Download large electrostatic forces would exist, for which the potential energy

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1. INTRODUCTION
The discussion in Chapter II was concerned primarily with the behavior
and properties of individual particles in a partially ionized gas. We wish
now to turn our attention to the macroscopic behavior of collections of
chargedparticles.Theseconsiderationswill lead to the introduction of
two related fundamental parametersassociatedwith the electrical properties
of a partially ionized gas, namely, the Debye length and the plasma
frequency. As noted in Sec. II 8, the collective behavior of neighboring
charged particles during a collision between two charged particles plays an
essentialrole
in section.
the calculation
of the
particle
momentum
collision
cross
The notion
ofcharged
shielding
involved
here alsotransfer
enters
the description of the ionized gas region, called a sheath, immediately
adjacent to a solid surface.
The last three sections of this chapter are concerned with several topics
which involve applications of the fundamental concepts introduced earlier.
We discuss first the classical theory of electrostatic probes and their use in
making measurementsof the properties of low-pressure ionized gases.We
then discuss some of the concepts involved in the description of collision-
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dominated ionized gasesadjacent to solid surfaces.Finally, we discuss the
elementary theory of the propagation of electromagnetic radiation through
an ionized gas and how diagnostic information about ionized gasescan be
inferred from experiments which employ electromagnetic waves.
2. ELECTRICAL
NEUTRALITY-THE
DEBYE LENGTH
A basic property of a partially ionized gas is its tendency towards electrical
neutrality. If over a macroscopic volume the magnitudes of the charge)
densities of the negative and positive particles differed just slightly, very
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largeelectrostatic
forceswouldexist,for whichthepotentialenergyper
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126
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Section 2
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Electrical Neutrality-The
Debye Length 127
1
particle would enormously exceed the mean thermal energy. Unless very
special mechanisms were involved to support such large potentials, the
charged particles would move rapidly in such a way as to reduce these
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potential differencesand therebyrestoreelectricalneutrality.
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To obtain a quantitative estimate of the dimensions over which deviations
from charge neutrality may occur, let us consider the following simplified
model.. Let us sup~ose the ga~ is initia~ly .electrically neutral and that
the electrons and Ions are unIformly distributed throughout space, as
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indicatedschematicallyin Fig. l(a). Initially, the electronand ion number
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(0) Initial condition
Yo
(b) Intermediatecondition
Figure 1. Work necessaryto create a region of net positive charge.
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densitieshave the common value n = ne = nj' We wish to calculate the work
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necessary to displace the electrons bodily to the right through some
distance, d. This will then be ,the work necessaryto create a region of net
positive charge density pc = ne between the planes y = 0 and y = d.
Let us supposethat Yo representssome intermediate displacement of the
electrons, as shown in Fig. 1(b). The electric field E set up by a distribution
of charge density pc is determined quite generally by Gauss' equation
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For the one dimensional distribution in Fig. l(b), we have
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dE"
y::;O,y=O=E,,=O,
y
( 2.2a)
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O~y~yo,-=-=>E,,=-y,
dy
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(2.2b)
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In obtaining
that
the
electric
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contInuous.
field
is zero
prior
to
the
displacement
and
that
E"
is
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In the region to the right of Yo, the electric field acting on each
electron is E,,(yo).The work necessaryto move each electron an additional
distance dyo is
dW = eE,,(yo)dyo.
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Therefore,
the total
work
that
must
be done
on each
electron
to produce
a
total charge separation of distance d is
~
fd
ne2 d2
W = eE,,(yo)
dyo= - _
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2.
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(2.3)
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In particular, if this energy is to be derived from the mean thermal
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= ADis
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energy in the y-direction
kT /2, then the corresponding
distance d
called the Debye length and is given by
ne2Af, kT
--=-,
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In MKS units,
AD = 69.0
( T ) 1/2
n
m.
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(2.4b);
As an example,for the characteristic conditions in an MHO generator T =
2500oK and n = 1020m-3, we have AD~ 3.4 X 10-7 m. This distance may
be compared to the value of the electron mean free path Ie ~ 1.3X 10-6 m,
for the conditions specified in Exercise II 8.2. In this case, therefore, the
Oebye length is about a factor of five less than the electron mean free
path. Values of the Oebye length for various other conditions of interest
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are shown in Fig. 12 of Chapter II.
T
129
In 1929,Langmuir (1961)introduced the term plasmafor a partially ionized
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gas in which AD is small compared to other macroscopic lengths of
importance (for example, the macroscopic scale of change in electron number
1
density). Under
such circumstances one may make the assumption
of
electrical neutrality, i.e.,nj ~ ne.The word plasma derivesfrom a Greek word
.
meaning"to mold" and was suggested
to Langmuir by his observationsof
the manner in which the positive column of a glow discharge tended to
mold itself to the containing tube.
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Exercise 2.1. Consider a neutral plasma of charged particle number
densityne= nj
= 1014cm-3 and temperature2500oK:
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1. Supposethat in some manner all the electrons present in a sphere of
radium 1 mm were suddenly removed. Calculate the resulting electric field
(in voltsjm) at the sphere'ssurface.
2. Calculate the potential difference through which an electron would
need to be accelerated in order to acquire a kinetic energy corresponding
to the mean thermal energy of an electron in this plasma.
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3. What is the maximum fraction of electrons that can be removed from
the sphere such that the resulting potential difference between the center and
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the surface of the sphere shall not exceedthe potential difference calculated
in part 2?
4. Calculate the Debye length for this plasma.
3. SHEATHS
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One of the most important situations in which charge neutrality does not
prevail is in the region of a partially ionized gas immediately adjacent to
a solid surface. Such regions are referred to as sheaths. The relevant
macroscopicscalehere being the distancefrom the surface,we may anticipate
that the assumption of charge neutrality will be violated in a region whose
extent is of the order of AD' The detailed structure of sheaths may be
quite varied, depending on many factors. The discussion of this section will
be limited to the simplest of models and is aimed at bringing out the
salient features of sheaths most expeditiously.
An important aspect of the sheath problem concerns the disposition of
charged particles which strike the solid surface. For many situations where
the surface is cooled, it is possible to regard the surface as nonemitting and
catalytic. Under such conditions the incident charged particles are either
retained on the solid surfaceor they recombine and are returned to the gas
as neutral particles.
Let us consider the special caseof a floating electrode suddenly immersed
into a stationaryplasma.We shall assume,for simplicity,that the electron