Download Characteristic Properties of Plasma

Characteristic Properties
of Plasma
The study of the behaviour of the plasma is based on the description
of the collective behaviour of the particles that compose it. It passes
from the microscopic description of the plasma characteristic
phenomena to a macroscopic description.
One of the main properties of plasma is its tendency to charge
neutrality. When in a volume of appreciable dimensions the density of
electrons and the density of positive charges (ions) differ, an
electrostatic force with a potential remarkably higher than thermal
energy, is induced:
∇ ⋅ E = (n i − n e )
e
ε0
where n e and n i are the electron density and the ion density.
Unless special mechanisms will not support this high electric field,
rapidly the charged particles moves to reduce these fields, and
rebuild electrical neutrality.
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ΔE ≈
e
ε0
n Δx
For regions of dimensions of the order of a centimetre
and deviation from neutrality with densities of 10 18 m-3,
fields of the order of 100 MV/m are created.
Usually deviations from neutrality can be created in very small regions,
such that the energy required to support the corresponding electric
potential is the thermal energy.
When perturbations of charge neutrality occur in those regions,
immediately the charged particles tend to restore neutrality by crossing
again the mentioned regions with thermal speed. The phenomenon
goes on up to the constitution o a charged region in the opposite
direction. Then the charges travel back again and an oscillatory
motion is induced.
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Debye length
In unperturbed conditions it is n e = n i = n (n e, n i
electron and ion densities). For a disturbance
causing a charge separation region of size x0
(see figure), the force on to charge - e at the
position x within the range (0, x0), is expressed
by:
The word to move the particle from 0 to x0 is:
The Debye length λ D is the dimension of the region of charge
separation that can be created using the medium thermal energy. In
the x-direction it is:
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In 1929 Langmuir introduced the term plasma to
totally or partially ionized gases for which λ D is
sufficiently small compared to the other macroscopic
lengths of interest (for example, the characteristic
length of the electron density variation).
In these conditions it is possible to assume charge
neutrality for which ne = ni (ne electron density, ni
singly ionized ion density).
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Plasma frequency
A plasma of density n = n e = n i with a
one-dimensional geometry is
considered. For a displacement x of
the electrons for Gauss law it is:
(motion law of an electron)
This is the equation of a harmonic motion with
angular speed ω p, that is said plasma frequency: ω p =
e2 n
ε 0 me
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Debye length and plasma frequency
When the electron density and the ions density differ,
electrostatic forces are produced. They induce a motion of the
charged particles so as to reduce quickly these forces and
reconstitute electrical neutrality.
The dimension of the shift from charge neutrality in a plasma is
given by λD. The shift from charge neutrality produce a force that
tends to restore neutrality. Electrons are pushed backwards to
retrace the distance λD going ahead to constitute a shift from
neutrality in the opposite direction. Thus an oscillatory motion is
originated with a frequency ωp. The speed of the motion through λD
is given by:
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Shielded Coulomb
potential
A charge within an ionized gas attracts in
its around charged particles of opposite
sign. The electric potential which it
induces, is given by Poisson's law:
In a plasma at thermodynamic equilibrium the electrons and the ions
are subject to an energetic distribution that depends on the electric
potential. This distribution is given by the Boltzmann distribution of. In
an electrically neutral plasma it is n i = n e = n and:
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Shielded Coulomb potential
When it is set g(r) = r f(r), we obtain:
For a charged particle radius r0, when position r near the particle given by
r-r0 is sufficiently small the shielding effect must be negligible:
φ (r) =
(
q
exp - 2 r/λD
4πε 0 r
)
Shielded Coulomb potential
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Sheaths
Solid surface 0φ0-
ne = ni = n
x
φ(x)
A region where there is not charge neutrality, is in the vicinity of solid
surfaces. The thickness of this region is of the order of λD. The region
is called sheath or plasma sheath.
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Sheaths
In the plasma around a conductive
element (an electrode) when is not
heated and does not emit charged
particles, and when it is floating (it is not
referred to ground):
T = Te = Th
and for x >> λD:
0φ0 -
T = Te = Th
ne = ne = n
x
φ(x)
n = ni = ne .
Because Te = Th , initially the electron flow towards the surface is
much bigger than the ion one as <v e> ≫ < v i >. The floating
electrode becomes charged negatively and acquires a negative
potential in order to reduce the electron flow that must exactly
balance the ionic. Indeed, under stationary conditions, there must
be no further accumulation of charge.
The potential on the floating electrode becomes φ0 and
potential distribution in the sheath becomes φ (x).
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Sheaths
Only the electrons with speed directed
toward the surface, with sufficient
kinetic energy to overcome the
potential barrier given by φ0 (εc =
mev2 ex0 /2 = -e φ0 ), come to the electrode.
0φ0 -
T = Te = Th
ne = ni = n
x
φ(x)
Near the wall, within the sheath, the electron and ion fluxes in the
x-direction are:
From the condition Γ ex = Γ ix , it follows:
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Sheaths
From the Poisson’s equation it is:
0φ0 -
T = Te = Th
ne = ni = n
x
φ(x)
Therefore it follows:
where
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Electrical conductivity
•
•
Partially ionized plasma with electrons, ions and neutral particles
with densities: n e, n i, n n .
elastic collisions with frequencies: νei, νen , νeH = νei+νen
• v e ≫ v i, v n; v i ≈ v n ≈ u.
The law of motion of an electron accelerated by the electric field E
and decelerated by collisions is
+
d ve
= - e E + FC, e (t)
dt
-eE
_
-
+
me
eE
FC,e(t) is the force due to the
collisions of electrons with heavy particles.
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Electrical conductivity
The value of the collisional force FC,e(t) is averaged in a time much
higher than the time between two consecutive collisions ( characteristic
time Δt ≫1/νeH) . <FC,e(t)> is given by the number of collisions
(assumed elastic) that an electron undergoes in the time unit, times the
momentum loss per collision (total momentum loss of an electron per
unit of time due to collisions):
FC,e (t) =
By assuming:
Δpeh
Δt
= -m e
Q(1)
ei
Q
(e)
ei
(U e -U i )ν ei - m e
Q(1)
en
Q(e)
en
(U e -U n )ν en
Qei(1) ≈ Qei(e), Qen (1) ≈ Qen (e) , v e ≫ v i, v n; v i ≈ v n = u:
FC,e (t) = - m e U e ν eH
For t≫ Δt and sufficiently large such that equilibrium is reached
between the electric field accelerating force that and the decelerating
collisional force:
me
d ve
= - e E + FC,e (t) = 0
dt
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Electrical conductivity
From the definition of current density due to the electrons (usually the
ionic contribution to the current is neglected) it results:
where σ e is the electrical conductivity due to the electronic
contribution. Hence σ e is given by :
σe
e2 ne
=
m e ν eH
and, when the ionic contribution to the current is neglected, Ohm’s law is. :
J = σE
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Electrical conductivity
In a plasma, dominated by Coulomb collisions, the electron-ion
collision frequency νei and the electrical conductivity σe due to the
electron flow are:
ν ei = 3.64 × 10 -6 lnΛ
σe =
e2 ne
m e ν ei
ni
Z
T3
= 7.739 × 10 -3
ne
Z
n i lnΛ
T3
un plasma
ioni ionized
ionizzatiions
una sola
(Z =
1) ed elettricamente
In Per
a plasma
withcon
singly
(Z =volta
1 ) and
electrically
neutral
neutro (ne ≈ ni):
(n e = n i) it is:
σ e = 7.739 × 10 -3
1
lnΛ
T3
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Hall Parameter
For the fields E and B uniform and stationary, on a free charge a drift
motion perpendicular to E and B is generated. In a plasma, the
comulative effect of the drift depends on collisions. Immediately after a
collision the particle is accelerated by E along its direction and starts
its revolution around B. Therefore the motion in the direction of E
depends on the number of collisions which the charge undergoes
during its revolution and becomes more important for increases of this
number.
The radians done two consecutive
of collision during the revolution is
given by the Hall parameter:
ωq
βq =
νq
where ωq is the angular velocity of
the revolution and νq is the collision
frequency between the particle q
and the other particles.
z
x
Ÿ
y
B Ÿ
E
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Hall Parameter
Electron and ion Hall parameter
In a plasma the order of magnitude of the Hall parameter of
electrons is much higher than that of ions. This depends on the
difference of the masses of these two particles.
where βi and βe are the Hall parameters
of electrons and ions.
The collision frequency of a charged particle νq (electrons or ions) may
be with a good approximation by:
This is an expression valid for the order
of magnitudes where it is assume that
<Qe> ≈ <Qi>.
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Hall Parameter
E = Ey j, B = B k (E and B constant and uniform)
Collision dominated plasmas: β e ≪ 1
The drift motion induced by B in x-direction is
negligible compared to the motion in y-direction due to
E. This is due to the many collisions during one
revolution. Hence it follows that:
z
x
Ÿ
y
B Ÿ
E
Jx ≪ Jy, Jy ≈ σ Ey.
Collisionless plasmas: β i ≫ 1
The drift velocities of electron and ions in x-direction ExB/B2 are
equal. The velocities of ions and electrons in y-direction is negligible
as between two consecutive collisions B induces many revolutions of
the charged particle. Hence it follows that:
Jx = Jy = 0
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Hall Parameter
Ions collision dominated,
electrons not: β i ≪ 1, β e ≳ O(1)
The average diffusion velocity of the ions will
remain in the y-direction, but because of the
E × B drift, the average electron diffusion
velocity will exhibit a component in the xdirection. Only a fraction of the full E × B
drift velocity wD,E, is attained, because of the
interruptions caused by collisions.
|Jey|
βe
|Jex|
For sufficiently large βe, this fraction is nearly
equal to 1. Hence:
J ex ! - e n e w D,E
en
e2ne
σ
= - e Ey = Ey = - e Ey
B
m eω e
βe
βe
The current component Jex which flows in the direction mutually
perpendicular to both E and B is called the Hall current, after Edwin
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Herbert Hall who discovered this phenomenon in solid conductors in 1879.
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Hall Parameter
Electrons collisionless,
ions not: β e ≫ 1, β i ≲ O(1)
The ion current Jiy provides the dominant
contribution to the y-component of the
current, since Jey → 0. This phenomenon,
called ion slip. Ion slip can occur only in a
partially ionized gas.
From the definition of mean mass velocity:
u=
ρ eue + ρ iui + ρ n un
ρ
ρ eU e + ρ iU i + ρ n U n = 0
|Jey|
βe
|Jex|
for a fully
ionized gas
βe
ρ eU e = - ρ iU i
Therefore, for a fully ionized gas, as the mass density of electrons ρe is
much smaller that the mass density of ions ρi, |Jey | always is much
greater than |Jiy |, independently of the driving mechanisms.
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Generalized Ohm’s law
•
•
Partially ionized plasma with electrons, ions and neutral particles
with densities: n e, n i, n n .
elastic collisions with frequencies: νei, νen , νeH = νei+νen
• v e ≫ v i, v n; v i ≈ v n ≈ u.
The law of motion of an electron in a region where E and B are
present, becomes:
for t ≫ 1/νeH
and if J ≈ Je
Generalized Ohm’s law
or Ohm-Hall’s law
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Generalized Ohm’s law
The generalized Ohm’s law may be written as:
where
for B = B k:
And when defining E* = E + u × B, the generalized Ohm’s law
becomes:
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MHD Approximation
The Magneto-fluid-dynamics (MHD) studies the behaviour of a
conducting fluid in a field of fluid-dynamic and electromagnetic forces.
The main forces and the main energetic processes acting on a
conducting fluid are:
Forces:
• Fluid-dynamics: pressure gradients, frictional forces
• Electromagnetics: electromagnetic fields, Lorentz forces
Energy processes:
• Thermal flows, work due to pressure and friction
• Ohmic heating (the Joule effect)
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MHD Approximation
MHD model
Fluid dynamic equations
• Conservation equations of mass, momentum, and energy (Euler
equations or the Navier-Stokes equations)
• Equation state of the fluid (for ideal gas: p = RρT)
Electromagnetic equations
• Maxwell equations
• Generalized Ohm's law
For the definition of this model it is necessary to establish the
conditions at which it has to operate. These conditions are in the field
of engineering interest and leads to the determination of a model with
a very wide validity field. The name of MHD approximation.
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MHD Approximation
Conditions
a. Speed of the fluid u is much smaller than the speed of light: u ≪ c.
b. Mean free path between two consecutive collisions is much smaller
than the characteristic plasma dimension: LC ≫ < v s>/ν (where <vs>
average speed of species s and ν is collision frequency of the
species).
c. Validity of Ohm's law: tC ≫ 1/νeH
d. The displacement current is negligible with respect to the
conduction current.
e. The convection current is negligible with respect to the conduction
current
f. The electrostatic force is negligible with respect to the Lorentz force.
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MHD Approximation
Conditions
From condition c. through f. it follows:
c.
tC ≫ 1/νeH
d.
=
=
=
=
tC ≫
≪1
From c. and d. it follows:
tC ≫ 1/ωp
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MHD Approximation
Conditions
e.
≪
tC ≫ 1/ωp
f.
≪ 1
tC ≫ 1/ωp
The MHD approximation considers plasmas at speeds much lower
than the speed of light, pressures that allow the use of the model of
continuum ( LC ≫ <v s>/ν ) and describes the behaviour of plasmas
with time resolutions far greater than the period of oscillation of the
plasma around the charge neutrality configuration ( tC ≫ 1/ω p ).
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Magnetic regimes
Equations of Electromagnetism
(Collision dominated plasmas approximation - βe ≪ 1 )
Magnetic regime
From the above equations and from
=
the magnetic field equation describing the magnetic field behaviour,
results to be:
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A. Regime
magnetico diffusivo:
Magnetic
regimes
A. Magnetic diffusive regime (u = 0)
For a steady fluid (u = 0), from the equation of the magnetic field
behaviour in a medium with an electrical conductivity σ, given by:
∂B
1
=
∇2B
∂t
µ 0σ
This equation has the form of an equation of diffusion. It shows that
in a medium of finite conductivity the magnetic flux density tends to
spread in space and to decay in time. It is the equation of
magnetic diffusion.
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A. Regime
magnetico diffusivo:
Magnetic
regimes
A.
B. Magnetic convective regime (σ = ∞)
For a moving fluid (u ≠ 0) and a medium with infinite electrical
conductivity from the equation of the magnetic field behaviour in a
medium with an electrical conductivity σ, given by:
∂B
= ∇ × (u × B)
∂t
This equation is the equation of magnetic convection. It shows
that in a conductor of infinite conductivity the magnetic flux density
is moving with the fluid.
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Magnetic regimes
C(t+Δt)
S(t+Δt)
A.
B. Magnetic convective
regime (σ = ∞)
uΔt
In a time variation Δt the magnetic flux
ds
S(t) C(t)
through the surface S with moving with
the fluid. The variation of the magnetic flux ΔΦB in the time interval
Δt is given by:
The flux ΦB (t) changes for two reasons; first, because B = B(t) is
changing, and second, because the area S = S(t) bounded by the
curve C = C(t) is changing. In an interval of time Δt, each fluid particle
on the curve C(t) moves distance uΔt, so that an element of length ds
along C(t) sweeps out a vector element of area [ds x (uΔt)] (the
surface vector direction is perpendicular to the surface of it).
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Magnetic regimes
C(t+Δt)
S(t+Δt)
A.
B. Magnetic convective
regime (σ = ∞)
uΔt
The decrease in the flux associated with the
motion of the fluid during the time Δt is therefore:
ds
C(t)
S(t)
∫∫ B(t + Δt) ⋅ n! dS - ∫∫ B(t) ⋅ n! dS
Δ ΦB =
S(t + Δt)
S(t)
For the first order Taylor series approximation it is B(t+Δt)≈ B(t)+𝜕B/ 𝜕t.
Moreover, as B is solenoidal, its flux through S(t) is equal to its flux
through S(t+Δt) plus its flux through the lateral surface obtained by the
series of the vector area elements [ds x (uΔt)] along C(t). Hence it is:
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Magnetic regimes
C(t+Δt)
S(t+Δt)
A.
B. Magnetic convective
regime (σ = ∞)
For the Stokes theorem it is:
uΔt
ds
S(t)
C(t)
=
=
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Magnetic regimes
C(t+Δt)
S(t+Δt)
A.
B. Magnetic convective
regime (σ = ∞)
For Δt → 0 and from:
uΔt
ds
S(t)
C(t)
When the equation of the magnetic convection is satisfied, it results:
Therefore the magnetic flux through a
surface moving with the fluid is equal to
zero. Hence the magnetic lines of force
move with the fluid.
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Magnetic regimes
Magnetic Reynolds number
(effect of the flow on B)
For a non-stationary fluid of finite electrical conductivity, the magnetic
induction will change as a result of both convection with the fluid and
diffusion through the fluid.
The quantity called the magnetic Reynolds number Rm, evaluates
this effect as is given by the ratio of the convective and the diffusive
terms in the equation magnetic field equation. If L C denotes the
characteristic length of macroscopic change, it is:
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Magnetic regimes
Magnetic Reynolds number
(effect of the flow on B)
From the Maxwell equation (𝛻×B = µ0 J), it BC (ind.) indicates the
contribution of the magnetic field induced by the currents flowing
within the plasma to the total magnetic field BC , it follows:
For flows with small Rm, the convection of B lines by the fluid is
negligible, and the magnetic induction produced by currents in the
fluid/plasma can be neglected.
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Magnetic regimes
Magnetic interaction parameter
(effect of B on the flow)
The magnetic interaction parameter is a measure of the ratio of
the J × B force to the inertia force:
Magnetic interaction parameter
The Hartmann number is a measure of the ratio of the J × B force
to the viscous force:
H = L C BC (σ C /ηC ) 2
1
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Magnetic regimes
Magnetic interaction parameter
(effect of B on the flow)
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