Papers in Physics, vol. 14, art. 140004 (2022) Received: 20 May 2021, Accepted: 7 October 2021 Edited by: A. B. Márquez Licence: Creative Commons Attribution 4.0 DOI: https://doi.org/10.4279/PIP.140004 www.papersinphysics.org ISSN 1852-4249 Experimental study on the similarity of gas discharge in low-pressure Argon gaps Prijil Mathew1∗, Sajith T. Mathews1, Paul Issac1, P. J. Kurian1 Through experiments and theoretical analysis, we investigated the similarity of gas dis- charge in low-pressure Argon gaps between two plane-parallel electrodes. We found that the breakdown voltages depended not only on gap length and the product of gas pressure and gap length, but also on the aspect ratio of the gap, i.e. Ub = f(pd, d/r). When we considered similar discharge gaps, the radius r, gap length d and gas pressure p fulfilled the conditions of p1r1 = p2r2 and p1d1 = p2d2. In this situation the reduced field E/p was also constant. The voltage-current characteristic curves of similar gaps were approximately the same, which is a novel experimental result. Comparison of the discharge physical param- eters of the scaled-down gap and prototype gap shows that the proportional relations can be derived from the similarity law. Our experimental results provide some instructions on extrapolating two similar gaps and their discharge properties. Application of the similarity law is straightforward when we scale the discharges up or down if they are too small or large. I Introduction Paschen’s famous law states that the breakdown voltage of a gas gap does not depend individu- ally on the gap length d and gas pressure p, but depends on their product pd; i.e. Ub = f(pd) [1–9]. According to Townsend Paschen’s law is a unique case, with a uniform electric field, of a more general similarity theorem which can be used for breakdowns in non-uniform fields if they are depen- dent on ionisation by electron collision with neu- tral particles [2, 3, 10, 11]. Von Engel has discussed and summarised the similarity theorem successfully [6, 12–14]. He specified that under certain condi- tions similar discharges can be produced in gap- ∗prijilmk@gmail.com 1 Department of Physics, St. Berchmans College Campus, Mahatma Gandhi University, Kottayam, 686101 Kerala, India. sthat have the same geometrical shape but differ- ent linear dimensions. It is also notable that the similar discharges have all the physical properties, such as density of the charged particles and current density, in the correct proportions, and also display similar voltage-current characteristics [2,10,15–19]. It is also possible to use the known properties of the discharge in one gap to derive the charac- teristics of the discharges in another geometrically identical gap due to the similarity of gas discharge; this is useful in cases where experimental studies may not be practical or even possible [2, 10, 16, 20]. One pre-condition for such an experiment is to ver- ify that there is a similar discharge in the speci- fied geometrically similar gaps. Progress with sim- ilar discharge has been made recently in micro- discharge situations with the huge glow discharge of the International Thermonuclear Experimental Re- actor (ITER) and the picoseconds pulse discharge [2, 5, 10, 21, 22]. 140004-1 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. In this paper we have used experiments and the- oretical analysis to investigate the similarity of gas discharge in low-pressure argon gaps between two plane-parallel electrodes. The results show that the breakdown voltages of these gaps depend not only on the product of gas pressure (p) and gap length (d) but also on the aspect ratio of the gap; i.e. Ub = f(pd,d/r). Theoretically, it has also been proved that Ub = f(pd,d/r), the non-uniform elec- tric field between plane-parallel electrodes, is a spe- cial case of the similarity theorem of gas discharge [2, 10, 16]. The experiments show that similar glow discharges exist only in two gaps with a limited scaled-down factor k. Similarly, the theoretical analysis shows that processes such as stepwise ion- isation and inelastic collision of the second kind vi- olate the similarity of the discharge as k increases. The voltage-current (V −I) characteristics of the glow discharge region studied in similar conditions also confirm the similarity in the gas breakdown. II Conditions for similar discharges The first necessary condition for similarity dis- charges in two geometrically similar gaps is that the product of gas pressure (p) and gap length (d) for these two gaps should be the same [2, 10, 19], i.e. p1d1 = p2d2, which ensures that the total num- ber of collisions for one electron to cross the gap is the same [23]. The second condition is that the reduced field in these two gaps should be the same [2, 19, 24–27]; i.e. E1/p1 = E2/p2 for the uniform electric fields or E1(p1x1)/p1 = E2(p2x2)/p2 for the non-uniform fields at the corresponding points where p1x1 = p2x2, thus ensuring that the average energy of the electrons is the same [16, 23, 28]. One additional condition is required for the sim- ilarity discharges in two geometrically similar gaps: the discharges in these two gaps should be domi- nated by the physical processes allowable for a sim- ilar discharge, known as allowed processes [2, 29]. Many physical processes are happening in the gas discharge, such as stepwise ionisation, ionisation by single collision, diffusion, photoionisation, Penning ionisation, recombination and electron attachment [2, 10, 30]. In appendix 1 of his book von Engel has shown how to test whether a process is for- bidden or allowable for a similar discharge. An al- Figure 1: Experimental setup. lowed process is any process in which the change rate of particle density fulfils the conditions stated in Eq. (1) [2, 29, 31]. Otherwise it is a forbidden process that is not forbidden for gas discharge, but not allowed for similar discharge [2, 10, 29]. It is not possible to distinguish whether a discharge is dominated by forbidden or allowed processes, and this is not controllable.( dN dt ) gap1 = 1 k3 ( dN dt ) gap2 , (1) where k is a scaled-down factor or the ratio of the linear dimension of gap 1 to 2 [2, 10, 32]; N is the particle density. III Experimental setup The setup consisted of a cylindrical vacuum cham- ber made of stainless steel, about 30 cm in diam- eter and 80 cm in length. An aluminium stand was placed at a height of about 100 cm from the ground, on which the chamber was mounted hor- izontally. A digital pirani gauge (model IVDG – 1000) was also attached to the aluminium stand, showing the pressure inside the vacuum chamber in millibars. A rotary pump was connected to the cylindrical chamber to evacuate the pressure inside the chamber. A gas inlet was used to fill the cham- ber with gas. A glass discharge tube was placed inside the vac- uum chamber. The electrodes (anode and cathode) 140004-2 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. 0 5 10 15 20 25 30 35 40 350 400 450 500 550 600 650 700 B re a kd o w n V o lt a g e (V ) pd (Torr - cm) d=30cm r=4cm d/r=7.5 d=40cm r=4cm d/r=10 d=50cm r=4cm d/r=12.5 Figure 2: Paschen’s curves for constant electrode radius (r) and varying inter-electrode distance (d). were placed inside the discharge tubeusing a Wilson fed through arrangement from the end of the glass tube. This arrangement enabled us to change the separation during the experiment. The electrodes were made of stainless steel of about 1 mm thick- ness and a diameter of a few centimeters (5 cm, 8 cm and 10 cm, see Table 1). Thin circular mica sheets of about 7 cm diameter were placed around the electrodes to prevent field lines beyond the electrodes. We used a DC voltage supply that varied over a range of 0 to 1000 V, with a maximum output current of 1 A. To measure and limit the discharge current we connected a resistor (variable) in series. Table 1: Parameter values are chosen for similarity ver- ification. Gap d [cm] r [cm] p [mbar] d/r a Gap 1 50.0 5.0 0.20 10 1.00 Gap 2 40.0 4.0 0.25 10 1.25 Gap 3 25.0 2.5 0.40 10 2.00 Gap 4 25.0 5.0 0.20 5 1.00 Gap 5 20.0 4.0 0.25 5 1.25 Gap 6 12.5 2.5 0.40 5 2.00 Gap 7 5.0 5.0 0.20 1 1.00 Gap 8 4.0 4.0 0.25 1 1.25 Gap 9 2.5 2.5 0.40 1 2.00 0 5 10 15 20 25 30 35 40 450 500 550 600 650 B re a k d o w n V o lt a g e ( V ) pd (Torr - cm) d=25cm r=2.5cm d/r=10 d=40cm r=4cm d/r=10 d=50cm r=5cm d/r=10 Figure 3: Paschen’s curves for the same d/r values. IV Similarity in gas breakdown In 1928 Townsend revealed that the breakdown voltage Ub for a longer gap was higher than that for a shorter gap, even with an equal value of pd [5, 11, 28, 33–36], i.e. Ub = f(p,d) 6= f(pd). Conse- quently, Paschen’s curves for the gaps with differ- ent d values do not superimpose onto each other. In this paper, this phenomenon was investigated by measuring the breakdown voltages of low-pressure Argon gaps between two plane-parallel electrodes. A schematic representation of the experimental set up is shown in Fig. 1. A DC voltage was used in the electrodes. Figure 2 shows typical results with different d/r ratios, where r is the radius of the electrodes. From Fig. 2 we see that as d/r in- creases the Paschen’s curves move to the right and upwards. In Fig. 3 we observe that the curves with an equal value of d/r superimpose onto each other. From the experiments we conclude that the break- down voltage of these gaps depends on two factors: the product of gap length and gap pressure, and the aspect ratio of the gap, i.e. Ub = f(pd,d/r). The electric field that exists in the gap between two parallel electrodes is determined by d/r, and in the case of a non-uniform electric field break- down voltage would be a function of not only pd but also d/r. The value of E/p is the same for the same value of d/r, and when the d/r value is different the field distributions are also different [16]. In fact, the dis- tribution of the electric field is a function of d/r. A 140004-3 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. mathematical expression is obtained by the polyno- mial fit of the profile of the electric field [35, 37, 38]. E Eav = f ( x d , d r ) or E p = U pd f ( px pd , d r ) , (2) The breakdown criterion, the self-sustained condi- tion for Townsend discharge can be expressed as γ [ exp (∫ d 0 α(x)dx ) − 1 ] = 1 or ∫ d 0 α(x)dx = ln ( 1 + 1 γ ) , (3) where γ is the coefficient of a second electron emis- sion from the cathode by ion bombardment; α is the electron impact ionisation coefficient and is a function of the reduced field E/p, i.e. α = A p exp ( −B E/p ) , (4) where A and B are constants. By substituting (2) into (4), and then into (3), we obtain A ∫ pd 0 exp   −B UB pd f ( px pd , d r )  d(px) = ln (1 + 1 γ ) . Now, dividing both sides by A and substituting px = y -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -100 0 100 200 300 400 500 600 700 800 900 Gap 2 Gap 5 Gap 8 E le c tr o d e V o lt a g e ( V ) Discharge Current (mA) Figure 4: V −I characteristic curves for different gaps. ∫ pd 0 exp   −Bpd UB f ( y pd , d r )  dy = ln ( 1 + 1 γ ) A , ∫ pd 0 exp   −Bpd UB f ( y pd , d r )  dy = f (UB,pd, d r ) , f ( UB,pd, d r ) = ln ( 1 + 1 γ ) A . (5) From Eq. (5), theoretically, we prove that the breakdown voltage is a function of d/r and pd. The same results are also observed in our exper- iments. From Eq. (5) we see that, for any two gas gaps, if p1d1 = p2d2 and d1/r1 = d2/r2, the break- down voltage for these two gaps will be the same, i.e. Ub1 = Ub2. Substituting these three equations (p1d1 = p2d2, d1/r1 = d2/r2, Ub1 = Ub2) into Eq. (2), we know that the reduced field E/p in these two gaps at the corresponding point p1x1 = p2x2 will be equal, i.e. E1(p1x1)/p1 = E2(p2x2)/p2. Here, Ub = f(pd,d/r) is also a special case of the simi- larity theorem, with non-uniform electric fields be- tween plane-parallel electrodes [39,40], and extends Paschen’s law to this special case. It should be indicated that Ub = f(pd,d/r) also applies to the uniform electric field where d/r → 0, and it reduces to Paschen’s law. -2 0 2 4 6 8 10 12 14 16 -100 0 100 200 300 400 500 600 700 800 Gap 1 Gap 2 Gap 3 E le c tr o d e V o lt a g e ( V ) Discharge Current (mA) Figure 5: V −I characteristic curves for similar gaps. 140004-4 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. V Voltage-current characteristic curves of the similar gaps The voltage-current (V −I) characteristics of DC glow discharge plasma can be obtained either by gradually increasing the external voltage or by low- ering the external resistance [3,4,28,37,41–47]. Ex- ternal high resistance can be introduced to limit the amount of discharge current produced [47]. The operation region of glow discharge can be identi- fied by studying the voltage-current characteristics. The nonlinear nature of glow discharge plasma can be analysed by studying the V −I characteristic [48, 49]. Moreover, the V −I characteristic is the primary step that enables us to find out whether two discharge gaps are similar or not. The distance between the electrode (d), electrode radius (r), gas pressure (p), and external resis- tance (R) were kept fixed and the applied voltage (VA) was varied in equal steps over a wide volt- age range. High resistance was introduced into the circuit to limit the amount of current produced. For each voltage applied, the corresponding volt- age across the resistor (VR) was measured. The discharge current (I = Vr/R) and electrode volt- age (V = VA−VR) were calculated at each step and the forward characteristics obtained. After reach- ing the maximum voltage, the voltage was reduced in equal steps as before, the discharge current and electrode voltage were calculated and the reverse characteristics obtained. Placing the discharge cur- rent (I) on the x-axis and electrode voltage (V ) on the y-axis gives the typical V−I characteristics, as shown in Fig. 4 and Fig. 5. For any two gaps arranged to fulfil the relation- ships p1d1 = p2d2, p1r1 = p2r2 and E1/p1 = E2p2, the gaps are said to be similar [16]. Along with the physical quantities mentioned, the voltage-current characteristic is approximately the same. Experi- mentally, the validity of this similarity law for V−I characteristics of a large discharge tube is verified here for three discharge gaps satisfying the above similarity relation. The external resistance chosen for all the three cases is 10 kΩ. In a physical sys- tem, the occurrence of hysteresis refers to the para- metric dependence of a state on its history. Hys- teresis is a clear sign of nonlinearity in the system [43, 47]. The jump phenomenon and hysteresis in discharge current are very well known phenomena in gas discharge, due to the variation in discharge -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -100 0 100 200 300 400 500 600 700 800 900 Gap 2, Reverse Gap 2, Forward Gap 5, Reverse Gap 5, Forward Gap 8, Reverse Gap 8, Forward E le ct ro d e V o lta g e ( V ) Discharge Current (mA) Figure 6: V−I characteristic and hysteresis for different gaps. voltage [43]. A gradual increase in discharge volt- age causes a sudden increase in discharge current. This is called the jump phenomenon. The current increases gradually with an increase in voltage, con- firming the operation of glow discharge plasma in the abnormal region [4–6]. After reaching an ap- plied voltage of 900 V, the voltage is decreased in steps of 10 V. The characteristic curve does not re- trace through the forward path. There is a decrease in the amount of current discharged in the reverse direction. The current lags behind the voltage and hysteresis is observed; the jump phenomenon can also be observed in the reverse direction. The V−I characteristic and hysteresis for gaps having differ- ent d/r is shown in Fig 6. Figures 7, 8 and 9 show the V −I characteristic and hysteresis for similar gaps and confirms the similarity experimentally. When p1d1 = p2d2 = p3d3, the total number of collisions undergone by one electron to cross the gap will be the same for the three gaps. The electric field in a gap between two plane-parallel electrodes can be obtained using the ratio d/r; the distribu- tion of the electric field is a function of the d/r ratio. The E/p ratio for the three gaps is found to vary almost constantly, by making the d/r ra- tio constant . The E/p ratio signifies the energy gained by the electron between two consecutive col- lisions [23]. By fixing the parameters E/p and pd, the electron multiplication rate of the gaps becomes fixed [23]. The rate of electron multiplication de- 140004-5 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. -2 0 2 4 6 8 10 12 14 16 -100 0 100 200 300 400 500 600 700 800 Gap 1, Reverse Gap 1, Forward Gap 2, Reverse Gap 2, Forward Gap 3, Reverse Gap 3, Forward E le ct ro d e V o lta g e (V ) Discharge Current (mA) Figure 7: V −I characteristic and hysteresis for similar gaps (d/r = 10). termines the rate of ionization, which in turn deter- mines the rate of discharge current [23]. For voltage varying constantly for the three gaps, the discharge currents produced become equal, as the ionization rate due to electron multiplication is the same. The three curves overlap, and the occurrence of forbid- den processes in this gap can be discarded. -2 0 2 4 6 8 10 12 14 16 18 0 200 400 600 800 Gap 4, Reverse Gap 4, Forward Gap 5, Reverse Gap 5, Forward Gap 6, Reverse Gap 6, ForwardE le c tr o d e V o lta g e (V ) Discharge Current (mA) Figure 8: V −I characteristic and hysteresis for similar gaps (d/r = 5). 0 5 10 15 20 25 0 200 400 600 800 1000 Gap 7, Reverse Gap 7, Forward Gap 8, Reverse Gap 8, Forward Gap 9, Reverse Gap 9, ForwardE le ct ro d e V o lta g e ( V ) Discharge Current (mA) Figure 9: V −I characteristic and hysteresis for similar gaps (d/r = 1). VI Conclusions In a special case, Ub = f(pd,d/r) is the breakdown voltage between two plane-parallel electrodes with low-pressure gaps, and is the non-uniform electric field between plane parallel electrodes which is con- nected with the similarity theorem of gas discharge. Similar glow discharge was observed only in two Ar- gon gaps which had a limited scaled-down factor k, and in the case of forbidden processes such as the inelastic collision of the second kind and the step- wise ionisation which tend to violate the similarity of the discharge as k increases [2, 10]. From the ex- periments we observe a clear cathode fall layer, a positive column between the electrodes, and a neg- ative glow zone. These findings indicate that the discharge is a typical glow discharge [6, 37]. The comparison of discharge physical parameters be- tween the scaled-down gap and prototype gap en- ables us to find the proportional relations derived from the similarity law. The same voltage-current characteristic curves of the two similar gaps are also obtained. Studies have been carried out on DC glow dis- charge at low pressure for more than 100 years; the mechanism of the discharge is well studied. The area of DC glow discharge has many applica- tions, but some of the issues remain unsolved. The glow discharge cleaning of the International Ther- monuclear Experimental Reactor (ITER) is consid- ered one of the unsolved problems. In the case 140004-6 Papers in Physics, vol. 14, art. 140004 (2022) / M. Prijil et al. of ITER, a huge tokamark device, the fusion re- action takes place inside a toroidal chamber. The fusion reaction should be stopped after a period of operation, and once stopped the inner wall of the toroid needs to be washed with DC glow dis- charge plasma . This cleaning involves inserting small electrodes that function as the anodes for the glow discharge on the inner wall. The inner wall serves as the cathode for the glow discharge be- cause it is electroisolated from the small anodes. The question before designers of ITER is whether the DC glow discharge plasma made up of small anodes can uniformly cover the huge wall of the toroid. Unfortunately, it is not possible to showcase the full-scale experiment at present. In this paper, we tried to answer the ITER designers’ question using a scaled-down experiment, and investigated whether the glow discharged plasma consisting of small anodes can uniformly cover the wall of the scaled-down chamber or not. The affirmative an- swer obtained from the scaled-down experiment can be extrapolated to ITER. Acknowledgements - The experimental work- was carried out in the Plasma laboratory at St. Berchmans College, Mahatma Gandhi University, Kerala, India, set up under the project of the Board of Research in Fusion Science & Technology (BRFST), India. 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