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The Maxwellian nature of free-electrons' gas spectrum of noble gases at low pressure
Vacuum 110 (2014) 19e23
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The Maxwellian nature of free-electrons' gas spectrum of noble gases at low pressure Malisa Alimpijevi c, Koviljka Stankovi c*, Milan Ignjatovic, Jovan Cveti c Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 July 2014 Received in revised form 1 August 2014 Accepted 2 August 2014 Available online 12 August 2014
A free-electrons' gas spectrum shape in gases under the influence of the electrical field is considered in the theoretical part of the paper. Based on the nature of electrons' interaction with ions and molecules (or atoms) of gas, it was concluded that the spectrum is Maxwellian, if noble gases and subpressures are used. This conclusion was experimentally verified in the Townsend's region of electrical discharge. The gases used were Helium, Neon, Argon, Xenon and Krypton. The utilized pressures ranged from 0.3 Pam to 10 Pam. Interelectrode distances were in the range of 0.1 mme1 mm. It has been assumed that the freeelectrons’ gas spectrum is of the Maxwellian type by means of the mathematical analysis of macroscopically measurable consequences, and this assumption has been verified through the experiment. New constants enabling electrical discharge parameters' calculation in the Townsend's region have been presented according to Townsend's, Takashi's and Maxwell's expressions. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Free electron gas Maxwell spectrum Noble gas Low pressure
1. Introduction Gases in nature represent a gas mixture of neutral atoms (or molecules), ions and electrons. Ions and electrons, created by photoionization, positive-ion ionization and metastable ionization, form a free-ions gas and a free-electrons gas. Each of these components represents a partial “gas” to which the Gas Mixture Laws is valid. The free-electrons' gas has a decisive role in electrical breakdown in gases [1e3]. Namely, the electrical breakdown of gases occurs as a result of collision of free electrons, accelerated by an electrical field, and atoms (or molecules) of gas, which in turn creates new free electrons and ions, which possibly leads to forming the self-sustaining mechanism (positive feedback). Selfsustained discharge forming is controlled by the collision processes' relations comprising charged particles and their transport properties. It can be concluded that the process development of gas electrical breakdown depends on the relative efficiency of mechanisms for generation and annihilation of free electrons, which determine both spectral distribution and free-electrons' gas density during the electrical discharge [4e6]. In normal conditions, i.e. without the electrical field influence, free-electrons' gas spectral distribution (as well as other gas mixture components) is of the Maxwellian type [4,7]. Nevertheless, when such a gas mixture is found within a direct electrical field,
* Corresponding author. Tel.: þ381 113370186; fax: þ381 113370187. E-mail address: [email protected] (K. Stankovi c). http://dx.doi.org/10.1016/j.vacuum.2014.08.005 0042-207X/© 2014 Elsevier Ltd. All rights reserved.
one drift component is superpositioned in the direction of the field by thermal speeds of its charged constituting components [8,9]. Generally speaking, in such a case, free electrons and ions' gas spectrum is no longer of the Maxwellian type. The reason for this is the non-elastic nature of these collisions in the field direction. Namely, a part of free-electrons and free-ions gas energy, taken over from the electrical field, is lost due to the excitation of neutral molecules' rotational and vibration quantum mechanical states. In addition to this, the deviation from the Maxwellian shape of freeelectrons and free-ions gas spectrum also occurs due to their Coulomb interaction with other free electrons and ions. However, these phenomena do not occur in the case of atomic (i.e. noble) gases at low pressures (100 Pae104 Pa) (atomic gases do not have rotational and vibration states, the ionization degree is small at low pressures, and consequently the interaction between the charged particles is negligible). As a result of this, the condition may be considered as the free electrons' interactions are of elastic type, so that the Maxwellian spectrum is not disturbed owing to the electrical field influence. The aim of this paper is to verify the statement on the Maxwellian nature of free-electrons gas spectrum, and to check the possibility of its application in the electrical discharge modeling process for noble gases at low pressures.
2. Electrical breakdown of gases As mentioned above, electrical breakdown of gases is a selfsustaining process based on the primary and secondary processes
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23
20
of electrical discharge in gases. The basic primary process of gas discharge is electron ionization. Electrons ionization occurs when a free electron on a single free path takes over the energy from the electrical field that is greater than (or equal to) the electrons' bond energy in neutral atoms (or molecules). More precisely, this electron becomes an initial electron. This enables electron to perform ionization in the ensuing collision. Once the ionization has been initiated, it leads to the free electrons' avalanche. When the free electrons' avalanche, created in this way, reaches the anode, the process would stop if there were not secondary gas discharge processes, which provide a positive feedback leading to a sequence of avalanches which sum results in the breakdown. Secondary gas discharge processes are divided into cathode active processes (ionic ejection, photoemission and metastable ejection) and the gas active processes (positive-ion ionization, photoionization and metastable ionization). If the dominant secondary processes are active at the electrodes, breakdown occurs according to the Townsend mechanism. If the dominant secondary processes are active in the gas, breakdown occurs according to the streamer mechanism [10e13]. Townsend's (primary) ionization coefficient a and secondary ionization coefficient g are introduced in order to mathematically formulate conditions for the Townsend and streamer mechanisms of breakdown. Starting from the assumption that a single electron creates a new electrons on the path length in the field direction, we can conclude that the free electrons number increase by n free electrons on the path dx is:
while p is the gas pressure, d is the interelectrode distance, and E is the electrical field, and C1,C2,Z1 andZ2 are constants which depend on the type of gas. Secondary ionization coefficient g does not depend on the electrical field and pressure, but it depends on the electrode material and the electrode surfaces' processing type. Due to that, g may be, under constant experiment conditions, considered constant [17,18]. With ionization coefficients defined in such a way, it can be demonstrated that the condition for gas breakdown according to the Townsend mechanism is:
dn ¼ aðxÞndx
Zd
(1)
Based on the expression (1) and the analysis of experiment results [14], two expressions of the a coefficient dependence on the electrical field E and pressure p are provided:
pi h a ¼ C1 p exp C2 E 0
2
(3)
Z∞ si ð3 Þnf ð3 Þd3
(4)
31
where n and 3 are the speed and kinetic energy of free electrons; si is the cross-section for ionization; 3 i is the ionization potential (bond energy) of the observed gas, and f(3 ) is the free-electrons gas spectrum. Under the assumption that the free-electrons' gas spectrum is of the Maxwellian type, and originating in the expression (4) the following expressions are obtained:
p3 þ 2 exp i ZE ZE
p3
i
(6)
and
Z ¼ Z1 ðpdÞZ2
(7)
Zx adx
Zd aðxÞe 0
g
dx ¼ 1;
(8)
0
while the condition for gas breakdown according to the streamer mechanism is:
aðxÞdx ¼ 10; 5:
(9)
0
Expressions (8) and (9) are significantly simplified in case of the homogenous electrical field [19,20]. 3. The experiment and the experiment results' processing
13
where Ci(i ¼ 1,2) and ki(i ¼ 1,2,3) constants depend on the gas type and the range of the pd product (pressure multiplied by interelectrode distance), to which the expressions (2) and (3) are applied. The expression (2) is often called the Townsend expression, while the expression (3) is often called the Takashi expression [15,16]. However, the a coefficient dependence on the pressure and electrical field is most precisely obtained by starting from the definition according to statistical physics [2]:
a ¼ C$p
C ¼ C1 expðC2 pdÞ
(2)
BEp k2 C7 6 B C7 a ¼ k1 p6 41 exp@ k A5 3
aðp; EÞ ¼ n0
where
(5)
The measurements have been performed in the well-controlled laboratory conditions. During the measurement, a disassembling gas chamber and the corresponding vacuum-gas circle have been used. Experiment parameters comprised gas types and the interelectrode distance. Applied gases were noble gases (Helium, Argon, Neon, Xenon and Krypton). The electrodes that provided a pseudohomogenous electrical field were of the Rogowski shape and were made of copper. Before each measurement series electrodes were sandblasted. By sandblasting electrodes, it was enabled the statistic sample of the dc breakdown voltage random variable to be independent from the irreversible changes interelectrode topography during a single measurement series. The U-Test has demonstrated that 1.000 random variables of the dc breakdown voltage statistic sample belong to a unique statistic distribution with the statistic uncertainty of 0.5% (in case of polished electrodes, for the same conclusion, statistic uncertainty equals 5%). After multiple chambers' vacuuming up to the pressure of 104 mbar and 1 bar pressure working gas filling, the chamber pressure was calibrated by a special doses valve. This pressure enabled a spectroscopic purity of working gas during a single measurement series. Namely, with the aim of testing gas purity in the chamber, and utilizing the same procedure, a glass chamber was filled. After developing a stable electrical discharge the presence of any other gas was not detected by means of spectral analysis. Chamber sealing was good. Measuring pressure of Helium at 10 mbar (reduced to 0 C) during a 24-h period indicated no variations in pressure, which means that it was constant during a single measurement series for each of the noble gases (He has the smallest
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23
atom out of all noble gases, so if the pressure was constant for He, it was also constant for other noble gases). Gas pressure in the chamber varied from 102 to 104 Pa. Interelectrode distance was calibrated by the electronic micrometer screw from 0.1 mm to 1 mm (a corresponding pair of electrodes was made for each interelectrode distance). Supply source had a voltage rise rate of 8 V/s. Its ripple was smaller than 1%. Breakdown voltage measurement was performed by the Ohm ratio 1:273. During the experiment, measurement instruments were placed in the protection booth comprising the level higher than 100 dB. Measurement signal was fed into the booth by a twice protected, grounded, 50U cable. Combined measurement uncertainty (which was dominantly of type B, since the dc breakdown voltage is considered as a deterministic quantity and therefore the dispersion originating from the measurement repeating procedure can be neglected) was estimated at less than 3% [21e24]. The procedure of determining the Paschen curve for a single gas type comprised the following steps: 1 e the interelectrode distance was calibrated and the chamber was sealed; 2 e the chamber was plugged into the gas-vacuum circle, vacuumed and filled with gas and again vacuumed up to 104 mbar (this procedure of the socalled rinsing was repeated 5 times); 3 e working pressure in the chamber was calibrated and reduced to 0 C; 4 e the chamber was plugged into the electrical circuit; 5 e the chamber conditioning was performed through 50 consecutive dc voltage breakdowns; 6 e a measurement of 50 consecutive dc breakdowns was performed with a one-minute break between two breakdowns (for the U-test requirements, 1.000 values of the dc voltage breakdowns were measured); 7 e a new vacuuming and rinsing of the chamber were performed, and the procedure was repeated with next working pressure value (without chamber disassembling and interelectrode distance calibrating, i.e. utilizing a single interelectrode distance, a single series of measurement was performed); 8 e upon the completion of a single measurement series, new electrodes, made for a new interelectrode distance, freshly sandblasted and mounted onto the gas chamber, were utilized. The statistic samples set of the dc voltage breakdown random variables, obtained in such a way, was processed in the following steps: 1 e a statistic sample was subjected to the modified Chauvenet's Test [25] with the aim of rejecting the spurious measurement results; 2 e the dc voltage breakdown mean value was determined, as well as standard deviation; 3 e a significance level was determined according to the Student's distribution for each statistic sample (it has been demonstrated that it is always small and may be disregarded) 4 e based on the dc voltage breakdowns' mean values for certain pressures and interelectrode distances, an experimental Paschen curve was determined for all tested gases (in the region of pd product values, in which measurements were performed); 5 e experimental Paschen curves were fitted by the expression (8), in addition to utilizing the primary ionization coefficient obtained by expressions (2), (3) and (5); 7 e based on the constants obtained by fitting in expressions (2), (3) and (5), relevant dependences of the a/p ratio in the pd product function were drawn.
21
Fig. 1. Paschen curves obtained by using: equation (2) - - - ; (3) $ $ $ ; and (5) d for calculating the a coefficient along with experimentally results A obtained for Argon.
Fig. 2. Paschen curves obtained by using: equation (2) - - - ; (3) $ $ $ ; and (5) d for calculating the a coefficient along with experimentally results A obtained for Xenon.
has been observed. The constants C1(i ¼ 1,2) and Zi(i ¼ 1,2) in the expression (5), obtained through fitting, are given in Table 3. Figs. 1 and 2 show that the fitted curves are in agreement with the experimentally obtained values. However, for all fitted curves, the
Table 1 Constants applied to the equation (2). Gas
C1[1/Pam]
C2[V/Pam]
Helium Neon Argon Xenon Krypton
0.763 0.9 6.649 12.7 12.4
29.154 46.1 233.147 290.5 298.7
4. Results and discussion Figs. 1 and 2 show the Paschen curves for Ar and Xe, respectively, obtained by using equations (2), (3) and (5) for calculating the Townsend a coefficient along with experimentally results. Constants Ci(i ¼ 1,2) and ki(i ¼ 1,2,3) used in expressions (2) and (3) and obtained by fitting are shown in Tables 1 and 2, respectively. Constants values shown in Tables 1 and 2 differ from the corresponding constants found in the literature. The difference is the consequence of the fact that a relatively narrow region of small pd product values
Table 2 Constants applied to the equation (3). Gas
K1[1/Pam]
K2[V/Pam]
K3[V/Pam]
Validity range E(x)/p [V/Pam]
Helium Neon Argon Xenon Krypton
2.23 3.14 10.45 20.88 12.88
6 7.05 18.38 31.35 19.88
120 150 341.25 585 382.5
6e225 7.05e300 18.38e1200 31.35e1800 19.88e1500
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23
22 Table 3 Constants applied to the equation (5). Gas
C1
C2
Z1
Z2
Helium Neon Argon Xenon Krypton
0.4066 0.4565 3.2600 6.5560 6.2530
0.03423 0.02594 0.08472 0.14080 0.15920
0.42200 0.24150 0.04656 0.02880 0.03439
0.2065 0.1750 0.1374 0.2209 0.1997
best fit is obtained when expression (5) is used for a ionization coefficient. Figs. 3 and 4 show the a/p ratio dependence on the pd product calculated according to expressions (2), (3) and (5) obtained for Argon and Xenon, respectively. Based on the results obtained in Figs. 1e4, it can be concluded that a coefficient, obtained under the assumption of the Maxwellian free-electrons gas spectrum (equation (5)), is qualitatively and quantitatively compliant with the corresponding dependences obtained according to Townsend (equation (2)) and Takashi (equation (3)). In addition to this, since it has been shown, in Figs. 3 and 4, that a ionization coefficient implementation, obtained under
the Maxwellian free-electrons gas spectrum assumption, leads to the best dependence fit of the experimentally obtained Paschen curve, it can be concluded that this assumption has been justified. The same conclusion is valid for all other noble gases. The limitation on using both Townsend and Takashi equations determined by validity range E(x)/p (shown in Tables 1 and 2) has been overcame by derived equation (5) since the constants C and Z are functions dependant on the product pd. In engineering practice obtained results are significant for modeling and designing the gas filled surge arresters [26]. 5. Conclusion Based on the obtained results, it can be claimed with certainty that free-electrons gas spectrum of noble gases in the Townsend's discharge region is of the Maxwellian type (or very similarly shaped to it). The conditions in which this was proved are also valid for the streamer discharge region at low pressures. Apart from the fact that recognizing free-electrons gas spectrum enables the calculation of a sequence of electrical discharge physical parameters in gases, it is also significant in engineering practice. Namely, gas filled surge arresters used for the insulation coordination at low-voltage level have been designed as two-electrode systems isolated by a noble gas at subpressure and small interelectrode distances. Under these conditions, recognizing free-electrons gas spectrum enables a simulation of the gas filled surge arresters characteristics, as well as optimization of its characteristics in the construction phase. Acknowledgment The Ministry of Education, Science and Technological Development of the Republic of Serbia supported this work under contract 171007. References
Fig. 3. Dependence of the Townsend coefficient to pressure ratio (a/p) on the pd product for Argon according to equations equation (2) - - - ; (3) $ $ $ ; and (5) d.
Fig. 4. Fig. 3. Dependence of the Townsend coefficient to pressure ratio (a/p) on the pd product for Xenon according to equations equation (2) - - - ; (3) $ $ $ ; and (5) d.
[1] Pejovi c MM, Zivanovi c EN, Pejovi c MM, Nesi c NT, Kova cevi c D. Vacuum 2012;86:1860e6. [2] Meyer OE. The kinetic theory of gases: elementary treatise with mathematical appendices (classic reprint). Charleston: Forgotten Books; 2012. [3] Von Engel A. Ionized gases. Oxford: Clarendon Press; 1965. [4] Raizer YP. Gas discharge physics. Berlin: Springer-Verlag; 1991. [5] Brown SC. Introduction to electrical discharges in gases. New York: Wiley; 1966. [6] Meek JM, Craggs JD. Electrical breakdown of gases. New York: John Wiley and Sons Ltd; 1978. [7] Despotovic D, Cvetic J, Stankovic K, Osmokrovic P. Synergetic effect in a mixture of noble gases around the Paschen minimum. Phys Plasmas 2014;21. art. no. 013507. [8] Alimpijevi c M, Rajovi c Z, Brajovi c D, Stankovi c K, Vujisic M. Influence of the breakdown mechanism to the left of the paschen minimum on the breakdown temperature of the free electron gas Maxwell spectrum. Vacuum 2014;99: 89e94. [9] Uhm HS, Byeon YS, Song KB, Choi EH, Ryu HY, Lee J. Plasma Sources Sci T 2010;17. art no. 113510. [10] Todorovi c R, Vujisi c M, Kova cevi c D, Stankovi c K, Osmokrovi c P. Vacuum 2012;86:1800e9. [11] Mari c R, Stankovi c K, Vujisi c M, Osmokrovi c P. Vacuum 2010;84:1291e5. [12] Osmokrovi c P, Mari c R, Stankovi c K, Ili c D, Vujisi c M. Vacuum 2010;85: 221e30. Vujisi [13] Osmokrovi c P, Ili c G, Stankovi c K, Doli canin C, c M. Determination of pulse tolerable voltage in gas-insulated systems. Jpn J Appl Phys 2008;47: 8928e34. [14] Latham RV. High voltage vacuum insulation e the physical bases. London: Academic Press; 1981. [15] Townsend JS. Electricity in gases. Oxford: Clarendon Press; 1915. [16] Osmokrovi c P, Ðeki c S, Stankovi c K, Vujisi c M. Conditions for the applicability of the geometrical similarity law to gas pulse breakdown. IEEE Trans Dielectr El Insul 2010;17:1185e95. [17] Osmokrovi c P. IEEE Trans Plasma Sci 1993;21:645e54. [18] Osmokrovi c P, Vasi c A. IEEE T Plasma Sci 2005;33:1672e6. [19] Milanovi c Z, Stankovi c K, Vujisi c M, Radosavljevi c R, Osmokrovi c P. IEEE Trans Dielect El In 2012;19:648e59.
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23 [20] Jovanovi c B, Stankovi c K, Vujisi c M, Osmokrovi c P. IEEE Trans Dielect El In 2011;18:954e63. [21] Kova cevi c A, Despotovi c D, Rajovi c Z, Stankovi c K, Kova cevi c A, Kova cevi c U. Nucl Technol Radiat 2013;28:182e90. [22] Jaksic U, Arsic N, Fetahovic I, Stankovic K. Nucl Technol Radiat 2014;29: 123e7. [23] Vujisi c M, Stankovi c K, Osmokrovi c P. Appl Math Model 2011;35:3128e35.
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[24] Stankovi c K, Vujisi c M, Kova cevi c D, Osmokrovi c P. Measurement 2011;44: 1713e22. Stankovi [25] Doli canin C, c K, Doli canin D, Lon car B. Nucl Technol Radiat 2011;26: 164e70. [26] Brajovi c D, Vujisi c M, Stojkanovi c M, Kova cevi c U, Vasi c A. Nucl Technol Radiat 2011;26:164e70.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
The Maxwellian nature of free-electrons' gas spectrum of noble gases at low pressure Malisa Alimpijevi c, Koviljka Stankovi c*, Milan Ignjatovic, Jovan Cveti c Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 July 2014 Received in revised form 1 August 2014 Accepted 2 August 2014 Available online 12 August 2014
A free-electrons' gas spectrum shape in gases under the influence of the electrical field is considered in the theoretical part of the paper. Based on the nature of electrons' interaction with ions and molecules (or atoms) of gas, it was concluded that the spectrum is Maxwellian, if noble gases and subpressures are used. This conclusion was experimentally verified in the Townsend's region of electrical discharge. The gases used were Helium, Neon, Argon, Xenon and Krypton. The utilized pressures ranged from 0.3 Pam to 10 Pam. Interelectrode distances were in the range of 0.1 mme1 mm. It has been assumed that the freeelectrons’ gas spectrum is of the Maxwellian type by means of the mathematical analysis of macroscopically measurable consequences, and this assumption has been verified through the experiment. New constants enabling electrical discharge parameters' calculation in the Townsend's region have been presented according to Townsend's, Takashi's and Maxwell's expressions. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Free electron gas Maxwell spectrum Noble gas Low pressure
1. Introduction Gases in nature represent a gas mixture of neutral atoms (or molecules), ions and electrons. Ions and electrons, created by photoionization, positive-ion ionization and metastable ionization, form a free-ions gas and a free-electrons gas. Each of these components represents a partial “gas” to which the Gas Mixture Laws is valid. The free-electrons' gas has a decisive role in electrical breakdown in gases [1e3]. Namely, the electrical breakdown of gases occurs as a result of collision of free electrons, accelerated by an electrical field, and atoms (or molecules) of gas, which in turn creates new free electrons and ions, which possibly leads to forming the self-sustaining mechanism (positive feedback). Selfsustained discharge forming is controlled by the collision processes' relations comprising charged particles and their transport properties. It can be concluded that the process development of gas electrical breakdown depends on the relative efficiency of mechanisms for generation and annihilation of free electrons, which determine both spectral distribution and free-electrons' gas density during the electrical discharge [4e6]. In normal conditions, i.e. without the electrical field influence, free-electrons' gas spectral distribution (as well as other gas mixture components) is of the Maxwellian type [4,7]. Nevertheless, when such a gas mixture is found within a direct electrical field,
* Corresponding author. Tel.: þ381 113370186; fax: þ381 113370187. E-mail address: [email protected] (K. Stankovi c). http://dx.doi.org/10.1016/j.vacuum.2014.08.005 0042-207X/© 2014 Elsevier Ltd. All rights reserved.
one drift component is superpositioned in the direction of the field by thermal speeds of its charged constituting components [8,9]. Generally speaking, in such a case, free electrons and ions' gas spectrum is no longer of the Maxwellian type. The reason for this is the non-elastic nature of these collisions in the field direction. Namely, a part of free-electrons and free-ions gas energy, taken over from the electrical field, is lost due to the excitation of neutral molecules' rotational and vibration quantum mechanical states. In addition to this, the deviation from the Maxwellian shape of freeelectrons and free-ions gas spectrum also occurs due to their Coulomb interaction with other free electrons and ions. However, these phenomena do not occur in the case of atomic (i.e. noble) gases at low pressures (100 Pae104 Pa) (atomic gases do not have rotational and vibration states, the ionization degree is small at low pressures, and consequently the interaction between the charged particles is negligible). As a result of this, the condition may be considered as the free electrons' interactions are of elastic type, so that the Maxwellian spectrum is not disturbed owing to the electrical field influence. The aim of this paper is to verify the statement on the Maxwellian nature of free-electrons gas spectrum, and to check the possibility of its application in the electrical discharge modeling process for noble gases at low pressures.
2. Electrical breakdown of gases As mentioned above, electrical breakdown of gases is a selfsustaining process based on the primary and secondary processes
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23
20
of electrical discharge in gases. The basic primary process of gas discharge is electron ionization. Electrons ionization occurs when a free electron on a single free path takes over the energy from the electrical field that is greater than (or equal to) the electrons' bond energy in neutral atoms (or molecules). More precisely, this electron becomes an initial electron. This enables electron to perform ionization in the ensuing collision. Once the ionization has been initiated, it leads to the free electrons' avalanche. When the free electrons' avalanche, created in this way, reaches the anode, the process would stop if there were not secondary gas discharge processes, which provide a positive feedback leading to a sequence of avalanches which sum results in the breakdown. Secondary gas discharge processes are divided into cathode active processes (ionic ejection, photoemission and metastable ejection) and the gas active processes (positive-ion ionization, photoionization and metastable ionization). If the dominant secondary processes are active at the electrodes, breakdown occurs according to the Townsend mechanism. If the dominant secondary processes are active in the gas, breakdown occurs according to the streamer mechanism [10e13]. Townsend's (primary) ionization coefficient a and secondary ionization coefficient g are introduced in order to mathematically formulate conditions for the Townsend and streamer mechanisms of breakdown. Starting from the assumption that a single electron creates a new electrons on the path length in the field direction, we can conclude that the free electrons number increase by n free electrons on the path dx is:
while p is the gas pressure, d is the interelectrode distance, and E is the electrical field, and C1,C2,Z1 andZ2 are constants which depend on the type of gas. Secondary ionization coefficient g does not depend on the electrical field and pressure, but it depends on the electrode material and the electrode surfaces' processing type. Due to that, g may be, under constant experiment conditions, considered constant [17,18]. With ionization coefficients defined in such a way, it can be demonstrated that the condition for gas breakdown according to the Townsend mechanism is:
dn ¼ aðxÞndx
Zd
(1)
Based on the expression (1) and the analysis of experiment results [14], two expressions of the a coefficient dependence on the electrical field E and pressure p are provided:
pi h a ¼ C1 p exp C2 E 0
2
(3)
Z∞ si ð3 Þnf ð3 Þd3
(4)
31
where n and 3 are the speed and kinetic energy of free electrons; si is the cross-section for ionization; 3 i is the ionization potential (bond energy) of the observed gas, and f(3 ) is the free-electrons gas spectrum. Under the assumption that the free-electrons' gas spectrum is of the Maxwellian type, and originating in the expression (4) the following expressions are obtained:
p3 þ 2 exp i ZE ZE
p3
i
(6)
and
Z ¼ Z1 ðpdÞZ2
(7)
Zx adx
Zd aðxÞe 0
g
dx ¼ 1;
(8)
0
while the condition for gas breakdown according to the streamer mechanism is:
aðxÞdx ¼ 10; 5:
(9)
0
Expressions (8) and (9) are significantly simplified in case of the homogenous electrical field [19,20]. 3. The experiment and the experiment results' processing
13
where Ci(i ¼ 1,2) and ki(i ¼ 1,2,3) constants depend on the gas type and the range of the pd product (pressure multiplied by interelectrode distance), to which the expressions (2) and (3) are applied. The expression (2) is often called the Townsend expression, while the expression (3) is often called the Takashi expression [15,16]. However, the a coefficient dependence on the pressure and electrical field is most precisely obtained by starting from the definition according to statistical physics [2]:
a ¼ C$p
C ¼ C1 expðC2 pdÞ
(2)
BEp k2 C7 6 B C7 a ¼ k1 p6 41 exp@ k A5 3
aðp; EÞ ¼ n0
where
(5)
The measurements have been performed in the well-controlled laboratory conditions. During the measurement, a disassembling gas chamber and the corresponding vacuum-gas circle have been used. Experiment parameters comprised gas types and the interelectrode distance. Applied gases were noble gases (Helium, Argon, Neon, Xenon and Krypton). The electrodes that provided a pseudohomogenous electrical field were of the Rogowski shape and were made of copper. Before each measurement series electrodes were sandblasted. By sandblasting electrodes, it was enabled the statistic sample of the dc breakdown voltage random variable to be independent from the irreversible changes interelectrode topography during a single measurement series. The U-Test has demonstrated that 1.000 random variables of the dc breakdown voltage statistic sample belong to a unique statistic distribution with the statistic uncertainty of 0.5% (in case of polished electrodes, for the same conclusion, statistic uncertainty equals 5%). After multiple chambers' vacuuming up to the pressure of 104 mbar and 1 bar pressure working gas filling, the chamber pressure was calibrated by a special doses valve. This pressure enabled a spectroscopic purity of working gas during a single measurement series. Namely, with the aim of testing gas purity in the chamber, and utilizing the same procedure, a glass chamber was filled. After developing a stable electrical discharge the presence of any other gas was not detected by means of spectral analysis. Chamber sealing was good. Measuring pressure of Helium at 10 mbar (reduced to 0 C) during a 24-h period indicated no variations in pressure, which means that it was constant during a single measurement series for each of the noble gases (He has the smallest
M. Alimpijevic et al. / Vacuum 110 (2014) 19e23
atom out of all noble gases, so if the pressure was constant for He, it was also constant for other noble gases). Gas pressure in the chamber varied from 102 to 104 Pa. Interelectrode distance was calibrated by the electronic micrometer screw from 0.1 mm to 1 mm (a corresponding pair of electrodes was made for each interelectrode distance). Supply source had a voltage rise rate of 8 V/s. Its ripple was smaller than 1%. Breakdown voltage measurement was performed by the Ohm ratio 1:273. During the experiment, measurement instruments were placed in the protection booth comprising the level higher than 100 dB. Measurement signal was fed into the booth by a twice protected, grounded, 50U cable. Combined measurement uncertainty (which was dominantly of type B, since the dc breakdown voltage is considered as a deterministic quantity and therefore the dispersion originating from the measurement repeating procedure can be neglected) was estimated at less than 3% [21e24]. The procedure of determining the Paschen curve for a single gas type comprised the following steps: 1 e the interelectrode distance was calibrated and the chamber was sealed; 2 e the chamber was plugged into the gas-vacuum circle, vacuumed and filled with gas and again vacuumed up to 104 mbar (this procedure of the socalled rinsing was repeated 5 times); 3 e working pressure in the chamber was calibrated and reduced to 0 C; 4 e the chamber was plugged into the electrical circuit; 5 e the chamber conditioning was performed through 50 consecutive dc voltage breakdowns; 6 e a measurement of 50 consecutive dc breakdowns was performed with a one-minute break between two breakdowns (for the U-test requirements, 1.000 values of the dc voltage breakdowns were measured); 7 e a new vacuuming and rinsing of the chamber were performed, and the procedure was repeated with next working pressure value (without chamber disassembling and interelectrode distance calibrating, i.e. utilizing a single interelectrode distance, a single series of measurement was performed); 8 e upon the completion of a single measurement series, new electrodes, made for a new interelectrode distance, freshly sandblasted and mounted onto the gas chamber, were utilized. The statistic samples set of the dc voltage breakdown random variables, obtained in such a way, was processed in the following steps: 1 e a statistic sample was subjected to the modified Chauvenet's Test [25] with the aim of rejecting the spurious measurement results; 2 e the dc voltage breakdown mean value was determined, as well as standard deviation; 3 e a significance level was determined according to the Student's distribution for each statistic sample (it has been demonstrated that it is always small and may be disregarded) 4 e based on the dc voltage breakdowns' mean values for certain pressures and interelectrode distances, an experimental Paschen curve was determined for all tested gases (in the region of pd product values, in which measurements were performed); 5 e experimental Paschen curves were fitted by the expression (8), in addition to utilizing the primary ionization coefficient obtained by expressions (2), (3) and (5); 7 e based on the constants obtained by fitting in expressions (2), (3) and (5), relevant dependences of the a/p ratio in the pd product function were drawn.
21
Fig. 1. Paschen curves obtained by using: equation (2) - - - ; (3) $ $ $ ; and (5) d for calculating the a coefficient along with experimentally results A obtained for Argon.
Fig. 2. Paschen curves obtained by using: equation (2) - - - ; (3) $ $ $ ; and (5) d for calculating the a coefficient along with experimentally results A obtained for Xenon.
has been observed. The constants C1(i ¼ 1,2) and Zi(i ¼ 1,2) in the expression (5), obtained through fitting, are given in Table 3. Figs. 1 and 2 show that the fitted curves are in agreement with the experimentally obtained values. However, for all fitted curves, the
Table 1 Constants applied to the equation (2). Gas
C1[1/Pam]
C2[V/Pam]
Helium Neon Argon Xenon Krypton
0.763 0.9 6.649 12.7 12.4
29.154 46.1 233.147 290.5 298.7
4. Results and discussion Figs. 1 and 2 show the Paschen curves for Ar and Xe, respectively, obtained by using equations (2), (3) and (5) for calculating the Townsend a coefficient along with experimentally results. Constants Ci(i ¼ 1,2) and ki(i ¼ 1,2,3) used in expressions (2) and (3) and obtained by fitting are shown in Tables 1 and 2, respectively. Constants values shown in Tables 1 and 2 differ from the corresponding constants found in the literature. The difference is the consequence of the fact that a relatively narrow region of small pd product values
Table 2 Constants applied to the equation (3). Gas
K1[1/Pam]
K2[V/Pam]
K3[V/Pam]
Validity range E(x)/p [V/Pam]
Helium Neon Argon Xenon Krypton
2.23 3.14 10.45 20.88 12.88
6 7.05 18.38 31.35 19.88
120 150 341.25 585 382.5
6e225 7.05e300 18.38e1200 31.35e1800 19.88e1500
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22 Table 3 Constants applied to the equation (5). Gas
C1
C2
Z1
Z2
Helium Neon Argon Xenon Krypton
0.4066 0.4565 3.2600 6.5560 6.2530
0.03423 0.02594 0.08472 0.14080 0.15920
0.42200 0.24150 0.04656 0.02880 0.03439
0.2065 0.1750 0.1374 0.2209 0.1997
best fit is obtained when expression (5) is used for a ionization coefficient. Figs. 3 and 4 show the a/p ratio dependence on the pd product calculated according to expressions (2), (3) and (5) obtained for Argon and Xenon, respectively. Based on the results obtained in Figs. 1e4, it can be concluded that a coefficient, obtained under the assumption of the Maxwellian free-electrons gas spectrum (equation (5)), is qualitatively and quantitatively compliant with the corresponding dependences obtained according to Townsend (equation (2)) and Takashi (equation (3)). In addition to this, since it has been shown, in Figs. 3 and 4, that a ionization coefficient implementation, obtained under
the Maxwellian free-electrons gas spectrum assumption, leads to the best dependence fit of the experimentally obtained Paschen curve, it can be concluded that this assumption has been justified. The same conclusion is valid for all other noble gases. The limitation on using both Townsend and Takashi equations determined by validity range E(x)/p (shown in Tables 1 and 2) has been overcame by derived equation (5) since the constants C and Z are functions dependant on the product pd. In engineering practice obtained results are significant for modeling and designing the gas filled surge arresters [26]. 5. Conclusion Based on the obtained results, it can be claimed with certainty that free-electrons gas spectrum of noble gases in the Townsend's discharge region is of the Maxwellian type (or very similarly shaped to it). The conditions in which this was proved are also valid for the streamer discharge region at low pressures. Apart from the fact that recognizing free-electrons gas spectrum enables the calculation of a sequence of electrical discharge physical parameters in gases, it is also significant in engineering practice. Namely, gas filled surge arresters used for the insulation coordination at low-voltage level have been designed as two-electrode systems isolated by a noble gas at subpressure and small interelectrode distances. Under these conditions, recognizing free-electrons gas spectrum enables a simulation of the gas filled surge arresters characteristics, as well as optimization of its characteristics in the construction phase. Acknowledgment The Ministry of Education, Science and Technological Development of the Republic of Serbia supported this work under contract 171007. References
Fig. 3. Dependence of the Townsend coefficient to pressure ratio (a/p) on the pd product for Argon according to equations equation (2) - - - ; (3) $ $ $ ; and (5) d.
Fig. 4. Fig. 3. Dependence of the Townsend coefficient to pressure ratio (a/p) on the pd product for Xenon according to equations equation (2) - - - ; (3) $ $ $ ; and (5) d.
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