JETP Letters, Vol. 72, No. 2, 2000, pp. 34–37. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 72, No. 2, 2000, pp. 49–53. Original Russian Text Copyright © 2000 by Lisovsky, Yakovin.

Scaling Law for a Low-Pressure Gas Breakdown in a Homogeneous DC Electric Field
V. A. Lisovsky* and S. D. Yakovin**
*Kharkov State University, pl. Svobody 4, Kharkov, 310077 Ukraine **Physics and Technology Center, National Academy of Sciences and Ministry of Education of Ukraine, Kharkov, 310145 Ukraine e-mail: lisovskiy@ftf.univer.kipt.kharkov.ua
Received March 15, 2000; in ﬁnal form, June 5, 2000

Gas breakdown in nitrogen, air, and oxygen in a dc electric ﬁeld at various interelectrode distances L is studied experimentally. A scaling law for a low-pressure gas breakdown Udc = f(pL, L/R) is deduced. According to this scaling law, the breakdown voltage Udc is a function not only of the product of the gas pressure p and the gap length L, but also of the ratio of the gap length L to the chamber radius R. It is shown that, for any dimensions of the cylindrical discharge chamber (in the range of L/R under investigation), the ratio of the breakdown electric ﬁeld to the gas pressure p at the minimum of the ignition curve remains constant: (Edc /p)min ≈ const. A method for calculating the ignition curve in a cylindrical discharge chamber with arbitrary values of L and R is proposed. © 2000 MAIK “Nauka/Interperiodica”. PACS numbers: 51.50.+v; 52.80.Hc

As is known [1–7], the ignition curves of a glow discharge are described by the Paschen law Udc = f(pL); i.e., the breakdown voltage Udc is a function of the product of the gas pressure p and the interelectrode distance L. The Paschen law implies that the ignition curves Udc(p) measured for various distances L must coincide if they are drawn as the function Udc(pL). However, the measurements of the ignition curves of a glow discharge in neon [8] showed that, with equal values of the product pL, the breakdown voltage for a long discharge gap with planar electrodes is signiﬁcantly higher than that for a short gap. More recent studies [9–14] conﬁrmed this conclusion for some other gases (neon, argon, nitrogen, hydrogen, etc.). In spite of a great number of experimental and theoretical papers devoted to low-pressure gas breakdown in a dc electric ﬁeld, a method for calculating the ignition curve at arbitrary values of the interelectrode distance L and the radius of the discharge chamber R is still lacking. This paper is devoted to the experimental study of a breakdown in nitrogen, air, and oxygen in a dc electric ﬁeld in a discharge chamber with a variable interelectrode distance L. It is shown that, in the range of the ratio L/R under study, the ignition curves shift toward high values of the product pL and discharge voltage Udc as the gap length L increases. In this case, for any values of the gap length L, the ratio of the breakdown electric ﬁeld to the gas pressure (Edc /p)min at the minimum of the ignition curve remains constant. A generalized scaling law for the low-pressure gas breakdown Udc = f(pL, L/R) is deduced. A method allowing one to calculate the ignition curve for a glow discharge in a cylin-

drical chamber with arbitrary dimensions from the known ignition curve for a narrow discharge gap (for L/R 0), i.e., from the usual Paschen curve, is described. We measured the ignition curves for a glow discharge in the range of dc voltages Udc ≤ 1000 V and pressures of p ≈ 10–2–10 torr. A discharge tube with an inner diameter of 63 mm was used. The interelectrode distance L was varied in the range 0.5–10 cm; consequently, the studies were conducted in the range L/R = 0.16–3.2. Planar parallel electrodes spanned the entire cross section of the discharge tube. Both the anode and the cathode were made from stainless steel. The breakdown voltage was measured accurate to ±2 V. When determining the ignition voltage, the growth rate of the discharge voltage did not exceed 1 V/s. In all cases, our procedure for measuring the ignition curves was as follows. We ﬁxed a certain distance L between the electrodes and then, for various gas pressures p, measured the breakdown voltage Udc . Below, we explain why only this way of measuring the ignition curves of a glow discharge is correct. Figure 1 shows the ignition curves measured by us in nitrogen for different distances L between the electrodes. It is seen from the ﬁgure that, as L increases, the ignition curves shift not only toward higher ignition voltages Udc (as was obtained in [9–14]), but also toward higher values of pL. Apparently, such a shift of the ignition curves toward higher values of Udc and pL with increasing interelectrode distance L may be attributed to an increase in losses of charged particles on the

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SCALING LAW FOR A LOW-PRESSURE GAS BREAKDOWN (V/(cm torr)) (a) (b)

35

pL (torr cm)

(torr cm)

Fig. 1. (a) Experimental ignition curves of a glow discharge in nitrogen for the interelectrode distances L = (1) 0.5, (2) 2, (3) 4, (4) 6, and (5) 8 cm. (b) Dependences of U * dc and Edc /p on (pL)* obtained from curves (1)–(5) in Fig. 1a and the data from (6) [11], (7) [15], (8) [16], and (9) [18].

cylindrical wall of the discharge tube due to diffusion across the electric ﬁeld. Figure 2 shows the dependences of Umin and (Edc /p)min on the value of (pL)min obtained from our experimental ignition curves measured in a nitrogen discharge by varying the distance L. The solid line Umin = 407(pL)min and the dashed line (Edc /p)min = 407 ± 5 V/(cm torr) describe the results obtained fairly well. Hence, for any distance L between the electrodes, the ratio (Edc /p)min at the minimum of the ignition curve remains constant. This is also true if we change the value of the ion–electron emission coefﬁcient γ of the cathode material (as was noted in [2, 3]). Note that, by properly choosing the reference axes, we can make all of the obtained ignition curves almost coincide. For example, if we take ( pL )∗ = pL ⁄ ( 1 + ( L ⁄ R ) ) ,
2 a

Based on Eqs. (1) and (2) and the values of breakdown voltage given in the ﬁgures, we can calculate, to a high accuracy, the ignition curves for any cylindrical discharge chamber for arbitrary values of the distance

(1) (2)

(torr cm)
Fig. 2. Dependences of Umin and (Edc /p)min on (pL)min for nitrogen. The solid line corresponds to Umin = 407(pL)min, and the dashed line corresponds to (Edc /p)min = 407 V/(cm torr).

U* dc = U dc ⁄ ( 1 + ( L ⁄ R ) ) ,

2 a

as the abscissa and ordinate, respectively, where a ≈ 0.12 for nitrogen, then the ignition curves presented in Fig. 1a coincide accurate to ±5 V (Fig. 1b). It is evident that, for L/R 0, we have the usual Paschen curve Udc = f(pL). It follows from Eqs. (1) and (2) that U* dc /(pL)* = Udc /pL = Edc /p; i.e., the dependences Edc /p = f((pL)*) for different ignition curves must also coincide (which is seen in Fig. 1b). Here, we also see a reasonable agreement between our results and data from [11, 15–18]. For air, we have a ≈ 0.09 (Fig. 3) and for hydrogen, we have a ≈ 0.03 (Fig. 4). Note that, in Figs. 3 and 4, the dependences Edc /p = f((pL)*) obtained from experimental results [2, 9, 15, 16, 19] agree satisfactorily with our data. From our results, it follows that the scaling law for the gas breakdown can * = be written in the form Udc = f(pL, L/R)) or U dc f((pL)*).
JETP LETTERS Vol. 72 No. 2 2000

(torr cm)
Fig. 3. Dependences of U * dc and Edc /p on (pL)* for a glow discharge in air for the interelectrode distances L = (1) 0.5, (2) 1, (3) 2, (4) 5, and (5) 10 cm and the data from (6) [15], (7) [19], (8) [16], and (9) [20], and (10) [21].

(V/(cm torr))

(V/(cm torr))

36

LISOVSKY, YAKOVIN (V/(cm torr))

U dc 1 = U dc 0

1 + ( L1 ⁄ R1 ) ------------------------------, 2 1 + ( L0 ⁄ R0 )

2 a

(6)

(torr cm)
Fig. 4. Dependences of U * dc and Edc /p on (pL)* for a glow discharge in oxygen for the interelectrode distances L = (1) 0.5, (2) 3, (3) 6, (4) 8 cm and (5) the data from [15].

L and radius R. As an example, we ﬁnd the position of the minimum in the ignition curve for a nitrogen discharge for R = 3.15 cm, L = 5 cm, and a stainless-steel cathode. From Fig. 1b, it is seen that U * min ≈ 280 V and ( pL ) * min ≈ 0.6 torr cm. From (1) and (2), we obtain
2 ( pL ) min = ( pL ) * min [ 1 + ( L ⁄ R ) ] 0.12

,

(3) (4)

U min = U * min [ 1 + ( L ⁄ R ) ]

2 0.12

.

For L/R = 1.59, we have Umin ≈ 323 V and (pL)min ≈ 0.7 torr cm. From our experimental results, it follows that Umin ≈ 320 V and (pL)min ≈ 0.73 torr cm; i.e., there is a good agreement between the coordinates of the maximum in the ignition curve obtained experimentally and those calculated using Eqs. (1)–(4). To calcu* and late any other point on the ignition curve U 1 ( pL ) * 1 , we use the same procedure. In the general case, to calculate the ignition curve in a cylindrical discharge chamber with arbitrary dimensions, one should have the Paschen curve (i.e., the ignition curve measured in a discharge chamber such that L/R 0 and with the same cathode material). Then, from Eqs. (1) and (2), one can determine the values of Udc and pL. If the initial ignition curve has been measured in a discharge chamber with L0 and R0, such that the condition L0/R0 0 does not hold, one should ﬁrst calculate the dependence U* dc = f((pL)*) using Eqs. (1) and (2) and then, from the same relations, calculate the ignition curve for a discharge chamber with the given dimensions L1 and R1. This may be written more concisely in the following form: 1 + ( L1 ⁄ R1 ) ( pL ) 1 = pL 0 ------------------------------, 2 1 + ( L0 ⁄ R0 )
2 a

where the index “1” stands for the ignition curve to be sought and the index “0” stands for the initially known ignition curve. The ignition curve of a glow discharge is usually measured by two methods: (i) the distance L is ﬁxed, and the breakdown voltage Udc is measured at different values of the gas pressure p; and (ii) the value of p is ﬁxed, and the breakdown voltage Udc is measured at different values of the distance L. However, the results obtained in this study show that the latter method of measuring the ignition curve (at a ﬁxed value of p and variable L) is incorrect. The ignition curves obtained in this way are close to the Paschen curve only at small values of L but shift toward higher breakdown voltages with increasing L. Hence, the ignition curve of a glow discharge must be measured by varying the gas pressure p, the distance L between the electrodes being ﬁxed. In summary, the ignition of a glow discharge in nitrogen, air, and oxygen at a variable interelectrode distance L have been studied experimentally. It is shown that, at any interelectrode distance L, the ratio (Edc /p)min at the minimum of the ignition curve remains constant. In the range of L/R under study, the scaling law for gas breakdown Udc = f(pL, L/R) holds; i.e., the breakdown voltage Udc is a function not only of the product of the gas pressure p and the gap length L, but also of the ratio L/R. A method for calculating the ignition curve in a cylindrical discharge chamber with arbitrary values of L and R is proposed. REFERENCES
1. F. Paschen, Ann. Phys. Chem., Ser. 3 37, 69 (1889). 2. M. J. Druyvesteyn and F. M. Penning, Rev. Mod. Phys. 12, 87 (1940). 3. J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Clarendon, Oxford, 1953; Inostrannaya Literatura, Moscow, 1960). 4. Yu. P. Raœzer, Gas Discharge Physics (Nauka, Moscow, 1987). 5. V. A. Lisovsky and V. D. Yegorenkov, J. Phys. D 27, 2340 (1994). 6. A. V. Phelps and Z. Lj. Petrovic, Plasma Sources Sci. Technol. 8, R21 (1999). 7. M. Sato, Bull. Yamagata Univ. 25, 119 (1999). 8. J. S. Townsend and S. P. McCallum, Philos. Mag. 6, 857 (1928). 9. H. Fricke, Z. Phys. 86, 464 (1933). 10. S. P. McCallum and L. Klatzow, Philos. Mag. 17, 279 (1934). 11. H. C. Miller, Physica (Amsterdam) 30, 2059 (1964).
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SCALING LAW FOR A LOW-PRESSURE GAS BREAKDOWN 12. L. Jacques, W. Bruynooghe, R. Boucique, and W. Wieme, J. Phys. D 19, 1731 (1986). 13. M. Yumoto, T. Sakai, Y. Ebinuma, et al., in Proceedings of the 8th International Symposium on High-Voltage Engineering, Yokohama, 1993, p. 409. 14. G. Auday, P. Guillot, J. Galy, and H. Brunet, J. Appl. Phys. 83, 5917 (1998). 15. M. J. Schonhuber, IEEE Trans. Power Appar. Syst. 88, 100 (1969). 16. T. W. Dakin, J. Gerhold, Z. Krasucki, et al., in Proceedings of the International Conference on Large HighVoltage Electric Systems, Paris, 1977, p. 1.

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17. B. Held, N. Soulem, R. Peyrous, and N. Spyrou, Trans. Inst. Electr. Eng. Jpn., Part A 116, 925 (1996). 18. B. Held, N. Soulem, R. Peyrous, and N. Spyrou, J. Phys. III 7, 2059 (1997). 19. J. A. Pim, Proc. Inst. Electr. Eng., Part 3 96, 117 (1949). 20. S. C. Brown, in Basic Data of Plasma Physics (Wiley, New York, 1959), p. 240. 21. H. Ritz, Arch. Elektrotech. (Berlin) 26, 219 (1937).

Translated by N. Larionova

JETP LETTERS

Vol. 72

No. 2

2000