Lecture notes on «Physics of gas discharge»: educational-methodical tool

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Yu. V. Arkhipov A. Askaruly L. T. Yerimbetova


Almaty «Qazaq university» 2015


UDC 530.1 (075.8) LBC 22.365 я 73 А 87 Recommended for publication by Academic Council of the department of physics and technology al-Farabi KazNU and Editorial and Publishing Council of KazNU Reviewers: D.Sc., professor O.Yu. Prikhodko D.Sc., professor V.M. Somsikov

А 87

Arkhipov Yu.V. Lecture notes on «Physics of gas discharge»: educationalmethodical tool / Yu.V. Arkhipov, A. Askaruly, L.T. Yerimbetova. – Almaty: Qazaq university, 2015. – 58 p. ISBN 978-601-04-0791-6 The educational tool presents lectures notes on «Physics of gas discharge», developed by the authors, according to the curriculum of the course. The lectures cover questions about the processes of an electric current flow through the ionized gas, the formation of gas discharges. The process of gas breakdown which leads to the formation of various types of discharges is discussed in details. The models of discharges used to describe the latter in the scientific literature are outlined. This work was performed under a grant of Ministry of education and science «The best lecturer of HEI in 2013». For students of «Physics» 5B060400 and «Nuclear Physics» 5B060500. В пособии представлены лекции по курсу «Физика газового разряда» на английском языке, разработанные авторами, согласно учебной программе данного курса. В лекциях освещены вопросы о процессах протекания электрического тока через ионизованный газ, формировании газовых разрядов. Подробно обсуждается процесс пробоя газа, который приводит к образованию различных типов разрядов. Описаны модели разрядов, используемые для описания последних в научной литературе. Настоящая работа выполнена в рамках гранта МОН РК «Лучший преподаватель ВУЗа 2013 г.». Для студентов специальностей «Физика» 5В060400 и «Ядерная физика» 5В060500.

UDC 530.1 (075.8) LBC 22.365 я 73 ISBN 978-601-04-0791-6


© Arkhipov Yu.V., Askaruly A., Yerimbetova L.T., 2015 © Al-Farabi KazNU, 2015


LECTURE 1 Topics to be discussed in the lecture. Plasma and its main characteristics. Purpose of the lecture: relationship of plasma and gas discharges. Key words: plasma, gas discharge.

It can be argued that all around the world, surrounding us, is 99% of plasma. All bodies are composed of molecules and atoms, the last consist of positively and negatively charged particles (electrons, protons). Gas, in which the ionization occurred and the charge separation arose, is called «ionized gas». Such gas is called «plasma», if the condition of quasi-neutrality was hold, the conditions of sufficient spreading and the lifetime of the plasma. It should be noted that in recent years an interest in the study of plasma with dust particles formed during the development of a discharge due to erosion of the electrode material or the walls of gas-discharge volume has increased. Such plasma possesses specific properties associated with the presence of microparticles that cause ionization and recombination in the medium. The quasi-neutrality condition in such plasma has the form:

ne  Zi ni  Z d nd . Here it should be noted that the dust particle charge Zd may vary with time. The latter may lead to a significant change in the spectra of low-frequency oscillations of such medium, its thermodynamic properties and transport processes. Determination of the Langmuir frequency and Debye screening length can be introduced similar to 3

the plasma without dust particles. In order to obtain plasma it is necessary to heat a neutral gas, or pellet of plasma-forming substance. There are the Joule heating in the tokamak devices with further increasing of temperature during the high frequency heating, or beam warming when the beam of high-energy atoms is injected, by their recharge and stopping processes in plasmas. In the case of the inertial fusion plasma a heating is carried out by light (laser) radiation impact on the D-T pellet thereby forming a dense high-temperature plasma. Let us formulate briefly the types of plasma, which exist in the nature and in experimental installations. Plasma may be ideal and non-ideal, when a parameter of coupling (ratio of a potential energy of an interaction between particles to their kinetic energy) is less or greater than one, respectively. Plasma may be isotropic (without external magnetic field) or anisotropic (magnetic). Also, there are other types of plasmas: hot and cold, dense and rare. Questions to Lecture 1: 1. What is a plasma? 2. Formulate the types of plasma.

LECTURE 2 Topics to be discussed in the lecture. Classification of gas discharges. Purpose of the lecture: study of the classification of gas discharges. Key words: ionization, gas breakdown, avalanche.

Gas discharge is a process of a current flow through a gas. In order for the passage of a current the existence of ionized particles and an electromagnetic field, in which they move are needed. The latter can be created even without any electrodes by an induction. A gas discharge occurs in a wide range of pressures and frequencies of the electromagnetic field. Characteristics of a gas discharge depend on parameters of the electric circuit. Values of currents, flowing through the volume of gas discharge, vary from 10-8 to 106 А. At the present time, the physics of gas discharge processes is experiencing new growth associated, on the one hand, with the inte4

rest in construction facilities for the thermonuclear reactions in hightemperature plasma, and on the other, with creating of powerful lasers, continuous and pulsed. In the installations thermonuclear plasma is generated with the help of powerful gas discharges. And the low-temperature plasma of the positive column of a glow discharge is an active medium for laser generation. In addition, the gas discharges are used in industrial plasma torches utilized in metallurgy, chemical industry, welding and cutting of materials, etc. It is possible to classify gas discharges, first, by the nature of their flowing, and, secondly, by the nature of the state of the ionized gas and by frequency of an external electromagnetic field. The first classification was proposed by V.L. Granovsky, and the second was proposed by P. Reizer. According to V.L. Granovsky, the gas discharges are divided into self-discharges and non-self-discharges. Non-self-discharges take place only in the presence of external ionization source, using, for example, the phenomenon of thermionic emission, photoelectric effect, etc. Let us consider the classification of gas-discharge processes in according with Yu.P. Reiser. The boundary effects may be neglected and one can pay attention to the spatial processes. Non-equilibrium discharge plasma is a rarified (pressure is about 133 Pa), weakly ionized gas in which electrons are «hot», and ions (molecules and atoms) remain «cold». Typically, the temperature of electrons, Te is about 104 К or 1 electron-volt, the temperature of ions and neutrals, Ti , Ta are less than 1000 K. An example of such plasma is plasma of positive column of a glow discharge. Equilibrium plasma is dense plasma at atmospheric pressure and at the same temperatures of electrons and ions, constituting about 10 4 K. The degree of ionization in such plasmas corresponds to the specified temperature and pressure, the degree of which varies from 10-3 to 10-1. An example of equilibrium plasma is the plasma of positive column of an arc discharge at high pressure. Of course, it is often difficult to find the boundaries between one and another state of the ionized gas.


It turns out that the physics of breakdown in a quickly oscillating microwave field is less complex than in a slowly changing or even in a constant field. In the case of a constant field, it is explained by the nonlocal nature of the development of electron avalanches in the discharge gap. Electrons move along the direction of an applied electric field with drift velocity, and this motion is superimposed on the chaotic motion of electrons. The number of electrons in the process of breakdown increases continuously from the cathode to the anode. The field contributes to the charge transfer to the anode and to their compensation. In the case where the amplitude of the oscillation of electrons in the field is much lesser than the characteristic size of the system (high frequency), the formation of electron avalanches localized in space and there is no need for compensation of electrons. In this case, the e-characteristic of ionization velocity is the frequency of ionization by an electron impact  i . This is the number of electrons, which are produced by an electron in a given region of space per second. i 

N ne

 f      d . i



Here N – density of ionized atoms, ne – electron density,  – electron energy, i () – the ionization cross section, depending on electron energy, I – ionization potential of the atom f () – electron energy distribution function. In a constant field it is rather convenient to use not a time i, but a spatial characteristic, so called a first Townsend ionization coefficient . It shows a number of electrons, which are produced in the path of 1 cm by an electron, moving in the direction of the electric

field E . It is easy to find a relationship between the coefficients i and . If during the time dt, idt particles are formed then during this time the electron will traverse the path dx = Ddt and forms Ddt particles. It follows thence:

I = D. 6


Let us consider a formation of an avalanche under the influence of the microwave field. Also, let us discuss the possibility of energy losses of electrons and losses of electrons themselves, which are slowing down the development of the avalanche. Energy losses occur due to elastic and inelastic collisions. The energy, which is lost by elastic collisions, is proportional to the relative masses of colliding particles m / M (electron and atom). Inelastic losses occur due to the excitation or ionization of atoms. In certain conditions, the excitation of atoms may contribute to the formation of avalanches. As an example there is a process that develops in a gas, composed of atoms of two varieties, where the potential excitation of one of them is greater or equal to the ionization potential of the other (e.g., excitation potential of the helium atom  I He  19,8 eV, ionization potential of mercury atoms I Hg  10, 4 eV). In this case, the excited atoms of one kind, colliding with atoms of another type ionize them with a high probability (Penning effect). The formation of avalanches can slow down and even cease, due to losses of electrons. It is obvious that recombination affects only the latest stages of the discharge development, when there a sufficiently large number of positive ions are accumulated. Therefore, we can say that recombination has virtually no effect on the breakdown. Questions to Lecture 2: 1. What is an electronic avalanche? 2. What is the Penning effect? 3. What is the basis for the Raiser's classification of gas discharges?

LECTURE 3 Topics to be discussed in the lecture. The balance equation of electrons. Purpose of the lecture: to bring the balance equation of electrons. Key words: ionization, diffusion, diffusion length.

  Ionization frequency  i ( E ) can be obtained from (1) ( E – electric field strength). Let’s presume that there is a gas in the limited volu7

me, whose molecules electron affinity energy value is negative. The walls are electron absorber; therefore the density of electrons, ne near them is small. As a consequence, there is the diffusion flux of the gas-discharge volume to the walls. Thus, the desired equation is the diffusion equation with source,   ne  di e   i ne ,  e   Dne . t


For the case when the number of electrons is very low (breakdown), the coefficient of a free diffusion of electrons can be estimated by the usual formula of the kinetic theory of gases

1 2 2 D   m   . 3 3 m 3m m


where  m is a free path of electrons,  – average velocity of random motion, m – effective frequency of collisions and   m 2 2 .   On condition that the field E is uniform in space,  i ( E ) does not depend on the coordinates and the equation (3) can be solved by the   method of separation of variables. Suppose ne (r , t )   (t ) (r ) , then 1 d   i  D   d .  dt 


The equation, describing the evolution of the total number of electrons Ne in time Ne ~ Ф, is

 dN e  N e  i    d  .   dt

 


In equation (6) i describes the birth process, and d describes diffusion losses, i.e.  d ~ 1 ~ D 2 . If we consider the effect of d

trapping, then (6) will be rewritten as 8

 dN e  N e  i    d   a  .   dt

 


The rate of diffusion d is determined by solving equation (5)

D   d ,


with the boundary condition  = 0 at the walls and with natural condition of the absence of a diffusion flux on the axis of symmetry  = 0. In cylindrical geometry, (8) is rewritten as 1    r r r  r

2 2 1  1    2   2 .  2 2     r z 

(9)    0  , we   

Considering that the problem is axially symmetric 

solve the equation (9) by the method of separation of variables  r, z   Ar  B  z  , 1 1     1 1  2 r     C1 .   A r r  r   2  z 2


Constant C1 is determined from the boundary condition B  0 at Z = L / 2 (L – length of the gas discharge tube) and finally we can get   cos

 z. L


The equation for A  r  transforms to Bessel-type equation, r 2

2   r  r 2   0 . 2 r r



Here r  r    2  C1  . The solution of this equation with the conditions A  0 at r  R ( R – radius of the gas-discharge volume) has the form  r     I 0    2,4   R  


and, consequently, the general solution of (2.11):  r     ne ~   I 0    2,4   cos   z  . L   R  


Questions to Lecture 3: 1. What is the diffusion length? 2. What is the balance equation?

LECTURE 4 Topics to be discussed in the lecture. Criteria of breakdown. Purpose of the lecture: define the breakdown criterion. Key words: breakdown, the ranges of high and low pressures.

Assume, that a sufficiently long signal with steep front is provided to a gas. The duration of the signal (front) rise is small in comparison to the ionization time, and the duration of the signal itself is greater than the time of ionization. Then,  i  t   const for t ≥ 0 and from (7) we have: t

Ne  Neoe  , where    i   d   a 



– is the constant of the time of ava-

lanche. Obviously, if  i   d  a the avalanche cannot develop. The


development of an avalanche will occur only when  i   d  a . But

  i   i ( E ), therefore i reaches a value d + a at a certain threshold

value of the field Et. This field is determined from the equality:

 i  Et    d  a ,


which is called as «fixed criterion of breakdown». Let us consider how the frequency of ionization depends on the field. Assume that the losses in elastic and inelastic collisions are small or equal to zero. Then the energy of the electrons  in the microwave field varies as:

d   E  , dt m


where E ~ E2 is an energy, acquired from the field, during the time between collisions. During time 1

 I   d  E  I   m    dt    E 


electron gains energy I (I is an ionization energy of an atom). Since 1  i ~  E  then  i ~ E 2 . Now consider the case when the duration of the microwave pulse is not great:  N   i E   d   a  t11 ln  1  . (19)  N0 

 

Here N1 is a finite and N0 is an initial number of electrons in an avalanche, and N1 must be sufficient, so that when the gas was ionized, its conductivity had significantly increased. The dependences of intensities of threshold fields from a pressure of a gas were obtained during experiments on microwave break11

down. At low pressures, the threshold field decreases with the increase of pressure and at high pressures – vice versa. The threshold field decreases with an increase of sizes of the discharge chamber or with a decrease of a frequency in the range of low pressures. Let us discuss these relationships. Suppose that we have a mixture of helium and mercury, so-called Heg-gas, which is placed in a microwave field. This mixture allows neglecting the presence of inelastic losses (Penning effect). Consider the case of low pressures. In this case the diffusion losses are large, and in order to compensate them we need to have a large value of   i E , i.e. a strong field. In such kinds of fields, when the energy of

 

the electrons does not exceed the excitation energy of the helium  atom I He , the elastic energy losses of the electrons are small. 2m    I He . M 

   max  


e2 E 2 m  d   1    i ~ E       E  m  .  I He m 2 I He  dt  I He Using the stationary breakdown criterion (16), with a = 0, we have:

 D i E  d  2 . 

 

Then, from (20), in the limit of low pressures we obtain the following expression for the intensity of the threshold field: 1

 m D I H* e 2  2 Et   . 2 2    me   1



   I  m 2 m 2 m 2    I He   1  I He   m  He   t  . (21)      ~ e  m  3  e 3 e  N 3       m  


Consider the case of high pressures  m   . Diffusion losses are small and the main role in the development of the breakdown is played by elastic losses. A dynamics of a change of the electron energy in the microwave field in a presence of an elastic loss is described by an equation:

 e 2 E 2 2 m  2 m  d     E    m .  m   2 dt  M  M   m m


It is possible, using (22), to find the maximum value of energy, which is determined by the condition of stationarity:

d 0. dt 2

 max


M e2 E 2  E   E  ~  ~  .  2m m m2   m    


* In the case when  max  I He , an electron does not excite the helium atoms and there is no breakdown. Therefore, the threshold

* field could be found from (23)  max  I He :

t  2 M m

m ~ P. e

Questions to Lecture 4: 1. What is a dependence of a threshold field on a pressure at high pressures? 2. What is a dependence of a threshold field on a pressure at low pressures?


LECTURE 5 Topics to be discussed in the lecture. Breakdown in the constant field. Purpose of the lecture: study of breakdown processes. Key words: breakdown, Paschen`s law. electrons, emerged

Let us study the effect of breakdown in a constant field. Assume that there are two electrodes in a gas-discharge volume at a distance L and the potential difference maintained between them is U. In this interelectrode space, due to both external ionizator and as a result of collision ionization of atoms by electrons accelerated in an external field there appear electron-ion pairs. Suppose that the appeared ions are singly ionized, the rate of electron-ion pairs formation in the volume is constant S  const , the mean free path  m is much smaller than the distance between electrodes L. Drift velocity of electrons and ions is proportional to the electric field

De  x   e   x  

e   x , m m


Di  x   i   x  

e  x . M m


Here e (i ) is the mobility of electron or ion,  ni is the frequency of collisions of ions with neutrons. At the cathode, under the action of bombarding with ions, the secondary electron emission occurs. It is assumed that the emission from an anode does not occur. Taking into account all mentioned above, one can write the equation of balance of particles in the volume:


ne   S   i ne    e ne E  x   , t x


ni   S   i ne    i ni E  x   . t x 


Here n i is the density of ions. The boundary conditions to equations (26) and (27) can be written as

x  0 (cathode),  i ji (0)  je (0)


x  L (anode), ji  L   0


je  eneVDe


ji  eniVDi .

je  eS   je x


ji  eS   je , x


je  ji  j  const .


x  je  x   c  x  exp     x   dx   ,   x     E  x   . (34) 0  x   x   x  je  x    c2  eS  exp      x   dx   dx   exp     x   dx   . (35) 0  0   0  

c 2  je  0   j i  0   i   i  j  j e  0  

c2 

i j 1   i  ,

(36) 1

L  x     L       j  eS   exp      x  dx  dx   exp     x  dx   i  . (37)  0  0     0   i  1 


L   i exp     x   dx     i  1 0 


  Pf (eElm )


Taundsend got the following result (assuming approximately that lm ~

1 ): P

  A exp   BP / E  , P


e L i   i  1 ,


which can be interpreted as a condition for beginning of the reproduction of electrons or the threshold of a breakdown. Now, from (32) and (33), exluding α, one can obtain the Paschen`s law:

Et B  ln PL  C 0 P


L . ln L  C0


Ut 

Questions to Lecture 4: 1. Specify the Paschen`s law. 2. Specify the Townsend`s criterion of breakdown.


LECTURE 6 Topics to be discussed in the lecture. Optical breakdown. Purpose of the lecture: study of the processes of self-dependent gas discharge. Key words: Current-voltage characteristics of the discharge.

First reported observation of a «laser spark» – an optical breakdown of a gas, appeared in 1963. The possibility of such phenomena occurred after the creation of lasers, which are able to emit the pulses of megawatt power. Position of the minimum can be estimated from the condition of proportionality of the frequency of the laser radiation and the effective collision frequency of electrons. Some interesting breakdown is a breakdown of air at atmosphere pressure. The breakdown of air may interfere with the propagation of radiation, since the formed plasma absorbs it. A threshold field for CO2-laser significantly reduces in the presence of dusty particles in the breakdown region, since it promotes the appearance of the seed electrons, which are necessary to initiate the avalanche. In the case of ruby or neodymium laser usage, which has a shorter wavelength radiation (high frequency), the seed electrons occur without the presence of dust particles. The specific feature of the optical breakdown is a possibility of the existence, along with the usual mechanism of impact ionization, of a mechanism for the separation of electrons from atoms as a result of many quantum effect – the absorption of several quanta of light at an instance. For ionization of an atom one need to take a few quanta, for example, for a ruby laser  =1,78 eV, and the ionization potential of argon is I=15,8 eV – therefore, for the separation of an electron one need to take at least nine quanta. The density of such multi-photon processes increases significantly with increase of light intensity. It is known that in a discharge volume one can initiate a field of high- and low-frequency ranges in a variety of ways: inductive (using a solenoid, mounted on the discharge volume made of dielectric), capacitive (electrodes are in contact with the plasma discharge) and capacitive with isolated electrodes from plasma.


Let’s choose the characteristic scales of values – they are a free path of electrons  m , depending on the pressure, the size of a volume L, an oscillation amplitude of free electrons rE, or drift electrons RE.   Since a displacement of a charged particle in the field   0 sin t has a form:  r

 eE0

 m eE0  sin t  cos  t  r0 , 2 2 2 2  m    m  m    m 


then for the low-frequency or high-pressure  2   m2 an electron oscillates with an amplitude: RE 

eE0 , m m


while for high frequencies (low pressure) the amplitude of oscillation is: eE0 rE  . (46) m 2 The case of a breakdown at low frequencies was relatively studied a little. When the frequency decreases to a certain value and at a constant pressure, a breakdown voltage increases sharply. This frequency is called the limiting frequency, it corresponds to the case of oscilL lations with an amplitude of RE , which approximately equals to 2

2eE0 2 RE  ~ 1. L Lm m


At frequencies, which are lower than boundary, the threshold Et is almost unchanged. This and other vague effects that occur during the development of low-frequency discharges are studied a little, and is


still far to a full understanding of the physics of the processes occurring in them. In conclusion, let us consider a case of a strongly rarefied gas  m  L . This case can be found in the experimental installations. Reproduction of electrons occurs as a result of secondary electron emission from the walls. In this regard it necessary to electrons that hit the wall to obtain enough energy for extracting not less than two ones, i.e.   50 eV. At the moment when electron hit the wall, the field must change its direction in way that the ejected electrons will be able to accelerate in the opposite direction. Thus, there a critical frequency exists – the frequency of cutoff and if the field is changed with less frequency, the electron would reach the wall at a time when the acceleration force is directed in a direction of its motion. Therefore, the emitted electrons can not get into the volume. If the frequency of the field is greater than the cutoff frequency, an electron is slowed down before reaching the wall and loses energy. However a breakdown is possible in principle, but it is necessary to increase the intensity of the electromagnetic field. Now, let us consider the self-maintained discharges. After the breakdown passed there is a possibility to obtain one or another type of gas discharges. Glow discharge and its structure. In various electric and radio devices, the plasma (discharge) devices are used. As it was mentioned above a gas discharge is a process of flow of an electric current in a gaseous environment. For the passage of a current it is necessary, on the one hand, the presence of an external electromagnetic field, and on the other – the processes leading to ionization of atoms and molecules of gas. The latter can be provided by the current itself (self-maintained discharge), and by additional ionizer (non-self-maintained discharge). Self-maintained discharge gas discharge occurs in a wide range of gas pressures, after reaching the breakdown voltage, the discharge current can vary from 10 -11 to 106 or more amps. Properties of the normal glow discharge, produced by the pressures ~ n 102 Pа and currents of 10-4 - 10-1 А, are used in the electrical devices with the current-voltage characteristic in the form of a straight line, parallel to the currents axis. That is for the given range of values of the current, 19

the voltage, applied between the electrodes, does not depend on a current. This effect is used in a gas-discharge diode. If a potential difference across the electrodes of a gas discharge is reduced below the breakdown voltage, then a self-maintained discharge is immediately terminated. This property is used for creating of relaxation oscillators. Current-voltage characteristics of different types of a discharge are shown schematically in Figure 1.

1 – corresponds to the dark, Townsend discharge 2 – the transition to glow discharge 3 – normal glow discharge 4 – abnormal glow discharge 5 – the transition to arc 6 – arc discharge Figure 1. Current-voltage characteristics of a self-discharge

A glow discharge is sufficiently well studied at a constant external field and low pressure. Its prevalence is explained by the way, with which such a discharge can be obtained and maintained with ease. Its hallmark is a potential distribution between the cathode and anode, which is characterized by a region of a significant cathode fall. The cathode fall is the potential change in the magnitude of 90% of the potential difference, applied to a discharge interval (Fig. 2), due to the presence of a space charge in the given area. The space charge is formed by positive ions.


Let us consider the characteristic structure of a glow discharge (Fig.3). The latter consists of 8 regions, where 1-5 are cathode ones, and 6-8 are anode. Experiments show that if one brings electrodes closer, the cathode regions of the discharge will not vary in size and the length of the positive column 6 is reduced; positive column disappears due to the further rapprochement of electrodes, then Faraday dark space disappears and with the disappearance of a glow 4, the discharge stops. If the distance between the cathode and anode is not sufficient for the dark Kruks space 3 and the beginning of a glow 4, the discharge can exist only at a high potential difference and it is called as abnormal. Thus, cathode regions play the primary role in ensuring of the existence of a glow discharge.

Figure 2. Distribution of potential along the axis of the glow discharge

Figure 3. The glow discharge at low pressure

Let’s consider the processes, taking place in various parts of the glow discharge. Electrons, formed by the bombardment of the cathode by positive ions, are moving in the Aston dark space with 21

low speed, which are insufficient to excite the gas atoms, so there is a glow in that area. Illuminated region (cathode glow 2) occurs because the electrons in the dark Aston space, are accelerated and gain enough energy to excite atoms. Ionization in this area does not exist, because the probability of it is small due to the slow energy of moving particles. In the Kruks`s dark space 3, the gas glow is absent because a strength of accelerating field is large (see Fig. 2) and the electron energy reaches the value of the order of the ionization energy. Flying through 3, electrons enter the region of anegative glow 4, in which the potential gradient is small. This leads to the fact that the accelerated electrons partially lose their energy and get the opportunity to excite the atoms. Field in region 5 increases slightly, and the electrons regain the opportunity to ionize the atoms. After dark space 5 there is a region of the positive column 6. This part of the discharge is a plasma with a small electric field strength. Review Questions: 1. What is multiquantum processes? 2. What is happening in a glow discharge volume? Recommended readings [1-4]

LECTURE 7 Topics to be discussed in the lecture. Structure of the glow discharge. Purpose of the lecture: study of the structure of the glow discharge. Key words: strata, stratification

Plasma in the positive column of a glow discharge is maintained by the processes, occurring in the same area, and its properties are not affected by processes, occurring at the electrodes. As a rule, the positive column is the longest and the brightest part of the glow discharge. Since there is plasma in this area, i.e. the quasi-neutral set of electrons and ions, then, in contrast to the breakdown with the normal diffusion, there is an ambipolar diffusion in the positive column. The electrons are faster than ions. They depart from the ions, and ions may catch up the electrons and slow them down. Thus, 22

the electron-ion system moves in a consistent manner that can be described in the framework of the ambipolar diffusion. The value of the coefficient of the ambipolar diffusion is greater than the ratio of free diffusion of the ions, but less than that of electrons. The positive column of a glow discharge may have a layered structure in the longitudinal direction, i.e. it may be stratified. Strata are visible layers (disks in cylindrical geometry), which can stand still or move at speeds up to tens of meters per second. Also note that at high pressures and sufficiently strong current, a discharge can form contraction, i.e. it can gather in the form of a cord, along the axis of the discharge tube. Thus, there is a bright light near the axis of the discharge tube, where the sharp increase in the concentration and temperature of the charged particles occur. Questions to Lecture 7: 1. What is a stratification of a discharge? 2. What is a contraction of a discharge?

LECTURE 8 Topics to be discussed in the lecture. Arc discharge. Purpose of the lecture: study of the arc discharge. Key words: structure of the arc discharge, channel model

Arc discharge is a self-sustained discharge, in which the cathode fall of a potential has a relatively low order of magnitude of ionization potentials or excitation of atoms, i.e. an order of 10 eV. This make the difference of an arc discharge from a glow discharge, whose cathode fall is hundreds of volts. The small cathode fall is the result of the cathode emission’s mechanisms. These mechanisms can provide a large amount of electric current from the cathode, which is close to the full current of the discharge. There is no need for considerable strengthening of e-current, which is a function of a large cathode fall in a glow discharge. Cathodes emit e as a result of the thermo-electronic, auto-electronic and thermo-auto-electronic emissions. Perhaps, there are more complex processes of the e-birth from the cathode. 23

Arc discharge has large currents (i ~1-10 5 A), significantly larger than the currents in the glow discharge (i ~10-4 – 10-1 A). There is also a high density of current on the cathode. In some forms of arcs they are jK ~ 10 2  10 4 A / cm 2 , in others they are jK ~ 104  107 A / сm2 . When a pressure is equal to 1 atm (it is normal for the glow discharge), the normal density (j = 155 A/cm2) in the air at the copper cathode corresponds to the lowest limit of the arc range. The voltages of the arc burning are mostly low. In the short arcs they are equal to 20-30 V, in other cases they are equal to several volts. CV-graphic of arc discharges is falling, but not always. Cathodes of the arc whether completely or partly and during short time receive enormous energy from the current and have a high temperature. They are become destroyed due to erosion and evaporation. If the spectrum of the radiation from a cathode-region of a glow discharge is like the spectrum of a gas, in the case of the arc it contains lines of the vapor of a material of electrodes. And we say vacuum arcs burn in the evaporation of the vaporized metal. Concerning the plasma of the positive column – regions between near-electrode layer, – in this case along with the equilibrium arcs, there are many non- equilibrium ones. This depends on the pressure of a gas. So we can say, that the plasma equilibrium in the discharge of a direct current is typical only for the arc, and non-equilibrium is peculiar to the glow and arc discharges, when the latter occurs at low pressure. The plasma temperature. CV-characteristics of the arc’s column of the high pressure. The dense equilibrium plasma with low temperature is more interesting for research than the non-equilibrium and weakly-ionized plasma. It is being studied in laboratories, used in the experiment and technology. The arc method is the simplest, the most available and the most widespread way of receiving such plasma. One of the major characteristics of plasma is its temperature, and the task consists in knowledge, how it is connected with current and power, and how these relations are reflected in the CV-graphics. Thermal ionization. The nature of the ionization process is differrent from what is happening in the weakly-ionized non-equilibrium plasma, where molecules are ionized by electrons, which have received enough energy directly from the field. In the highly-ionized dense plasma, the impact of the field takes no role. The field provides the energy to the whole electron gas altogether. The electrons are 24

thermalized as a result of collisions with each other, acquiring the Maxwell distribution. Electrons, which received enough energy during the collisions, ionize the gas. Thermal ionization is independent of the ways the energy arrives into plasma. The equation of the plasma column equilibrium. Let us consider a  long cylindrical column of the arc in the longitudinal field E . Let the arc light constantly in a stationary gas, closed in the cooling pipe. We well be interested in the atmospheric pressure at which the plasma T=11000-12000 K. Losses on radiation in this case are less than the outflow of energy from the column, and we will not take them into account. Since rot E=0, the electric field in the column, which is uniform along its length, is constant in the cross section. The radial distribution of the conductivity σ, current density j=σE, the sources of the Joule heat w=jE=σE2 W/cm3 is defined by the distribution of temperature through the dependence  (T ) . The balance of the plasma energy is described by the heat conductivity equation:

1 d dT rJ   (T ) E 2  0 , J   . r dr dr


Boundary conditions: if r=R then T=Tw. The temperature of current-conducting plasma is higher, than the T of the wall, so, T of the wall TC  0 . The discharge current is (4.2): R

i  E   2rdr .



Current is regulated during an experiment, and therefore it is a set parameter. The field can be found as the result of the solution of the assigned problem, which is well defined at known characteristics of a matter  (T ),  (T ) . So we obtain a CV-characteristic of the column T

E (i ) . Introducing a potential of the heat flux    dT , it allows 0

us to restrict with one material equation () instead of two. The equation (48) is called as Elenbaas-Geller’s equation. 25

The channel model and a principle of minimum of a power. The non-linear character of real functions σ(() does not allow to solve the equation (48) in a general view analitically. There are some methods to solve this equation: linearization of the function σ((), splitting of the integration’s region into separate zones, numerical methods. One of the best is a channel method, proposed by Steenbeck in 1932. When the temperatures are not so high the conductivity is vanishingly small. At T = 4000-6000 K it becomes noticeable and it grows rapidly with increasing of T. Thanks to the action of the heat flow, T falls from the axis to the wall quite uniformly. The current flows only in a paraxial part of the tube, where the T is high. It is illustrated on figure 4. Introducing the effective radius of the channel r0, and assume approximately, that outside of the channel (at r  r0 ) there is no current and   0 . Inside the channel (at 0  r  r0 ), the conductivity is high and is close to k(Tk), which to corresponds to the temperature on axis TK  T0 . The channel model is reduced to an approximate replacement of the actual distribution  (r ) by a stepped, shown on the figure 4 by a dash-line.

Figure 4. Schematic distribution of T and conduction  along the radius of the arc column.


Then it is possible to evaluate the difference between the temperature on the axis and the point with coordinates r0. Let us assume, that the ratio of the conductivity at these points is equal to an exponent e, then, referring to the Saha ratio, one can obtain an equation for the temperature difference:

T  Tk  T0  2 k BTk2 / I .


Questions to Lecture 8: 1. How does the transition from a glow discharge to an arc discharge occur? 2. What is the channel model?

LECTURE 9 Topics to be discussed in the lecture. High-frequency induction discharge. Purpose of the lecture: study of the induction discharge. Key words: an induction discharge, the model of the metal cylinder

The high-frequency field in the discharge volume can be aroused by a variety of procedures. These procedures are separated according to different features, whether the magnetic field lines are closed or not, or in other words whether the field is a vortex or potential. The first group includes induction methods based on the using of the phenomenon of electromagnetic induction. In the methods of the second group, the voltage fed to the electrodes, bare or insulated from the plasma by a dielectric. High Frequency induction discharge is usually used for frequencies 0.1-100 MHz. The electric field is excited by an alternating magnetic field, such as a solenoid, dressed on to plasma volume. The magnetic field inside the coil is directed along its axis. The vortex electric field is induced under the action of alternating current in the coil. Its lines of force are circles, concentric with the turns of the coil. This field can ignite and maintain the discharge; the currents are closed and directed along the magnetic lines of force. In the coil is inserted a dielectric tube with a plasma-forming gas. On the other hand, it is more difficult to implement the RF induction discharge. Because one needs to coordinate the generated plasma, as a load, 27

with the RF generator. The electrical parameters of the plasma load (ohmic resistance, self-induction) affect the operation of the entire electrical system in general.The equation of heat balance in the highfrequency inductive discharge has the form:


d rI   E2  0 ; I r   dT dr . 2 dr r

Using the conditions H  H z , E  E , we can write the Maxwell's equation in the cylindrical coordinates (at small values of the bias current): Er 1   H r H z  4  z  r   c  E ; r  r rE     

 i  H ,  c

which can be are solved in the approximation I r  0,  r  0  E  0 .

r  R, T  Tc  0 . The Magnetic field in a non-conductive cold environment around tube H z  R   H 0   4 / c  I 0 n  , n is the number of turns per unit of length, I0 is the current in the coil. Model of a metal cylinder. Plasma conductor is like metal, but the radius and the temperature at the boundary are not known. Conditions are as in channel model  k /  0  e . Let’s write the solution for the case:   r0 ,   c / 2 a (+) is a thickness of the skin layer. Then the skin layer is thin and the problem can be solved in Cartesian coordinates, counting the x coordinate from the surface into the depth of the layer ( against the radius r ), the y axis will send on the surface, then we can rewrite equation: dE y dH z 4 i   Ey ,   Hz , dx c dx c H  H0 , x  0 , E y , H z  0 , x   .


Solving the system, as well as with the consideration of the topic «skin effect» in the «Electricity and magnetism» course, we have:

H z  H0e

 x /  cos t  x /  


1/ 2

E y  H 0  / 4  e  x /  cos t  x /    / 4  .

The flow of electromagnetic energy is directed into the depth of the conductor, normal to the surface and equals to: 1

S  S0e

2 x / 

 c    2 2 S0     H0 .  16  2 

In the unit of length of a cylindrical conductor, a power is allocated: T W  2 r0 S0   .  /2 From this equation it is possible to estimate the temperature in the center of the plasma. Questions to Lecture 9: 1. What is the RF induction discharge? 2. What is the model of a metal cylinder?

LECTURE 10 Topics to be discussed in the lecture. Stable and unstable states of the HF induction discharge. Purpose of the lecture: study of the induction discharge. Key words: microwave discharge, optical discharge

The model of the metal cylinder does not allow us to find the main thing – the conductivity of the plasma conductor or plasma temperature. Temperature is determined by the energy balance in the 29

very area of an energy output, and for consideration, how the heat comes out from this area, one need to take into account the real existing temperature drop in the cylinder model and to quantitatively distinguish the meaning of the concept of a conducting from a nonconducting medium. In the middle of the plasma region, where the RF field does not penetrate and where the heat source does not exist, we have T  r   const  Tk . The entire temperature drop between the axis and the surface of the equivalent conducting cylinder, T  Tk  T0 falls on the surface layer with thickness  / 2 , where the energy is released. A heat flux, coming out from the conductor is about J 0  2k T /   k   Tk   , and it coincides with the flux of the electromagnetic energy S 0 from the inductor. Hence we obtain the energy balance equation of conducting plasma: 2 k  T /   S 0 , 4 k  T  W  / r0  .


Since the function  T  is the same as for a permanent field of the arc, the condition (50) remains valid; this condition determines the temperature drop T . If we combine this relation with (51), we will find the temperature of the plasma of induction discharge. Thus, using (51) with an effective ionization potential I and with the formula for the H0, we can express the plasma conductivity through an adjustable parameter, which is a current ampere-turns of an inductor: 2 (52) k  2 kT 2 / I   k   I 0 n / 2  . Because the  T  is the sharp function, Tk, and k vary within 2

fairly narrow limits and, approximately,  k   I 0 n  . The temperature slowly increases with an increase of the current in the inductor: Tk 


I / 2k


ln 4k T k C / I  ln  I 0 n 

const , const  ln  I 0 n 


in complete analogy with the dependence for the DC arc. So as in there, it is difficult to reach a high temperature: one need to have the large ampere-turns and power:

W  2 r0 S 0  2 RS 0  H 02 k1/ 2  I 0 n   1/k 2 .


Plasma temperature does not depend on the frequency of the field (if the skin layer is fine). The threshold conditions for the existence of an equilibrium plasma. If we decrease the current in the inductor, ranging from those values for which the skin layer thickness is   r0  R , then a temperature and a conductivity of the plasma, according to (52) and (53), will decrease, and  will increase. When  – reaches a value of the order of r0 , R , the effect of the skin effect will disappear and these formulas will lose their power. In the opposite case   R , a magnetic field inside the solenoid, so as in the absence of plasma, is uniform and equal to H0; an electric field is E  r   i H 0 r / 2c . In the unit of the plasma cylinder, a power released is: 2


W   E 0


3 2 4   2 H 2 r 4   k  r0  I 0 n  2 rdr  k 2 0 0  . c4 16c


Now the temperature falls from Tk to T0 , which corresponds to an effective plasma boundary, then

W  4 r0  k  T / r0  4k  T  8k kTk2 / I .


The plasma temperature cannot drop too much, otherwise the conductivity will disappear and energy will not release. Hence, the value of the power (56) is now more or less stable, even if the conductivity decreases. Plasma radius decreases with a temperature.


Figure 5. Qualitative dependence of ampere-turns of the plasma conductivity

Figure 6. Qualitative dependence of the contribution of power in the plasma on the conduction region

Consequently, the current in the inductor or I0n is inversely proportional to the conductivity or temperature of the plasma and to the field frequency (low conductivity). Over a large interval of conductivity change, covering the two cases (   R ,   R ), the relations between I0n , W ,  k and  will be as shown in Fig. 5 and 6. Dependence of I0n on  k or Tk passes through a minimum, which corresponds to the place of the joining of curves for two considered cases, i.e. to condition   R . There is a minimum threshold current in the inductor, at which it is still possible to maintain the equilibrium of plasma in the inductive discharge. It can be estimated by extrapolating of the formula (53) upto a threshold temperature Tk  Tt at which   R : 1/ 2

 I 0 n t   I 0 n min

 4 t kTt 2 c 2  .  2    I R 


For example, for air p = 1 atm., R = 3 cm, f = 13,5 MHz (I0nt  10 (AB) / cm ). Threshold temperatures are Tt  7000 – 8000 K. Stable and unstable states. Fig.5 and 6 show another property of the equilibrium discharges of various types. At a given value of a current inductor I 0   I 0 t and 32

other similar conditions, there are two steady states of the equilibrium of a discharge. One of them corresponds to a high conductivity and to a significant skin effect, another one corresponds to a low conductivity and to a lack of skinning. However during the study only the first condition is realized. States on the left branch, when T  Tt , are unstable. Suppose, for example, for some random reason, the temperature increased slightly. In order to maintain this new state it would be enough to have a smaller current in the inductor than the actual. And the heating of plasma will begin until it reaches a state on the right branch of the Fig.5, 6, when T  Tt . It is easy to verify by similar treatise, that it is stable. Microwave discharges Discharge in the waveguide. The long-existing high-pressure microwave discharges have been obtained in the early 50's, when the necessary generators of power were created. There are various ways of supplying energy of the microwave field to the plasma. Rectangular waveguides are permeated by the dielectric tube, which is transparent to a microwave radiation. The plasma is being maintaining at the intersection due to the dissipation of electromagnetic wave energy. The heat is carried away by a gas, which blows through the tube. The latter option represents a microwave plasma torch. Typically, H01 -mode is used. It is the mode, running through the waveguide. Electric field vector is parallel to the narrow wall of the waveguide. The field strength in this direction is constant and along the wide wall varies according to the cosine law. The discharge is organized in the middle of the cross section of the waveguide where the electric field has a maximum value. Plasma column is stretched along the vector E. The size of the waveguide is related to the applied frequency. For f = 2.5 GHz (wavelength in a vacuum 0 = 12 cm) the wide wall has a width of 7.2 cm, narrow wall has 3,4 cm. The diameter of the dielectric tube is equal to 2cm. The diameter of the plasma column is of the order of 1 cm. Continuous optical discharge. Peculiarities of an optical method of maintaining the plasma. Discharge in the optical range of the electromagnetic field is a relatively new phenomenon. Even the combination of words «Optical discharge» is not yet completely familiar. Nevertheless, it 33

captures an essence of a physical process just the same as the longentrenched terms such as a microwave discharge, RF discharge. Dense equili-brium plasma can be permanently maintained by an optical radiation, as well as others, by constant and oscillating fields. In the same way, blowing through the region of the discharge by a cold gas, we can make a plasma generator – optical plasmatron. The feasibility of these processes has been theoretically proved and in 1970 a con-tinuous optical discharge was obtained in the experiment. Scheme of experiments. A Ray of the CO2 is being focused by lens (or mirror); Fig. 7. The lens is made of NaCl , or KCl , because an ordinary glass is opaque to infrared radiation  10,6 microns. To ignite the discharge, one must create an initial plasma chamber. This could be done by the gas breakdown in the focus region, using the auxiliary system or by putting for a short time a tungsten wire in the focus. Some metal from the surface will evaporate, vapors will be ionized and begin to absorb the laser beam. Then the wire is removed, and the discharge continues to burn in the atmosphere of the gas. Plasma will be shifted from the focus towards the laser radiation till the cross section of the converging light channel, where the intensity of the beam is still enough to maintain it. Its dimensions vary from 1 mm (the existence limit) to 1 cm and more at high-power of the laser.

Figure7. The experimental device for obtaining a continuous optical discharge. Plasma (shaded) is shifted from the focus towards the laser radiation Questions to Lecture 9: 1. Which states are called stable and unstable? 2. Describe the process of optic gas discharge.


LECTURE 11 Topics to be discussed in the lecture. Gas breakdown at high pressures. Purpose of the lecture: study of gas breakdown at high pressures. Key words: avalanche, ion-molecular processes.

Spark discharge takes place at atmospheric pressure or higher, when voltage, applied to the electrodes, is above the breakdown voltage. For a breakdown of the gap with such a big value of pd, one needs to have the considerable voltages, about tens or hundreds of kilovolts. The discharge passes rapidly, in non-stationary manner, and represents a phenomenon, which in everyday talk is called «skips a spark». Spark is accompanied by a characteristic crackling like lightning – a thunder peals. The sound is caused by the shock wave. Its source is a sharp increase of pressure due to the intense Joule heat release in the spark channel during the passage of a strong discharge current. Sparks occur even in a homogeneous field of plain gaps, being related to the arbitrary locations of the electrodes, and in strongly inhomogeneous fields: between the tip and the plane, between a thin wire and concentric cylinder, etc. In the latter cases, the spark discharge, if one moves to ever higher electric field strength, is preceded by a corona discharge. Corona is a low current discharge, which appears in the vicinity of a tip and a wire, where the field is dramatically enhanced. Ionization occurs only in this zone, and a gas glows. Electric current is closed by the flow of charges of either sign (depending on the polarity of the tip), which are produced in a self-maintaining zone near the tip and stretched by a relatively weak field to the other electrode. In the outer region there is no glowing. Corona discharge usually occurs at atmospheric pressure, in the air around the wires of high voltage lines, near lightning rods and masts of ships («the lights of St. Elmo»). For the ignition of corona one need to require a certain, relatively high voltage, which depends on the specific conditions. At even higher voltage the rest of the gap between the electrodes breaks down there, and skips a spark in the interelectrode region.


A strong current runs through the formed spark channel, its value is about 104 – 105 A. Due to the voltage drop across an external resistor or due to a fast capacitor detente (if the latter provides power to the discharge), the voltage across the electrodes decreases rapidly and the discharge is extinguished. If as the result of the discharge extinction the voltage across the electrodes is restored, then the breakdown repeats. If the power supply has sufficient capacity and can provide a flow of high current for a long time, then due to the current of a spark a cathode spot and the arc lights are formed. In general, the plasma state in established channel is like a state in the arc column, so that sometimes the final stage of the spark discharge can be regarded as an impulse arc (pulsed arc discharge). Breakdown mechanism, based on an avalanche multiplication through the secondary emission cathode, operates mainly at low pressures, approximately at pd  200 tor  cm. The corresponding theory, the principles of which were formulated by Townsend at the beginning of the century, explains many things. It gives a convincing interpretation of the Paschen breakdown voltage dependence V on pd with its characteristic minimum, and even agrees with the experiment quantitatively. However, with an improvement of experimental techniques there appeared, more and more new facts, which do not fit into the framework of a Townsend scheme. For large pd and significant over voltages, the breakdown in a plain gap develops much more quickly than required by the propagation of avalanches through the cathode emission. Ion-electron emission generally discounted, since during the time of the breakdown ions simply do not have a time to «budge». But even the mechanism of a photoemission is not fast enough in this case, because the conductive channel under these conditions is formed during the time of the electron pass from the cathode to the anode, or even sooner. And there is no time for avalanche repeating through the cathode emission. High-speed photography has allowed us to observe such a glowing ionized channel, which covers the period, after the passage of the first powerful avalanche.


The last most accurate measurements testify that cathode processes do not take part in the mechanism of a breakdown evidenced by the independence of the breakdown spark voltage from the cathode material. Field at the cathode is too low, and a multiplication of electrons does not occur. Streamer theory. The foundations of new theory, designed to explain the phenomenon of spark breakdown, were laid in the works of Leba, Mika, Raether about 1940 [6]. It is based on the notion of germination between the electrodes of a thin ionized channel – streamer, which makes its way through the positively charged track of the first powerful avalanche. And the electrons of many secondary avalanches follow this track. Leader. In the broad air gaps, in the lightning discharge there is a breakdown which occurs due to the sprout of a leader from one electrode to the other and also of a thin channel, but which is ionized and is conducted stronger than the streamer. This process is larger than the streamer: it includes streamers as an integral element. The breakdown, which is a real danger in high voltage engineering, one should consider it as the effect of a short circuit: the formation of the highly conductive spark channel, which transmits such a strong current that the voltage on the discharge gap decreases rapidly. An individual avalanche is a primary and essential element of any mechanism of a breakdown. Let us consider an avalanche in a uniform external field E0 between plate electrodes. Let it start from a single electron emitted from the cathode at time t = 0. The x-axis is directed from the place of the cathode to the anode. Radial distance from the x-axis is denoted as r . Diffusion spatial distribution of charges. Taking into account the possible formation of negative ions, the total numbers of electrons and ions increase as the avalanche move forward: dN e


 (  a) N e ,

Ne  exp[(  a) x], Ni  

dN i 


  Ne ,

dN i 


 aN e ,

 a ( Ne 1), Ni   ( Ne 1),  a  a




where  and a are the coefficients of ionization and attachment (Townsend coefficients). All new electrons fly to the anode like one group with the drift velocity. However, due to the diffusion of an electron, a cloud spreads around a central point x0  vDt , r  0. Electron density in the cloud ne ( x, r, t ) obeys the diffusion equation, which must take into account the drift and the birth of electrons. Solution of the equation has the form:

ne  (4 Det )3/2 exp[

( x  vDt )2  ( e  a )vDt ] . 4Det


The density ne decreases with a distance from the moving center in according to the Gaussian law. A radius of a sphere, where the density e times less than the central density ne ( x0 ,0, t ) , increases with the time rD ( x0 )  4 De t  4

8 x 0 De xe  , 3eE0  e E0


where  is a mean chaotic energy of the electrons. In the absence of attachment in the limit t   and not far away from the axis the approximate solution to (58), (59) for ions give

ni  ( x, t ) 

 r2  x  exp{ },  [rD ( x)]2 [rD ( x)]2


where rD (x ) is defined by (61). The contour rD (x) is not parabolic, rD  x , but wedge-shaped: rk  rD  x  8 / eE 0 x



Figure 8. Schemes of the fields in the presence of an electron avalanche: the left – the line of the external field E0 and of the field of a space charge of the avalanche E are presented separately, on the right – the line of the electric field strength of the resulting field E = E0 + E; circles show schematically centers of space charges

When a number of charges Ne is large, the diffusion spreading of an electron cloud give way to their electrostatic repulsion. And the velocity of this last process increases with Ne, i.e. t , x, while the diffusion speed drD / dt  t 1/ 2  x 1 decreases. Expansion velocity of the charged sphere which takes place due to the repulsion is determined by the electron drift in the self-charge field:

dR / dt  e E  ee R 2 exp( x), x  e E0t.

Figure 9. Scheme of the electric field in the gap after the avalanche has reached the anode and the electrons leave to the metal: left – the line of the field space-charge of the avalanche track E and its electrical image in the anode; on the right - the line of the electric field strength of the resulting field E = E0 + E


By integration we can find the expansion of the sphere R (t) or R (x), then the field Eand electron density ne = 3Ne / 4R3: 1/3

 3e   E0   x  3E   R , ne  .  exp   4 e  3   E0   E0 


The diffusive spreading is replaced by repulsion by some estimation at N e  e x ~ 10 6 ,  x  14 . The field E in this case is 23% of the external field. The measurement results are consistent with the estimates of the given type. Questions to Lecture 11: 1.What is the streamer and leader? 2.What is an avalanche? Recommended literature [1,3,5]

LECTURE 12 Topic title: The concept of a streamer. The cathode and anode streamers. The theory of Loeb and Meek. Backward wave of the strong field and ionization. Expansion of the spark channel. Corona discharge. Trichel pulses. The objective of the lecture: the concept of the cathode and anode streamer, corona discharge. Keywords: streamer, corona discharge.

A streamer is a moderate, weakly ionized thin channel, which is formed from the primary avalanche in a strong enough field and germinates in one or another or both sides of the electrodes. Having some conductivity, it can transform a field that open up the possibility of a sharp increase in ionization and current, and this will eventually lead to a spark discharge in the gap. Appearance of the streamer and the complete of the gap are not necessary, but sometimes it’s a sufficient condition for the breakdown. For the transformation of the avalanche into a streamer, the field must reach a sufficiently high amplification. Space-charge field 40

should increase to values of the order of the applied field; otherwise there will be no reason for the disruption of the normal flow of the avalanche development. This takes place in the not-too-long plain gaps, with not very high overvoltage (compared to the breakdown), this happens when the avalanche exhaust the entire reserve of the amplification, i.e. reaches the anode. Then the streamer originates near the anode, in the most dense part of the space charge, and grows toward the cathode. Such streamer is called a cathode streamer or positive. Along long gaps, at high overvoltages, the number of charges in the primary avalanche becomes large enough. And the avalanche is reborn into a streamer before reaching the anode. In this case, the streamer grows to the both electrodes. If the streamer is formed when the avalanche is still not far away from the cathode, it grows mainly to the anode. This streamer is called anode streamer or negative. Formation mechanism of the cathode streamer. It is explained in Fig. 10. The main role, according to the hypothesis, belongs to the energetic photons, emitted by excited atoms in the avalanche and produce photo ionization near the primary avalanche. Electrons, extracted by photons, initialize the secondary avalanches, which are drawn into the track, just because the resulting field is directed in this way (Fig. 10). Electrons of the secondary avalanches, mixing with the primary avalanche ions, form a quasi-neutral plasma. They also excite atoms, which leads to the emission of new photons. Ions of the secondary avalanches increase a positive charge at the cathode end of the plasma channel. This positive charge creates a field that attracts electrons of the secondary avalanches, etc. so streamer grows. The process of neutralizing of the ion track of the primary avalanche starts from the place where a positive charge and the field have the biggest value – from the anode, unless it reached a condition of rebirth    0 . This situation is shown in Fig. 10. If the source of the photons and the seed electrons for the secondary avalanches is strong enough, and this, appears to be a case, the growth rate of the streamer is not limited by the rate of a avalanches generation, but by the rate of neutralization of the positive space charge near the cathode end of the streamer. But the electrons are drawn into this area with the drift velocity corresponding to the field existing there. The field significantly exceeds 41

the external one, and its strength increase with the height of the of the «needle»; this is dictated by electrostatics.

Figure 10. Scheme of the cathode streamer: left – a streamer in two consecutive moments of time; there are the secondary avalanches, tending to the positive streamer head, wavy arrows – photons, due to which seed electrons occur for avalanches are shown; on the right – the electric field strength lines around the streamer head

Since photons are emitted and absorbed randomly, there may be situations where at some point there is a new preferential direction, along which there is a lot of secondary avalanches will occur. So, apparently, zigzag kinks of the streamer and the spark channel are formed, which are observed in the experiment. As it follows from the above, streamer forms from an avalanche if the field of its space charge reaches the order of the external. The Corresponding approximate equality,   eR 2 exp    0  x    0 ,


can be regarded as a criterion for the streamer formation. Thus a known Meek breakdown condition is obtained, which in simplified form is: x 8    0  d  18  20 , Ne  e ~ 10 .


In the theory of Loeb and Meek, the onset of the breakdown was identified with the fact of the streamer formation. In fact, it is not always so. 42

Mechanism of its germination to the cathode remains the same. But the character of the process of its expansion to the anode is a bit different from the previous one, since in this case electrons drift in the same direction, in which the front of the plasma streamer move, and not in opposite, when the germination to the cathode takes place. Secondary avalanches originate under the influence of photo radiation in front of a negatively charged streamer head, facing to the anode (Fig. 11). Front electrons of the head move quickly in a strong total field 0   and enter the ion track of the secondary avalanches and together form a plasma.

Figure 11. Scheme of anode streamer: left – photons and secondary avalanche in front of the head of the streamer in two consecutive moments of time; on the right – electric field lines around the streamer head

The potential of the peak or of the streamer head, germinating from the anode to the cathode, is less different from the anode potential comparing to the potential of the unperturbed field at the same point. Ideally conducting streamer, which is in a contact with the anode, would be at the anode potential. This was discussed earlier, and this is causing the field enhancement at the streamer head. As the head approaches the cathode the fraction of the applied voltage to the electrodes, which falls on a non-conducting gap between the head of the streamer and the cathode, and a field in the gap both increase. By the time when the head contact with the 43

cathode, the field becomes so strong that electrons are ripped from the cathode, or from atoms with photons, and their number increase with great intensity. There is a front propagating from the cathode, along the channel of the initial streamer in the opposite direction to the anode, behind which there is more highly ionized plasma. It is like a backward streamer, but now with a significantly higher ionization than the original. The current flow of large density is accompanied by concentrated release of a Joule heat. This leads to a strong heating of the plasma and its thermalisation, and probably, to increment of the thermal ionization. The channel is now expanding due to the radial expansion of the gas, carried along by a shock wave and due to the process of the heat conduction. An indispensable condition for the corona discharge is a sharp inhomogeneity of the field. About one or both of the electrodes, the field should be much stronger than in the rest of the gap. Exact solutions of the equations of electrostatics for simple geometries, are indispensable for the construction of theories of the corona and for the interpretation of experiments. In the space between coaxial cylinders with radius r (inner) and R, on some radial distance x from the axis, the field is   V x ln  R r  ,  max  V x ln  R r  ,


where V is the voltage between the cylinders. For a single wire above a plane ( b  ) and for just two wires ( d   ), we have respectively  max  V r ln  2d / r  ,  max  V r ln  b r  .


When the applied voltage V is less than the ignition voltage of the corona for the conditions Vc in the circuit, one can register a nonself-maintained current of the order of 10 -14 A. It means ions start to stretch, which appear under the influence of the cosmic rays and natural radioactivity. For example, in the air in 1cm3 there are 10 pairs of ions. The ignition of the corona in the laboratory is exhibited not only by the glow near the corona discharge electrode, which may 44

be not seen, but also by an abrupt increasing of the current up to the value of the order of 10-6 A. Corona discharge belongs to the number of self-maintained discharges, and the condition of its emergence reflects the physical mechanism of reproduction of the electrons in the region of enhanced field, where ionization occurs. The mechanism of electron multiplication depends on the polarity of the corona electrode. If it is the cathode (so called negative corona) then a multiplication of avalanches occurs. The secondary process is emission from the cathode, and possibly, a photoionization in the gas volume. Ignition of a negative corona, in principle, has no different from the Townsend breakdown and ignition of dark Townsend discharge. Taking into account the effects of attachment and nonuniformity of the field, we can write the relation x1

   x   a  x dx  ln 1    , 1



where  is an effective secondary emission coefficient. If it is the tip then the wire becomes an anode (a positive corona), and the remote large cathode (around which the field is weak), does not participate in the multiplication. Multiplication of electrons provides secondary photo processes in the gas in the area of the tip. In contrast to the smooth glow of a negative corona, one can observe glowing filaments in the positive corona, running away from the tip. It is assumed that there are streamers. As the criterion of the ignition of the positive corona one can take the condition of the streamer (3.9) formation generalized also to the case of non-uniform field: x1

   a dx  18  20.



The intermittent phenomena were discovered in the laboratory by Loeb Trichel, Keane (1938). Review Questions: 1. Types of streamers. 2. Trichel Pulses.


LECTURE 13 Topics to be discussed in the lecture. The leading mechanism of the breakdown. Purpose of the lecture: study of lighting. Key words: leader channel, the back stroke.

Along the path that is made by previous streamer there sprout up a thin highly ionized conductive channel, which initiates from the active electrode from the strong fields area, and to a far greater degree, comparing to the streamer, transfers the potential of the electrode to its top point. This is called a leader. The channel of the leader somehow lengthens the tip of the electrode moving it together with the high potential in respect to the other electrode to this electrode. Being, like the tip of a metal edge, a source of a particularly strong field, the head of the leader exudes streamers, which prepare the initial number of electrons. Electrons intensively ionize the gas in the strong field of the leader head, creating a «new» head and thereby ensuring the propagation of the highly ionized channel (Fig. 12). When it loops the interspace, a backward wave propagates from the second electrode (plane), and this is the beginning of the transformation of the leader channel into a spark one.

Figure 12. Scheme leader, germinating from the positive edge of the path traced by the streamer, which, in turn, involve an avalanche

We have to repeat almost the same words that we had previously used about the streamer breakdown mechanism. Indeed, the difference between the streamer and the leader is not so much qualitative as quantitative: it depends on the degree of ionization, on 46

the power generated by the field. Streamer captures avalanches, the leader captures streamers. In the chain of processes that initiate a spark: avalanches – streamer – backward wave, another link is included: avalanches – streamers – leader – backward wave. Hypothesis about the mechanism of the leader formation. We do not have a clarity here, and the theory of leader processes is even more far from perfection than streamer ones. It was conjectured, that the current in the streamer channel, which extends from the anode, with its Joule heat warms up the plasma at the mouth of the channel to a sufficient temperature for thermal ionization. In this case plasma starts to ionize heavily and along the primary weakly ionized streamer channel a highly ionized leader grows out from the anode. However, according to estimates, which are made for defense of the hypothesis, the streamer current heats the air just till 3000 K, and it is not enough to initiate thermal ionization of air, which requires 8000 K. Some information about the leader can be extracted by decoding the data of slow-motion photography, which are taken with an electron-optical apparatus, simultaneous electrical measurements, etc. The head of the leader is charged with the same sign as the electrode, from which the channel is stretched. And the whole leader path is charged with the same sign. This is dictated by electrostatics. After all the side surface of the long conductor, nibbed from the electrode being at more or less same potential close to the electrode one, is the source of the radial electric field. Germination of the leader is accompanied by the injection of a charge in it. But any movement of charges in the gas gap, even when the leader head is isolated from the opposite electrode, causes a current in the external circuit. This current can be detected on the oscillograph. The field in the channel can be found, using an auxiliary electrode probe. Leader propagates with a speed of 2 • 106 cm / s, its channel expands with a radial velocity  r  104 cm / s. A streamer zone stretches before the head at a distance about of 1 m (d ~ 10 m). There are also radial streamers which move away from the side surface of the channel, where there is a strong radial field. They take to the side some part of the charge, so that the leader channel is surrounded by a cover of a weakly conducting plasma and volume charge. For the leader propagation less external field is enough comparing to the streamer propagation. Considering the breakdown voltage in air we come to that there only 200 – 500 V/cm. 47

When the streamers of the leader head reach electrodes (plane), the leader begin to accelerate. An intermediate stage, before leader head touches the electrode, is called through stage. Sometimes, when the leader comes close to the electrode due to the high field appearing between them there is a leader which starts from the opposite electrode. Backward wave (return stroke). Here we will not consider the through stage and collision between main leader and the opposite one. A collision of the leaders, in principle, is not different from collision of the leader with the electrode. When the head of the leader make a contact with the electrode, its charge immediately neutralizes, and the top channel acquires the potential of the electrode. If this is the cathode, then electrons are pulled out of it by photons or strong field, and the cathode spot is formed, if this is the anode – the leader electrons move to the metal. Along the leader channel there is a backward wave which propagates in the direction of the tip and is responsible for potential relieve and disappearance of the linear charge of the leader channel and its surrounding («cover»); charge may be neutralized completely or partly. The physics of a backward wave effect could be better understood if we consider a process similar to a discharge of a charged long line when faulting it on the ground. Exactly such treatment of the phenomenon was used for numerical model calculations.

Figure 13. Scheme of the discharge on the ground perfectly positively charged line, simulating a backward wave after coming to the cathode of the positive leader; t – the time of the closure line on the ground, t1, t2 – wave motion. Showing diagrams of current and potential


Lightning and thunderclouds. The initial cause of electrical discharges in the atmosphere is the formation and spatial separation of positive and negative charges, due to which there appear an electric field. If the field reaches a value sufficient for a spark breakdown in air, then a discharge occurs. Approximately in 90% of cases, the negative charge is located at the bottom of the clouds, but main positive charge is at its top. For the cloud, shown in Fig. 3.9, the potential differences V between the centers of positive and negative charges q, and between the negative bottom side of the clouds and the earth have the order of V ~ q / L ~ 108 V (q = 40 C = 3,6 • 1013 V • cm, L  2  5 km), the average electric field is of the order of  ср ~ V / L ~ 300 V / cm. Such fields of the same order are measured by the ground under thunderclouds, and inside the clouds (by aircrafts). Modern ideas about the origin of an atmospheric electricity were laid in 40-ies by Ya.I. Frenkel. Charges in the atmosphere appear as the result of ionization of molecules or due to extraction of electrons from the macroscopic particles under the influence of cosmic rays.

Figure 14. Scheme of the thundercloud (7); probable distribution charges. The circles denote the centers of gravity of the charges

According to the measurements of electric fields near the clouds in the top p = +40 C, in the bottom p = +10 C, N = - 40 C. It is also 49

possible to electrify water droplets when they spray during their fall. Tear off spray take away predominantly a negative charge. And negative ions formed during the ionization of air tend to stick to the water droplets. It is known that the polar water molecules in the surface area of the droplet are oriented with positive ends inward, and negative – outward, therefore there an electric double layer is formed by the surface. Inside of the layer there is a field, directed toward air. Outside the layer there is no field. Consequently, there is a potential jump  and in the droplet  is higher than in the environment; according to measurements  = 0.26 V. A droplet captures negative ions, until a potential difference between the liquid and air disappears. Number of absorbed charges N is defined by an equality, eN / r   , where r is a radius of the droplet. When r = 10-3 cm then N =2 • 103. Charge separation in the scale of the cloud is the result of the subsidence of the negatively charged droplets under influence of gravity; many positive charges remain in molecular form and do not tend to move down. Lightning flash lasts in the average 200 ms. It consists of several pulses of 10 ms with intervals of about 40 ms. Each pulse begins with the germination of the leader channel from the cloud to the ground (Figure 15). The channel lights poorly, with the exception of the head part. The leader carries a negative charge (from a negative cloud), while the current is about 100 A. While approaching the ground the channel branch out and their paths have zigzag form. When the main leader reaches the ground or collides with the opposite leader, then in the opposite direction, towards the cloud along its path with an enormous speed about 0.1-0.3 of speed of light there propagate a glowing channel – a backward wave. This is called a return stroke or the main stage of lightning. In this case a lightning current reaches a maximum value of 100 kA. It is a peak current the most dangerous effects of lightning are related to (overvoltage in transmission lines, etc.). Then, through the formed spark channel, within 40 ms with a small current of 200 A, the negative charge flows down to the ground. Outflow of an electricity from a large volume clouds with dimensions of about 1 km is possible only thanks to the release of electrons from the negatively charged


macroscopic particles and ions, which occurs under the influence of a strong field. Due to the heat release in the channel pressure increases, becoming a source of impact waves. On long distances the impact wave transforms to an acoustic wave which we hear as a thunder. During the lightning qV / 2 ~ 109 1010 J energy is released, which correspond to an explosion of a ton of explosive material. According to spectroscopic measurements, the temperature in the spark channel is approximately 25 000 K, the density of electrons is 1  5   1017 cm-3, which corresponds to a general complete single ionization of atoms.

Figure 15. Scheme of lightning: a – first (step), the leader goes to the ground at speeds υ1; b – a wave of return-stroke goes up with velocity υ2; in – there was in clouds breakdown of the return stroke channel on the left side of the cloud, the charge of the right side of the stack to the spark channel d – the second leader of the (sagittal) is a rate υ3 partially decomposed by the plasma of the spark channel, etc.

Charges from distant parts of the cloud come to the top of the channel as a result of local inner clouds breakdowns, and when the path to distant regions is laid, the next pulse of lightning begins. At this point, the conductivity in the spark channel of the first stroke falls down and by a residual channel a new leader extends from the clouds to the ground. It increases the ionization in the old channel, second and subsequent leaders do not branch out. When the second leader reaches the earth, the second back stroke takes place, and it repeats several times until the entire negative charge from the distant parts of the cloud will not flow to the ground. A positive charge located far away (very high), apparently remain, since for this huge distance from the ground a voltage between it and earth is too small for a breakdown. 51

Back stroke. This process has been discussed above. A strong current flows after the wave, which is directed upward from the earth and carry the charge away from the leader channel and its surrounding into the land. A potential there is close to the potential of the ground. A potential in front of the wave same as leader’s one, which is close to the cloud’s potential, but there is practically no current, since the leader current finishes with the stopping of the leader movement. In according to the photoscan, the backward wave of a lightning sweeps upward at a velocity   (0,1 – 0,3) c  3  104 – 105 km / s, i.e., in 102 μs. Questions to Lecture 13: 1. The structure of a thundercloud? 2. Lightning?

LECTURE 14 Topics to be discussed in the lecture. Plasma heating. Purpose of the lecture: demonstrate the application of gas discharges in science and technology. Key words: devices of fusion power, dusty plasma.

As it is known, gas discharges are used for (preliminary) heating of a gas. a) Plasma of a positive column of a glow discharge was used in Langmuir experiments (in the first experiments for the study of plasma). b) In order to fill the open trap (magnetic mirror) with plasma Marshall guns were used, in which plasma is formed due to the pulsed arc discharge and fired in the required direction. The trap is filled with plasma through «caps», and then «caps» are closed. c) The formation of plasma in Tokamak begins with the passing of the RF induction gas discharge, in which, due to the release of Joule heat, plasma is warmed up. Further, the heating is performed by injection of fast atoms beams to the plasma. d) Generation of plasma in various experimental and industrial plasma torches is connected with the formation of gas discharges. 52

Such plasma torches can be used for cutting, welding of steel structures, destruction of rocks, etc. Glow discharges are used as lighting devices, the positive column could be used as a medium with an inverse levels population (CO2 laser), arc discharges are used for welding and cutting of metals, corona discharges are used as electrostatic precipitators in the funnels for removing large particulates. Gas discharges are used in order to create the different coatings, different tasks in the field of nanotechnology. It was found that in the process of passing of the gas discharge, the dust particles are formed (particles from the walls of a discharge chamber, a material of electrodes). They can significantly change the properties of the plasma, which are used in the installations of the fusion power. Questions to Lecture 14: 1. Where gas discharges are used? 2. What is dusty plasma?



1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.


RF-induction gas discharge its formation and use. Energy balance in the arc discharge. Model of the metal cylinder. RF induction discharge – the cases of high and low conductivity. Stable and unstable states of the rf induction discharge. Features of gas breakdown at high pressures. General representations of spark and corona discharges. Unacceptability Taundsend model breakdown for high pressures. Leader and the appearance of breakdown. The distortion of space-charge field. Anode and cathode streamers. Backward wave and the strong field ionization. Corona discharge. Trichel pulses.


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

Raizer Yu. P., Gas Discharge Physics (Springer-Verlag, (1991) 1st ed.). Granovskiy V.L. Elektricheskiy tok v gaze (ustanovivshiysya tok). – M.: Nauka, 1971. – 472 s. Rayzer Yu.P. Fizika gazorazryadnyih protsessov. – M.: Nauka, 1987. – 450 s. A. M. Howatson, Introduction to gas discharge (Pergamon Press, (1976) 2nd ed.). Lozanskiy E.D., Firsov O.B. Teoriya iskryi. – M.: Atomizdat, 1975. Velihov E.P. i dr. Fizicheskie yavleniya v gazorazryadnoy plazme. – M.: Nauka, 1987. – 156 s. Smirnov B.M. Problema sharovoy molnii. – M.: Nauka, 1988. – 124 s. Metodicheskaya razrabotka k spetskursu «Osnovyi fiziki gazovogo razryada» / sost. Yu.V. Arhipov. – Izd. KazGU, 1989. – 37 s. Metodicheskoe posobie k spetskursu «Fizika gazovogo razryada» / Yu.V. Arhipov. – Izd. KazGU, 2005. – 63 s. Smirnov B.M. Fizika slaboionizovannogo gaza (v zadachah). Izd. 2-e. – M.: Nauka, 1985. P. Lynch and A. Nicolaides. Worked examplesin physical electronics. HARRAP LONDOM, 1972 Metodicheskoe posobie – Elektronnyiy uchebnik – UMK «Fizika gazovogo razryada», sost. Arhipov Yu.V., Nikiforova O., 2005. Baimbetov F.B., Zhotabaev Zh.R, Ramazanov T.S., Arhipov Yu.V., Dzhumagulova K.N., Mukusheva M.K., Davletov A.E. Osnovyi fiziki upravlyaemogo termoyadernogo sinteza. – Almatyi-Kurcha-tov, 2004. – 232 s. A. Anders, Cathodic Arcs: From Fractal Spots to Energetic Condensation (2008). Springer, New York. R. L. Boxman, D. M. Sanders, and P. J. Martin (editors). Hand-book of Vacuum Arc Science and Technology (1995). Noyes Pub-lications, Park Ridge, N.J.

Additional literature: 16. Levitskiy S.M. Sbornik zadach i raschetov po fizicheskoy elek-tronike. – Izd. Kiev: Kievskogo universiteta, 1964. 17. Bronshteyn I.N., Semendyaev K.A. Spravochnik po vyisshey matematike. – M.: Nauka, 1981. – 719 s.



1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.


Plasma is a partially or fully ionized gas, in which the densities of positive and negative charges are virtually identical. Avalanche is the increase in the number of electrons, moving in the field due to the impact ionization. The equation of balance of electrons allows describing the distribution of electrons in space, during the process of breakdown, and obtaining the criteria of breakdown. Stationary and non-stationary criteria of breakdown allow estimating the possibility of a breakdown. The Townsend`s breakdown criterion is used for a constant field. It is based on the assumption that, after an avalanche, there must be more number of electrons (compared to the original number). Paschen law describes the dependence of the threshold field on geometrical and physical conditions in the gas-discharge volume. Optical breakdown is explained by the presence of multiquantum effects. Glow discharge has a complex structure, a number of areas, each of which is responsible for certain physical processes. Glow discharge potential distribution has a specific feature – the potential drop near cathode, where a positive volume charge is formed. Stratification and contraction are instabilities of the discharge relative to the longitudinal and transverse perturbations. Channel model is used for describing of arc discharge. RF induction discharge is formed under the influence of the inductor discharge volume. Model of metal cylinder is used for describing of the inductively coupled RF discharge at high temperatures. The anode and cathode tape drive is a tape drive, moving to the anode and the cathode respectively. Corona discharge is a discharge, appearing in thunderstorm weather on masts of ships. Lightning is a spark discharge, having a large length.


Lecture 1............................................................................ 3 Lecture 2............................................................................ 4 Lecture 3............................................................................ 7 Lecture 4............................................................................ 10 Lecture 5............................................................................ 14 Lecture 6............................................................................ 17 Lecture 7............................................................................ 22 Lecture 8............................................................................ 23 Lecture 9............................................................................ 27 Lecture 10.......................................................................... 29 Lecture 11.......................................................................... 35 Lecture 12.......................................................................... 40 Lecture 13.......................................................................... 46 Lecture 14.......................................................................... 52 Colloquium ........................................................................ 54 Referencies ........................................................................ 55 Glossary............................................................................. 56


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