Electric field non-uniformity effect on dc low pressure gas breakdown between flat electrodes

Accepted Manuscript Electric field non-uniformity effect on dc low pressure gas breakdown between flat electrodes V.A. Lisovskiy, R.O. Osmayev, A.V. Gapon, S.V. Dudin, I.S. Lesnik, V.D. Yegorenkov PII:

S0042-207X(17)30186-0

DOI:

10.1016/j.vacuum.2017.08.022

Reference:

VAC 7548

To appear in:

Vacuum

Received Date: 11 February 2017 Revised Date:

16 August 2017

Accepted Date: 16 August 2017

Please cite this article as: Lisovskiy VA, Osmayev RO, Gapon AV, Dudin SV, Lesnik IS, Yegorenkov VD, Electric field non-uniformity effect on dc low pressure gas breakdown between flat electrodes, Vacuum (2017), doi: 10.1016/j.vacuum.2017.08.022. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Electric field non-uniformity effect on dc low pressure gas breakdown between flat electrodes V.A. Lisovskiy 1),2), R.O. Osmayev1), A.V. Gapon1), S.V. Dudin1), I.S. Lesnik1), V.D. Yegorenkov1) 1) Kharkov National University, 61022, Kharkov, Svobody Sq. 4, Ukraine 2) Scientific Center of Physical Technologies, Kharkov, 61022, Svobody Sq., 6, Ukraine

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This paper presents the results of studying the gas breakdown in a non-uniform direct current electric field. The breakdown curves have been measured in nitrogen between flat electrodes of 6 mm in radius spaced 3 to 300 mm apart and placed inside the discharge tubes of 6.5 mm and 28 mm in radius. The effects of the non-uniform distribution of the electric field inside the inter-electrode gap and of the diffusion loss of charged particles to the discharge tube walls on the gas breakdown have been studied separately. A conclusion is drawn from the experimental data that the general form of the gas breakdown criterion must be as follows U = f ( pL, L Rel , L Rtube ) in which the L/Rel ratio of the inter-electrode gap value to the electrode radius describes the electric field nonuniformity inside the discharge tube whereas the L/Rtube ratio characterizes the diffusion loss of electrons on the discharge tube walls. It has been found that the breakdown curves for different electrode radius values and a fixed gap L value intersect at such value of the gas pressure that corresponds to the location of the inflection point of the breakdown curve for a uniform electric field. PACS: 52.80.Hc

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Introduction

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Direct current glow discharge is widely applied in a multitude of technological processes and devices, e.g. for pumping gas discharge lasers [1, 2], for plasma nitriding [3–5], in xenon and mercury high pressure lamps [6, 7], in surge protectors / transient voltage surge suppressors [8, 9], for plasma sterilizing medical tools and equipment [10, 11], in dc diode sputtering systems [12, 13] etc. A breakdown curve is one of the important discharge characteristics because it determines the conditions for plasma production in a given research or technological chamber. Therefore for many years a great attention has been devoted to gas breakdown in discharge chambers of different design. Actually all monographs devoted to low temperature plasma outline the Paschen's law describing the ignition of the direct current discharge. This law has been established by Paschen in the course of preparing his thesis (under August Kundt as a supervisor) devoted to studying the gas breakdown in the dc electric field [14]. In his experiments Paschen has employed the spherical electrodes with variable spacing L and measured the breakdown voltage U at different gas pressure values as well as at different spacing values. He has demonstrated that the breakdown voltage U depends not on pressure р and spacing L separately but it is a function of the pL product. In order to preserve historical justice one should remark that the U(pL) dependence was first observed by De La Rue and Müller [15] 9 years before Paschen. Actually Paschen's law means the following. Let us measure the breakdown curves U(p) for any two different distance L1 and L2 values. These curves possess a U-shape with the minimum pressure values of pmin1 and pmin2, respectively, for which the minimal breakdown voltage values Umin1 and Umin2 have to be close to each other. When we now plot these breakdown curves to the U(pL) scale, they would match each other. Under such plotting not only minima values Umin1 and

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Umin2 coincide. When the products p1L1 and p2L2 are identical, then the corresponding breakdown voltage values U1 and U2 also have to be identical. However, soon after its establishment it has appeared that Paschen's law is valid only in a very limited range of inter-electrode distance values. Already in papers by Townsend and McCallum [16], McCallum and Klatzow [17] the deviations from Paschen's law have been discovered. The authors of papers [16, 17] have measured the breakdown voltage values at different gas pressure and inter-electrode spacing values keeping the pL product value unchanged and demonstrated that in order to ignite the discharge in a short gap one requires remarkably smaller voltage values than in a long one. Similar deviations from Paschen's law have also been observed by the authors of papers [18–26]. As a result, it has been established that Paschen's law can only be applied to the description of the gas breakdown in narrow discharge gaps when the inter-electrode discharge spacing L does not exceed the tube radius Rtube i.e. when the condition L/Rtube < 1 holds. In the opposite case when the condition L/Rtube > 1 holds the breakdown curves U(p) are shifted with the growth of the spacing L to higher breakdown voltage values and lower gas pressure ones [20–23]. Again, Paschen's law also has to hold for breakdown curves measured in a geometrically similar discharge tubes for which the following condition L1/Rtube1 = L2/Rtube2 is met. At the same time, when the condition L/Rtube > 1 holds, then the breakdown voltage value at the minima of these breakdown curves would be higher than ones for narrow tubes when the condition L/Rtube < 1 holds. The most detailed studies of the glow discharge ignition have been presented in the recent paper [25], which reports the breakdown curve measurements in the broad range of interelectrode gap and discharge tube radius values. It demonstrates a new phenomenon not observed earlier, that in long tubes (at L/Rtube > 10 ÷ 20) the breakdown curves are shifted with L growing to the region of higher breakdown voltage values, the minima of the breakdown curves being observed at the actually unchanged gas pressure. The gas breakdown in long tubes may be affected by the escape of charged particles to the tube walls due to diffusion as well as to the non-uniform distribution of the electric field strength within the inter-electrode gap. While many papers are devoted to diffusion loss of charged particles to tube walls (see e.g. [20–23, 25–27], the effect of the non-uniform axial electric field on the discharge breakdown still remains not to be investigated enough. Quite a number of other papers are devoted to the investigation of the gas breakdown in the dc electric field (see, e.g. [28-42]). The present paper reports the study into the gas breakdown in the non-uniform dc electric field. We have investigated how the charge particle escape to the tube walls due to diffusion as well as how the electric field non-uniformity affects the discharge ignition between the electrodes. We have demonstrated that the general form of the gas breakdown criterion has to be written as U = f ( pL, L Rel , L Rtube ) where the ratio of the inter-electrode gap width to the electrode radius L/Rel describes the degree of the electric field non-uniformity in the discharge tube and the L/Rtube ratio characterizes the escape of electrons to the discharge tube walls due to diffusion. Experimental details

Experiments have been performed in two discharge chambers with the design shown in figure 1. We have employed flat stainless steel electrodes of 6 mm in radius with the interelectrode distance L varying from 3 to 300 mm. The cathode was fixed and the anode was movable along the tube axis. One breakdown curve was measured for the electrode radius of 27.5 mm and it is presented in Fig. 8 (see also the corresponding text). In all other cases it is assumed that the electrode radius is 6 mm. In one case the discharge has been ignited in the tube of 6.5 mm inner radius located inside the tube of 28 mm inner radius. At the cathode end the narrow tube was vacuum sealed to

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prevent the undesirable breakdown along the long path inside the large tube. The anode end of the narrow tube was open to level the pressure inside it with one in the large tube. In another case the inner tube was removed, and the gas breakdown was produced in the large tube in the space limited by the inter-electrode gap L. Here the anode was located inside the glass disc of 27.5 mm inner radius to prevent the breakdown behind the anode. The surfaces of the anode and the disc looking to the cathode were located in one plane. Experiments have been performed in the nitrogen pressure range of p = 0.05 − 100 Torr with the breakdown voltage values in the range of Udc ≤ 4000 V. The gas pressure was measured with the capacitive manometers (MKS Instruments) with the maximum registered values of 1000 Torr and 10 Torr. We have employed a conventional technique of breakdown curve measurement. For a fixed value of the gap L a certain gas pressure value is fixed, and then the voltage across the electrodes is slowly increased in value until a gas breakdown occurs. The increase rate has not exceeded 1 V per 5 s. The appearance of the glow within the discharge gap and of the active current in the circuit together with the lowering of the voltage across the electrodes have indicated the presence of the gas breakdown. At each gas pressure value the breakdown voltage has been measured three times with a subsequent averaging. Then the measurements have been performed at other values of the gas pressure. Note that we have studied the ignition of a self-sustained discharge without the application of any sources of ultra-violet or ionizing radiation. Experimental breakdown curves

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We have remarked above that the discharge ignition in a long tube may be affected not only by the escape of charged particles to the tube walls due to diffusion but also by the nonuniform distribution of the electric field within the gap between the small radius electrodes. If one neglects the influence of the surface charges deposited on the tube walls on the gas breakdown then one and the same profile of the electric filed between identical electrodes will be not only in a large tube but also in a narrow one. Therefore near the cathode and the anode the electrons will gain the largest energy and the ionization rate will be at maximum, and at the gap center both the electron energy and the ionization rate will be at minimum. Correspondingly, in a narrow tube as well as in a wide one the non-uniformity of the electric field will have the same influence on the gas breakdown development. However, the escape of the charged particles to the walls of a narrow tube due to diffusion will be more intense than the escape in a wide one. As the rate of the escape due to diffusion is inversely proportional to the tube radius squared [38], then the 4.3 times increase of this radius will involve the 18.6 times decrease of this escape. Consequently, the comparison between the breakdown curves in narrow and wide tubes would permit to evaluate the role of the electric field non-uniformity within the gap as well as the electron loss to the tube walls due to diffusion. Now consider the breakdown curves which have been measured in a narrow and a wide tube with different inter-electrode distance values and depicted in figure 2. One observes in the figure that in the narrow tube of 6.5 mm in radius increasing the distance L between the electrodes from 3 to 6 mm has shifted the breakdown curves to the range of lower gas pressure values with a slight change of voltage at their minima. However, a further increase of the distance L causes the shift of the breakdown curves to the lower gas pressure range as well as to the range of higher breakdown voltage values with the voltage at their minima growing fast. Such a behavior of the breakdown curves obeys the modified Paschen's law U(pL, L/Rtube) (where Rtube is the discharge tube radius) and it has been described earlier, e.g. in papers [20–23, 25]. In a wide tube (Rtube = 28 mm) increasing the inter-electrode distance shifts the breakdown curves to the range of lower pressure values down to the gap value of 48 mm, and for large L values the pressure at the minima of the breakdown curves remains actually unchanged. At the

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same time the minimum voltage of the breakdown curves grows weakly with the gap L increasing. The breakdown curves for narrow gaps in both discharge tubes plotted to the U(pL) scale (see figure 3) match each other indicating the validity of Paschen's law. For a narrow tube increasing the L gap shifts the breakdown curves to the range of higher voltage and pL product values, the right-hand side branches of the breakdown curves coinciding at large gas pressure values. A fast shift to higher breakdown voltage values with the L gap increase is associated with an enhanced escape of electrons to the tube walls due to diffusion. In a wide tube the breakdown curve minimum for the gap of 18 mm shifts to the range of higher breakdown voltage values whereas the left-hand and right-hand branches remain close to the corresponding branches for narrower gaps. At L = 30 mm one observes the displacement of the right-hand branch of the breakdown curve to the range of higher pL values. A similar behavior of the breakdown curve prevails for the gap values up to 48 mm (the right-hand branch is displaced still stronger to the range of higher pL values whereas the left-hand branch and the minimum experience practically no changes with the L values growing). With subsequent increase of the inter-electrode distance the breakdown curve shifts as a whole to the right along the pL axis, the voltage at the minimum experiencing weak changes. The strongest displacement of the right-hand branch to the range of higher pL values is observed for the 72 mm gap. For longer gaps the breakdown curve minima experience fast displacements to higher breakdown voltage values as well as to the range of higher pL product values. Here the behavior of the breakdown curves for long gaps and small electrodes becomes similar to one for a narrow tube but the right-hand branches of the breakdown curves do not match each other but shift to the range of higher voltage values. Coordinates of the breakdown curves minima

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Now consider the behavior of the coordinates of the breakdown curves minima for a narrow and a wide tube for different inter-electrode distance values. Figures 4 and 5 depict, respectively, the products of pressure and inter-electrode distance (pL)min and the voltage Umin at the breakdown curves minima versus the ratio of the inter-electrode distance L to the electrode radius Rel and to the tube radius Rtube. For a narrow tube the (pL)min product remains constant up to the value L/Rel = 3, and then for higher L/Rel ratio values the electron loss to the tube walls grows and the (pL)min product experiences a linear increase. As for the narrow tube its radius (6.5 mm) and the electrode radius (6 mm) are close to each other, then for it the graphs of the (pL)min products versus L/Rel and L/Rtube differ weakly in figure 4. For the same electrodes placed in a wide tube the (pL)min parameter at the breakdown curve minima experiences weak changes in the range L/Rel ≤ 12, and only at higher L/Rel ratio values it grows fast. However, the dependences of the (pL)min parameter on the L/Rtube ratio for the narrow and the wide tubes in figure 4 have happened to be close to each other. From this fact one may draw a conclusion that the (pL)min parameter weakly depends on the circumstance whether the gas breakdown occurs in the uniform or the non-uniform electric field. At the same time the fast growth of the (pL)min parameter for the L/Rtube ratio value exceeding 3 for both tubes indicates that the presence of the enhanced escape of charged particles to the tube walls due to diffusion affects substantially the gas pressure value at which one observes the breakdown curve minimum. In contrast to the product of the pressure and the inter-electrode distance (pL)min, the voltage at the breakdown curves minimum Umin is not conserved for moderate L values but it increases slowly with the parameter L/Rel growing. At the same time the Umin values are actually the same for the narrow as well as the wide tubes (see figure 5). The voltage Umin increase for moderate gap values with the distance L growing for the narrow tube is probably associated with a moderate but still increasing diffusion escape of electrons to the tube walls. In the case of the wide tube a portion of electrons escapes due to diffusion spread of the electron avalanche but not

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to the tube walls but rather to the surface of the dielectric, i.e. a glass plate inside which the anode is installed. However, under the condition L/Rel ≥ 3 the behavior of the curves for Umin experiences substantial changes: for the narrow tube the quantity Umin becomes directly proportional to the L/Rel ratio (it is readily observed when plotting figure 5 to a linear and not to a semi-logarithmic scale). For a wide tube the quantity Umin first becomes almost constant and at L/Rel ≥ 12 it also assumes a linear growth with the L/Rel parameter increasing. The part with the constant Umin corresponds to the breakdown curves within the distance L ranging from 18 to 72 mm for the electrodes in the wide tube. Probably within this range of inter-electrode distance values the diffusion escape to the tube walls is not large but the non-uniform distribution of the electric field already affects the discharge ignition. If one plots the voltage at the breakdown curve minimum Umin against the L/Rtube ratio, the curve for the narrow tube runs somewhat lower than one for the wide tube. Besides, in the curve for the wide tube one clearly observes the portion with the actually constant voltage at the breakdown curve minimum within the inter-electrode distance values where the non-uniformity of the electric field distribution makes the discharge ignition easier. However, for the ratios satisfying the condition L/Rtube ≥ 3 one requires identical minimum voltage values Umin for the gas breakdown in the narrow as well as in the wide tubes. Let us now compare the breakdown curves for the narrow and the wide tubes at different inter-electrode distance values. The breakdown curves presented in Fig.6 for the distance of L = 3 mm match each other. This is an indication that the discharge ignition throughout the total gas pressure range we studied takes place only within the narrow gap between the electrodes, and the diffusion escape of electrons to the walls of the narrow as well as the wide tube does not play a remarkable role. However for a sufficiently large gap of L = 72 mm (see Fig.7) the breakdown curve for the wide tube runs remarkably lower than one for the narrow tube (370 V and 583 V, respectively).

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In Fig.8 we compare between the breakdown curves for the wide tube, namely, the above described curve for the electrodes of 6 mm in radius and the breakdown curve for the electrode radius of 27.5 mm (solid triangles). One observes in the figure that the minimum of the breakdown curve for the electrode radius of 27.5 mm is located at the lower voltage value (329 V) as well as at the lower gas pressure (0.09 Torr). For the electrodes of 6 mm in radius the minima are observed at higher gas pressure values, namely, at 0.17 Torr for the narrow tube and 0.23 Torr for the wide tube (see Fig.7). Consequently, the employment of the narrow tube (solid circles in Fig.7) with a high diffusion loss of electrons on its walls has involved approximately 1.8 times increase in voltage and 1.9 times increase in pressure at the minimum compared with the breakdown curve for large electrodes in the wide tube (solid triangles in Fig.8). There attracts attention practically the same increase in voltage and pressure values at the minimum. It suggests that for a fixed gap between the electrodes L in the tubes with the cross section completely or almost completely occupied with the electrodes, the minima of the breakdown curves are observed at the same average reduced electric field (E/p)min = Umin/(pminL) (so called Stoletow's constant [43–45]), despite the fact that the local electric field may be non-uniformly distributed between the electrodes. Besides, at large gas pressure values the discharge breakdown occurs at the one and the same voltage value as the right-hand side branches of the breakdown curves match (solid circles and triangles in Fig.7). The same effect has been observed for the breakdown curves measured with different inter-electrode L distance values [20–23, 25]. Measuring the breakdown curves with moderate size electrodes (of 6 mm in radius), placed not only in the narrow but also in the wide tube, has enabled us to reveal the role of the nonuniform distribution of the electric field in the process of the gas breakdown. Increasing the tube radius has led to a considerable decrease of the diffusion loss of electrons on its walls. The minimum of the resulting breakdown curve for the wide tube is shifted to the range of higher gas pressure and lower voltage values. Besides this minimum is not expressed so abruptly as it is for

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the narrow tube or the electrodes of 27.5 mm in radius in the wide tube. To the left of the curve minimum for the wide tube the breakdown voltage increases remarkably slower with the gas pressure decreasing than it is for other breakdown curves presented in Figs. 7 and 8. Perhaps it is related to the phenomenon that in the wide tube the breakdown between moderate size electrodes occurs at low gas pressure along longer lines of force of the electric field (near the edges of the electrodes). Analytical modeling

 1 αL = ln1 +  ,  γ

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Conventionally, one employs Townsend's criterion [46] for describing the dc gas breakdown (1)

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where α is the first Townsend coefficient for ionization, and for molecular gases it can be written in the form [43, 46]  B⋅ p  , α( z ) = A ⋅ p ⋅ exp − (2)  E (z )  where γ is the ion-induced electron emission coefficient, A and B are the constants depending on the gas species, p is the gas pressure. This criterion is valid only for the gas breakdown in a narrow gap L between large radius electrodes Rel >> L, i.e. for the uniform electric field E(z) = const. Besides, it is also assumed that a single mechanism of the loss of charged particles is their drift motion in the electric field whereas their diffusion escape to the tube walls does not play a remarkable role. It is difficult to get an analytical expression describing the gas breakdown and taking into account the non-uniform electric field as well as the diffusion escape to the tube walls. Conventionally, one writes the differential equations of the balance between electrons and ions with boundary conditions on the cathode and the anode as well as on the discharge tube walls [43, 47]. Integrating these equations one may get the analytical expressions for the distributions of electrons and ions along the axis as well as along the radius of the tube. Then these expressions are substituted into the boundary condition on the cathode leading to the gas breakdown criterion. However, the term for the rate of production of charged particles due to ionization contains the ionization rate νi = α⋅Vdr in which the electron drift velocity Vdr = µe⋅E(z) (here µe is the electron mobility) and the first Townsend coefficient α depend on the axial distribution of the electric field strength E(z). Exactly this circumstance impedes the derivation of the criterion for the gas breakdown in the general case. For the smallest distance of L = 3 mm the electrode radius exceeds the gap L only twice, therefore practically in the total range of distance values L = 3 – 300 mm we studied the electric field strength has to be non-uniformly distributed between the electrodes. In this case the gas breakdown criterion must be written in the form [43, 48–52] L  1 (3) ∫ α[E ( z )]dz = ln1 +  .  γ 0 Let us obtain the formula for the axial distribution of the electric field strength E(z) between two electrodes having the form of the discs. The book [53] contains the formula for the potential produced by a single charged disc (with the total charge q) with the radius R at the point located at the distance z from the disc surface and at the radius r from its axis

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q ⋅ tan −1 R

(

2 ⋅ R2

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12

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(4)

  r 2 + z 2 − R2 +  r 2 + z 2 − R + 4 ⋅ R2 ⋅ z 2    For the axial distribution of the potential (at r = 0) we have q R (5) ϕ( z ) = ⋅ tan −1   . R z The rigorous solution of finding the electric field strength from two identical oppositely charged discs has been obtained in two papers by Love [54, 55]. But for our purpose it is much simpler to use formula (5) and to proceed as follows. Then for two oppositely charged discs with the difference of potentials U and the distance L between them it is easy to get a convenient approximate formula for the axial distribution of the electric field strength  U ⋅ R  1 1 . E (z ) = ⋅ + (6) π  z 2 + R 2 (L − z )2 + R 2  Of course, this formula may be applied when the distance between the discs exceeds the electrode radius. Axial distributions E(z) calculated with the help of formula (6) for different electrode radius values are shown in Fig.9. For small electrodes the electric field near their surface is the largest and at the central part of the gap it is the smallest. The less is the electrode radius and the larger is the inter-electrode distance, the stronger is the non-uniformity of the electric field distribution. Naturally such a non-uniform distribution E(z) affects the discharge ignition considerably. Substituting expression (6) into formula (3) produces the breakdown criterion for the nonuniform electric field in the form:       L  1 B⋅ p⋅π dz = ln1 +  . A ⋅ p ⋅ ∫ exp − (7) γ     1 1 0  +  U ⋅ R ⋅  2 2 2 2   (L − z ) + R   z +R  Now let us compare in Fig.8 the theoretical breakdown curves calculated with equation (1) for the uniform electric field and equation (7) for the non-uniform field. Ion-induced electron emission coefficient γ = 0.009 has been determined from the coordinates of the breakdown curve minimum for the narrow gap of 3 mm employing the values of the constants A = 21 cm−1 Torr−1 and B = 469 V/(cm Torr) found in paper [45] in the range of high reduced field values E/p = 200 − 1000 V/(cm·Torr). It has been assumed for simplicity that the coefficient γ is the same throughout the total range of gas pressure values and it does not depend on the value of the reduced electric field E/p. Calculations have been performed for the inter-electrode distance of L = 72 mm and the electrode radius of Rel = 6 mm. One may observe in figure 8 that the curve calculated with formula (7) agrees well with the experimental breakdown curve near and to the right of its minimum. The divergence of the lefthand branches of the theoretical and experimental curves is related, first, to the assumption on the constant ion-induced electron emission coefficient γ (at larger reduced electric fields E/p the coefficient γ grows). Second, at sufficiently large distance L between moderate radius electrodes the gas breakdown may develop not only along the electrode axis but along longer lines of force near their edges. This circumstance increases the number of ionizing collisions which an electron may produce during its motion from the cathode to the anode and the effect plays a large role at low gas pressure values. The breakdown curves for the uniform electric field in wide tubes with the electrodes of large radius (when one can neglect the diffusion loss of electrons on the tube walls) obey

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Paschen's law U = f(pL) [14, 15, 25, 43]. If we know a breakdown curve for any distance, e.g. L1 = 3 mm, which minimum is observed at Umin1 and pmin1, then one may obtain the breakdown curve for a larger distance, in our case for the distance of L2 = 72 mm, with the minimum at Umin2 = Umin1 and pmin2 = pmin1⋅L1/L2. That is, this breakdown curve is shifted L1/L2 times to the left along the pressure scale compared with the curve for the narrow distance L1. Exactly in this way we have got the breakdown curve with the subscript ‘Short gap’ in figure 8. One observes in the figure that the breakdown curve calculated from equation (1) for the uniform electric field is in good agreement with the experimental curve for the short gap L = 3 mm shifted to the lower pressure range. On treating the experimental breakdown curves of the discharge, one may draw the conclusion that in the general case the gas breakdown criterion must have the following form: U = f ( pL, L Rel , L Rtube ) , (8) in which the L/Rel ratio describes the non-uniformity degree of the electric field between the electrodes and the L/Rtube ratio expresses the loss of electrons to the discharge tube walls due to diffusion escape.

Breakdown curves intersection at different electrode radius values

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Note that in Fig. 8 the breakdown curves for the ‘short gap’ and the electrodes of 6 mm in radius intersect at the nitrogen pressure of p = 0.23 Torr. Besides, the ratio of this pressure value to its value of p = 0.086 Torr at the curve minimum for the ‘short gap’ is equal to 0.23/0.086 = 2.67 ≈ e (the base of natural logarithms). The intersection point of the breakdown curves corresponds to the location of the inflection point on the curve for the ‘short gap’ indicated in papers [43, 56, 57]. The phenomenon of the breakdown curves intersection measured for a fixed distance L but for different electrode radius values has been described earlier in papers [58–60]. To the right of the inflection point the breakdown curve runs lower than one for the ‘short gap’. At lower pressure than one corresponding to the inflection point the gas breakdown occurs at a higher voltage value than one corresponding to the ‘short gap’. This observation may be explained as follows. Let us consider an electron avalanche propagating from the cathode to the anode. For the gas to be broken down, according to criteria (1) and (3), a certain number of ion-electron pairs have to be born. This number is determined only by the secondary ion-electron emission coefficient γ and it does not depend on the circumstance whether this breakdown occurs in a uniform or a strongly non-uniform field. The integral in formula (3) is exactly equal to the value of the product α·L which corresponds to the breakdown voltage for the given gap in the uniform field between large flat electrodes. With the increase of the reduced electric field E/p the first Townsend coefficient α (2) grows with the increasing rate and at high values of the E/p parameter the α growth rate decreases. The α(Е) inflection point is located at the electric field strength value Е = Вр/2 [43, 56, 57, 61]. To the right of the inflection point (at higher gas pressure values) the non-distorted breakdown field is Е < Вр/2. Besides, the redistribution of the potential between the electrodes, when the voltage between the electrodes

L

U = ∫ Edx 0

is identical

for the uniform and non-uniform distributions of the electric field, makes the conditions for the gas breakdown easier because the increased field adds up to the integral (3) more than the decreased one subtracts from it. To the left of the inflection point (Е > Вр/2, lower pressure values) due to the electric field redistribution the process of ionization multiplication within the discharge gap is impeded and the breakdown voltage increases what one can observe in Fig.8. Let us demonstrate the effect of the potential redistribution between the electrodes on the ionization process. Compare the multiplication coefficient Mu = exp(αL) values for the uniform

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dc electric field, in which the first Townsend coefficient α is determined by formula (2) with E = U/L = const, and for the non-uniform field L  L  B⋅ p  dz  (9) M = exp  ∫ α( z )dz  = exp  ∫ A ⋅ p ⋅ exp −  E (z )   0  0 with the axial distribution of the electric field strength (7). It is clear from Fig.10 that those two dependencies intersect near the inflection point Е = Вр/2. At low strength values of the reduced electric field Е/p = В/2 (large gas pressure case, right-hand branches of breakdown curves) the coefficient of ionization multiplication for the non-uniform field may considerably exceed one for the uniform field. In the opposite case of low pressure values (strong electric field values, left-hand branches of breakdown curves) the nonuniform distribution of the electric field deteriorates the ionization process, for it the coefficient of ionization multiplication would be much less than one for the uniform electric field. Therefore, the breakdown curves measured for large electrodes (which radius exceeds the inter-electrode distance considerably, the electric field is distributed uniformly) and small electrodes with the non-uniform distribution of the electric field must intersect at the gas pressure values close to the inflection point of the breakdown curve for a uniform field. Exactly this we observe in Fig.8.

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Calculations with OOPIC Pro code

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Taking into account that the dimension of the electrodes is less than both the tube diameter and its length, one can expect the presence of significant radial fields in the vicinity of the electrodes, so 1-D consideration is insufficient. In order to reveal the physical mechanism of the gas breakdown in non-homogeneous electric field we have carried out a series of numerical simulations using two dimensional particle-in-cell (PIC) model implemented by Tech-X Copr. in OOPIC Pro code [62]. The model includes Monte Carlo collision algorithms for modelling collisions of particles with background gases that might result in the ionization of the gases and the production of a pre-defined species of particles. The used code considers elastic, excitation and ionization types of the electron collisions with neutral gas and elastic and charge exchange types of the ion collisions with neutral gas. The simulations have been performed for nitrogen with the crosssection set [by Phelps] taken from LXcat resource [https://fr.lxcat.net/]. Each macro particle has included one physical particle excluding the simulations of electric charge accumulation on the dielectric surfaces (see below) where the macro particle has contained 2·103 electrons. In a typical simulation ten thousand particles were used. The uniform gas temperature was set at 0.025 eV. The simulation grid and time steps were chosen reasonably small to ensure authentic physical results with minimum computing time. The simulation time step should be significantly less than the period between electron collisions, and less than the grid cell crossing time by the fastest particles. Thus the time step for the simulation was set in inverse proportion to the gas pressure in the range 10–10 – 10–12 s. The grid step was 0.5 – 3 mm depending on the tube length. We have used electrostatic simulation field solver, so the electromagnetic fields generated by particle currents are ignored. As an initial condition of the breakdown, we have accounted for natural background ionization using random generation of ions appearing uniformly over the entire volume of the chamber. To determine the breakdown voltage, we have used the condition of balance between production and loss of ions. For each value of the gas pressure we have searched for the voltage providing unchanged mean number of ions over the period of few ion transition times through the whole system. Obviously, the particle balance in the discharge chamber is affected both by the appearance of electrons in the chamber volume and the electron emission from the walls. For nitrogen ions the ion induced secondary electron emission coefficient was assumed to be constant. The

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secondary electrons are generated in the model according to Maxwellian distribution with 2 eV temperature. The important question is the charged particle loss at the dielectric surface, especially for long tubes. For exact calculation one should account for secondary electron emission as well as for the reflection and scattering of primary electrons with complex energy dependence. However, in the described simulations we have aimed to clarify the field non-uniformity impact on the gas breakdown rather than to obtain an exact solution. Thus we have used an effective electron reflection coefficient Reff representing integral effect of secondary emission, reflection and scattering. It is supposed that all the ions are absorbed at dielectric surface. Systematic simulations for different tube lengths have been performed using the described model. The calculated breakdown curves for the short tubes (L/Rtube < 1) are in good agreement with the experimental data and with the results of analytical calculations. However, the main reason for using the 2-D model is the accurate description of the particle motion in highly nonuniform electric field with a significant radial component and the charged particle loss on the tube walls. Below we describe the results of the gas breakdown simulation in long tubes (L/Rtube > 1) in more detail. The calculation results for the tube length of 72 mm are shown in Figure 11 in comparison with the experimental data. Two curves have been calculated with different conditions of the electron loss. The first calculated curve represents the test case of high magnetic field (100 T) superimposed in the model in axial direction that excludes radial electron motion and the diffusion electron loss. The second calculated curve shows simulation results for a regular case without magnetic field using the effective reflection coefficient Reff = 0.5. It is seen from Figure 11 that both curves are quite consistent with the experimental data, whereas accounting for the particle loss at the walls shifts the breakdown curve to higher voltage values. The most challenging issue is the case of a very long tube (300 mm in our study). One can expect there the most significant effect of both the field non-uniformity and the wall particle loss on the breakdown curve. Thus, let us consider this case in more detail. Fig. 12 shows the spatial distributions of electric potential and electric field (axial Ez and radial Er) disregarding the surface charge on the tube walls. One may observe that the electric field in the vicinity of the cathode and anode is much higher than in the middle of the tube with peaks at the electrode edge. Note the presence of significant radial electric field near the electrodes. The field is “defocusing” for electron flow near the cathode, and it is “focusing” towards the anode. Such configuration of the electric field causes the ionization rate increase near the cathode edge and especially near the anode edge. Electrons and ions adsorbed by a dielectric surface can form the surface electric charge changing the electric field distribution inside the discharge tube. In order to study the role of this phenomenon we have performed long-time simulations (up to 0.1 s of discharge time) with increased number of electrons. Each superparticle has contained 2000 electrons that cannot disturb the potential due to space charge, while the space charge accumulated during such long time can be significant. The stationary potential and electric field distributions after the wall charging are shown in Fig. 13. One can see some redistribution of the electric fields: the anode field becomes higher while the cathode field decreases. Fig. 14 illustrates the evolution of the electron avalanche in time as well that of ions produced under ionizing collisions between electrons and gas molecules. The avalanche begins from a short bundle of electrons emitted from the cathode. The electrons originated from the cathode are accelerated in the high non-uniform electric field in different directions. The majority of the electrons is not directed exactly towards the anode and will be lost on the tube wall due to diffusion and the motion in the radial electric field. One observes in the figure that at small distances from the cathode the electron escape to the tube walls is weak, and it becomes playing a role when the electron avalanche is located sufficiently far from the cathode. Due to the high electric field the electrons are accelerated to the energy exceeding the ionization threshold, and new electrons born due to ionization come to the middle region of the discharge

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tube where the electric field is much more uniform and its radial component is close to zero. Then the avalanche moves along the tube up to the anode region where the high electric field focuses the electron flow to the anode accelerating the electrons to higher energy leading to the ionization rate increase. The effect of such field redistribution is presented in Fig. 15. It is obvious that the wall charging is able to change the breakdown curve dramatically. However, it should be mentioned, that the exact accounting of this physical phenomenon requires the exact knowledge of secondary emission, reflection and scattering parameters of the dielectric surface. In the present simulations we approximate all the emissive factors by the effective coefficient Reff = 0.9, aiming to reveal a qualitative physical picture of the breakdown process in long tubes with a nonuniform electric field. We will present the results of our detailed research into the effect of different mechanisms participating in the gas breakdown in long tubes in our future papers.

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This paper outlines the results of our research into the effect of the electric field nonuniform distribution on the dc breakdown in nitrogen. We have measured dc breakdown curves for two cases: i) the electrodes of 6 mm in radius are located in a narrow tube of 13 mm inner radius; ii) the same electrodes are located in a wider tube with 22 mm inner radius. In the first case the discharge ignition is affected by both the electric field non-uniformity and the diffusion escape of charged particles on the tube walls, and it is impossible to study them separately. With narrow gaps L between the electrodes (when L is less than the tube radius Rtube) Paschen's law holds, and the breakdown curves measured for different L values match when one plots them to the U(pL) scale. For L/Rtube > 1 increasing L causes the enhanced diffusion loss of electrons thus shifting the breakdown curves U(pL) to the range of higher breakdown voltage U values as well as pL product ones. In this case the contribution of the electric field nonuniformity is expressed weakly. In the second case (electrodes in a large chamber) we have found the range of interelectrode distance values (from L = 30 mm to 72 mm) in which Paschen's law does not hold but the role of diffusion loss is still small. Therefore we have managed to perform separate studies of the effect of the non-uniform electric field distribution on the dc gas breakdown. Under these conditions, increasing the L distance shifts the breakdown curves to the right along the pL scale with a slightly varying breakdown voltage at their minimum. As far as we know, such an effect of the electric field non-uniformity on the breakdown curves has not been observed earlier. And only with large L gap values, when the diffusion loss of electrons to the tube walls becomes considerable, the behavior of the breakdown curves is similar to the case of L/Rtube > 1 in a narrow tube. We have demonstrated, as a result, that in a general case the gas breakdown criterion assumes the form U = f(pL, L/Rel, L/Rtube), where the non-uniformity degree of the electric field between the electrodes is described by the L/Rel ratio whereas the diffusion escape to the tube walls is described by the L/Rtube ratio. The breakdown curves for a fixed gap L and different electrode radius values intersect at one point corresponding to the location of the inflection point of the breakdown curve for the uniform electric field. To the right of this inflection point (at high gas pressure values) with long gap values, the right-hand branches of the breakdown curves in a wide tube at large L/Rel values run in the range of lower breakdown voltage values than in the case of the uniform electric field. Under these conditions the redistribution of the potential between the electrodes makes the conditions for the gas breakdown easier. At low gas pressure values (to the left of the inflection point) the gas breakdown in the non-uniform electric field occurs at higher voltage values because the electric field redistribution impedes the development of electron avalanches within the inter-electrode gap.

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We have also presented the formula for calculating the breakdown voltage values taking into account the non-uniform axial distribution of the electric field between the electrodes when one may disregard the diffusion loss of electrons on the tube walls. Here we have neglected a possible enhancement of the electric field near the electrode edges considering the breakdown only near the tube axis. The breakdown curves calculated with this formula are in good agreement with the experimental data near to and to the right of its minimum. In order to evaluate the breakdown in longer tubes we have performed calculations with the OOPIC Pro code. 2D profiles of the potential, axial and radial electric field in the absence and in the presence of surface charge on the tube walls, as well as corresponding breakdown curves have been obtained. We have demonstrated that a considerable number of charged particles may be generated in the regions of high electric field near the edges of the cathode and especially the anode. The calculation results are in satisfactory agreement with experimental breakdown curves for L = 72 mm and L = 100 mm.

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Fig.1. Designs of discharge chambers employed in this paper.

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Fig.2. Breakdown voltage versus the gas pressure for different inter-electrode distance values and the inner radius of the discharge tubes of 6.5 mm and 28 mm.

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D = 56 mm

U56(L=3mm) U(L=6mm) U(L=18mm) U56(L=30mm) U56(L=72mm_new) U56(L=150mm) U56(L=300mm)

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Fig.3. Breakdown voltage versus the pL product for different inter-electrode distance values and the inner radius of the discharge tubes of 6.5 mm and 28 mm.

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Fig.4. Product of the pressure at the breakdown curve minimum and the inter-electrode distance pLmin versus the ratio of the inter-electrode distance L to the electrode radius Rel and the tube radius Rtube for the tube inner radius of 6.5 mm and 28 mm.

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Fig.5. Voltage at the breakdown curve minimum Umin versus the ratio of the inter-electrode distance L to the electrode radius Rtube for the tube inner radius of 6.5 mm and 28 mm.

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Fig.6. Breakdown curves for the inter-electrode distance of L = 3 mm and the inner radius of the discharge tube of 6.5 mm and 28 mm, the electrode radius of 6 mm

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Fig.7. Breakdown curves for the inter-electrode distance of L = 72 mm and the inner radius of the discharge tube of 6.5 mm and 28 mm, the electrode radius of 6 mm.

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Fig.8. Breakdown curves for the inter-electrode distance of L = 72 mm for the discharge tube of 28 mm inner radius: empty circles are for the electrode radius of 6 mm, solid triangles are for the electrode radius of 27.5 mm, the solid curve is calculated with equation (7) for the tube of 28 mm inner radius and the electrode radius of 6 mm, the broken line is the calculation with equation (1) for the uniform electric field, empty squares are for the experimental breakdown curve for the uniform electric field.

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Fig.9. Axial distributions of the electric field strength for the electrodes of unlimited radius (uniform field), and for the electrode radii of 2.75 cm, 1.5 cm, 0.6 cm and 0.3 cm. The interelectrode distance is 72 mm.

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Fig.10. Coefficient of ionization multiplication M for the uniform and non-uniform electric field distributions across the electrodes versus the reduced field E/p.

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Fig.11. Breakdown voltage versus the pL product for the inter-electrode distance of L = 72 mm for the discharge tube of 28 mm inner radius. The results are obtained by OOPIC simulations. Black circles are for our experimental data, red triangles are for our calculation data accounting for the diffusion loss of electrons on the tube walls, blue squares are for our calculation data disregarding the diffusion loss.

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Fig.12. 2D profiles of the potential, axial Ez and radial Er electric fields disregarding the surface charge on tube walls.

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Fig.13. 2D profiles of the potential, axial Ez and radial Er electric fields taking into account the surface charge on the tube walls.

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Fig.14. Temporal evolution of electrons and ions in the tube of 300 mm in length.

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Fig.15. Breakdown voltage versus the pL product for the inter-electrode distance of L = 300 mm for the discharge tube of 28 mm inner radius. Black circles are for our experimental data, red triangles are for our calculation data accounting for the surface charge on the tube walls, blue squares are for our calculation data disregarding the surface charge on the tube walls.

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We investigated the gas breakdown in a non-uniform dc electric field. The effects of the non-uniform electric field and of the diffusion loss were studied separately. The general form of the gas breakdown criterion must be as follows U = f ( pL, L Rel , L Rtube ) .