Кафедра методики фізики
Педагогічний Університет ім.Драгоманова
Ministry of Education and Science Department of Physics Graduate student Danilova IV
1 INTRODUCTION 3
1.1 Definition of Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Characteristic of Plasma:Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Criteria for Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Nonlinear Plasma Theory and Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 DUSTY PLASMA 6
2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Charging of Dust Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Forces on a Dust particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Dust Acoustic Waves 10
4 SAGDEEV POTENTIAL 12
4.1 Plasma Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Nonlinear ion acoustic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 STUDY OF THE DUST ACOUSTIC WAVES IN A DUSTY PLASMA SYSTEM 16
5.1 Current Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Existence of Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Моя теорія Плазми буде розкритой ,та обгрунтованой ,і висвітленой та доведеной з точки зору теогетики та митодики викладання.
Методика викладання даного предмета лежить доцільно в вивченні ядерної фізики з курсу загальної фізики та виклажання предмету в середній школі та вищих навчальних закладах.
My theory of Plasma will be discovered, grounded, and illuminated, and proven in terms of theogetics and mythology The technique of teaching this subject is appropriate in the study of nuclear physics from the course of general physics and teaching the subject in high school and higher education.
1.1 Визначення плазми Слово "Плазма" має грецьке походження, що означає щось "ліпне або виготовлене". Два американських фізики Лангмюр і Тонкс у 1929 р. Вперше ввели термін "плазма". Як назва «Плазма» придумала історію сама, Тонкс і Ленгмур, вивчаючи позитивну область стовпчика в розрядній трубці, в якій міститься майже однакова кількість атомів та іонів, виявлено, що за певних обставин існує якась згуртованість серед частинки іонізованого газу, схожого на желе або те, як червоні та білі тільця рухаються через плазму крові. Звідси і назва "Плазма". Плазма - четвертий стан речовини. Коли молекулярний газ достатньо нагрітий, кінетична енергія молекул перевищує їх молекулярну енергію зв’язку, і вона дисоціює на атомний газ. При ще більш високій температурі, тобто поблизу енергії іонізації атомів або перевищує її, атоми газу дисоціюють на електрони та іони і утворюється іонізований газ або плазма. Електрони та іони в плазмі не обмежені, і коли вони рухаються, вони створюють локальні концентрації позитивних та негативних зарядів, що спричиняють електричні поля через рух цих зарядів, створюючи струми і, таким чином, породжуючи магнітні поля . Ці електромагнітні поля є невід'ємною частиною плазмової системи і призводять до дії навіть на великих відстанях, тобто плазма проявляє колективну поведінку. Плазми, отримані в результаті іонізації газів, містять майже рівну кількість позитивних і негативних іонів. Отже, за шкалою макроскопічної довжини плазма є електрично нейтральною, і ця властивість, виявлена плазмою, відома як квазінейтральність. Взагалі стан плазми речовини може бути визначений як "квазінейтральний газ заряджених і нейтральних частинок, який проявляє колективну поведінку і є макроскопічно електрично нейтральним провідником, здатним взаємодіяти з електричним і магнітним полями".
1.1 Definition of Plasma
The word “Plasma” has a Greek origin, which means something “molded or fabricated”. Two
American Physicists Langmuir and Tonks in 1929 first coined the term “Plasma”. How
the name “Plasma” came has a story itself, Tonks and Langmuir while studying the positive
column region in the discharge tube, which contained almost equal number of atoms and ions
found that under certain circumstances some sort of cohesiveness exists among the particles
of an ionized gas which is similar to jelly or the way red and white corpuscles move through
blood plasma. Hence the name “Plasma” was given. Plasma is the fourth state of matter.
When a molecular gas is sufficiently heated the kinetic energy of the molecules exceeds their
molecular binding energy and it dissociates into atomic gas. At even higher temperature i.e
near or exceeding atomic ionization energies, the atoms of the gas dissociates into electrons
and ions and an ionized gas or plasma is formed. The electrons and ions in a plasma are not
bounded and as they move around they create local concentrations of positive and negative
charges giving rise to electric fields whereas the motion of these charges generates currents
and thereby give rise to magnetic fields. These electromagnetic fields are an integral part
of the plasma system and results in action even at large distances i.e the plasma exhibits
collective behavior. The plasmas resulting from ionization of gases contains almost equal
number of positive and negative ions. Hence in the macroscopic length scale the plasma is
electrically neutral and this property exhibited by the plasma is known as quasi-neutrality.
In general, the plasma state of matter may be defined as “A quasi-neutral gas of charged and
neutral particles which exhibits collective behavior and is macroscopically electrically neutral
conductor capable of interacting with electric and magnetic fields”.
1.2 Characteristic of Plasma:Debye Shielding
Apart from the quasi-neutrality and collective behavior exhibited by plasma, one of the most
important characteristic shown by plasma is its ability to shield out electrostatic fields applied
to it or to defend the perturbations applied to the plasma system. Suppose we perturb the
plasma system by introducing a positive charge +Q, then the highly mobile electrons will surround the positive charge, forming a sheath called Debye sheath and will effectively shield
the effect of the +Q charge so that the plasma remains electrically neutral. This is known
as Debye shielding and the distance upto which the effect of the perturbation is shielded is known as Debye length. The expression for debye length is given as-
λD =r εoKTe ,ne2
The Debye shielding is valid if and only if there are large number of particles in the shielding
1.3 Criteria for Plasma
For an ionized gas to be called plasma it has to satisfy the following three conditions-
1. λD L; where λDis the Debye length of the plasma and L is the dimension of the system
2. ND 1; where NDis the number of charged particles in the Debye sphere
3. ωτ >1 ; where ω is the frequency of plasma oscillation and τ is the mean time between
collisions with neutral atoms. 
Plasma constitutes 95-99% of the matter in the universe. Stars, nebulae, and even the
interstellar space are filled with plasma. The solar system also comprises of plasma in the
form of solar wind and the Earth is surrounded by plasma trapped within its magnetic fields.
Plasmas also occur in lightening, fluorescent lamps, in many of the laboratory experiments
and in numerous industrial processes. Thus in recent past “Plasma effects” have found ever
increasing applications in astrophysics, solid state physics etc and has been widely studied.
Plasma is a highly complex system and for its study many theories are put forward which
are based on approximations. For the sake of simplicity we can divide plasma theory into
• Linear Plasma Theory
• Nonlinear Plasma Theory
1.4 Nonlinear Plasma Theory and Nonlinear Effects
In describing most of the plasma phenomenon we assume that the amplitude of perturbation
is small and can be linearized i.e higher order terms can be neglected. For simplicity let us
consider the function f (ax + by); if we can write the function as below-
f (ax + by) = af (x) + bf (y) (1.2)
then the function can be called as a linear function. But in most of the cases the amplitude
of perturbation grows exponentially and the linearity breaks down and the function can not
be written as a superposition of two different functions as seen above. In linear theory, we
assume that the dependent quantities have a harmonic wave solution which is proportional
to exp (iωt + ik.r) but in the nonlinear theory this linearization cannot be taken. Now the
question which arises is Does the amplitude of the wave goes on increasing? If we observe the
wave after an interval of time, it is observed that the wave amplitude saturates and attains
large values. It means that some effects have limited the growth of the wave amplitude and
these effects are known as Nonlinear Effects. Plasma is a highly rich nonlinear medium.
The nonlinearities arises from the harmonic generation involving the trapping of particles in
the wave potential, ponderomotive force, nonlinear Lorentz force etc. These nonlinearities
in plasma contributes to the localization of the waves leading to different type of intensity
coherent structures viz solitary structures, shock waves, vortices etc. These structures are
studied both theoretically and experimentally
Nonlinear effects in plasma can be divided into three categories-
• Basically nonlinearizable problems such as anomalous diffusion, hydromagnetic equilib-
rium, the plasma sheath etc.
• Wave particle interactions such as trapping in wave potential, plasma wave echoes, etc.
• Wave-Wave interaction such as beating of waves, resonant interaction between waves,
parametric instabilities etc.
Here we have mainly focused on the characteristics of the Solitary Waves structures formed
due to the nonlinear effects in a dusty plasma
Dust is one of the omnipresent ingredient in the universe and most of the dust particles in
our solar system comes from the collisions of asteroids, comets, meteoroids etc. The captured interplanetary dust particles show traces of silicon, oxygen, calcium, iron and nitrogen.
Thus in most cases a plasma is found to coexists with dust particles whose size ranges from
nanometer to micrometer. These dust particles are heavy and large compared to ions and
electrons. They are not neutral but are positively or negatively charged according to the sur-
rounding plasma environment. Thus a mixture of charged dust or macroparticles, electrons,
ions and neutrals forms a “Dusty Plasma”. The presence of dust particles makes the plasma
system much more complex. The history of dusty plasma is quite old (Mendis 1997). Dusty
plasmas are low temperature fully or partially ionized electrically conducting gases. Dusty
plasmas are widespread in astrophysical situations like in the rings of Saturn, in cometary
tails or in interstellar clouds. 
If rd = is the dust grain radius, a = intergrain distance, λD =the plasma Debye radius, then
the situation rd a λD corresponds to a Dusty plasma and the charged dust grains
participate in the collective behavior.
Presence of dust particles in plasma:
• Alters the density and charge distribution,
• Modifies plasma instabilities
• Introduces new dust driven waves.
The measurement of dust particles provides us information regarding:
• The forces operating between the particles of the plasma.
• The forces arising due to their interaction with the plasma.
The electrostatic potential of the plasma.
• Velocity distribution and thermodynamics of the dusty plasma.
• Phase transitions
• Self organizing properties of matter like crystal structures.
• Waves and many more phenomena in the kinetic particle level.
Hence Dusty plasma have become a new line of research in the field of plasma. Many
theoretical and experimental studies have been going on in this new section of plasma.
2.2 Charging of Dust Particles
The charge on a dust particle cannot be assigned beforehand and mainly depends on the surrounding plasma parameters and the size of the dust particle. The charging processes mainly depends on the:
• Interaction of dust grains with gaseous plasma particles
• Interaction of dust particles with energetic particles (ions and electrons)
• Interaction of dust grains with photons.
Different theoretical and experimental methods/models have been deviced in order to predict the charge on a dust particle and one of them which is most used is the “Orbit-motion-
Limited” model which is basically valid on the assumption of collisionless ions. When finite
sized neutral dust grains are immersed in an unmagnetized plasma, the plasma particles
(electrons and ions) are collected by the dust grains which act as probes. The dust grains
are therefore charged by the collection of the plasma particles flowing onto their surface. If
electrons are collected by the dust particles, then the dust particle acquires a negative surface potential; on the other hand if absorption of positive ions takes place then the surface of the dust particle acquires a positive potential. The current depends on this surface potential.
When the surface potential is negative electrons are attracted and ions are repelled as such the dust grain current carried by the electrons reduces and that carried by the ions increases.
The situation will be reversed if the surface potential is positive. The OML theory predicts
these currents and conversely by knowing these currents one can predict the charge on the dust particle effectively.
The dust grains are basically negatively charged due to the high mobility of the plasma
electrons compared to the ions. When energetic plasma particles are incident onto a charged
dust particle they may undergo back scattering/reflection by the dust grains, some of them may even pass through the dust grain. During the passage they may lose their energy fully or partially. A part of this energy excites other electrons which in turn may escape from
the material. These emitted electrons are called secondary electrons. The emission of these electrons makes the surface of the dust grain positive. There are other processes too by which
dust particles can be charged like Thermionic emission, Field emission, Impact ionization etc.
2.3 Forces on a Dust particle
A dust grain is subjected to a number of forces like gravitational forces, ion and neutral drag
forces, radiation pressure forces, electric field force etc. Focusing on the electric force field;
we see that the electric force field acting on a charged dust particle due to an external electric
field is given as:
FE = QdE ,where Qd = is the charge on the Dust particle
E = is the external electric field
The electric force is the governing force for charged dust particles. As mentioned earlier the charged dust particles acts like Langmuir probes and due to the shielding property, a Debye sheath surrounds the dust particulates which shields its long range coulomb field. In most of the cases the Debye sheath comprises an excess of positive ions and a deficiency of electrons due to the negatively charged nature of the dust particle (as the mobility of electrons is more than ions).
Hamaguchi and Farouki analyzed that the shielding cloud is formed due to the presence of the electric field of the dust particle and the external electric field. The shielding cloud acts
according to these fields and is an effect observed due to the presence of these charged dust particles. When the dust particulates move from one part of the plasma to the other the
shielding cloud doesn’t move along rather the concentration of ions decays at the former
place and reforms at the new site. The shielding cloud is not attached to the dust by a force and thus no counter force acts on the dust from the cloud. Thus full force (indicated by equation acts on the dust particle. The shielding cloud doesn’t hinder the action of
the electric field force. This is the case for spherically symmetric shielding clouds. However the spherical symmetry of the shielding cloud may get distorted if there is a density gradient or an external electric field; generated by direct charging processes.
In case, of non-uniform plasma the Debye sheath is a function of position and no longer
spherically symmetric but gets deformed as one side of the dust particulate may be sub-
jected to more positive ions, thus creating an excess of charge on one side of the Debye
electron and ion pressure and the inertia is being provided by the massive dust particles.
Moreover the phase velocity of the DAW is much smaller than the electron and ion thermal
speed. The dust plasma frequency can be given as ωpd =s z 2,d e,2ndo,εomd
where zd= is the number of charges residing on the dust grain surface.
md =is the mass of the dust particle.
ndo= is the equilibrium dust density.
The dust acoustic wave speed can be given as CDAW =r KTi md z 2 d ε
Like in the ion-acoustic wave the the sound speed depends on the electron temperature and
on ion mass. In DAWs the dust wave sound depends on the temperature of the lighter species
i.e the ions (T)iand mass by the heavier species i.e the dust particles (md).
Dust Acoustic Waves
Experimental studies have proved that plasma is highly rich in waves and by studying these
waves we can get information of the plasma parameters. The electrons, ions in a plasma
move on their own and during their motion they interact with each other through their elec-
tromagnetic forces and also responds to the perturbations applied to the system externally.
Thus different variety of waves are seen to exist due to the coherent motions of large plasma
particles. An electron ion plasma sustains both longitudinal and transverse waves.
Example of longitudinal waves in unmagnetised plasma are Langmuir waves and ion-acoustic
The situation changes when dust particles come into play. Presence of the dust particles
modify or even may dominate the wave propagation. The change in wave propagation takes
place due to the inhomogeneity associated with the random distribution of the charged dust
particulates, the deviation from the quasi-neutrality condition because of the presence of dust
and also due to the consideration of the dust particle dynamics. The waves in dusty plasma
can be broadly divided into two classes-
1. Weakly coupled plasma
2. Strongly coupled plasma.
The former do not require strong coupling of the dust particles, whereas the later does require
strong coupling of the dust particles.
The Dust Acoustic Waves (DAW) were theoretically proved by Rao et al in a multicomponent
collisionless plasma in 1990 whose constituents were electrons, ions and negatively charged
dust particles. The DAW are low frequency waves with wave frequencies of the order of the dust plasma frequency; which is less than the ion and electron frequency due to the presence
of the dust particles. The DAW is complete analog to the ion-acoustic waves observed in
ordinary plasmas. In DAW the dust particles play the role of the ions; and the electrons
and ions take the role of the electrons in the ion acoustic waves. The DAW is driven by the
of the ions at any arbitrary point in the sheath region, m is the mass of the ions, nois the
density of the main plasma and ni
is the density of the ions in the sheath.
From conservation of energy we have,
1,2mu2,o =1,2mu2 − eφ (x)
⇒ u =u2,o −2eφ,m1,2
where, eφis the coulomb barrier and for the electrons to reach the walls they have to cross the
sheath potential barrier and for this additional amount of energy of eφ has to be supplied.
From equation of continuity we have,
nouo = niu(x)
nouo = niuo
o,1,2⇒ ni = no1 −2eφ(x)mu2,o− 1,2
In case of steady state, electrons follow the Boltzmann relation,ne = no exp eφ,KTe
From Poisson’s equation,o,d,2φ,dx2= e(ne − ni) .
Substituting Equations(4.6)and (4.5) in equation o,d,2φ,dx2= e(no exp eφKTe− no1 −2eφmu2 o 2− 1)⇒ o,d,2φ,dx2= eno exp ,eφ,KTe− (1 −2eφmu2o)− 1,2
Which is the plasma sheath equation.
Let us now re-scale our variables into their respective dimensionless forms-χ →eφKTe
, ξ →xλd=x
KTeo noe 2
1,2, M → uo(KTem)1 2
With the help of these the L.H.S of equation (4.7) can be written as o eno d 2φ dx2 =2εo eno d 2
In case of a real particle, the particle will start from rest at x = 0 and will suffer reflection
from the right side of the potential well and will again move to its origin making a single
transit. Likewise a quasiparticle in the above case will make a single excursion to positive
χ and return to χ = 0. Such a single pulse is called a Soliton: it is a potential and density
disturbance that propagates in the plasma medium. A Soliton is equivalent to a plasma
sheath as we have already seen above, but it is not stationary i.e it is a moving sheath in
the plasma medium. A Soliton can be considered to be a wave-packet consisting of waves
of various frequencies and amplitudes and their superposition gives the single pulse which
moves with its height and width intact. Example of these wave is tsunami, which travels
long distances retaining its shape. A Soliton is shown in the Figure
5.2 Existence of Solitons
In the left panel of Figure (5.2),the formation of the pseudo potential well is shown for various
• From figure, it is seen that as M increases, the pseudo potential starts to form, signifying
existence of solitary waves.
So, the lower limit of the Mach number ML is indicated by the critical Mcr, when ∂
changes sign form positive to negative. For negligible σ and γ = 3, we can determine ML as ML =1 − R0(1 − β) 1 + z
0 d (0) ½ where z 0 d(0) = 1
1 + µe(1 + β)β
β − W(z) 1- 1 + W(z)
z = δmβeβ µe1 + µe
In deriving the above expressions, we have used Equations (5.50),(5.53),(5.56)and (5.63)
The upper limit of M is determined by the maximum value of M, for which V (φ) = 0,
φ 6= 0.
The existence of MU can be seen by an inspection of the value of nd(φ) as given by Equation which will be real as long as M <
p−2Id(φ) − 3σ. We note that Id(φ) < 0 for φ < 0.
So, we need to determine the limiting value of φ = φlimit for which,
M =p−2Id (φlimit) − 3σ (5.67)
which after substituting in the equation V (φlimit) = 0 (from equation) must be solved for
M = MU .
The upper limit in our case must be determined numerically as analytical solution
of φlimit from Equation which is not possible. The numerically calculated upper limit
is shown in the right panel of Figure (5.2).
5.3 Summary and conclusion
We have studied the Non-linear Evolution of the Dust Acoustic Waves (DAWs) taking into
account the Polarization Force. The Polarization Force is usually neglected in Plasma Dynamics and is found to have minimal effect in the common electron-ion plasma. However
as the Dust particles are extended objects the Polarization Force considerably effects the
evolution of the Plasma dynamics. Their study is found to highly affect the Dust plasma
In our analysis we have theoretically shown, that the Polarization Force considerably
effects the formation of Non-linear structures in the Dust Acoustic regime by using the
pseudo potential approach. We have noted how the Dust density varies as the Polarization
force approaches unity. And have also highlighted the formation of the pseudo potential for a specific Polarization force with varying Mach numbers and have found that with increasing M the Sagdeev potential begins to form, thereby indicating the existence of solitary waves.
We have also determined the upper and lower limits of the Mach number for the existence
of Nonlinear structures in Dusty plasma systems.
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