AP10_1.vp 1 Introduction So far it has been believed [1] that electron populations n of avalanches crossing a discharge gap d in a homogeneous electric field are governed by Furry statistics [2–3] w n d n n n n n n n 0 1 11 1 1 1 ( , ) exp� � � � � � � � � � � ��� � � � � �� ��� � � � � � � � � � � � � , exp ( ) ,n x x d � d 0 (1) where w0 and � are the probability density function and the first Townsend ionization coefficient, respectively. The valid- ity of this law used to be accepted [1] independently of the size of the avalanches, i.e., regardless of the magnitude of their mean electron content n, in spite of the fact that there were strong indications [1], [4–5] showing different statistical be- havior. Especially with highly populated avalanches n �105, clear deviations from the Furry law (1) were often observed [4–5]. It has been illustrated many times [6–12] that the population statistics of such highly populated avalanches and streamers obey Pareto (fractal) statistics w n d a D1 1( , ) ( )� � � �n , (2) where a is a constant and D is the so-called fractal dimension [13]. Convincing examples of different statistical behavior of highly and lowly populated avalanches can be found in the earlier work of Richter [5]. He measured population statistics in ether under discharge conditions that were favorable for creating a mixture of pre-streamer and streamer avalanches, with a majority of the latter. Such highly populated ava- lanches (n �108) provided distribution functions with a very deep bending in the semilogarithmic co-ordinate system, where the population statistics of avalanches should show linear be- havior in accordance with the Furry law. When analyzing Richter’s curves in the bilogarithmic system, one can easily rec- ognize two neighboring different regions (see Fig. 1). There is a longer linear part and a shorter non-linear (bent) part. The bending of the latter corresponds to the exponential Furry behavior typical for less populated avalanches, whereas the linear part represents the Pareto behavior characteristic for highly populated avalanches and streamers. This figure clearly illustrates that the exponential and power fitting func- tions in the Furry and Pareto regions, respectively, represent a good choice among possible analytical candidates, since both the fits follow the experimental data well. Another instructive example showing how well the Pareto power function represents the experimental data is given in Fig. 2. In this case, avalanches were detected across a resis- tance as short voltage pulses with random heights u. (The re- sistance (R �100 k�) was connected in series to the discharge gap (C) so that the two components formed a classical RC- -circuit – for more details see [14].) Since the voltage pulses u were not calibrated against the number of electrons n, the resultant distribution curves w1 are dependent on u instead of n. Assuming linear proportionality u c n� � , the curve w u c u D1 0 1( ) ( )� � � � will preserve the same shape as w n1( ), i.e. they will both possess the same slope � �( )1 D . Recently a new statistical pattern has been developed [14] w n d G n K N N n G n K N d j j J j d n d ( , ) ( ) ( ) � � � � � � � � � � � � � � � � 0 1 1 j j J j d n N n � � � � � � ����� � � ����� 0 exp , (3) w n d G F n d J d N K D K N ( , ) ( , ), , , , ln ln . � � � � � � � � 1 1 1 (4) Pattern (3) unifies both Furry (1) and Pareto (2) statistics into a single analytical form. For example, if J � 0, the Furry distribution results from (3), whereas for J � 0 a superposi- tion of Furry/exponential functions creates Pareto behavior, i.e. the linear section on the graph w(n, d) plotted in biloga- rithmic co-ordinates, as shown in Fig. 1. In addition, a rigor- ous mathematical proof was presented in Ref. [14], showing equivalence between statistical forms (2) and (3). In order to explain the meanings of parameters G, K , N, and nd used in generalized analytical form (3), it is necessary 40 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 50 No. 1/2010 Statistical Distributions of Electron Avalanches and Streamers T. Ficker A new theoretical concept of fractal multiplication of electron avalanches has resulted in forming a generalized distribution function whose multiparameter character has been subjected to detailed discussion. Keywords: fractal multiplication of electron avalanches, Furry and Pareto statistics, fractal dimension. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 41 Acta Polytechnica Vol. 50 No. 1/2010 Fig. 1: Population statistics of a mixture of lowly and highly populated avalanches. Statistical relevancy: Furry fit R2 0 9975� . and � 2 0 00018� . , Pareto fit R2 0 998� . and � 2 0 0001� . . Data taken from [5]. Fig. 2: Avalanche statistics (voltage pulses) in air at normal laboratory conditions. After [14]. to make a brief description of the scenario of fractal avalanche multiplication (Fig. 3). Let us consider a parent avalanche started at the cathode. After crossing a critical distance �, it may gather a certain number of electrons N � exp ( )�� (� is the first Townsend ionization coefficient), and the uv-radia- tion associated with collisional ionization may initiate K new displaced avalanches (K is the so-called multiplicity). The dis- placed avalanches continue their own independent tracks, and after passing a distance � they may (or may not) generate K 2 new displaced avalanches of the second generation. In general, the j-th generation of displaced avalanches contains K j avalanches and the highest generation J is limited by the length d of the discharge gap, critical distance � and the value of �, i.e. J d� � �1. The populations of displaced ava- lanches of the j-th generation that reach the anode can easily be expressed: � �n d jd j, exp ( )� �� � . The term n nd d� ,0 is a population of the parent avalanche ( j � 0). Each generation has its own distribution function w n K N n N nd j j d j d n , ( ) ( ) � � � �� � � � � � � � 1 1 and their sum over all generations ( j J� 0 1, , ,� ) leads to sta- tistical pattern (3) – more details can be found in Ref. [14]. The symbol G represents the normalization constant. Since the process of fractal avalanche multiplication is highly sto- chastic, average values K N nd, , � and are used. Creation of additional smaller avalanches inside the dis- charge gap is a rather different process from that of branch- ing with streamers. While multiplication of avalanches results in creating a set of separate avalanches, streamer branching leads to a connected network of plasma channels. A common item of these two processes may be photoionization. Photo- ionization is the “driving force” necessary for streamer propa- gation, but in the case of avalanches it represents one of the possible creation mechanisms. However, streamer branching, as described by recent research papers [15–18], seems to be a complex process. Kulikovski [15] interpreted streamer branching as an instability that transforms the non-standard streamer into a number of standard streamers. Pancheshnyi [16] described the effects of streamer branching on the basis of background ionization and photoionization. Arrayas et al. [17] described the splitting at the streamer tip as a Laplacian instability. Montijn, Ebert and Hundsdorfer [18] compared this instability to the branching instability of fluid interfaces in viscous fingering. There are many other works that nu- merically simulate propagation and branching of streamers. Nevertheless, to our knowledge there is no work that numeri- cally simulates fractal multiplication of avalanches as a basis for explaining the Pareto behavior of avalanche population statistics, though existing numerical models of streamer growth could determine these statistics, including avalanche multiplication. The present paper focuses on a recently published statisti- cal model [14] concerning populations of electron avalanches that undergo fractal multiplication within the discharge gap. The discussion focuses on the model parameters and their connections with physical processes underlying the phenom- enon of fractal avalanche multiplication. The discussion ex- 42 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 50 No. 1/2010 Fig. 3: Scheme of fractal avalanche multiplication. After [14]. plains in detail the role of each particular parameter, and pro- vides a deeper insight into the model operation. 2 Discussion of model parameters The probability density function (3) has the character of a fitting function containing four parameters J, K N, and nd which, together with dimension D, deserve further discussion and explanation. 2.1 Parameter nd This parameter represents an average number of electrons n dd � exp ( )� in the parent avalanche. For a given discharge gap, gas content and physical conditions, the value of nd is a fixed number. However, from the viewpoint of fractal theory nd cannot be considered as a fixed reference population scale that will “anchor” all remaining avalanche components (displaced av- alanches), since each generation ( j) of displaced avalanches has its own scale (n Nd j ) and all these population scales are as important as nd . Thus, the statistical set of all avalanches is a mixture of many mutually different but equally important population scales, none of which stands for a unique, basic scale defining a reference. This is one of the basic properties of all fractal objects. Now that the principle of the lost refer- ence scale has been mentioned, it is clear that it has no sense to change the only parameter nd at fixed others to model the transition from Furry statistics to Pareto statistics. The only item that will be influenced by changing nd is the position of the linear section on the graph log (w) versus log (n). But if a complete set of avalanches is considered, i.e. n � �( , )0 , nd � � (for d � �), the linear section will be infinitely long and there will be no need to change nd . In such a case the parameter nd will lose its analytical role, which consists in shifting the linear section along the graph when an incom- plete avalanche set is fitted. According to the adopted statisti- cal concept, it is not the value of nd that governs the transition between Furry and Pareto statistics. The high values of nd that can be observed when such a transition occurs seem to be only an accompanying effect (not a primary effect). 2.2 Parameter N The value of this parameter is intimately connected with the critical distance � at which the primary avalanche may generate the first displaced avalanche, i.e. N � �exp (� � . In other words: the average distance � is necessary for the primary avalanche to assemble a certain number of electrons N that are capable of generating sufficient uv radiation (for- mation of photon sources) to facilitate the creation of dis- placed avalanches. It is assumed that the effective ionization length � [15], [19], which is passed by photons prior to their absorption (photoionization events), is independent of �. (The photoionization process is assumed to be effective in the case of molecules that have been excited to their higher energy states in previous collisions with electrons. The dif- ferent values of the first ionization potentials of N2 and O2 molecules in air make this process still easier.) Although the values of these two parameters can be arbitrarily different, both the processes, i.e. the appearance of the critical popula- tion N and the appearance of the first displaced avalanche, are almost synchronous, due to the very high speed of the photons. However, it should be mentioned that not all pho- tons are capable of performing photoionization, and not every photoionization terminates by starting new avalanches, and, of course, not all newly-created avalanches propagate independently of the parent avalanche (some of them may be integrated into the body of the parent avalanche). The quantity N actually represents a measure of the capa- bility of parent avalanches to generate displaced smaller ava- lanches by means of the complete photoionization process. From this viewpoint, it is clear that the number of electrons adequate for this purpose must be higher than one, i.e. N �1. 2.3 Parameter J This parameter determines an extension of the fractal re- gion (linear section of the graph log (w) versus log (n)). If the whole region is measurable by the experimental device that is used, parameter J can be estimated accurately and represents so many avalanche generations – i.e. “humps” [14] on the graph – that are capable of covering all the linear region mea- sured. This is quite easy to realize, e.g., heuristically (trying various numbers of generations). In such a case, the value J provides a right number of displaced avalanche generations and is equal to its upper limit J dmax � �� 1. However, experimental devices are sometimes not capable of measuring the whole extent of possible data. They usually measure in some restricted interval (measuring window). The result is a population distribution restricted to a certain extent which is narrower than the real extent, and parameter J, when fitted to the length of such a linear section, will not represent an exact number of avalanche generations but, instead, it will be smaller J J d! � �max � 1. This is the case for the statis- tics in Fig. 4. Inserting the length d � 0 7. mm of the discharge gap used and the value � " #� 57 56 m into the formula of the upper limit J dmax � �� 1, one can obtain eleven genera- tions Jmax �11, but the fitting to the measured linear section in Fig. 4 gives J J� !7 max, which seems to be a consequence of constrained measurements. Therefore, in both the cases – © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43 Acta Polytechnica Vol. 50 No. 1/2010 in constrained and unconstrained measurements – the quick- est way to determine J is heuristic testing. 2.4 Parameter K Parameter K , which has been termed “multiplicity”, speci- fies multiplication (splitting) of displaced avalanches within all their generations. Its value can hardly be predicted, be- cause the number of the displaced avalanches is a matter of purely stochastic processes, and there is no rule for estimating this quantity. However, some limitations can be specified: K cannot be zero, since in this case lim D K� � �� 0 , which is an unacceptable result. In addition, K must be a positive number and larger than or equal to one, i.e. K �1 because a fractal dimension in our case must satisfy the following relation 0 3$ � $D K Nln ( ) ln ( ) . It is clear that the magnitudes of K and D cannot be deter- mined directly from discharge parameters �, d and �, just be- cause K is a result of an unpredictable, stochastic process and D is dependent on K , i.e. D f K� ( ). However, statistical pat- tern (3) has been proposed as a fitting pattern, and all its pa- rameters, including K and D, can be determined either by using an optimizing procedure or heuristically. For example, if the whole distribution is measured, the entire linear section is available, from which Jmax can be determined. As soon as Jmax is known, the parameter � �N d J� �exp ( )max� 1 is available. Parameter D (fractal dimension) can be estimated from the experimental data by fitting the linear section of the graph by a straight line – a regression line possessing slope s – i.e., D s� � �( )1 . Then parameter K can be estimated by us- ing K N D� . 2.5 Parameter D Although the fractal dimension D does not explicitly occur in the statistical pattern (3), it does have a fundamental mean- ing. For this reason it will be convenient to discuss its physical interpretation and properties in connection with the fractal production of displaced avalanches. The value of D is defined (4) by the values of parameters K and N, i.e. D K N� ln ( ) ln ( ) where K �1 and N �1. A frac- tal dimension, like a topological dimension, must be re- presented by a positive number, and because splitting of displaced avalanches occurs in the discharge gap, which is a three-dimensional Euclidean space, the fractal dimension D cannot be larger than three, i.e. 0 3$ $D . An increase in the 44 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 50 No. 1/2010 Fig. 4: Fractal pattern fit of population statistics (voltage pulses – data taken from Fig. 2). The quantity ud � 10 1 555. mV is a voltage scale corresponding to the average voltage pulse created by the largest avalanches detected. The quantity g is not a normalization con- stant but only indicates the shift between unnormalized data and the unnormalized distribution F n d( , ). After [14]. D-value is possible only if K increases or N decreases, or if both these changes act simultaneously. An increase in K means that the “splitting” is more effective (higher multi- plicity means splitting into a larger number of displaced avalanches). A decrease of N (at fixed �) means a smaller critical distance �, and, therefore, a larger number ( Jmax) of avalanche generations ( )maxJ d� �� 1 , which implies a numerous set of displaced avalanches. Thus, large D means that the discharge is accompanied by an abundant swarm of displaced avalanches, which can be interpreted as a tendency to delocalize the discharge over a larger portion of the inter- -electrode space. In short, a higher D-value means a higher discharge delocalization and, conversely, a lower D-value indicates a more localized discharge with sporadic appear- ance of displaced avalanches. Since the effect of discharge lo- calization/delocalization is undoubtedly limited, among other things by the electric field E used in experiments, it will be no surprise that such a field dependence D(E) has been observed previously [9], because of the acting space charges of the par- ent avalanches, i.e. due to the dependence D E( ( ))� . The variability of parameter D when going from less pop- ulated to highly populated avalanches is well observable by comparing Figs. 4 and 5. Less populated avalanches, whose statistics are given in Fig. 4, generate lower values of dimen- sion D and also smaller N and K, in comparison with big ava- lanches with a prevalence of streamers (Fig. 5). This means that streamer-like avalanches split more easily into side ava- lanches. They are more delocalized (higher K) and are capa- ble of filling better in the discharge gap (higher D). 3 Fitting procedure Function (3), like any other multiparameter function, must be handled carefully when performing a fitting proce- dure. It is the starting values of the fitting parameters that have an essential influence on the results of an optimizing procedure. An inappropriate set of starting values may lead to final values that satisfy the mathematical conditions but may be physically completely unacceptable. To avoid such a fail- ure, a proper choice of input values is necessary. In the case of function (3), there are several useful aids for establishing a proper choice. Firstly, the value of nd should be estimated directly from the statistical data rather than from the relation n dd � �exp ( )� , especially when the measuring device pro- vides only a narrow acquisition range. It is suggested to esti- mate the nd values as the horizontal asymptote of the ‘lowest hump’ on the graph w n d( , ). If only a linear section (without © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 45 Acta Polytechnica Vol. 50 No. 1/2010 Fig. 5: Fractal population statistics of a mixture of lowly and highly populated avalanches. Data taken from Fig. 1. G is a normalization constant. any ‘humps’) is available, the ‘corresponding asymptote’ can be simulated by using w n d n n nd d ( , ) exp� � � ���� � � ����� 1 . However, if only a restricted measuring range is available, such a value may not have the meaning of the populations of the largest parent avalanches. Instead, it may correspond to the populations of some displaced avalanches. However, from the viewpoint of the fitting procedure it will be a right choice in either case. The second parameter D can also be reliably estimated prior to starting the optimizing procedure. Linear regression of either the linear section or the ‘humpy section’ will provide a slope s and, thus, the value D s� � �( )1 . Parameter D deter- mines the ratio ln ( ) ln ( )K N , which may assist in estimating K and N – the value of K does not usually exceed 4. As soon as the estimate of nd , K and N has been made, one can try to draw the graph w(n, d) using only the first two terms ( J �1) of sum (3), and compare the result with the ex- perimental data. After a re-adjustment of K and N, further terms ( J �1) of sum (3) can be added until a full cover of the linear section of the measured data is accomplished. If necessary, final corrections of some parameters are recom- mended in order to ensure a good accord with experimental data. Only such ‘pre-processed’ values can be successfully used as proper input data for the chosen optimizing (fitting) procedure. If the scenario described above is followed, it will be easier to find results that may satisfy both the mathematical and physical conditions. 4 Transition between Furry and Pareto statistics The proposed statistical concept (3) is based on creating displaced avalanches formed most probably by photoioni- zation during the initial stages of the collision process in the parent avalanches if a sufficiently high electric field (and high E p) is present. The term “sufficiently high electric field” refers to such a field as is capable of ensuring the production of highly populated avalanches (for example, in air at nor- mal atmospheric conditions n �105). In such a case, displaced avalanches may accompany the highly populated parent avalanches that terminate either without converting into streamers or as regular streamers that may or may not undergo channel branching. Therefore, the condition for the transition from Furry statistics to Pareto statistics of electron populations is a sufficiently strong electric field facilitating the creation of displaced avalanches. There is also a simple mathematical condition ensuring the transition between these two statistics. Furry’s distribu- tion can be expected if � � d. As a consequence of this condition, one can find J dmax � � �� 1 0 and, in addition, N d� �exp ( exp ( )� �� . This means that the parent avalanche will reach the anode without starting displaced avalanches and, thus, K loses its sense, and also D cannot be rigorously defined. The condition Jmax � 0 excludes the presence of displaced avalanches and, on the other hand, ensures the participation solely of parent avalanches. The parent avalanches cross the whole discharge gap and form Furry statistics (1) with an average population n d� exp ( )� , which represents a fixed non-fractal reference scale. 5 Conclusion Instead of simple photoionization that acts solely within the primary (parent) avalanche, a new concept of displaced avalanche splitting has been proposed that allows for photoionization going beyond the parent avalanche channel, and for creating new smaller independent avalanches. The new displaced avalanches modify the overall population dis- tribution of avalanches and cause a transition from Furry to Pareto statistics. Such a transition may occur especially when the critical distance � for initiating displaced avalanches is es- sentially smaller than the discharge gap itself � !! d . Furry and Pareto statistics can be unified into a single generalized analytical pattern that is capable of following the experimen- tal data faithfully in both the statistical regimes (Fig 5). The main limitation of the pattern consists in its restriction to ho- mogeneous or quasi-homogeneous background electric fields. Acknowledgments This work has been supported by Grant no. 202/07/1207 of the Grant Agency of the Czech Republic. References [1] Raether, H.: Electron Avalanches and Breakdown in Gases. London: Butterworths, 1964. [2] Furry, W. H.: Phys. Rev., Vol. 52 (1937), p. 569. [3] Wijsman, R.: Phys. Rev., Vol. 75 (1949), p. 833. [4] Frommhold, L.: Zeitschrif für Physik, Vol. 150 (1958), p. 172. [5] Richter, K.: Zeitschrift für Physik, Vol. 158 (1960), p. 312. [6] Ficker, T.: J. Appl. Phys., Vol. 78 (1995), p. 5289. [7] Ficker, T., Macur, J., Kliment, M., Filip, S., Pazdera, L.: J. El. Eng., Vol. 51 (2000), p. 240. [8] Ficker, T., Macur, J., Pazdera, L., Kliment, M., Filip, S.: IEEE Trans. Diel. El. Insul., Vol. 8 (2001), p. 220. 46 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 50 No. 1/2010 [9] Ficker, T.: IEEE Trans. Diel. El. Insul., Vol. 10 (2003), p. 689. [10] Ficker, T.: IEEE Trans. Diel. El. Insul., Vol. 10 (2003), p. 700. [11] Ficker, T., Macur, J., Kapička, V.: Czech. J. Phys., Vol. 53 (2003), p. 509. [12] Ficker, T.: IEEE Trans. Diel. El. Insul., Vol. 11 (2004), p. 136. [13] Mandelbrot, B. B.: The Fractal Geometry of Nature. New York: Freeman, 1983. [14] Ficker, T.: J. Phys. D: Appl. Phys., Vol. 40 (2007), p. 7720. [15] Kulikovski, A. A.: J. Phys. D: Appl. Phys., Vol. 33 (2000), p. 1514. [16] Pancheshnyi, S.: Plasma Sour. Sci. Technol., Vol. 14 (2005), p. 645. [17] Arrayas, M., Ebert, U., Hundsdorfer, W.: Phys. Rev. Lett., Vol. 88 (1998), p. 174502. [18] Montijn, C., Ebert, U., Hundsdorfer, W.: Phys. Rev. E, Vol. 73 (2006), p. 065401–1. [19] Penney, G. W., Hummert, G. T.: J. Appl. Phys., Vol. 41 (1970), p. 572. Prof. RNDr. Tomáš Ficker, DrSc. Phone: +420 541 147 661 e-mail: ficker.t@fce.vutbr.cz Department of Physics Faculty of Civil Engineering Brno University of Technology Veveří 95 662 37 Brno, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 47 Acta Polytechnica Vol. 50 No. 1/2010