THE MUTUAL INT'1
AUTIONS OF PLAS1A ELEC•RONS
.E.,
ciusett·
a
ZSELTI ,E
R.
fILLIa6
s LIntitute
T.
of Tec 1o-oy
1934
XUBlIrT TD IT-RTI.L
REuitlH
FULFILL2ZNT O: T
i:'iS FOR THE DEuREE OF
DOCTOR OF PHIL2SOPHY
From:
the
eS CT-1USETeSyLOGY
IXSTI'tT
'ienature
F TzC
of autLor .
D~e alrtment of >-ysics,SI~nature
E1
01:
ay.12,L13'd
Prolessor
in-1i_:r e of Resea, c .?
Snt i
of
Cono.ittee on
C-_Lirm
rdu
n of DePartment
te
-tudents
A
AiZ
kn&
C'le LMent
The autilor wishes to ex•resj
his Gratitude and -is
debt for the u-•ailing encouragement,
aI-on gii-ve nghim
advice,
by ProfesCsor W7llia rP.
0222764
and co-
A~is.
Abstract
Te scat te ri" f electrt
s b1 the J.ialds of other
.j
ectrons in a plasma has been investigat
ed, and a method
devised
.andlin.g it•
for
cess has
*a-n
has bee- found that this pro-
important effect an the velocity distribution
der certain
nder
certain coiton,
aondltlon,
Sce
when
!AL
%A--- is
amey:
.....
small, and
Such conditions actually occur ep erime-etally
rd for them "thi theory predicat
that the 3 stributi•"
n
will be nearly maxwellian.
The scattering at small angles is
more important than
that at large angles.
Or the velocity distribution is
m:aintained principall
byby many small changes in energy
rather than by less frequent large changes.
size,
(,
The exact
chosen for the potential hill used in computing
the scattering cross section is of relatively small importance.
Fa.rl.
u.d
fr
This approxi
'atin,
then, is nt eritical.
satisfactory distribution functions have been
dschLarges in whichinelasti
c coliisiors are un-
important z•d . also for some in which ionization cannuot be
neglected.
Table of Contents
Section
Page
- - - - - - - - -
I
Introduction
II
General Equations
III
Constant Cross Section
IV
Calculation of Cross Sec.
V
Integration over Angles
VI
Special Cases
VII
Conclusions
1
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4
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14
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20
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23
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32
50
After
Page
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9,L0
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14
15
- ables
I
II
II
Iv
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36
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42
I-
Introduction
Many measurements made with probes in gas discharges
have shown the existence of electronic velocity distributions of Maxwell - Boltzmann form, with very high average
energies. Such results are, in particular, found when the
electron density is high, and the net space charge (electrons plus positive ions) is low, as in the plasma of a
low pressure are.
This form of distribution requires a
very high rate of exchange of energy among the electrons.
In theoretical studies, this electronic interaction
has been considered only by Langmuir (/1)
and Gabor (9 ).
Of the two, Gabor's account is the more complete. It is not,
however, satisfactory.
He takes into account only the
electrostatic phenomena, and is thus unable to compare their
relative importance with that of collision processes and
the macroscopic field.
In his calculations, Uabor consit-
ers the electrons to be scattered by the fields of partially shielded,
stationary positive ions. Because of the great
mass of the positive ions, this cannot lead to as rapid a
change in $he velocity distribution as will the scattering
of electrons by electrons.
Furthermore,
his methods of aver-
aging are somewhat obscure.
At low electron densities, the electronic interactions
can be neglected, as is shown by the quite good agreement
of theory with experimental data. (7,2/)
Since these interactions increase with the square of the
density, while the number of collisions with atoms increases with the first power, at some density the two effects willbe equally important.
The object of this thesis is first, to find a means
of calculating the mutual interactions of the electrons;
second, to discover the exact conditions under which they
become important; third to find approximately the form of
the velocity distribution in the cases in which neither
process can be neglected.
The discussion willbe limited to discharges between
large, plane-parallel electrodes, so that only one direction
in space can be singled out.
It will be assumed that the
positive ion and electron densities are equal, and that
the drift current is small compared to therandom electron
current..It will be assumed further that the discharge is
homogeneous in space, so that diffusion can be neglected.
The simplest mode of exchange of energy among the
electrons is that between individual pairs, or by "collisions".
Among more complicated processes is that sug-
gested by Langmuir of "plasma oscillations".
It is the
mutual collisions which will be investigated in this paper.
The problem is simplifiedb y considering only the
following processes:
a
acceleration by the field,
b
elastic collisions with atoms,
c
inelastic collisions with atoms, (excitation and
ionization)
d
mutual collision.
Collisions between ellectrons will be infrequent com-
pared to collisions with atoms, but they are important.
When two electrons collide, each is likely to suffer a coniderable change in energy, while when one collides with an
atom elastically,it loses on the average, only afraction m
of its energy.
Collisions with ions will be as infrequent
as those with other electrons, and will result in as little
an energy change as a collision with an atom.
Accordingly,
electron-ion collisions will be neglected.
Let7) be the electron density, and 7J fdy the number
of electrons per cm! in an element dý' of velocity space.
f will depend on only the absolute velocity u, and the angle m that this velocity makes with the field.
f= f(u,cosa)
As f is nearly spherically symmetrical, it can be develop,
ed in terms of spherical harmonics, the first two terms
giving:
f = f (u2 ) + f (u')cosa
and f
will be much smaller than r .
Following
ioltzmann and Lorentz, the usual procedure
to determine the distribution function would be to find
the number of electrons entering and leaving each small element of velocity space per second on account of each type
of process, and to set the sum equal to zero.
in our prob-
lem, this involves too many successive integrations orthe
unknown function f,
and is unmanageable.
two equations are necessary to determine f
and f
.
These will be an energy balance and a momentum balance equation.
The first can be obtained by requiring the net num-
ber of electrons leaving each central sphere of velocity
space to be zero.
The second is found by placing a similar
requirement on the number leaving a region on one side of a
plane perpendicular to the direction of the field.
The
first will mean that there will be just as many electrons
having an initial velocity u, less than any given value w,
which acquire in unit time avelocity greater than w, as
there
are having initial velocity greater than w and final
velocity less than w.
The second equation expresses a
similar balance of the changes in velocity component u
x
allel to the field.
II
-
par-
The General Equations
Consider the energy balance equation, inv6lving processes a,b,c,and d.
5
a
Electric field E,
In time dt,
an electon moving in a field E increases
its velocity by an amountEdt cosD.
m
The number per sec.
that leave the central sphere of radius w will be:
2n7_E7V (fo + f•cosm)
m 0
cosm sino dm
or
4•h7eE wat (we)
3m
b
Elastic Collisions,
An electron of velocity u, colliding elastically
with an atom at rest, loses speed in amountsu =
u (1 - cos
), where 0 is the angle between the initial
and final velocity directions*.
Let:
N = atomic density
w = angle u makes with field E
J= corresponding azimuthal angle
cTa( u, 8) = atomic cross section for elastic scatering at angle e
Then the number per sec. of electrons originally having a
vectorial velocity in the element
u' sinm dw dY) which
are scattered by angles between 0
and e + dG
will be:
N
n u"
f sino
od 9V.
2n u
cr(u, e) sineaed
The only electrons able to cross the sphere w
tic collisions will be those for which
due to elas-
w<u<w + Su. The
number per sec. entering such a sphere will be:
If
2n N
•o wTra f sine dm" 2# / f u
-a sinBd4
Q = 2s , CT- a ( 1cois)
sinT
d
,
this is
21TNflm w4 Q /
(f + f
o
M
coso ) sinm dw
Or, finally,
4nNTm •
M
o
q
o
Inelastic collisions,
Let:
Letm us = excitation
energy
e
2
m us = ionization energy
i
Q.(u)
=
cross section for ionization
Q (u)
a
cross section for excitation
e
It is assumed that when an electron excites an atom
by collision, it loses just an energy 1 m us .
with velocity greater than / wi + us
e
sphere w in this way.
It
is
Electrons
will not enter the
also assumed that there is
just one ionization energy, and that the surplus kinetic
energy in an ionizing process is equally divided between
the two resulting electrons.
average, N u Qe
per sec.
An electron will make,
exciting, and
N u Qi
on the
ionizing collisions
Then the number per sec. entering the sphere w is,
/wa + ua
2r N
+ 2 N
1w
W
7
=n N)?
Let
equal to
d
Qeeo(u) u 8 du
f
2Q (u)u" du
I
e
7w 14+ u
w
w
+ u's
W
J(w)
(f
T
+ f
cosaO)
sine dn
f + f coso)
Qe(u) fo u du + 2/
w-2+ ul
sin
dW
Q fl u3 du
be defined by setting the above expression
4n NNJ(w).
Mutual collisions,
The mutual collisions of electrons are much more com-
plicated,
Let:
u = initial velocity of electron considered,
V
t
It
scattering electron,
U = velocity of center of gravity,
u = velocity of first electron relative to C.G.
u'~
'
corresponding quantities after
collision,
0 = scattering angle reffered to C.G.
= corresponding azimuthal angle,
0-(i,8) = mutual scattering cross section for
angle 8 ,
S=
angle between directions of u
and
v,
= corresponding azimuthal angle,
= angle between directions of
U
It
A•
U and
u,
'
and
= corresponding azimuthal angle,
7/
= angle
v
makes with the field
EXi
Fig. 1 shows the geometry of a collision of an electron of velocity u
velocity is 2 U.
with one of velocity
v.
Their relative
Vectorially:
U= 11( u * v )
1
=
1u
)
u = U* T
Algebraically:
I
4 tui = u' + v -2uv coa (
2
4 U4 = us
+ 9+2uv cos 9(
After collision the velocity of the first particle is:
U +
u=
.
u = u',
T •. 0 and
sphere of radius
about
or u, u ,
1 as an axis, and /)
about the axis
u
are
v
alllie on the same
99 are the polar angles of i'
and
the same velocity about the axis
v
V,
and
A4 are the polar angles of
U .
2
The polar angles of
. a-depends on the
relative velocity.
I f
and
v
u
lies in the element of velocity space dd"u ,
in the element drv , the number of such collisions
per sec. is:
2iOcXr, 8)
sin( dod
'9f(u,cosc)
f(v,cosY)
dyu d~yv
To find the rate at which elbctrons of velocity
u-
w
/ /·
~1
are given a velocity
u'> w
by mutual collision, this ex-
pression must be integrated:
and
first, over allvalues of G
on that portion of the sphere of radius
outside the sphere of radius
Yv
with
u
cuts the sphere
second, over a domain of
found by integrating over 0
of the sphere
;
w;
third, over a domain off u
The rate at which the reverse process takesa
u< w.
place is
u
inside the sphere
and over apart of ru
Because of symmetry, it
and ,
than 8
and
and B on that part
w;
with
over a part of the
u>w.
is more convenient to use A
9.
cos8 = cosScos
- e~n Ssinj
F'or the purposes of integration,
replaced by
sinA dý dA, .
and A from
0
Then '
to 3A , where ~
this circle, vectorially:
W
cos2wo
4
/
will run from
is
- u2 - v
4U T
Also,
= u4
can be
-rr to n,
the value of A
U + U = w , or
U4 + U2 + 2uU cos
cosL
sinGdOd
the. circle of intersection of the spheres
-
which lies
(velocity space of second electron) such that the
sphere
X
w;
u
u
and
w.
on
On
o + U" + 2u Ucos%
whence,
uM -•a
cos S
5
In the integration over XY,
v,%,
used as variables, if cosy in
cos
cosf+ sin5 sin/Tcosq
6
since
cosA
expression 4 for
u' + v*-
6
c(must be restricted.
4Ulv2co
u"
+ V;> w,
cos
-/>0.
When
w, (that is,
follows that
w)
<
v are on opposite sides of sphere
u
when u
and
and
T -
(6 = 0.
v
and
v
the fraction is
( (T
- •.
When
w, there is no re-
, as the fraction is greater than one.
this case it is possible to let
2(
)
(u + v' ,-a
less than one,and it
are always
±xpanding
r>,ý> 4W* + u* + v*1 -qw (u +. v
are both greater or both less than
striction on/
u,
,
are on the same side of sphere
and
is replaced by
1, and since, through U and
cosA
cos
Because
,
f(v,cos7)
can be
.
-l1< cos /)
depends on/
and
In
Then the limits:
2( .
The number of electrons leaving the sphere because of
mutual collisions is:
u
11
I± =
udau
n
fo
2nr
2X
f(u,coso)sin~dm fo d
-r
vdv j
sin
(d
(uT,
)d/0
o
TT
-rr
sifnl
2
d A fc
-
-TI
Let:
Z
B •.
o
0
Tsln•~
•I,(
Cfo
,
-n1
d.,,,
The order of integration can be changed:
TT -2o
XT
I
= 21' f u'du J vadv J
uB sinXd(
2rr
"of f(v,cosv)d
Expanding
f
2TT
and substituting for
iof(u,cosw)sinadm
2rr
ad
d/
cos 7-'
2rr
f(f(v,cos)v)d
= f
(va)
+
f (v2 )(cosýcosw
+ sin'sin•ocos)
= 2r(va) +if (v')cosX cosm)
Taking the product of this result with
,
fr.f.(ua) + f (uO)cos
1f.(v)a)
f(u,cosa),
sin(owdo
(+)iosfco
= 2f o fua)f 0 (us) + 2f
(u')f (v'a
3
.1
f
is much smaller than
is negligible.
f
, so the product
12
Then:
a
I = 16na h
(ua)uadu
kf
ft
uB sind
(v')vadv
Turning to the electrons entering the sphere due to
mutual collisions,
let:
IT
B =
Tn
sinA dA f9 (u,0 )da
The number entering is:
I
= 161·T ns
ff (u*)usdu if
10
o
0
(v')vadv j
uB sindX
oa
'
a
a,b,c, and d have now been cal-
the four processes
culated, and the energy equation can be written symbolically:
a
11
4
+
b
+
.M1týgf
+
d
-4~ýAQwt f
I
3m
0
M
=
o
0
or:
,
+ I -I
This must be true for all values of
a
- 4rrN
J = 0
w.
If the mutual scattering terms are omitted, and the
inelastic terms
considered constant, one can derive
J
from this equation, by one differentiation, Lq. 8 of Morse,
Allis and Lamar (with diffusion left out).
-, E
2w
4TrN2J
(waf)
dw
.
, ma d (w
M 2v dw
o
)
=O
is the number of electrons entering the sphere
due to inelastic impacts.
13
The mutual collisions of electrons will not be as important in the momentum balance equation as in tIhenergy
relation.
In collisions with atoms,electrons of energy H
undergo a change of energy of the order
H.* In correspond-
ing encounters with other electrons, the change is of the
order
H . On the other hand, this is not true of the
2
changes in momentum. Electrons are scattered by atoms nearly isotropically,
momentum.
and lose almost all of their directed
The pair interactions of electrons will be neg-
ligible for the purposes ofthis equation, because they are
so much less frequent than collisions with atoms.
The in-
elastic impacts, which are also much less frequent than
the elastic, will be neglected too.
An equation corresponding to eq.ll can be set up for
momentum balance.
-E d
d
12
which is
This leads to:
+ NQwf
= 0
equation 7 of Morse, Allis,
vation from the integral form is
and Lamar.
Its MIri-
given in the appendix.
So far, those integrations have been performed which
involve only the geometry of the mutual scattering process.
It
O-.
is now necessary to consideir the cross-section function
The true form of this is difficult to handle;
is worthwhile to postpone its treatment,
so it
and to consider
14
first the assumption that Cr is
true if
constant.
This would be
the electrons behaved like hard spheres.
'he
dif-
ficulties arising from the form of 0- will be avoided, and
it will be possible to show clearly the peculiarities of
mutual collisions, as well as the method of handling them.
First, the energy and momentum equations can be comf •
bined to eliminate
13
-4TQfr
M
uM
Ln
3Qf
o
3NQ
I- m
m
d
E w-f
d
dw o
+ (I
- I ) -4N
a
=
J
0
The four terms come from the elastic collisions, thefield,
mutual collisions, and inelastic collisions respectively.
III -
Constant Cross Section
it is convenient to represent mutual collisions on
a diagram of the v*,u
a
plane.
(Fig.2)
Each point stands
for the velocities of a pair of particles.
conservation of energy,
Because of the
every point PI'epresenting the re-
sult of a collision between a pair P, will lie on a 450
diagonal line through P. Points in the shadea region, for
which
us + va<w a , cannot lead to collisions in which a
particle crosses the sphere
v = w
w.
The lines
u = w, and
divide the plane into four quadrantg,each of which
must be treated separately.
and v<w), the angle
In 2 (u(W,v>w)
.o= 0 , because
and 4 (u>w
u and v lie on op-
V= w
4
U
A"iy
t= w
15
posite sides of sphere
w.
In 1 (u<w,v<w)
and 3 (u>w,vw),
o< X-< T
0
2
Integration
over quadrants 1 and 2 gives the electron
flux across the line u = w in an increasing direction; over
3 and 4, the flux in a decreasing direction.
Let :
Tr-o
B' =
u B sinXdX9
B' = S
Quad.
1
B sinX d?
14
B'3
B'
4
=
-
ff
B sinX dX
asnXX
From eq's. 7 and 9,
B
1
0
sin
ad~
C~ d•/A
-T
= 2 cr(l - cosAo)
B
= f?b
sin)d'
/fr--dA
-iT
Repeating eq's. land 2,
4u' = us + Vo - 2uvcos
4U* = u2 + v2 + 2uvcos X
Eliminating
15
cos
(X
4U" = 2u2 + 2v
- 4 u
= 2n~(1 + cos 0o)
16
When
u
and
v
are held constant,
4udu = uv sindd X
47d.
U V
sindXaX=
cos~o from eq.4,
Substituting for
B
B
= 21Td
.31
a
1 -
1
= 2O
v• •
- us-
2w
2f Uu
- us
+
4 u T
2
Thus the integration over
I
can be replaced by one over
U, if the proper limits are found.
be
u
.
and u
8a
,
Let the limits on a
u<u , sothat when,
U=u
O
U=U
awm,
For Quad's.2 and 4,
X. = 0 ,
0
4 g = (u - v)
I
Since
u
and thus :
, 4 U
(U + V)a
a
is essentially positive,
Quad.
(v - u)1/2
(v+ u)1/2
u
For quad's.l and 3,
cosa'
IV
from inequality 6,
8
= (u•" u vlv
W)
(u - v)1/2
(u + v)1/2
17
Using this value for
cosX.,
u2 + v -2wJ us + v
= u'
4
-
'
2
4 ua = u2 + v + 2wJ u" + Vt w
Let
s
= u
+ v
- wa
is the energy of the
1/2 ms"
two particles in excess of that necessary to produce a collision in which one electron crosses the level w,.
of
s,
In terms
4u= (W - 8
4 _U
a = (w + 8)
This gives:
(w- s)1/2
u
(s- w)1/2
(s +w)1/2
(w + s)L/2
8
The integrals of equations 14(over X) can now be
thrown into the forms:
B'- 8
1
uv
S(w+s)
8 - 1/2 _
(w s) /2
(~w-s)1/2
1
2uJ
-
"4
au
26/ 2u +2vO-4773
(mw-s)Jv 2ua*2va-4us
_U+
v8TrB'u8(ws)
3
uv
16a
1
1
UV3
u du•
(w]+s)1/2
1/2
(w-e)1/2
uv3
Similarly:
B'=
3
(s+w) /2
(s-w) 1/ 2
"g
2
a-
5a
2i~~4ZuLL
4
udu
18
ca
BJ3 = 8T___3u
3uv
16b
s
B'
(v+u)1/2
B' = 4 1T-u(3vQ - 3w*
S
3uv
16c'
-
uv lv-u)1/2
a
B'-
2u / 2u'+2Va-4iU
2u
Svu (u-v)1/2
3' = T y(3w-
16d'
J
+ 2u*)
-(u+v)1/2[_
8
udu
""
I
udu
2 / •u+2,'-4u
- v")
B' and B' can be simplified by taking advantage of
4
0
the symmetry in u and v.
I
-
I
38
, say
16ai-"•
w
I'= •
o
-
f
They appear eq.13 as terms of
, where
(u')usdu
An
f(u')usdu
f o (v')vadv B'
a
f
f(v')v'av B'
Substituting B' and B' from eq's.16',
I' =•0L
2)vdv u(3v' -3,~ +2u')
Jf(us)udu
fa (v
S4cfr- f(u)du
3
0
f (v') vdv v(3w
- vS)
19
Because of the symmetry of I'
in u and v, 2ua cab be sub2v a is
tracted from the bracket in the first integrand, if
subtracted from that in the second.
I' = 41r- 4
f (uO)u.du ff (vs)vdv (v'
0o
o
w)
V-
W
ra-f f (u')u'ndu f f (v')v'dv S. "* )
Wo
o
o
U
-
But this is equivalent to taking:
16c
16d
---
B' -
4
B' -=
4
-
( v a _ wa)
(wa - v)
u
The functions B'
derived here
are much simpler than
the ones to be calculated later for the shielded Coulomb
field. They do show, however, the same symmetry properties
in u and v.
If all processes except the mutual interactions are
neglected, the solution of the balance equations willbe a
maxwellian distribution.
f
=Ae
-us/wa
o
This fact can be used to chebk the calculations. The u',v9
plane can be divided into strips between the diagonal lines
u2 + v a = c S and u4 + vs = ca +
c a . The first
integration
may be carried out along the length of these strips, instead
20
of along lines parallel to the ua or v a axis.
The maxwellian
distribution function has the unnque property of being constant along such a diagonal line. In other words, for this
function and this path of integration, the f's
from the inner integral.
disappear
It turns out that the BI functions
can then be integrated. The result is zero for every value
of ca,
showing that the mutual collisions do not change this
distribution,
complicated
This was a valued check on the correctness
expressions derived 2rxm from the assumption
of a shielded potential hole.
IV -
Calculation of Cross-Section
The force between two electrons in the plasma is
when they are very close, where r is
separation. When
r
--
their distance of
is larger, this force is reduced. Be-
cause the space is macroscopically neutral, at very large
distances
There is
r
the force between electrons will be zero.
a space charge of positive ions,which, as far as
the faster moving electrons 6ze concerned, is nearly uniformly distributed.
Near each electron there will be,on
average, a defliciency of other electrons, leaving an
average net positive density which "shields" the electrons
from each other.
A suitable approximation to the interaction potential
is:
a
V = .. .
-r/l
r
This defines a potential "hill" whose radius is measured
21
The value to be assigned to ( is uncertain.
by a.
A low-
er limit is the mean distance of separation of the particle&,
_-L
-4
1 3. This may be of the order 10 cm. An obvious, but
certainly excessive upper limit is the size of the apparatus,
say
lcm.
It
turns out that this enormous range foro
is
reflected in the results by an altogether smaller effect.
o(
cannot,
however,
be infinite,
coulomb potential, and with it,
section.
for this would give the
an infinite collision cross
No value willbe specified for this parameter at
present.
The cross section-(U(,
8)
for this potential hill is
found with the aid of quantum mechanics.
The Born approx-
imation can be used. (29)
Let the wave function!
of the two electrons be a
product of two functions, one representing the motion of
the center of gravity, and the other,
lative motion.
plane wave,
,
ikx
Let:
If'
be approximated
V/,
representing
by the sum of a
and aspherically scattered wave,
-
e ikx
re-
, then
ikx
,
where x
and x I are the distances
ofthe two electrons from the center of gravity, and
k - 2nmu
h
Then the first apprwoximation to the siation of Schr6dinger' s
22
equation,
with
V =
e Le -r/w(
, gives:
r
A4m
7..=
A
.
+ (2
4k 2 s in g
=1 ,
Or since
',
2.
--
h
/
(4kA
sin2".9 + -J-
This can be written:
17
cr(iu6')
1
/6z-274
where,
h
32=
18
1/10
2 To
measures the size of the hill din terms of A,
the wavelength corresponding to the relative velocity 2u.
,132 is
L MU
-2=
=
a very small number. Measured in volts,
150 he
92
A••
16 n• Vm2Vr
-30
-
16
7= 0.95" 10
Vr oe
If the electronic density is
to
10
per cc.,
and d is set equal
77 3
-46
0T
and if 0 is larger, 13
by so much the smaller.
5
23
in the denominator of-,-
It is this term
that pre-
vents the cross section from becoming infinite at small
angles,as it would for a Coulomb field.
controlling part for distant encounters.
small,
It will be the
Since it
is
so
it will scareely affect g-for large angles. These
correspond to close encounters, and scattering by a nearly
purecoulomb field.
V -
Integration over the Angles
In this section, the functions B and B' are calculated.
Equation 17 capalso be written:
1
J=
-
4ma
4
(1+29
-cos)
iby equation 2 for coso , this becomes
,-=
..
Y
m
1
(1+2,8' -coss8cos•,+sinSsinjcos/)
When this is integrated over/, and 3 , we obtain the
functions
B, and B,
which are given on page 24.
The
details of this and the following calculations are given
in the appendix.
These functions B are to be integrated over %,
results are the B'
functions.
Exactly as was done in the
case of constant cross section, an integration over
substituted for that over X.
sin
d(
=
4u du
uv
T he
u
is
[
I509
Ax
5 0.ft
2o
a
fe
)hIV1
I
(Z V -f4 V) -1
-
)o
~a
-
oy
~'
~·(br'J
L(
',ý) (,C/ -/,-&
/,)
-
to e ,
15--~Z
UU~7
a LL 7
25
A new symbol "a" is defined,
a
=
2h
w
4nd mw
where Aw is the wavelength corresponding to the velocity w.
Since~3 is extremely small, "a" also is very small.
It
is
found upon evaluating the integrals over
B1
=
u,
that:
19a
2T #
,2
miw xauv
FS-
•A
B
19ce
u - W -u -w
1w-Iatuv
S
=
2 A
-~i uv
a
-I1
2was
tan
wx- ul +wa-aJ
-t
tan
2wa
=B Ž'wZa
2 T uv
B
I I
we-u A
+w a
2wa
w _U
V
2Wa a
W+wm
- u2-w -w a
2w a
2w au
wtuR +w4 a0
tan
2w a
tan
1
w
2wav
, which list the
These may be compared with eq's. 16, which list
the
corresponding quantities for 0Q = constant.
The functions B' are large near the line u = w, and
decrease rapidly away from it.
are independent of v.
In quadrants 2 and 3,
In quadrants 3 and 4 they are asymp-
totic to zero as u becomes large,
B' is
2
they
and B' is
zero when v=O.
zero at u = 0 , and B' = 0 along the line u + v = w
26
These functions are plotted in fig.3.
the value 6.1 was used for "a",
In this figure,
so that the peaks are not
so sharp.
The B' functions would,when multiplied by the distribution function, be very difficult to integrate. Therefore
approximations are made. Over most of the plane, it is
satisfactory to expand the arc-tangent functions, retaining
ao .
only terms through the order
breaks down for
But this expansion
utw4(wa . Hence, in a narrow strip about
the u = w line, some other method must be used.
width of this strip, the product
f(uO)f(v a )
Over the
does not
change anywhere nearly as rapidly as Bt . So the distribution product is expanded, and the exact form of BI is retained.
fWhen
w2 - u>> wa ,
20a
B'=2_. 4 s3
20c
B'
1
m"uv 3(wi-u)•3
ýa
2=
m~ uv
4u"
3(w '-U )
+
W
2u
'
4-3
20b
B' = 2rTA
20'
2n a4 3T(U4v
BI = muv
Yz-
+
2v
These expressions no longer containg "a",
which is as-
sociated with small angle scattering. They represent that
part of the flux across the level u = w ,whichis caused by
collisions in which there is a comparatively large exchange
07
5"/
07
5
27
of energy.
They become infinite at u = w , while the more
exaxt expressions do not.
Outside of a strip of width tw"
(fig.2) on each side of this
proximations. (t>Ž a)
mation is used.
line, they are very good ap-
Inside this strip the other approxi-
The number "t" must be so chosen that on
the one hand, t>> a, and on the other, so that the expan-
sion of f(u2)f(9V)
o
is sufficiently precise inside the strip.
0
I t is shown later that this choice is not critical.
In order that the approximation in the interior of the
under
strip may give a correct result fgrxkhz conditions giving
rise to a maxwellian distribution, the product f,(us) (va )
is expanded along lines us* va= constant.
'or this pur-
pose the variables x and y are introduced.
21
u+v
x =
us =
x +v
S2/2
y =
v
us, va
/2
=X
- r
/2
Let
u(
u)=(vs)
Then
(x,y)
where
y
(X,y)
(x,yo) +
= y - y
(xy
yj
, and y. is to be so chosen that the
point (x,y,) lies on the line u = w .
Let
rant.
Sj represent the strip integrals of the jth quad-
Below, symbols
Hj,k (j= 1,2,3,4 ; k= 1,2) are used
temporarily to show more clearly the structure of the Sj's.
( B1uv) willbe carried along as a single function,of either
28
u and v', orx and y , depending on which set is
in use at
the moment.
Sj
An
4=
=
) j9du'• dv 2L/V
f(u;)(v2
n=2
4
( BtL v)
(x,yo) + 9i
(X,yY
Ydy1
dx
For the first integration, x is held constant, and there-
fore g(x,y,) and [
are also constant.
('Yx,y)
Let
Hj,1 = ' (BCtv) dy,
y dy
(B
j,2
ff
Sj = 41f2,
Then
(x,y )Hj,1dx * J&
(,y)
Once having obtained this forfmfor Sj,
Hj,2dxJ
it is more con-
and S by returning to the
venient to evaluate H
j
variables us and v. j,k
y =
•u
S=
y -
- x
- x
y0 = J2w
,
o = /2(
u a-
wa)
In the first integration,
dy, = /2dus
22a
H = r2 / (Bjiv) duo
22b
HJ= 2f (Buv)(ua-
The limits on u s are:
- wa(1-t)
lso,
(x,,
hf
where
and wa;
a), dttd
for the first and second quadrants,
for the third and fourth, w2 and w4 (l+t).
) = f,(w')f(va),
Y(we
f"(ws)
I(w)ro(V)
f•
d_ ff(ws)
WL
d4
and
2) (v( )
(
29
As
us= wa when y = y
throughout the second integration,
,
dx =
+
x = We + ya
__w
drx=
/2
di9
/2
substituting allthis in the expression for Sj ,
j
Sj = 4rr"Tý2f(w)
f(v")-ii.
dv
+
f
23
f'(wa)f(vs) - f(vrs)f'('j
Neglecting terms of higher order thanthe second in
24
J-eH
o
1 a q -To
l
•
2wae
vhe-
a,
i
hta3w
77 7____
-7 -7..,
,
The explicit- calculations are in the appendix.
Because the integrals fromquadrqnts 3 and 4 are subtracted from those of quadrants 1 and 2 to find the net flxx,
all terms willdisappear which contain: a and t otherwise
than in the logarithm alone.
30
Assembling all terms from both approx imations, the
total contribution to the balance ecuation 13 is written
That form was used as
out explicitly on page 31, eq. 25.
of the numerical computations.
the frame
In order to verify that errors have not crept into
our rather complicated calculations, proof is given in the
appendix that both the values of B' listed in eq's. 19,
and the approximate forms, permit an equilibrium solution
of iaxwell-Boltxmann form.
II
Let
-
I2 = 8r9i"
1W K
m)
where
K
integrals of eq.25.
-NQm
wf
-
,representing the
The balance equation 13 is then:
d
1
3NQ
M
V6
is a factor of dimensions
f
-
NJ +
2_a__P
KW =
mA.
dw
m
0
Letting
Letti
= r
m2NQ
26
-m wA f
C
-
1
=
dw f
E-a
mNQ
,
- J + 2nri w 3 K
corresponds to the parameter
E
this becomes:
-o
= 0
of the Townsend
theory. "i" is a measure of the intensity of ionization.
The function K can be separated into two parts: those
terms containg the factor
In t ; and the remainder. The
2a
31
25"
t
-2 )2
2
2
Sf0
o
2
2
00.r2
l
z
I
+c
0
(',LZ)
-OtI
wa 7,_f
O~i
/-h~~r
0-
2co
C4- 2 K
1
(,a2}~'
3ýL6 2 -
--2,
2
7
2-1r
4-
I,.
2L M2ur
i
-"
ff-ccr 2
00
/
91,rz
)CIO
2J
2T
] O
c
2
10
eo&."?
32
former come from the narrow strip, and correspond roughly
to smallangle collisions, in which little energy is transfered.
u ,va
ihe latter come mostly from the regions of the
plane outside of the strip of width
tw', and repre-
sent close collisions with large energy exchanges.
K
contains a number of rather complicated integrals,
and a solution of equation 26 cannot be found directly.
is necessary to select a tenc
It
function having the properties
to be expected in a solution, and to test it by substituting it in the equation.
For most choices, the integrals
will have to be evaluated by numerical quadrature.
VI -
Special Cases
A
The first special case investigated was that in which
the effects of the field, the elastic collisions, and mutual collisions were included but the effects of inelastic
impacts neglected.
It
is
J = 0.
Then
known that if
the mutual interaction is
also neg-
lected, the distribution function has the form:
-w'4/2w,'
fo
=Ae
When only the pair collisions are considered, the distribution is maxwiellian:
-awe/wa
fo
=.A e
The function chosen as a trial solution is:
-
f= A
-(w/w)
e
-(l-g)(W
/e2w,
33
27
f
=
-'g(w'/w 2 )
A
(-g)(w4/2w*)
-
e
=
3
2niw
e
2nwt3
with 0< g,
-p
1 .
At g = 1 or 0 , this reduces to one or the other of those
above. g and w0
are parameters which are to be determined
so as to give the best approximation to a solution ofeq26.
g is a measure of the relative importance of electronelectron an electron-atom collisions.
trons having speeds between
The number of elec-
w and w* dw is:
4n
wff(wO)dw
Taking the derivative,
d
-P
= d
wtf(w')
weP
dw
dw
The derivative is
I
w
0
w'
g1-g)
gw -
= 2w -
zero at
-g (-g)
~T
w* =0
or when w- = wt
w
zW
is thenefore the most probable velocity. "A" is
a normal-
ization constant such that
a
1= 2A fw e - dw
7r
0
WO-
Letting
WW
0
A
is
dz .
Ze
,
And
-gz -(l-g) z
o
0
a function of g alone.
Define:
In t
2a
=
In t
2a.'
+ In w
w
,
or:
a
=
2h
4r0mwo
2nd
34
From pp.
22 and 25,
-U
a
= 2*0.975 * 10
o
V
o
expressed in electron
1/2 * mwe
is the energy
0
volts.
oV
-q @of the order/O , and V S a few volts,
o4
0
If o( is
•
1 * 10
is of the order
a
0
is
a
introduced to separate
the dependance of K on the parameters 0(,
g,,and
w
from
that on w ,
K' = 4~w
Define:
Then:
Al
o
and
K
w@o e -
1
K' = w KeP
W
4TrNawb
K'
6
A"
0
In
is a dimensionless function of only
w/w
t
2a
, g,
.
- mw'w-o
+ &
g * (l-g)wj+ AiK'
=
0
Re arranging:
0
Because 0( and
t
only apear in
the equation through
their logarithms, the results are not critically dependent
on the exact values given them.
lations,
t
equal to
was chosen
In:,
1
the numerical calcu.
25
larger than
expansion of
a, and is
This is
safely
small enough so that the error in the
f(ua)f(v4 )
willbe small.
35
It is seen that A i K' is linear in
A K' = k
+ k
I
0
may be a solution.
function of w*/wI
0
or
wa/w
B
o
This is an condition that K' must satisfy
f
,
•w/wa
In fig.
4 ana~
in order that
K8 is plotted as a
, for g = 0 , and g =0.732 . These curves
are seen to be fairly straight over the range containing
most of the electrons. (w8 <4w2 )
The agreement is best for
0
the larger values of g. The amount of agreement measures
the correctness of the function f
which has been assumed.
O
The deviation of the curves at high velocities indicates
that the function gives too many high speed particles.
The slopes and inter-cepts of the straight portions of
the curves similar to fig. 4 (but containg the factor A)
are plotted in fig's. 5 and 6 as the functions
g
k and k
of
and Ilna
From eq. 28 follow
:-
i
29
k
29'
equations 29 an 29'.
gel
= - 2Oa (l-g) + m w*
aO
These equations can be solved for g andw
the parameters i,e*,
and In ao .
in terms of
In particular, g and
40 /i only.
w*/C*
are functions of log a
This is
shown in fig. 7 , where lines of constant
and the ratio
constant w'/t a are plotted on the
log a
8
g and
*/i plane.
t~
't7 rl,
i
h
4ý-:"
ý
3/-
/7-
I
i
I
I'
r
i .
.
:
i. _j
Ii.
I
i I
: 1
Hi
i
11111111111111111111
.•
:
Pi
3/:.1
1>11
2
V
Z
ci
;i
i
i
5/
0\
36
Empirical equations for g and
30
g
=
0.26 + 2.4 6 log a,
2/i
0.26 * 2.4 log a.- 2
3
30'
W,
= M
are:
w
-2.92 - 239 logao
+
2](1-g)
It has been mentioned that K consists of two parts.
One does not contain the parameter a
That is, iu is indepen
aent of the size of the potential hill.
It corresponds main-
ly to collisions involving a large change of energy.
The
second part, that containg the factor In t/2a ,corresponds
to glancing collisions with small energy exchange.
These
glancing collisions are much more numerous than the first
kind, and it is interesting to see which type is themore
important.
Table I
gives data for two values of a0 that
fall in the experimental range.
the function
w7•K' e
,
Column 2 gives that part of
which is independent of In t/2a
uolumns 3 and 5 list that part of the same function
con-
ing the factor. Columns 4 and 6 give the sums of 2 with
3 and 5 respectively.
.
L.
0.6342
1.356
.7352
b.464
-G.ES
-
-0 .55
-. 785
-0.461
-0.742
-0.0091
-&,2695
-0.07562
+0.102
+0 .222
+034
+0.00201
+0.00513
iyl74 +0.
Table
I
-0I.97
-OU .Ot
+0.00729
+0.007319
-0.1080
+0.418
+0.00990
37
I t is seen that the larger part is that containing In a
or that representing small angle collisions.
It
is
interesting to see if
conditions can exist
which this theory would predlict a large value of
for
g.
A search through the literature shows that thare are
very few papers on th subeject of gaseous conduction which
present enough data
to determine allthree parameters. The
required data are:
a
the gas used
b
th~ atomic density
c
the field strength
d
the electronic density
e
an estimate of the average energy
Among the few papers giving all this dataare,one by
T.
J.
Killian (II),
and one by A.H. van Gorcum (8).
In
Killian's work, the gas pressure was so low (a few bars)
that*: the mean free path of the electrons was comparable
to the tube diameter or larger.
Plainly, this theory is
inapplicable to such a case.
Van Gorcum reports work done on neon at a pressure
of 4.7 mm.Hg. at 00 C .
In the well developed plasma, he
found by probe measurements that the electrons seemed to
have very nearly a Maxwell distribution. The values found
for the auantities listed above clustered about those in
table
.
-3
17
1.67.
1
cm
-3
10
9.5
*
10
cm
0.93 volt/cm
2.7 e.v.
,
38
77
Vav
-3
10
17
1.67* 10
-3
cm
-3
O10
9.5*10
cm
Table
For neon,
If
0.93 volts/cm
2.7 e.v.
E
Q• 0.28 cms
eV, =1 mw
then for a maxwellian distribution,
o
300
7 aav
V= 2
o
3
For this case,
2p8
=
9
.1
-4
2.19" 10
taking ci1
-
7)=
1.93,
a
= 6.6
31
then from equation
U
-5
10
g=0.-184...
30,
log a
,
0
we have;
= -4.18
.
This is quite a large value for g, and hence the mutual collisions are very important .
V l mwa
from eq. $0'
, and
calc
V
=
Ocalc
=1
mwO
2
calc
1.39
e.
V
1.8
v.
From the experiment,
o
=
e.v.
If
w
is calculated
39
These
values are in fairly good agreement, considering
the roughness of the approximations, and the fact that ionization has been neglected in the computations.
If
d
is
chosen to
g
=
equal
0.91 ,
and
1 am , it
is
V cal
1.45
Ocalc
found that:
e.v.
The agreement betweenthese two theoretical results is
astonishing
d
in view of the violently differing values of
that were used.
In discussing measurements made with probes in a plasma,
it is usual to plot the logarithm of the electron current
to the probe against the retarding voltage,
for a maxwel-
lian distribution, the resulting curve is a straight line:
log ip =
c + V/V
In the general case, the current to the probe can be
calculated from the equation:
1=
It
0o
(5 )
f(w') (wa - 2QV) w dw
SIVim
(Vm
)
may be mentioned here that probe measurements are no-
toriously treacherous.
In fig.8 , log ip
is plotted against V for
tions obtained by setting g equal to
the func-
0.5 , 0.8 , 1.0 .
The corresponding curves given by van Gorcum are qtite a
bit straighter than the one given here for
g = 0.8 .
This would indicate that the mutual interactions are of
evengreater importance than is predicted by this theory.
On the other hand, the conditions in his tube Were
mppe
~zzumed
than thoQ
much
-w2
O
complicated than those assumed here. ihis fact may
more
account for part of the divergence.
B-
can be taken into account by as-
Inelastic collisions
electron which acquires a velocity great-
suming tha tnevery
erntha a fixed value
w 1A
suffers such a collisin and loses
all its energy. This assumption is equivalent to setting
f(w') = O.for w> wi
Eq.
, and J w
= constant = j for w< wi
.
26 takes the form
*Zw'W t
-w
+ 2niw 3 K= j
w 4f
The assumption may be expected to
w = 0 *
andnear
f
(w<W)
be good except
must be sharply cut off
near
w = wi
at wi
. In an actual discharge, it is not zero at this
point,but only very small , and decreasing very rapidly for
w>w i
.
Near the origin
thb
function may be expected to
be large,gor the assumption that the electrons undergoing
aninelastic collision lose all their enrgy
cessive piling up
gives an ex-
there.
Two forms of distribution function were tested:
OW"/we
A
a
2nw,
in
wL
ew
Mw 4//W
b
-
f =
Consider function
w w0
=
a
w
e
Let
A
41
Then
-z
=f z In
1
e
A
dz
s
Eqation 32 becomes
mAw
M
-Z
z 2 in 1
e
-Z
+ 2
z
A (z In 1
Az
3 w
+ Aa•K
m
w.
Where
K
w3 K
=
a ~A
This is
and
)
=
a
K
trJ
* 1) e
III
is dimensionless.
of the form
-Z
e
+ F(z In 1
-Cz In 1
a
-Z
+ 1) e
* LK,=
2rj
The terms on the left represnt collisions with atoms,
acceleration bythefield, and mutual collisions,respectively.
K (z),,
In a )
a
a
Values of
A
wi/wa= 2.08
and
1/4.16 .
(i.e.,
used ware
1/2.08
or
Curves of K (z,A ) ,/za In I
z in i + 1
finite
was evaluated by numerical integration.
4.16)
e
are shown in fig's.9 and/O.
at z = 0, but is
U, F, and
it
is
-z
eZ
K
very large for smallvalues of
From the form of the curves,
the parameters
and
is
z
plain that no choice of
can satisfy the equation.
This
function is not, then, a satisfactory solution.
I t is,
however,
interesting, to compare the relative
O-h
or
a t~
fe7
Z-Z•
zw"*
74,o
~?/7~?
z~
/
,.afil·
/
"-f·~·d;
r9t~(14 z
7-T
Q'Z
07-
F 10-
U,-
go
07
0;
O'Z
- '/
go
1
zo
+0,
9'0
42
magnitudes of those terms of Ka which contain the factor
In t/2a
tabled f
and those which do not.
n
.
S+
9
3
these values are given in
The actual factor used was
In
t
2a
Column 2 contains thevalue of the
terms indepndent of this
quantity. Columns 3 and 5 , the values of the terms containing it, and Uol's. the sums.
Here , as in the case ma
where ionization is neglected,
the terms representing the effect of small angle collisions
are the more imporgant.
~nr;~
o/0t+ 3 3
W
IV
4
0.2~5
1A " 2.08
1.00
].00
+2.312
+4.188
+4 ;14
+U.
-0.1110
+0.639
-0.172
'bU
z- 4.16
'300
'0'o
.447±1
t
+0.57O
-VC 377
-0.5713
-0.0240
+0 ,006
.'-
1ý
7
+2.2(65
-0.008C>'
Li.~
20
15
i
0.50
/½
-C0.79
3
+5,817
+b.U09
-u.233
+0.517
-1,.143
-1.168
-(3• 132
-0.235
-0.301
-0.307
+A
+U.07539
+0.0868
+0.0937
1054
-0.437
-u.583
-0.675
-0.24
-0.919
-1.065
-0.0079
-0.244
-0
252
-0.3 66
-0.374
-0.487
-0,495
+0.032.
+0. 276
+0.308
+0.403
+0. 435
+0.530
+0,568
Table 11
It has been seen that the most important of the mutual
are greatly
collisions are the glancing ones. Tihe integrals
simplified if the others are neglected, that is, if only
are considered.
the terms containg the factor In t/2a
Now ;
~teZen
_& -t
&
'
Except very near w = 0, the term
small compared to
In t/2a' .
wil be
21n w/w i
It will be neglected.
Then (eq.Ze)
3-
2
7__r
_
toi3
j
4
I7~z,)
7_0-2),
fJ
~
p
0~f
.&.# Z
GZW
4f (/V2)
CýO
/0
~U-2~
2
4
:3
-..
3
2
T~f0
$
7he expr-ess/'on /7
'~4K7
";·,
S d~L~c7I
c
6- nec *ke s
Ifo
is
r
2 /742
(U2/
74
3
(/0
,Z
C,,2
)
I-e
Z
4We can now verify that
the Maxwell aistribution is
the equilibrium solution with the mutual interaction abbreviated in this manner.
-/.,1, 2) X,/I<r2) -
For,
let
2- e2
--.22
2e
A) 2
-wd
Then
/a
_
aJ·
ur t'2)j
OC
fidJ
or -I =o
Peh/w y Ao 9enera/
4
Pd
I-VJ
n
00
A
~ff~wi
/0
'4r2)c&Z
,
.
U
AAT
hy 00vo1
. •d
44Crg £,,.<..vl .- 31~r/••!C,-j
2<l/-·C
a, '7e~, /ePg-·
~
frc~c~r)B
/~4r27~
7a,
AA~r
fCrP~Cv
AA-r2.
z
~
.tva
t(;.•7,
SSuAs(
2 -jf
i71
33
7r de 2
IIt-
3
2r
on
BdAA
r
UIrlI
o 0
;A~jCr
1/ rs
45
We now make an assumption about
f
Let:
-wx/wa
f
A
l
-
e
wa
a = wa
Defingtg:
0
1
0
1w
w1
In terms of
*
z ,
-z
f
a
= f
(wO)
f
The upper limit is
o
=
(z)
0
(1- ) z) e
A
2rnw
-A
-z
1/A , since
e
d2z
f"
= 0
when
z >
Calculating the field term in eq. 32,
w d
dw f
(w2 )
2A
= 2vla
A
=
w w•
W-*
O
o
w /iw
-
w
-we
i
-Z
2_A -
;1 - Az) z e
1
-Wa /Wa
Likewise,
-A
1
o
e
+
+1 1
w 0w
2.
o
=
-A
o
.
0
0
a
oI (1 +
0
A
-z
-
z) e
/
.
46
Consider the integrals in
f ° (v=) d"v
we
- I
I
A
a
i-
fi
wa
2A
=
I
vT
w
o
(1 -
e
y)
v
e
dy
0
when we set
y = v-/wa
0
Under the same substitution,
f
a
V ) dv4 =
v3 f (T
A
w- o
2nw z
@o
J
z
0
-y
y
(1 -
y) e
y
a3r
and
v
Jo
=
f (-V)
-y
z,
A
y-"(1-
z
2naw
2,iow
3 y) e
dy
Evaluating these three integrals,
fJ
f
w
°
(v)
Svf
dv
0
= 2A
2inw
-
(1-
(va)d9v =
Az) e
z
2rnw z3/', L
Wa
f
v f (v3)dva
iva
Substituting these in
2nwAz
eq.
33 for I
+
-
1
-
I
2,
e
)]
]
47
=
331
2A
77
83rr
rn 2.Y
36
2al
L(/-A-z)e- 2
t /,- - Az) e -z
S
J3
7
UIz (4/2)
+ _L_,hiAz)
-
+,Iýj
(7(1)]
736 e b•4anc e e u7a t ic hn 32 then becomes:
-••Az^U -Az )e - +
+ Ai
T3
where
.3
A24i
3
z-
K3
3
.A9
-,<,21
-x"Gr3
t";
---
(0+3-ez )e-z
27t
2
#zpit
),__ . X3
4 rNQ
47rivQ
ft im es
= 271a/
exp ress ion
braces, and/ Is di'm ension/less.
E7. 36 Is of the form
ý2
'S
-Cz.2-Az
-
+F r
+)eZ
T-he functions z /-az)
A
K, are p/o 7 ed
l(/f,4-z)e-l"+
z
6*n fi•y
/AI
) Kj= 2fj
-,a4e
zge- nd
*
// and/2
rj
L
-PO./
1.0
'-U
4e6o
iA
0.3
i
,4;4~
i
i
t
i
i
e
YI-i)
r
j
·
i
i
r
i
0/
·:
·~
·3
i
i : ;
I
i
I
I .
i
0
r
I
i
,
9
z~./
r
i-02
C
C
1•
lZa
.0Z
-O
•o
i
2.0
48
i~,77, E, and a
Suppose that the values of
If the function
A (1
-
w
) e
are given,.
is the correct dis-
tribution function for this discharge, equation 36 will be
satisfied for all values of
was assumed ad hoc.
z
(or w'/wa).
But the function
hence the problem must be considered
from the opposite point of view.
values 6f the parameters N,
4,
we ask, " Are there any
a , E
for which equation 36
is satisfied for all or nearly all values of
z2".
If such
a set be found we will have a problem to which our
f0 is a
slution.
There are really only three
independent parameters
in eq. 36, as is seen when it is written as eq. 737. They are
A, u/F , and r/F
.
The problem is to find what, if any,
values of these parameters will make *k. the sum of the
terms on the left of 36 nearly constant -- and positive.
We can pickj arbitrarily, and then may be able to find,
by a lest squares method,a reasonably satisfactory pair of
values for
In
c/F
and)F . This procedure was followed.
/3',',AI
fig(s. the values of this sum are t)otted for the
S, U/F, C/F
tabulated below.
/2.2
4 .-
C/F
0.63
0.42
?V/F
6.56
Ta ble-lT
33.0
sets
~
·6/j
OF
i
07oz
?ov.
500
r
i
I
i.
I
0/0
Oi o
7'0
i.
0.3
51
~
t-Jl-
0f
so
a
FQ
/'0
E'O
49
The curves are fairly level except at the lower end.
The degree to which they approximate a horfizontal line is
ameasure of the correctness of the chosen function. It is
seen that the fit is reasoably good,except the distribution
function gives too few slow electrons.
zrom the babulated values of the ratios, the relations
among the parameters of the discharge can be found.
Semi- logarithmic curves of probe current vs voltage
are plotted for this type of distribution.
(fig./•).
,-r2
2,
1.o
-•o-
.o
3
.0
- -~--~;-L~L-r
-----i---~_11-----
~~~.~---r
~---
--
~
i--r:- _- ~
:-·-:---:.._.-·--------------
------ ------- ·---- ·- ----------
-----------.---rl------
~
----------l;-L
1-
~I
~-------r-- ------- L._~
1:
"'
----------
:--_ ·---- l:i-_
.. _ ~_ : _~:,--:II
--.-----~--
::----
-----
-------;-..~.~~.~.-.~~.~
----
------~--- . ---..~i~~-.-~_._...
.~.__
·- ------ 1-- ---------. -~
-· ---- i--:-- --- ,--_C-.-L~,i
__
i----------~
0.0
-------- :-:-- :. :::ll·iL::-::~:
: :~1__:~
-- '--·-:------"-;;;;--I;;i~l:;'i~
·- ·---------;-;
:-.t------ I-------i-,
---.--~---~-~I_._ r-~-t---~---I.~
.-.l___(_i._
~~~___f-
i
L;
·----i
L: -::
i
-~~---·--------------
-i--;--·- : 1
i · -·--~---~
--
I
r- ---
0.00
--- -----------------1 ·-- ---- -- _---.~~._
-~~~~~~~-~-'-~'------
i
i;
._ ···--:... ; .~
·---. -.~-..~..
-.·---·
·-.. · · ;II!.-:----:-:
i..
1!
---------------::::-:
i
~----^
1
0,00-,
"v•V
O-S
2
-4"14 Z
.a
4Yos
,;,
/5
/=/ýF / x
2.0
VII
Conclusions
"·e have considered some of the processes t~king
place in a simJpliflied model of the plasma or 'positive
cl
-C
region
c of a
n
low pressure.
c
c
•tz••
l••
dischtrge tough
gas at
The most difficult problem arose when
li sions of particles of equal mass were con2
sidered.
The first part of this paper deals with this question in a jen -era
way.
We have been interested pri-
marrly in the velocity distribution function of the electrqns.
Two balance equations in this unknbwn function
were set up.
rq
quire
One,
expressing conservation of energy,
that the number of electrons acquiring in unit
time a speed greater than any value, w, be equal to the
number losing energy and falling below this speed.
The
second expressed balance of the momentum parallel to
the
fiel d.
From these, one non-linear integral equation was
derived.
The geometry of the mutual scattering process was
examined and the necessary integrations performed insofar
as these depended only on this geometry.
This much of
the work applies to any type of cross section function
irrespective of its dependence on relative velocity or
scattering a nle.
As an example,
the calculation fo
the case of elastic spheres was carried out upto
to the
point where integrations over the distribution
function,
f, became necessary.
In the second part of this thesis the cross section
o.n oCunis derived for electron-electron collisisons.
This was done by means of a wave-miechanical approach.
For the interaction potential a shielded Coulomb
field,
-
-
was used.
tions depending on
Next, th ose integra-
r-but not on f were carried out.
In
view of the relative complexity of the resulting
expresslons, two different approximations were made.
One was
valid when the velocity v of the pariticle was
very nearly
equal the velocity w.
This corresponded to grazing col-
lisions or small angle scattering.
The other was used
when v was appreciably different fm
, and correspoonded
to close cllisions, or large angle scattering
by a pure
Coulomb field.
In the third part we attempt to find functions
which
ae so••tions of the problem.
"hen the term
representing inelastic Impacts is disregarded in
the equation,
it
is
found that the fl.nction
7(
is a fairly
3
ie
ood a -roximate
: s•lution.
Here the parameter
a .ueasure o0,
C'llisions.
the rat
e iOportace
f
.'
the mutal'U
hen the inelastle te rm 1 Ire the func-
tion
was acceptable,
though it
gave too few low velocity elec-
trons.
These trial functions were not expect• d to be exact
solutions but they served to test the ascul p tons
an. to
indicate the relative importarce of the various processes.
The result of the calculations on mutual collisions
can be cler~ly divided into two parts.
true
pror•ess with large angle scattering~b
Gollision
Coulomb field.
ad of t:h
It
f
ptential hill,.
te
e scattering and is
egl~
It contairs
the width of the
strip
A c7hane
trelattive
.
The
-( dependence
n th e value
ss
eona
chage
f o
uV
u
L
of a d.i-
,L
"
Here
lane
plstne,
aa
an
d
anda,
and on the assu~ptions cons not criteal,
however,
rom- 10-4 to i produced onl.
i the
out that this second part is
'he
•-e
h.ýe second
Thus this depen-s both o
-pro••
mation involvIn• t,
erning
the factor-;
--
is the electron wavelengti.
t:he
a
independent of the approximati-on
small a
f'raction natre,
is
is
sieO, Oh,
part represents
t
The first is a
fna
result.
It
a
tutrn
the largest over most of the
range of velocities.
tribution is
In other words, the velocity dis-
maintained mainly by many exchanfges of small
rather than by less fre;quent large
amounts of energ
ch anges.
The mutual interaction of electrons becomes import-
ant when
-I
NQ
is large and -E.
N Q
small, where 2/ is the
electron density, E the field strength, N the atomic density, and Q the atomic elastic cross section for electrons.
This can be pictured as follows.
The electron-
electron collisions tend to set up a Maxwell distribution.
This is
disturbed by the collisions with atoms.
The
number of encounters of a given electron with others is
7
proportional to
atoms to NQ.
, and the number of collisions with
The first fraction measures the relative
frequency, of the two types of collision.
The field also
tends to disturb the equilibrium distribution,
energy and imposes a drift on the electrons.
of its effect is
path.
the field strength
This is
It adds
A measure
times the mean free
E
The Townsend discharge,
with very small
92,
is
an ex-
amprle of the cases where this mutual interaction is certain.ly negligible.
In the plasma of an arc, on the other hand, the conditions are such that thIs mutual collision process is
54
very important
rndeed.
The cal3ulations
,•how that it
very largely determines the form of the distribution.
A
comparison with one of the rare complete sets of expeerimental data showed a fairly go.
check.
A si• gle electron-electron collision catnnot change
the mean energy of the distribution.
But the aggregate
of such encounters does change the fo.rm of the distribut ion and,
in particular,
locity.
In this way the rate is
alters the most probable vechanSed at which other
processes take place and the mutual encounters may very
well lead to a different mean energy.
When inelastic impacts were taken into account in
finding an approximate distribution function, it was considered that their effect was merely to cause electrons
acquiring a velocity greater than some value w, to lose
all their energy.
A more exact treatment is desirable.
This would be possble usg• • the complete balance equation 13 with approximate ionization and excitation probabilities.
In this way the form of the distribution above
the critical potentials could be found and the intensity
of ionization calculated.
An exiaination by power series
of the abridged equation 31 shows that f must have a logarithmic singularity at the
ritgn.
This is caused by the
piling up of low velocity electrons, which results from
the crude way in which the inelastUz
ollisions are
ý%V,
vj'. Collisions are
han dled.
".jLffj. /~R J
The more correct treatment wuld remove thi,
Lst oO
S-ymvbols
pear•n,
N.ormal
313 2
/
25
71
at
w = w , w
0
34,43
'11 ,
Integrals
=
on page:
zation fac'tor
=~
a
First a p-
,,,4)
47
47
Electric field strength
30
e/m NQ
=E
base of natural logarithms
charge of electron
distribution funct on
parts of f
47"
parameter- in f0
13
energy
j, k
Integrals,
(j
= 1,2,3,4; k = i,2)
Plank' s const4ant
degree of ionization
Ionization integral
0"
0constant
Stun of integrals
K
I,
If
¶I
#1
40
30
47
47
k
wave number
2i
on
k,,ka defined on
M
mass of atom
77n
47
m
mass of electron
N
atomic density
7)
electronic
5
"
elas "tic cross section
;e,4
excitation, ionization cros
6
section
6
r
distance
20
S1
integrals
27
s
velocity, s9
t
2t w' = width of strip
dt
element of time
5
U
velocity of center of gravity
7
u
velocity
3
u,u'
half relative ve>•3ity
"u',v,'
u
x
= u+
NT
17
27
velocities
velocity parallel field
u e,
e:cItation, ionization velocities
V
p.ot ntial
V
retarding
V
ener"y
&- Volts)
O
otential,probe
ener"gy of relative motion volts)
er
w, or/c r ve4olCO S+
39
ve.ott
par
in f
32
dum•my variable of inteSration
2
=
/
w 2
0
radius of potential hole
00
22
'eleent of veloc i ty space
angle betweenr, U and
polar angles about
LI
!I 0
e1 go
'V
A
AJAWA
w "/
2
av
wave functions
defined on
a-
mutual
cross •eog
section
II1....~
•~Cb.,
0I"
elastic partial cross sectio~
polNr
angles aecut E
(refer to u)
7
8
Bib.lio
Allen,
H W.,
gr
a nhy
Phys.Rev.
A!i, W17.P.,
5,
707
and -.W.A2en,
Phys.Rev.
2., 703
3.
Compton, K.T., and C.Eckart, Phys.Rev.
25, 139
4.
Davydov,
5.
., Phys.ZeitCchr.d Soy Un,. 3, 9
Druyestuyn, M.J., Zeitschr.f.Phys. 64, 781
6.
Goroux,
9.
10.
:.i
Physikal.Zeitschr. 33, 856
S"
Physika,
H. van,
Gabor, D.,
"
I"
Zeitschr.f.Phys.
"
65
Physika 3, 207
84, 473
Physikal.Zeitschr.
-nI Kill.an, T.J., Phys.Rev.,
and
Langmu.ir,
35,
34,
38
1238
K. Mott-Smlth,
G-en.Elao.Rev. 2J
T
13. Langmuir,
Phys.Rev. 26, 585
if
20
14.
3,
Nat.Acad.Sci.Proc.
II
it
Phys.Rev.
.3,
14, 627
954
Found and -
Phys.Rev.
39,
237
I.
Mott-Smith and 18.
19.
. Tornks and
UcCu.rdy,
W.H., and P.Dalton,
20. Morse, ,.M.
21. Morse,
Pidduck,
Phys.Rev.
727
33, 195
Phys.Rev. 27, 163
hys.Rev. 31, 1003
P .. , W. P.A11is
22. Nottingham,
23.
Phys.Rev. 28,
tW.B.,
F.B.,
24. Schott:y,, W.,
and E.S.Lamar,
Jour.Frank.Inst.
Proc.London Math.Soc.
Physikal.Zeltsechr.
Phys.Rev.
207, 299
1,
25, 342
89
48, 412
3j
Sloanhe, R ..
ayd
.G.ELeeus,
- • ....
jt
oP,
"lA*
He r.
f3?.. $91Sibo·,
H?-Pai
a
Phys.Rev.
"e5
Acta 9, 123
--
$S.
Smit, J.A.,
29
Sommerfeld, A., Atombau1und Spektra-linies
mechanischer ErSanzungsband
30.
Thomson, J. J.,
31.
Tonks, L. and W. P. Allis,
32.
Tonks, L. and
3$.
Townsend, J.S,,
Phil.Mag. 11
34.
Uyterhoeven,
Nat.Acad.Sci.Proc.
Physika
W.,
3, 543
The Conduction of Electric
Gases.
.
La.,muir,
Phys.Rev.
cit
2, 710
34, 992
1112
15,
-
32
Vellen-
Through
Bio graphy
Borne June 27,
911.
•-rdouated- from Prpon Hi.h School,
Attended La Chataignerais,
19O2
June 1928.
Coppet,
Switzerland, 3eptember
to June 1929.
Attend.ed
ilpon College, September 1929 to June 1930.
Entered M. I, T. September 1930.
Bachelor of ScLence,
M.I.T.,
June 1934.
Teaching Fellow in Physics, I. i. T., 1935-1936.
Assistant in
,atemuatics,
_i7ersity
of Wisccnsin, 1936-1937.
Member, American Physical Society.
APPEND/IX
4m94
2,3 2 - cosA'cosA
/
'I
8,=
/o
Sz
=
+ sin SPcosA CO05
1
)
SiiT
s/nA dA
Cc/U
JA
s iM A O
-/7
Se
//n
Ihe In/ey'ratl s of /e /orrr77:
z=/2g
-e4
,7aZ
r4n
-
W7
IV =
z
d1
4 f77
-cs0
Cos
,
o
sCs /n5
Plere 'c Sh7or/ 7,/e o/
Then31y
2r
,1m
//egfra/s
J#o
300:
ton
-4
rp
ý--q-z
Ian
iLL
-Z
2ff-s4~
(>Z_~zi4/
ilz~
//Now,
PZ2
Z
= 7 - 27Zc os ~~
50
S1- s/'n2 27-e 4
4mnZ
fyt-
+ cos'Z cosC
y27cos2cos
-
y 2_ cos5 cosA
,s
yCSO
sny5/2ccos
-
sn
cosZ-
Cs:
s5nZ
B&i
3,
at/,J
CA
Acs,
0
xXz- cos
cs A
5C blity
W-e 4 Cos AO
S -:7
where
B3
'=
p'= Y) 5/npzJ,
P/e ceis
/62,
cZ') +f2cosc
9( a' €'
(4p'.
2e 'f
ZZ,2 r-
7os)
-4
4m'I
4
+
Z) Y
ZA<
2 sX'
cos' (- 2zcsJ+Z
Cos 2sz
- 2os Cos XI
7 y4/.(y 9 -/+cosd'-
2/2 xcosS-i
xz
co
,Y'OSAO -t5
,?;Z-e4
4t~/I
Is
)2 cosS.
/
?2
B n4mu
-
/2(05
same wa y
#7 e
o7a ,,7lead
O/x
a5 /h1e //m/is
d o+ cos o,/
21Y-zcos
and 7i
&s/n
y zcos : - co•s
•Z
4m'
7(7r
z
~1'
//Xa
Rememder/ny
-
C05/o
=
/1 -/+
2
zc - u
2
zycoscos5, o
(eis. 4 and 5/
- -t.
4~Uu
7y= 2z/8 /
cC05S9-
=
Z
Cos05
zz
twa
4/fly
2
Scos,~)9'
Z Z
1 /1e6L/
n1a
4,
2-P =',2
/66
z
/6 m+U-2)
L, 2
2
%
4U&
r
Cc
-//uv
4z~ i+
2
a4 -
£
4UI i~i
I.,
•,
C4(-:
.41'
uz
- .:
2W-u-zr u
2W U/67 -2
I/
/SU
7Th upper s/§n /5 usedl or 3, the /ow -rfor
-v-ul 2 "',ýýz2
-
z/2 -?/2
(2w2- 2-
a-
4(
it , Z
.
/
a2)
.4l/Uz = 2 u + 2/ • • - 4 z
an
_44
I
/+1:
/62mf 2
/KL/j/U
2
Ofsiua.
b/- z(2'
LeA'.
ar z
2
2)
/
k
=
2)
Oz (-w 2
2ý
/6A
2%)(a)+a2
,
2••
J'12 _ 2• _
=~~cu
g (z.
(
)-
; 22
/6=
Then .t
p1,.2
/7,---,7,1
7w
w- u.
2
Z
A- 2 /)
u(-g-e
3.
I
z
2(Z~
z
~i
S Lz
/
'Zo
0B3 sIny,
ly
2
/6U'O'
L
-7 4
24z)(w-t
-
u. *
v
,Jla
jz
where For
I
=
/
/
I
20,
L-3,
-•4,
2w
r 4e /,
4mQ'u
3"
Se/Yn5
=
s-2t
Z uF
z 22f uurw'v7- ZLW -U Z(
n
+(
j
z"L79211
2
-
2)dU
I
-a7
- 2
4 nj aif/li,lit d
~t2/
2 -2
7+0
~
/2
Z2
/•_hZ-
2
' 7 -iz
Z)V
4 -1' y
-az
= 2-e4
L/U
--
4m-_zt
2z
1
A' ýeadran/.:
In order and
2/
(auadra
2 - U,
S
=
(
arc f/Igen/
U
fan d-
Then /it•s
z77
fan
'
5
erm decom•.e,
/an
(02-
- 2kz
/,
_n ltz
-,h-janyU
2l'VF/kV
2an2
I
21-k)z
7zz
9 -l't/h
2V'
~- Zk%4-e
Y
h z - k z,
~ V -y k
erms
uz
2Vk
/
e4
The firsA
7/-
49, /6/
Pierce s
Examine
LU2
.lI
d,
61j
2 Z
St w-
ZL }7u
i-/,-V,
£1
,
02
Z=
27e~4
,v- ·t1Ju
25,
2u,
_-Z,
1j= 2,
Ut+*S
-=
1= /, 2b,
S i/g
I,
Ay/ yea'rat77s,
Z
Z z,
(
uacadran/ /
44s
tzatr
-' 1
v z'az2t 4 4 2
@
3
II
/1
- G an-'
Co/ns /ler
+
4
-/Zv-• 4-
seco/cl/ arc l/l//el;
- 4g
=
S- Zklz =
/nserf/~ný 1he
86(u
r'9
/z
va/lus of
appropro ial
h- ,kz,
(fwiu 2 _/6 /f 2
r. / or 3
2for 4
Z /r4
(z,•
2/
--
I,
ano' O,
h - 2hz
'ws
/6--/6u
Gu
(We
/61
(?
Z
/6a
/it
A/so, /he ra'/ca/ is
84 4 (UL
S+hz- z =
2
z1
(2W
t§/
=
lnser zli.,
4(a
tP8M+
4/+.
v
a
Z
uz
for ý1aadr / /4is /ecomes
S'w-/•+4 2(t
1• ZI
5I
/ 2(u
- 2
I
--
-- ,
Now,
80- (2gZ-
4/'
4
.2=
S424
ws
-ws-
2
4
4-
4 (W - s)
4w
And
6
(v
SZu (2~ , sZ
(
(w 2-
5 2(t1_ 5()2
-
or
2( tv/
s)+
-(
5
s-
2ilCZ
2
(
2/
2 j7s- (Z -
L7Z
-
2)
)(! -
2~
hz, - kz /
ii
(j'-
W=zjw-S
=7
T/ (d
=
2W 24W
4
L
_a Z)(
S2)+
s z}{W , -u27 - 441,'Z - S)
4 Z(w-
5J
2
for /A,0rs/
pladran-.
5, rn//a ny for /he /ower /,mi/t
,£
7+hz,- A
/n
U2) (W
·4 I'"
&= ;'(W-S,
-S),
412(14*52/2
uad/ran,/ 2
2ý
414
ir hz - kz2 2J
W7-1
{
'vuf
2Uiv/(
(/r+
u)f
4
e/'
2-
2(r<'+~ U
l' Z.'(.,
(2W,' .2u-,2
i-
5of
2
-+
vair)-
2
Z
()
(;
2 IZv-U
2
z
2
2
z 2-
2
j
- -v
v')
tZ
2/Zi
ZZ)
Z/7 '-
Pzz
)
V+ &
~-v
(vv
:-
2
U)
X-
ere /ore ,
7/
z-
2r(,u+ /.-+u )2 - (2 2(•
-(
and
T
(V/+ U
/1j z-
)Za +--,
z
z
ztl-W7,- 2 !L#
,/1.
2)(
c)
2
u
likewiise, for f1e sarme ý7 a aran;,
+4 '
+/z
+A-j
The eporess/ons for
.4 as for 2,
u, and 2 eer
UZ on/-y/1Y
,*e/r sIle'/
u/7 /hey can be wr///en .
pow ers.
iAz,- kzZ 3
/
4kiA-
ýz //~~2 /72 4
S•a ýZ
4z,
2
ze/
2el +
where
when
-
/Ae
fou r //
2
a3
= -4
z
-
V/
912
-f-
=ur
2
are of 7%e form
aov0
4f/z 2z p
//e seJcotn'
Ouai.
ur /cu
-
2
a
T//en
it
z7
.
.
2z/
/
q((
-
exvpressi on 's
/(a2
4Ik/
/
_
2
acn a'
s- z + 42 " ,
'-, ,
- kz 27
res
2/
a-~U
kz,1 3
+ hz,,-
4
3 are M•e same
c&uaaran/
and for ,uadran/
forruaa'ran! /,
2
=
-c9=
2
arc- /algen/
arn /yivous
/
/erms i/ BI, Aeco/ e,
sg/n
Ovaa.
O
-I
e±2
/an
4, l6g4_ /l.t'
4/Az (w)
ws
Z j5
S ifr a;~
t1
/+4-//6z
FIp"4/fai( ar) + dt g' (-
a
- aan
/('
41
2
(w-s), 4/ (w,, s'j7
a- /6/• •
aj(
4#f
/4ff
w 4l fit)#
/
Ouad'
32/
0
4
4/16
884
rA
/?c/%'r
/o
These
z
ca
o?
/Gbfur
El
f-
af
4
4
I
Ur
a
fa
ý017
U UJ
t:
*C /6
/61
af-
u4
V
6feais
//-
VLY
- /T.lr
42ý C
la ble
e .5"/,
,
+hz- /'Z2
.-4 and/he A/ /forcan be reduced to
Qood.
-
a,
,e.
-
San
4
x,'z + 4'
- 4/?u
cz2t
2
4ý5
P
-
aZ + 4 1
(
-4euw
3
4•
4 z
rf2+
2
4.
t -an'
Consider
now
42/h- _f*
/_A
u/-rd
z/e
-
2y-
arc- tangent
~(
2Z
-
-
2
217+ (,4 2
r /I
6464+/61
C,64 os vj
+4(w 2 z
2(z/24?Z
(utý,o+
22
12
-
(2v
2/
( 4 2,<
•(
A)z
=
/
- (2 1(
2v v
V
GIZz•>U iStC 7
t z%//6lO
IC ftg 2//- /4
g
7erm,
wU-
z
Lrz2
=2W4
A /so.
z
a 4 + 44
2 2z7e/
LW -
32/ 47-2 z
(2u
2
_U:
24
z ,Y-2,L
9j--
/f now we l/e
2P,
/19/
2
P2
-1P
2
the n
1
2
21/
z
(a'-j
-/I2/
2
2'
a-
2
Iolr
fa//vdaran/s.
2y - .A +(A+ 2 2 k)z
-
g;"
T7-he
A
4I
62,
/e
2:
4f+q g)4 0
eif"2(a
+4
4'U
(v42
2
'
V'77h - q -/l
kt
aov/ ears both outs,,de
and under /he lan-' opera,or, oord can Ae repkcec/
in bo/h
expression
/oces AB4' GI'=
bering-7 1ha7l
/
ý /7z -_z
/1ia212
; a/so remnetr-
r~~W2
z
2v
+4
,
/he fhbrd arc- arn en/
4-
4i
2a{
l-,
4
4f,/7
4(t
1 42 z
(vuZIl-2
22. u L>(af
, f•<e4+/ /2M
40,,z2,,
•/42
2.a
where /be + s5/i/
and 1h½ - s gn
(a
.2,
•/,2r P8/7 +,-+4
-•49e
now app//es to
20,
uac/iran/5
2 aad 4.
qa adrcw/5
Th7ese can de reacead/o
~: 1~/~J17'
1,482(v
S(c~Z
4 j+
-
42
-)
+
/6
,
-2
/0
an
"4
=
Then, com,5n/n/.
56
2a2
'y**
-(
krV'
ie/lrt7s
o /7he R.',
t7?6k/tQ
(52, 53, 55, 56,)
2n4r
4 fu
31=
4rnef
8
272;e
=
?, /an
3f
4m3
4/P
4w
7
t(
z
a#z 4w9Z
~7na~';B·4duj
4~/e
/an
-
41
a7
-
fC
-"'
4·
2
4
, 4 z
zr4/ar'
4/-mu
4g-
ac,41Z
a
az
4
24M
e)v
Z
l/
_a-
zan
-
a~z-Lq2-4e4
4
,
an
-
2u
?-
- n
4s
,
/7
'I
a4 41
-4'
24
a7'
4l
4mt&!
/
az
46s
lan-'-
B =-
C7]
4 6'
4g
2
.42,/
Now
Startz &
-Ljo,
Zr.-
-Z
ruv
4--,
4a 4
4/
z
2
aZ_
t, i
Sz
,
4
7,- L
4
,z
z
as /1
i'c/ 7-, ana7
Tsese are syimme/r/ca/ /17 U
f e case wh/ere c=- cons/ana/, we are perm/I//ec/
5/iisfrom BI' /f we a/so s•vfrac/
ho s•u/rac/ the
the secona' /roo77 B/
Then finra//:
4
2
/
3,
se
-
4-m -u
I,
-
wu zi*4ý, IaI
v/
'J
Suz-
32
4A
4
22
47
f u ,
46u
w
27Y
4m fu
'3
4
-
an/7
2-7
equaG/ors 1/9
4
4
4w
z
4m7u
-
Since 24--
L+
r
S/
3
/n
uz, 44
U/
46%
71
7
145
Z-
l5
ze2
44 7
/hese are fhe samne as
of /be ma/n text.
u*a ,
/2
APPA/VD/I II
STRIP /NT§EGRALS
The fw-nch!o ns ±o he cac/ca/ofed
By fhe/r de/t/n/on,
2
I,;
Z/m7//s
0on
z'
B'-ra-u~
1(
22
e's
( u a'2
-
are
u
Ouadrants
So
4
3
anlId
/e
Scarr
/n76egra//0n
l/ine
1/Z
&iZ
carriea'
/i
z/)
=
70
Ut
S
z
2
(+/) 42
2
Z 227"
2
zU
2/2
=
U 4 V- - u
--
/77
Or
-halk of in/fera /ion,
(cZ+/ZZ
z
w(c
- Z/
Ile 2c z
/n7 ei?ua//Ons
/9
we -
tan
Z4/4C
/1e
ivariab/e
2'
/nse75 r/iny
or
w~t
,ZTZ
Then a/ony /e
rZ
Z2
Utr
Col a/ont7
Aep/a ce a:-by /he
Z=
H k.
are ihe
2a
o, z
-Z +a
i3
i,
Z-2
(-,Z +a2}
-
m u'cz
2a
2ac
,
tan
I
- z +a
, 2c,7
(-z- a2)
2ffe4
2
7-IlzT
-Zi-a2
t7? G~~r
U?/
2
-
z
a
tnz-'
2a
2a
2a
m? uwa
(z- a/n
u VB
34
iu~4
2
2a
2
m7 zua
2a_
2z
z+ a/
-
7
z~ a1
Then, wri///'l'
/
9/,
m2ruwzaz
177a'
43
22
ma
2 =
/
c/-
H,
2
/'
and also replacing z 6I
/ and 2, we Aove:
zC-
Za
o_
--o
/,2 =
/C72
t7
2,/
y7Ty2.
1
-,
-,
for ruao/ran/s
z = -z
zac
z' , a
z+al
z,
2ac
a2
Sfan-' Z /z
z
t C-
a 7 i?Lfln
0
Z 2
/
z-a 2 tar
)-a
267
z '- az
a
z- a
_Z
/z
2a k7Yn 'z'a'z
z/a
0
/4
Z-
a
Za
6
-3,
!
zea'
Za
2a Ian-/ Z7-2 z
3,2
0
74,
--z
/
z- a 2
4,
0z
z2
0
C~
Since z'
2a
oan '
7
2- z 'z
on/ly a a'mmy
var/ah/e
/he ltr/176
can be dropped.
/n legrat/ng by ioar/s :
z az
C-
Za
0
4'a
(z +a ZJ
=cz
A/
-4a
a
z = 0
_a3
ac /-- - cc
since
v
(Z+az
4
z
Sc
Zac
n_ /
/an -
z
(z)a
/
2ac
a
(
-
zaz 4a (z--a27*
an -'
Zac
cz
Z+a 2 ac/
-
2ac/dz t
4
a zc2
2ac2 te
-/2Lz+a
Z+al0
/1,s express/on7IS .
- aG
67-G (
,2c7
a is
very
/f t >> 2c, /he
2
-
) ,
2c
2111al
arc la ne7/ terrns can ;e
expanded whenr
approx/imoae/I
ct
c
+ýl
a'
z
The
.
=
2azc3
9-l
51 ve<
upper
za 2 c 3
40,3
6lZ·
3(t+a§)
3t
Retaining only second order /erms in a,
2
Sr
C
"I
Moww dý
It? /egra
2
lwc 22c
ur,
S=
SY ;
2iaw 2
I, /
a32
23
2w
@
(6)
3w
may be /ound Iro-? M/s
w3,,
or
/•in//e seco1706
we rep/a ce c 6y z2
Sand
we obh ain
i/Ct?
4o2c3
Jt
zA
3y
scbL//?
Zt -= Z
Ira
a
2 .Z* -
631,/
4a'
2
3t
Z-· -a21
?j/,Z
-~t
zc zdz
ToW':
z
I=2
_/z"
2z3+ a~zz
/2a
42a
2z'+'z
eo,
"
-/
z a
/y
/2a
ay
t
2ac
a
/§qaun
2ac)dz
'(2zz 3 3z a (Z12a a(z .
Zac
za
azC
tr
&
i
0/
re/a/aing only lermns 1f/ro•y
cz2 Zz' 3,zza2
Zac cz 2
-2
/2a
/an-
zz
a
,-
3
4aý2
4c
-)A-azz
second order o/ a,
4a
.
6 z~u+a
#0 ,
d-4
2
/
(main7
4a2a
c
/2
/6
2~t'Z3 t3 a
2'ac
/2a
ta+a
2ac
•
t 3 J_t2
ac'
3t>a
/2a
a
a-t
6
and
7
4 c-7a
'a
9
G
3
6t
At z = 0, the /oyar/hrn
4aJ
3 r/ -3 3
"7Y~
4a c3
ferl/r
3
da'v"
tc
3_-
//
Jul3 7j -3
Zac
Z//t zac ·
+
L kAewise,
3
3,2
t
/
4a'2
2a-/
3
Tonsiaer
t
4,1
Let
=a
Z
Z-La fan /
T
cr =
,/c2-Z
z = c (/- r-,
I/an
/
/
/n/egra//nry
S31
c
dz =-2c z. adr
C3C
/ 4,
20V" dz
27- a 2
2Z-a
a
Za
-3(c
2
2-rz (-2rar) .
/opr
/r-f
by parls"
4/
4
3
,
-/_____~
___
__
/7
$P7C
3
-t
c
c
41o
rz
2P
2
(/-r_ p}
2
Th/en
G.
C
_
Lp
I
2p(r
4p
¾
22,
3?(5
4"
2 r
/an
I
1-
-0r
//.ivZ
Sr
" c7
1__
_
_
,
/
2
z back,
Subs/iuing
-
/
cZP>7 I
J%A
4a
(/c~z-j; ar
4"-----
-t"
=+
cV2/
-
+
a20
~4a
(-
(C-
2YC
- ac/a/nI
az V2
1
za I/EsT 7t
77
2a1
Z-az2
4a3
k/•Ta
2_
e
0
a'c
a
-
Expanaiony Me
_
z~a
2
6
-('- t
-/J-2a2 z
(z -aaz z
/w
2
/
dr
"
'-+ 74r;o
Cal!/ /h15' integra/ /erm
4,1
/ r:)
arc /a7ngenr-s
2
2
7
aC2
2 Z77
2
2
3
4az(c
2
To second order
t)
Ct
//7
,V
tt
a
6Z
2
z
t V77
2
T 6 r
S2acP
acZc
a,.
2
a
fj
·4a 2(' (
<7-
a
",4,
7
ac
1
21/c
2
t
f
<<
rac
2
cz
3
4C3
3t
Then
1Facz
,
6 .
I
4,,
£
S,'
2u
2
Cc-
-9-
3
4a-c
3t
or
z
w
2-w2
7<
z
3wt
--
se /ling
/ i/kewise,
7ra
9---
-'-
Z
d4ac
a2
,1/
©
35
Now lake
/-
/-/t
-7 72/,o -
P' -Zdz
z/:
0
117legraýn hy parls,
-- - - -
: 4
1+//c
/5
2'/t-SL ,'3
3
rfz/-rp)2
2.
a // 1he
Infegra/l
-
70-~P
4d
/,ein /o second order i n p,
/9
Cs
zz
I~
/7
3
2
-/
A2/
'
z/
c'
Zrs 2p'r
2'
1
2
/5
3
Zr
/
(
r
'
-
953
i/7~pf/l'
#
Ca// 1f e /oa5 /nXegra/ G, . Then, se///ng
r-
Z(
fr%(,aI
S2
(,
:
-,Z)(
A2 ),
t
t~
A-7
_
I
'oJ ;
122~
t
c-/P
Az
s-
Now,/,
:pg,%
wit/ A re/
I-,-:
=
,4e
and /1, p,Z Ae
r/
i122
-idB
4e,
= 1/4'0e- ,
(z/-:2or (/~0/ 2cos L9 2/-
/- Is•Z
co. 8=
Z7
Ihen P'= (/A/e', A'2
2
a/lso p,z'+,
Pz -Az
,/Z
-,-,-
pc5/
/+2
5,1/7
Z'll n=
eO
I
COS
cc1,z
/, if+ ;. ,,'_/7+=
,5117 2
-
/0
rl70
1,:- I
p2;. _,-P
2
9/o1 (Ip?
20,
A = /lpi
PA-l=
-2/
-p
•
=
/-t rt
l-
~
z
z
/-P
e
/-
rt
bz- r
(/- r})p
lj3
V (/-
"
/~#-
r
lb,p,z-l/,ý b,
-P, 6,-, rr _-
p
r
i/anZa 7/- r2;r
z z,
p_
Srm'
i/C/7o
/2
-
Then, dropping ferns of
3-
2a7
;'Ian-I l,-+io
!2(/.jr2/
/-p z
S _-r
/a nr,2 rzl
2
•
/04t (Z+/
bz +r -P
J'z
--
hgher
In
arder
'f0
(/-
p.b //
(ij-5
Assem-nb/in 9 /he feris of /,
4ci1
-/41
4,2
/-It
/
3
52r2/,02
1
Zr
Z
3
9-+
/i1SUDS ;fl/Ii//n
-21 czt2•
4,Z
3
3
- -
3
e
3
/3-
21,o/ S/-
2
/4o/r ,'o '32
j
f -r P
3
,/l /7he
i/i/si.
4(czt)
jc
/5
-
/5
c
-,
, z
(7
I
3
I,/,7
• /and
(/j
r)
/p
p
1
t+az4 / ) 7az
! h-f,_a';
(c5-)
/5
23
/5
9
9
23
ac
Zz
a c Vc 2- Y
S
2
3
3
ac'6
3
3
3
Z
Siince c2 >» >
7
SaZc
c4c~-L c
0-
7
(c-$-
47
,
6
3
3
a
13
'
a
a,
(c+ -cz-t az
2 a2
(c- 'C
,•U • di z
1/ c 4
z
Iz
t
4
4c 2
-6z2
,
EIx and/ng /he arc-iangeni,
Za V•-7
4LZ
(iaa
t
8a:c
6a/t
az
3
4
t).
7
taz22
2
3
-
9
(o//ecl/ng ferns
3
4,2
3
9-
3w'
3
3
Za YZ
/3
4a
I7.1
3
/7n vew o/Ac 7e
///Y/dn/
6Y are
eavcoun
2 c
Zac
Se /nct
62 /Aroug
A
8oa4a
tA
Za
/
'3
oW1/
y69 7s
7l/ /o e4s
6/ //ie
£4 o/f7 e ma1/
e9 /a/w7ns
/e/d
22
rA22EI;=
e
The namber of electrone per
-~- du,
range
r ane e
.....
.elocitis
*
+
IIi
"..,
hazvinh
parallel
t
n ti~
di
da .
fl (a -zd)
(a
, the field increases the velocity of each
per unit time,
da
-
particle parallel to the field an amount
Then,
the number of particles for wihich this com-
ponent of velocity increases from leas
(
the field in the
is
2r 77
2?
than
speeds in the
than
to greater
_
is
27T7
T 00
--- /
/0
dtJ/ý
§~dt
Zne
2 77Y-nf-e
f
u i/-''ydli-na
gd
L-e
m
(nn
tnz
'V
0
(-ut ~f/y4Ja'.
In the case of elastic collisions with atoms, we neglect
the energy loss, and consider only the scatterinm,
Let
0, 9
Ie polar angles about
VP
"!
The number of electrons scattered
.
a
, of
"
er
c,
-U
cer second at
23
angles in range
$sin 6dBd
a7mwdwdy
is
N'77
'u
out of a velocity space element
*1fcosq, uca%,('up 8jsz 9d6a'p/ndcn
dw d .
The velocities are scattered over a sphere of radius U4.
The number of electrons leaving in this way the region of ve-
y
locity space to the left of plane
8
with respect to
is
found by integrating
for that part of the sphere to
and 0
and with respect to 60 ,
the-right of the plane,
L , and
U
for such values as make the initial velocity lie to the left
of the plane,
i.e,,
,
Now
si6lOded
(/, c-/
adfs u,
can be replaced by
for purposes of inteTration,
if
cos 9 = cos Wcos0
and
is
remembered that
+ si c)s/inA s/nU
and F
Then limits of
it
si/n/d•ad
are simpler than those of 0
:
0
A,
Now if
o 27r
,o fo
77,
where cos5. /
Z =
COS
.
we let
X: = COS OW
and write
a
=
x'z;
,
)
our expression becomes:
- Y
0
eor
z
tjc/zj2ff#
24
Similarly, the number scattered into the region to the
left of the plane is
The net number leaving by collision is the difference
of these two expressions,
change of the limits for
is symmetrical in
X
These differ only in the interZX , and
Z ,
and
Z , since
(X,
Z..J,
f
term
Therefore the
Requiring the total number leaving due to
disappears.
both field and collisions to be zero, we obtain
2'}u
(u
+
27r
-4 xdxJ dzj/u,
Now,
ai
J/
#4z; z/j}f
0.
zf,
a similar equation can be written for the plane
e eE o T,
Z 271
777
!w,/31/
+/
Z77
4V/
2i
f(Z,iI4au
V04'zA6
-M
•
But, since
1(-x,
we can replace
X
z
by -X
, and Z
by -Z
in the integ-
rals, and the collision term then becomes identical with
the correspondint
term in the equation for
+
25
Adding the two equations,
-- zdz
ddzjt,
xt,,
Z, , ,Jd- 0
the restriction
ivide
by
hi
-o
The absol1ute
. differentiate with respect to
, and
value eglas can nob
w be dropped, by making
-eEor
r
,
the transformation to
N=
COSU = --co
4-
is
I- i
o
n5/
COS
26
4;j~·B(Z11
I
Y
-o
i
-2
J 1
nos
-- i-s/0cos
U
(u
L2 o (/- cos
tL
/UijOk
•'
u..•
d
0 €
2
V,
The equation then is
and differentiate azain:
Divide by
ef d
rn
9Qýu dua = 0
(U
/yQ
'77
dy
quoted in the main paper,
This is equation
If the equations for
stead of added,
77?
is
+
and
- /
y
u
0.
impossible to satisfy this as'well as the pre-
vious equation and the energy equation
J
are subtracted in-
we obtain:
47r1ef
It
0.
/y/Qy,/yfoyZ
1yj
was approximated 'by 10f f,
what has been ne•lected,
COSi)
SIt enters because
and sho-:os just
27
APPEDIT•
TEI
AT
AS A CIECK SOLUTION
I•I•L LIT
We wish to show that the
zero along a path
Y
rZ
2
I1V
functions integrate to
(C /)w 2-
c 2 ,> /
There are two cases,
and
C < /
Along this path
S2
When
When
2
C 2..Z
Z
2
2-=
12
W
Z/
.
= Cu
2zr-e
Dropping the constant factor
4
, the expres-
z ZZ
/72 zra
sions to be evaluated are:
Case /.
/Z
c> I:
1)/a2u÷
z
oto-
2
2
u aa'w2
z
S2wa
Z01uv a.2Z
2wa
S2
LLrZ, /-I'
/cs
lura
S2
Case 2.
2zeta
us
c2 < /
U0011u
U+i13
2wa
W u Aaý
,.,• Zwa
2
Z
2 (kc/W i-
z
TWan
22
2
-/n
2
Z2
-•-u+a ur
1
2w- a,
up'
U2
U,
-
/
/d
ue Z•-ur*• aza ;k
2waZur-a
4U(--u-awIan-/
I•
u-w7a2d
ana z=--z,
?uta
'c
z
+/-
u-ar*-a
u
Cl
)d
-/a
vl ut,,a
uz
as /n Appen/r If
7ý-en
28
2
/
=j,977~
-R
al"'
0
2a Y/- i7,,
zla-
2a
z
z-a
zdz
0- 2a fa- z a
2 dz2
0
za-
z-az
2- an
-,-_
z
Za
2
C?,
d
a ,
z
and
I/
/"
3
Ri-
ac
,Z
1IcC
Za
0
Z+a /j'z
/s5 *n resu//5- from A4ppend/x
R,
2
z
.1:-
'zza' /f _" as a' fza_
<•Ia
4
z1-a
za
(/- zj/·*---
2
a T
CE,
- af
7-
4
2z
a2
a Z#2
---Ye
-I
or
A'
J
-Z'
3
a a/7/-,2 a
4-
/
( /#aI
1'
672·P
2a
--
c-/
2
(-/-a
44
4a
IZ
'a
c /#a2
za
[a
a 2z
29
3
z
///-A" f'/-'zz•4
-/aý
2
ten
/
2
Sz
Yca Z
2
47zao
, a2 c
ac"Z
( 44
·
2
z a
Zac '
Ta-zacz1447
2
2~ a
/-•-2
2
7L
a
4a
<
Z4ayz
2
Tcz/2Y/CZ'
zýý/, 2a /-7a
cz
,ZI/
-c
ac
-------9.
2/
ac
, 2ac
2
/-c *a2
4a
Zac
4a -aI YAt
-'
laj7-l
2j
4
a S
4
/L/' -
Zac
Iz
bnl/
-~'ac 7~
I5'~
7
-
O.
a