Download Definition of Plasma

Definition of Plasma
Generally speaking, the plasma is an assemblage of free positively and negatively
charged particles and neutrals, where the negative and positive charges nearly balance
each other at the macroscopic level. This definition is not accurate enough but later we
will give a more rigorous one. Plasma is often called the fourth state of matter. The
various states of matter occur as a substance is heated to temperatures corresponding to
thermal energies above the binding energies for particular state of matter. Structured
systems placed in a sufficiently hot environment decompose: e.g., crystals melt,
molecules dissociate. As an example, consider the states of H2O and its molecular,
atomic and elementary particle components at various temperatures. As known, at
atmospheric pressure, below 273 K it is in a crystalline form, i.e. ice. In this solid,
strongly coupled medium the binding energy is larger compared to the thermal one. In
the temperature interval 273-373 K the crystalline bonds are broken, but molecular
structures exist and the liquid state of H2O is called water. At temperatures greater than
373 K the long-scale molecular structure bonds are broken and independent molecules
form a gas. Further heating leads to dissociation of H2O molecules into hydrogen and
oxygen atoms. This third state is a weakly coupled medium because the interaction
energy between the particles is weak compared to their thermal energy. Further
increasing the environment temperature we finally reach the four state of matter, i.e., the
plasma state. At thermal energies near or exceeding atomic ionization energies, a
fraction of atoms can simply decompose into negatively charged electrons and
positively charged ions. These charged particles interact with each other via the
electromagnetic fields they create, which results in a variety of complex collective
phenomena. An ionized gas is in a plasma state if the charged particles interactions are
predominantly collective rather than binary. We say that these interactions are collective
because many charged particles interact simultaneously even though weakly, through
their long range electromagnetic fields. Therefore the plasma can be considered as a
collective but weakly coupled assemblage where the interaction energy is much smaller
than the thermal one.
The fraction of atoms that are dissociated relative to the total number of heavy particles
is called the degree of ionization. Unless otherwise specified, we refer here to the socalled “simple plasma”, which consists of neutral particles of one species (atoms),
singly charged positive ions of the same species, and electrons. The degree of ionization
η can be defined as the ratio of the ion concentration to the total concentration of ions
and neutral atoms
η=
ni
ni + na
1
The degree of ionization in a gas in thermal equilibrium is governed by the Saha
equation
ni ne  2π me KT
= 
na
h2

3
 2  2Gi
 
  Ga

U
 exp(− i )
KT

where ne is the electron density (usually ne = ni), T is the temperature in degrees Kelvin,
Ui is the ionization energy of the gas, i.e., the number of Joules required to remove the
outermost electron from an atom; K and h are the Boltzmann and Plank constants,
respectively, Gi and Ga are the internal statistical sums of the ions and atoms. Assuming
Gi ∼ Ga a simple estimation for air at room temperature shows that the fractional
ionization is extremely low. The fractional ionization is a very sensitive function of the
temperature. At ordinary pressures, as the temperature is raised, the degree of ionization
remains low until Ui is only few times KT. Then the degree of ionization rises sharply
and the gas becomes a plasma. This is the reason why plasmas exist in astronomical
bodies with extremely high temperatures. Note, however, that at extremely low
pressures high degrees of inonization can also be achieved even at relatively low
temperatures as can be seen from the Saha equation.
Concept of temperature in Plasmas
Before providing a more precise definition and discussing criteria for plasma let us
recall the concept of the main parameter in thermodynamics, i.e. the temperature, as
applied to classical plasmas. It is known that under the action of binary collisions an
ideal gas relaxes to equilibrium and the most probable distribution is the Maxwellian
distribution of velocities. Although plasmas usually never achieve Maxwellian
distributions exactly they may be close to it closely and it is useful in many theoretical
treatments to assume that the plasma is described by Maxwellian velocity distribution
functions. Because usually ions and electrons behave rather differently it is usually
necessary to consider distribution functions of each charge species with different
temperatures. The velocity distribution function is the number of particles per unit
volume in the velocity space at a given position in the configuration space. Let´s
consider a gas of particles in thermal equilibrium. For a one-dimensional Maxwellian
distribution, the number of particles per unit volume with speed between v and v+dv is
given by f(v)dv, where
2
1
f (v) = C exp(− mv 2 / KT )
2
1/2mv2 denoting the kinetic energy. The zero-th moment of the distribution function,
i.e., the density is given by:
n=
+∞
f (v)dv
∫
−∞
1/ 2
 m 
and the constant is related to the density by C = n

 2πKT 
. The shape of the
distribution is determined by T. The mean kinetic energy of the particles obeying this
distribution is
+∞
ε =
1
∫ 2 mv f (v)dv
2
−∞
+∞
k
∫ f (v)dv
−∞
1/ 2
and x = v / vth , the integral can be written as
Defining vth = (2 KT / m )
+∞ 2
1
2
x exp(− x 2 )dx
∫
2 −∞
ε k = mvth
+∞
∫
exp(− x 2 )dx
−∞
After integration by parts, one readily obtains
1
4
1
2
2
ε k = mvth
= KT
Therefore the mean kinetic energy is 1/2KT. It is easy to extend this result to three
dimensions:
3
2
ε k = KT
The general result is that
ε k is 1/2KT per degree of freedom.
temperature T is usually expressed in energy
KT = 1 eV = 1.6 × 10 − 19 J corresponds to a temperature T ~ 11600 K.
The
plasma
units.
For
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In a plasma in thermal equilibrium, the species concentrations and the temperature
completely characterize its state. All the species have the same temperature. The
temperature of such plasma determines not only the average energy of the particles, but
also their velocity distribution - a Maxwellian distribution. However, not always can the
plasma be considered in equilibrium. In particular, the gas discharge plasmas, which can
readily be produced in the laboratory, significantly deviate from equilibrium. One
occasionally comes across the so-called partial equilibrium, in which the velocity
distributions of charged and neutral particles are Maxwellian, but with very different
temperatures for electrons (Te,), ions (Ti,) and neutrals (Tg).
Quasi-neutrality of Plasmas
A fundamental characteristic of the plasma is its ability to shield out electric potentials
that are applied to it and to tend to quasi-neutrality. Plasmas are almost macroscopically
neutral with respect to the total electric charge. Even small differences between the
positive and negative charge densities result in the creation of strong electric field inside
the plasma. Potential energy of the particles in these conditions will be much larger than
their thermal energy. So, in case no external mechanisms are imposed to maintain these
fields the charged particles will mix up in such a way to decrease the existing potentials.
This neutrality is maintained because of the mutual compensation of the space charge of
the positive ions and electrons. This compensation, however, takes place only in terms
of averages, that is, over sufficiently large volumes and for sufficiently long time
intervals. Therefore the plasma is said to be a quasi-neutral medium. The dimensions of
the volumes and time intervals within which the charge compensation may be disturbed
are called the space and time scales of charge separation.
Let us estimate the space scale of charge separation. Consider that the neutrality is
disturbed in some plasma volume. For simplicity, we assume that this is caused by the
displacement of a plane layer of electrons, which produces layers of negative and
positive charge as shown in the Fig. 1.
The electric field E between the layers is equivalent to that of parallel-plate capacitor.
The electric field is determined by the surface density σ of the charge on the plates:
E = σ/εo = neex/εo
(1.4)
where e is the charge, ne is the electron density, εo is the vacuum dielectric permittivity
and x is the layer displacement. The distribution of the electric field intensity and of the
potential are shown also in the Fig.1.1. The total potential difference φl is equal to
4
x
l
E
x
φ
x
Fig. 1.1. Layers of positive and negative charge; distribution of the electric field intensity E;
distribution of the potential φ
φl ∼ El = neexl/εo
(1.5)
(l is the layer thickness). Obviously the deviation from neutrality due to the electron
layer displacement can be maintained only when the height of this potential barrier due
to the space charge field is less than the random motion energy of electrons and ions: eφl
< KTe, KTi. In fact, to create the electric field it is necessary to have energy. Every
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electron obtains some energy due to the charge separation. In case there is no external
mechanisms maintaining the electric field this energy can come from the thermal one.
Therefore, substituting the total potential difference and assuming x ∼ l, we obtain:
l < ε o kT/e 2 ne
(1.6)
where KT is the smallest of the values KTe and KTi. The quantity on the right-hand side
determines the maximum scale of charge separation in the plasma with accuracy to a
numerical factor. This quantity is called the Debye length.
Let us determine the time scale of charge separation and consider the electron motion
after the deviation from neutrality. The force acting on the electron layer is an attractive
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force from the ion side with intensity eE = nee x/εo. Therefore, the electron motion
equation has the form:
me d 2 x
dt
2
= −ne e 2 x/εo
(1.7)
This equation describes harmonic oscillations with a frequency
ω p = ne e 2 /εo me
(1.8)
It is easy to understand the nature of these oscillations of the electron layer. The electron
layer is attracted to the ion layer, overshoots it due to inertia, then it is attracted again,
and so on. These oscillations do not propagate or attenuate because thermal motion of
the charged particles and dissipation through collisions were not considered. Plasma
oscillations determine the quasi-neutrality restoration mechanism. Obviously, the
plasma can be considered neutral on the average over many oscillation periods.
Therefore, the time scale of charge separation in plasma is determined by:
td ≈
1
= εo me /ne e 2
ωp
(1.9)
.
Debye Shielding
Let us consider the disturbances of neutrality due to the effect of external electric fields.
If a charged body is introduced into plasma or moves near its boundary, charge
separation occurs in the vicinity of this body - unlike charges are attracted to the body
and like charges are repulsed from it. Plasma polarization results in the shielding of the
external field. The characteristic space scale of this shielding is the Debye length or
radius.
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Suppose we insert in the plasma a positive test charge (single charged) (see the figure).
Electrons and ions (with charge number Zi) are in thermal equilibrium with
temperatures KTe and KTi, respectively. Let us determine the test charge potential
distribution inside the plasma.
Neutral
plasma
~λ
λD
Fig. 1.2 Test charge in plasma
The positive charge attracts electrons (and repeals the ions) and a cloud of electrons thus
surrounds the positive charge, therefore the electric field created by this charge is
shielded out. Let us compute the radius of such charge cloud.
The field potential outside the charge must satisfy Poisson’s equation:
∆φ (r) = −
ρ
εo
(1.10)
For plasma containing positively charged ions and electrons, the space charge density is
determined by the difference of their concentrations:
ρ = Z i eni − ene
(1.11)
If the shielding sphere contains a great number of charged particles the instantaneous
values of the concentrations can be assumed to be practically equal to their timeaveraged values. The relationship between these average concentrations and the electric
potential, assuming that both types of particles have a Maxwell-Boltzmann equilibrium
distribution, is determined by the Boltzmann relation:
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ni = nio exp(
− Z i eφ
ne = neo exp(
)
KTi
eφ
)
(1.12)
KTe
Here, nio and neo are the charged particle concentrations in the unperturbed region
where the plasma is neutral, i.e. Z i enio = eneo . Substituting the expressions for ni and
ne into Poisson’s equation, the self-consistent equation for the plasma potential is
∆φ =
1
εo

− Z i eφ
eφ 
) − eneo exp(
)
 Z i enio exp(
KT
KT
i
e 

(1.13)
It is easy to find analytical solution of this equation for sufficiently large distances from
the probe charge, when eφ < KTe, KTi. At such distances, we can expand the
exponential function in a power series and restrict consideration to the first two terms of
this expansion:
exp(
Z eφ
− eφ
) ≈1− i
KTi
KTi
exp(
eφ
eφ
) ≈1+
KTi
KTi
Substituting these expressions into Eq. (1.13), the latter reduces to the form
∆φ −
1
λ2D
φ =0
(1.14)
where
 e 2 n   Z 2e 2 n 
eo  +  i
io 
=
+
=
2
2
2



λD λDe λDi  ε o KTe   ε o KTi 
1
1
1
Here the plasma quasi-neutrality condition is applied, i.e.
∑ q j noj
=0
j
⇒
Z i enio = eneo
For single charged ions Zi = 1, nio = neo = n and
 ε ( KTi ) × ( KTe ) 
λD =  2o

 e n ( KTi + KTe ) 
1/ 2
(1.15)
This is the expression for the Debye radius for an arbitrary ratio between KTe and KTi.
Considering spherical coordinates and assuming isotropy, Eq. (1.14) takes the form:
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φ
1 ∂ 2 ∂φ
(
r
)
=
∂r
λ2D
r 2 ∂r
(1.16)
The boundary conditions are:
r → ∞;
φ (r ) → 0
r → 0;
φ ( r ) → φo ( r ) =
1
e
4πε o r
Here, φo is the probe charge potential in vacuum at small distances close to the charge,
where there is no shielding.
Since
1 d 2 dφ
1 d2
r
(
)
≡
( rφ )
dr
r dr 2
r 2 dr
Eq (1.16) reduces to the form:
d2y
dr
2
−
y
λ2D
y = rφ
=0
Searching a spherically symmetric solution, which vanishes at infinity, i.e.,
y = A exp(−αr ) , the solutions can be easily obtained. Here, the constants A and α are
easily determined by applying the boundary conditions.
Finally, the solution for the potential distribution is obtained in the form:
φ (r ) =
1
exp(− r )
λD
4πε o r
e
It is seen that for r <<
(1.17)
λd the potential coincides with the Coulomb potential in free
space, while for r >> λd it is much less because of the plasma shielding effect. Thus,
the characteristic scale of the shielding region is determined by the Debye radius.
If we assume that the ion-electron mass ratio is infinite ( M i / me → ∞ ) so that the
ions do not move and just form a uniform background of positive charge around the
probe charge (so called “cold” ions approximation), the space charge density is
approximately:

ρ ≈ e n − n exp(

eφ
KTe

)

n = ni
(1.18)
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Substituting (1.18) into (1.10) the same solution (1.17) can be found but now with
λD =
ε o KTe
(1.19)
ne 2
Note that as the density n is increased
λD decreases, since the plasma contains more
electrons. Furthermore, λD increases with increasing KTe. Without thermal motion, the
charge cloud would collapse to an infinitely thin layer.
Let´s now determine the conditions for quasi-neutrality more accurately. The interaction
of space charges of electrons and ions maintains the electric neutrality of the plasma
over volumes larger than the Debye sphere and for periods longer than the reciprocal
plasma frequency t D ≈
1
ωp
. In order to fulfill these conditions, the following
inequalities must hold:
L >> λ D ;
τ >> t D
where L is the characteristic plasma dimension and τ is the characteristic time of
variation of its parameters.
The number density ND of particles in the “Debye sphere” is:
4
N D = n πλ3D
3
(1.20)
In order to have shielding many particles must be present in the “Debye sphere”.
Therefore, “collective behavior” requires that ND >>1.
Thus, we can define plasma as a quasi-neutral gas of charged and neutral particles
which exhibits collective behavior.
Determine the energy of interaction of the test particle with the charged particles in
plasma. Determine the total average energy of the charged particles in the plasma.
Assume eφ < KTe,. Assume that positive ions are cold and singly charged (Zi = 1).
The interaction energy of the test charge with all the charged particles in the plasma is
the sum of the energies of interaction of the test charge with each of the surrounding
charged particles. The energy of interaction of two charged particles in plasma (test
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particle + single particle with charge q in the Debye sphere) is determined by the
potential of the probe charge in plasma
: U (r ) =
eq 1
exp(− r )
λD
4πε o r
q = ±e
where r is the distance between the particles. The Debye radius is given by (1.19). If we
neglected the influence of the probe charge on the space charge distribution the average
interaction energy of the probe charge with plasma would be zero due to plasma quasineutrality. It is however finite when the influence of the probe charge potential on the
charge distribution is taken into account, since the probe charge attracts the plasma
electrons and repels the ions.
Taking into account the interaction with all the charged particles, the average of this
energy of interaction will be given by the following integral:
〈ε p 〉 = ∫ φ (r )[eni (r ) − ene (r )]dV
(V )
(1.21)
Here, φ(r) is the probe charge potential distribution given by (1.17). In spherical
coordinates and assuming isotropy, this integral becomes:
∞
e 1
exp(− r )[eni (r ) − ene (r )]4πr 2 dr
λD
0 4πε o r
〈ε p 〉 = ∫
(1.22)
As before we may assume that the charged particle distributions ni(r) and
ne(r) are
determined by the Boltzmann relation. By expanding the exponential terms in series and
taking only the first two terms, i.e.,
exp(
eφ
− eφ
) ≈1−
kTi
kTi
exp(
eφ
eφ
) ≈1+
kTi
kTi
and introducing a new variable x = 2r/λD , Eq. (1.22) reduces to:
〈ε p 〉 = −
e2
∞
e2
8πε 0 λ D
0
8πε 0 λ D
∫ exp(− x)dx = −
(1.23)
The total average energy of the charged particles is the sum of the average kinetic
energy <εk>=3/2KT and the average energy of interaction (1.23), i.e., the potential
energy,
11
e2
3
〈ε t 〉 = 〈ε k 〉 + 〈ε p 〉 =
KT −
2
8πε o λ D
(1.24)
If and when there are many particles in the Debye sphere, i.e., ND >> 1, it can readily be
checked that the second term in (1.24) is much less than the first one. In this case, the
plasma behaves as an ideal gas of charged particles, since the potential energy of the
particles is very small compared to their kinetic energy. This is called the “ideal
plasma”. In the opposite case, the interaction energy is dominant and the correlations
between particles determine the collective behavior of this “non-ideal” plasma.
Problem 1.2 Demonstrate that if and when the number of particles in the Debye sphere
is much larger than unity the kinetic energy of the charged particles is much larger than
the potential energy of interaction.
The number of particles in the Debye sphere is:
ε KT
4
4
N D = n πλ3D >> 1 ⇒ n πλ D ( 0 e ) >> 1
3
3
ne 2
Therefore
KTe >>
3e 2
4πε 0 λ D
⇒
3
9e 2
KTe >>
2
8πε 0 λ D
The average kinetic and potential energies are:
〈ε k 〉 = 3 KTe;
2
〈ε p 〉 =
e2
8πε o λD
Thus , when in a Debye sphere the number of particles is much larger than unity the
kinetic energy is much larger than the potential one, i.e.,
〈ε k 〉 >> 〈ε p 〉
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Problem 1.3 The half-space (x > 0) is filled with a homogeneous plasma with density n
= 108 cm-3. There is a uniform electric field with intensity E = 100 V/cm outside the
plasma which penetrates the plasma (see Fig. 3). The electron temperature is equal to
the ion temperature (KT = 0.8 eV). Demonstrate how does the plasma shield the field.
Calculate the field intensity at x = 0.5 cm. Assume eΦ(x)/KT<<1, where Φ(x) is the
electric potential.
E
Plasma
KTi = KTe
0
x
Fig.1.3
In a strictly steady-state situation, both the ions and the electrons will follow the
Boltzman relation (1.12). The field potential distribution φ(x >0) in the plasma must
satisfy Poisson’s equation:
∆φ (x) = −
ρ
εo
(1.25)
For plasma containing only single charged ions and electrons, the space charge density
is determined by the difference of their concentrations:
ρ = e( ni − ne )
After applying the assumption eΦ(x)/KT<<1 , Eq. (1.25) reads:
d 2φ ( x )
dx
2
=
φ ( x)
λ2D
(1.26)
The boundary conditions are:
x → ∞;
φ ( x) → 0
x → 0;
φ ( x ) → φ o ( x = 0)
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Searching for a solution in the form y = A exp(αx ) the solution is easily obtained:
φ ( x) = φ0 exp(− x
λD
)
where λD is given by Eq. (1.15). As far as E = −
dφ
and applying the boundary
dx
condition for the electric field, i.e., E0 = E ( x = 0) , the final solution is:
φ ( x) = E0 λ D exp(− x λ );
D
E ( x) = E0 exp(− x
λD )
(1.27).
The calculations show that λD = 6.65×10-4 m and E = 5.4×10-2 V/cm at x = 0.5 cm.
Problem 1.4 A charged conducting sphere of radius R is immersed in a plasma and
charged to a potential φ0. The electrons move to provide Debye shielding but the ions
are stationary and just form a uniform background of charge. Assuming eφ0/KT<<1
derive an expression for the potential distribution as a function of r in terms of R, φ0 and
λD.
In free space the potential distribution of a conducting sphere is
φ0 =
Q
4πε o R
φ (r ) =
Q
4πε o r
r=R
(1.28)
r>R
R
φ (r)
r
Fig. 1.4. Potential distribution of a conducting sphere in free space.
Inside the plasma the potential will be shielded and its distribution is determined by
Poisson’s equation:
14
∆φ (r) = −
ρ
εo
where the space charge density is
ρ = e( ni − ne ) .
The ions form a uniform background with density equal to that of the unperturbed
plasma, i.e., ni = n, while the density of electrons follows Boltzmann relation (1.12),
therefore

ρ ≈ e n − n exp(

eφ 
)
kTe 
n = ni
In spherical coordinates and assuming isotropy, Poisson’s equation can be written in the
form (see problem 1):
1 d 2 ( rφ ) φ
= 2
r dr 2
λD
λD =
ε 0 KTe
e2n
The boundary conditions are the following:
φ →0
φ → φ0
r →∞
r→R
Introducing a new variable y = rφ and searching a solution in the form
y = A exp(−αr ) one gets:
φ (r ) =
 (r − R) 
exp −
4πε 0 r
λ D 

Q
(1.29)
The total charge of the conducting sphere Q can be obtained by using Eq. (1.28), i.e.,
Q = 4πε 0 Rφ0 . Thus, the potential distribution of a conducting sphere in a plasma can
be presented in the following form as a function of r, in terms of R, φ0 and λD:
φ (r ) =
 (r − R) 
R
φ0 exp −
r
λ D 

Problem 1.5 Consider two infinite parallel plates at x = ± L set at potential φ(± L) = 0.
The space between them is uniformly filled with a gas of particles of charge q and
density n. Calculate the potential distribution. Calculate the energy needed to transport a
particle from the plane x1 = L to the midplane x2 = 0.
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y
φ(x)
-L
0
+L
x
Fig. 1.5
The potential distribution can be determined by applying Poisson’s equation (1.25).
d 2φ
dx 2
=−
nq
ε0
We are searching for a solution in a form
second derivatives are:
dφ
= 2 Ax + B;
dx
d 2φ
dx 2
φ ( x ) = Ax 2 + Bx + C . The first and
= 2A
At x = 0, due to the symmetry, the first derivative must be zero and consequently B = 0.
At x = ± L,
d 2φ
dx
2
φ = 0, therefore C = − AL2 . Since
= 2A = −
nq
ε0
A=−
1
nq
2ε 0
the potential distribution is:
φ ( x) =
1
nq( L2 − x 2 )
2ε 0
The energy needed to move a charge q from x1 to x2 is a change in potential energy:
∆ ( qφ ) = q[φ2 ( x = 0) − φ1 ( x = ± L)]
This energy is therefore E =
1
nq 2 L2 .
2ε 0
If we let L = λ D , then
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E=
1
2ε 0
nq 2
KTe ε 0
nq 2
=
1
KTe = ε k
2
for a one-dimensional Maxwellian distribution. Therefore, if L > λ D , E > ε k . The
main conclusion to be made is that a thermal particle would not have enough energy to
go very far in a plasma ( L >> λ D ) if the charge of one species is not neutralised by
another species of opposite charge.
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