiation from the plasma.
Another consequence of low values of the ratio j i /j em may be the necessity to take into account contribution of the emitted electrons to space charge in the near-cathode sheath [27], which is neglected in the model [20].Note, however, that the above-mentioned values of this ratio are still by an order of magnitude higher than the minimal value corresponding to the formation of virtual cathode, which equals the square root of the electron-to-ion mass ratio (e.g., [27]) and for gadolinium is approximately 0.2%.
The above results conform to experimental information and estimates available in the literature.In particular, the electron temperature measured in Gd is around 3-4 eV for the arc voltage V a = 10 V and increases to 7-8 eV for V a = 20 V [6, Fig. 9].The spectral line of doubly charged ions has been observed in Gd and its intensity was found to decrease with increasing V a ; a possible indication to the presence of triply charged ions (which are abundant at T e that high) [7].It should be stressed that only lines of atoms and singly charged ions have been observed in Cr plasma [7].The ion current to a gadolinium cathode does not exceed 5% of the electronemission current [4].

V. CONCLUSION
Qualitative reasoning and calculations have shown that the usual mechanism of current transfer to arc cathodes, characterized by a large electric power deposited by the arc power supply into the electron gas in the near-cathode space-charge sheath, in the case of chromium cathode cannot sustain current densities of the order of 10 5 -10 6 A • m −2 observed in the spotless attachment.The root reason is that the power deposited into the electron gas in the sheath is insufficient in this case, which originates in very low electronemission current.One should hypothesize therefore that the electrical power is supplied to the electron gas primarily in the bulk plasma, rather than in the sheath, and the current is transported to chromium cathode by ions diffusing to the sheath edge from the quasi-neutral plasma across the ionization layer; a regime similar to the ion current regime of electrostatic probes in thermal plasmas.Estimates of the ion current gave values comparable to the experimental current density, which supports the above hypothesis.
In contrast to the case of chromium cathodes, the spotless attachment of vacuum arcs to gadolinium cathodes should be viewed as a manifestation of the usual mechanism of current transfer to arc cathodes and cannot be explained by current of ions diffusing to the sheath edge from the bulk plasma.The results given for gadolinium cathodes by a model of near-cathode layers in vacuum arcs conform to experimental information and estimates available in the literature.The computed ratio of the ion current to the electron-emission current is about 0.04, which is significantly smaller than values of this ratio characteristic of current transfer to arc cathodes.Therefore, cooling of the cathode by electron emission prevails over heating by ion current and a possible mechanism of heating of gadolinium cathodes in the experiments with spotless arc attachment is radiation from the plasma.
Authorized licensed use limited to: b-on: Universidade da Madeira.Downloaded on September 02,2022 at 13:30:28 UTC from IEEE Xplore.Restrictions apply.

Fig. 1 .
Fig. 1.Calculations with account of conventional mechanisms of current transfer to cathodes of vacuum arcs.Chromium cathode, sheath voltage U = 20 V.

a
is the number density of the atoms at the edge of the ionization layer andQ(1,1) ia is the average cross section Authorized licensed use limited to: b-on: Universidade da Madeira.Downloaded on September 02,2022 at 13:30:28 UTC from IEEE Xplore.Restrictions apply.
Authorized licensed use limited to: b-on: Universidade da Madeira.Downloaded on September 02,2022 at 13:30:28 UTC from IEEE Xplore.Restrictions apply.