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Theses and Dissertations
Thesis and Dissertation Collection
1966-05
Elastic scattering of low-energy electrons
from oxygen
Johnson, William Lewis
http://hdl.handle.net/10945/9495
Downloaded from NPS Archive: Calhoun
NPS ARCHIVE
1966
JOHNSON, W.
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OF LOW-ENERGY
ELECTRONS FROM OXYGEN
.ASTIC SCATTERING
JOHNSON
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NAVAL POSTGRADUATE SCHOOL
MONTEREY, CALIF. 93940
for publio
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ELASTIC SCATTERING OF LOW-ENERGY
ELECTRONS FROM OXYGEN
by
William L. Johnson
Lieutenant, United States Naval Reserve
B„A,, University of Southern Mississippi, 1962
Submitted in partial fulfillment
for the degree of
MASTER OF SCIENCE IN PHYSICS
from the
UNITED STATES NAVAL POSTGRADUATE SCHOOL
May 1966
MPs
A<2£t*we
The cross sections for the elastic scattering of electrons from
atomic oxygen are calculated in the energy range between
electron volts.
and 13.6
Polarization of the atom is included by using a term
in the potential function which has the correct asymptotic behavior and
which vanishes at the origin.
results are discussed.
Previous theoretical and experimental
CI-
TABLE OF CONTENTS
Title
Section
Page
1.
Introduction
5
2.
Review of the partial wave method for computing
the cross section
7
3.
Potential function used for electron-oxygen
scattering
10
4.
Numerical methods
15
5.
Previous calculations and experimental data
19
6.
Results
22
7.
Conclusions and Acknowledgements
23
8.
Bibliography
24
LIST OF ILLUSTRATIONS AND TABLES
Figure
Page
1.
Radial Charge Density of Oxygen
26
2.
Behaviour of the Core Potential near the Origin
27
3.
Polarization Potential with q =
5
28
4.
Polarization Potential with q = 8
29
5.
Cross Sections with
ex
=
30
6.
Cross Sections with
ex
= 5.5
7
Theoretical Cross Sections
32
8.
Experimental Cross Sections
33
1.
Results of Calculation for
ex
= 5.5
34
2.
Results of Calculation for
ex
= 6.0
35
3.
Results of Calculation for
ex
=
36
and
6
31
Table
4-
1.
Introduction.
A knowledge of the elastic scattering cross sections for electron-
oxygen collisions is of practical importance in the study of the properties of air at high temperatures.
The experimental determination of
low-energy cross sections is especially difficult in the case of atomic
oxygen because it recombines easily to form molecular oxygen.
Only re-
cently have experiments been done to measure this cross section in the
energy range below 15 electron volts. [11,15]
retical treatments have been made.
Several previous theo-
[1,3,7,9,12,16]
The results are
divergent, and none agrees satisfactorily with the experiments.
The elastic scattering of low-energy electrons by neutral oxygen
is
treated here using the partial-wave method of Faxen and Holtzmark.[ 13]
This method, which is outlined in the next section, allows one to de-
termine the cross section if the potential field in which the electron
finds itself is known.
The first approximation to the scattering treats
the atom as a positive charge at r =
metric negative charge distribution.
surrounded by a spherically symThe charge density of this elec-
tronic distribution is calculated from the Hartree wave functions for
oxygen
This first approximation is inadequate because it ignores the dis-
tortion of the atomic electron distribution by the Coulomb field of the
incoming electron.
This polarization effect is included in this calcu-
lation by representing it with a polarization potential in addition to
the undistorted potential.
The polarization potential can be thought of
as the term which represents the difference between the true interaction
potential and the undistorted potential.
The form it assumes for large
separations is known; for small separations the form is uncertain and
will be discussed further in Section
3.
When the incident electron is within the electron cloud of the
atom there is an additional repulsive force due to the Pauli principle.
Hammer ling, Shine, and Kivel have treated this exchange effect by the
inclusion of a third potential to approximate the change in the electron's
energy due to the deficiency of electrons with like spin near it. [7]
Such an approximation is similar to Slater's treatment of exchange in
atomic wave function calculations.
The result of their cross section
calculation which includes this approximation shows that the exchange
effect makes a negligible contribution to the cross section in the energy
range considered here.
Furthermore our polarization potential will have
a parameter which is adjusted to fit experimental data.
The value of
this parameter is affected by the exchange contribution,
so no specific
account is taken of the exchange potential in this calculation.
The actual evaluation of the cross sections using the phase shift
method involves using a computer to integrate
numerically.
Schroedinger
's
equation
The details of the procedure are given in Section 4.
The following atomic units are used:
unit of mass = m = rest mass of the electron
unit of charge =
e =
unit of length = a
charge of the proton in Gaussian units
= h
2
2
/4tt
2
me =radius of the first Bohr orbit
2
Consistent with these, the unit of energy is e /a
,
which is twice the
ionization energy of hydrogen and the unit of angular momentum is
h/2rr.
It should be noted that one atomic energy unit equals about 27.2 electron
volts;
this is twice the value of another commonly used unit,
which is 13.6 ev.
the rydberg,
2.
Review of the partial wave method for computing the cross section.
The scattering of two bodies when considered in the center of mass
system is equivalent to the scattering of a single particle with the reFor the electron-atom problem the
duced mass from a potential, V(r).
reduced mass is very nearly equal to the mass of the electron.
In this
case the center of mass frame is identical to the lab frame.
Since the potential function, V(r), is assumed to be spherically
symmetric, the electron's wave function may be separated and written as
-2
Y(r,0)
In atomic units R.(r),
the radial wave function, satisfies the equation
^-
A_ -1
dr
2
RjCr) P^ (cos 0).
+ [2E
2V(r)
-
4i*±U
-
]
J
R (r) «
o.
^
a, beyond which V(r)
If we assume that there is a distance, r
is zero,
equation (1) may be solved exactly in the region r > a, giving
R^r)
= A^ [cos
^j^Ckr)
-
sin
6
£
n^(kr)
(2)
],
where A. and 6. are constants of integration,
j.
and n. are the spherical
=V2E.
Bessel and Neumann functions, respectively, and k
The asymptotic
form of equation (2) is
A
>
V
r)
~
2
[sin (kr
kr~
"
*
lTJ
+
V
]
'
so that
>
Y
(r,9)
~
2 P (cos
£
0)
H
~-
[sin (kr
Ju
•7-
-
\
Xtt
+
b
J
].
(3)
The physical meaning of 6. may be seen by realizing that if there had
been no potential anywhere, the asymptotic form of R.
—
— sin
(kr
-
\
4tt)
(r)
would be
Hence 6., the phase shift angle, is the quantity
.
which describes the scattering.
In a symmetrical scattering situation we expect an asymptotic wave
function to be in the form of a plane wave superimposed on a spherical
If the plane wave propagation is in the
scattered wave.
z
direction, we
express the asymptotic form of the wave function as
Y~
ikz
e
+1
r
f( 9 )
Since the asymptotic form of e
i k-z
e
ikr
.
is
00
ikz
e
~
„
S
4=0
.4
i
...
_
sin (kr-%*
(24+1)s P. t(cos 9)s
*r
kr
*
4rr )
,
the asymptotic scattered wave function may be written as
Y
~
i\u
2
+1) P,(cos 0)
8ln
*
1=0
< kr
kr
^
*n )
+ f(e)e
ikr
.
(4)
Equating (3) and (4), writing the sine function in exponential form,
and solving for f(9), we get
00
f(9)
E
=ttZlk
+ l)(e 2i6i -l) P,(cos
(24
*
4=0
9)
.
The differential scattering cross section is
f=|f(9)|
2
.
Integrating this over all solid angles, we get the total cross section
00
a
E
-f§k
(24
+
1)
4=0
8-
sinVl
.
(5)
The value of 6. will be very small for i > ka, so that for the range
of energies considered here values of & greater than 2 or 3 need not be
considered in calculating the cross section from equation
Hence in order to find a, we need the values of
6
.
5.
These are
i
found by solving the wave equation (1) numerically to evaluate R«(r)
and its derivative at r = a.
at r = a from equation (2).
R*(r) and its derivative are evaluated
These two wave functions are matched at
the distance a by equating the logarithmic derivatives.
the values of the phase shift angles, 6..
This gives
When all significant phase
shift angles have been calculated, equation (5) is used to evaluate the
cross section at the given value of energy.
This process is repeated
over the energy range of interest to find the spectrum of cross sections
versus energy.
Potential function for electron- oxygen scattering.
3
A calculation of the electron- oxygen elastic scattering cross
section is routine, although tedious, if V(r) is known and a computer
is available.
In this calculation we assume V(r) is the sum of the
undistorted atomic potential, V„(r), and the "polarization" potential,
n
V (r), which attempts to represent the difference between the true po-
tential and the undistorted potential.
V„(r) may be found in a direct electrostatic manner if the electron
n.
charge distribution is known. [14]
The charge distribution is calculated
from the Hartree wave functions of the oxygen electrons.
Hartree approximation treats each atomic electron
Since the
as moving in a spheri-
cally symmetric potential, the atomic electron wave functions can be
written as
U
=R n/ r )Y
'
ni m
e
m
(9
'-
'>'
where Y. (0',0') are the normalized spherical harmonics.
The values of
j&m
R
.
(r
1
)
have been tabulated by Hartree. [8]
In computing the electron
*
volume charge density one sums UU
for each electron.
For filled
n,jfc
shells this sum will be spherically symmetric; the only asymmetry will
be contributed by the outer shell which is not filled.
Hartree approximation to neglect this asymmetry.
It is
within the
Hence the charge density
of electrons is given by
^'>= S
N
nX
n,i
where N
3
„
nX
R
L (r
'>
P ground state of oxygen is considered here,
and N
Only the
represents the number of electrons in the n,Z shell.
=4,
-10-
for which N
10
= 2,
'
N
20
= 2,
V (r) consists of a negative term due to the nucleus and a posirl
tive term due to the electron cloud.
=°
„(,)..
H
i + f
r
2rr
tt
[2R
(r') + 2R
~
[ f
J
J
J
-
V
" [p 2
-
4R^(r') 1
.,..,.„.,
r'
sin 0' dO'dfl'dr'
r'
^(r'),
Letting r'R^Cr') =
-
(r') +
—
r
„
may be written as
It
(
-^
r «) + p
2
(r')+ 2P
2
(r
1
)
1
^-Tn
dr
.
The integration can be divided into the contribution of the electrons
and r' = r, and the contribution of those from r' = r
between r' =
to r'
=».
may be written as
Then V
rl
Vu (r) =
H
-
*8 +
r
8tt
r
f
[P
10
—&
(r )+ P
'
/
'
)+ 2P
^ (r,)
]
21
dr'
(6)
r
J
J
+8n r
^ (r
20
[
p
io
2
(r,
N . P .(r
The function S
n* nX
n,Z
>
1
)
>
+p ?
,)+2p
^
r '>
i
is called the radial charge density.
It
.
represents the probability per unit length of finding an electron in the
spherical shell between r' and r
1
+ dr
'
.
Figure
1
is a graph of this
function for oxygen.
The treatment of the polarization potential is less exact.
The pro-
cess of treating the atomic potential as the superposition of an undis-
torted potential and a polarized dipole potential is less than rigorous.
-11-
However, even if we accept this approach, the form of the polarization
potential, V_(r), is not certain.
If the incoming electron is at large r,
its electrostatic field at
the atom is nearly homogeneous and has a value of 1/r
2
.
This field in-
duces a dipole moment which is directly proportional to the field,
M<
= a /r
2
,
where a is the polarizability
.
This dipole moment then exerts
an attractive force back on the electron with a value of
So,
f = 2p«/r
3
= 2a/r
for large r,
V n (r)
-
-
f
2a/r
5
dr =
This form of V (r) for large r must be modified so that the function
does not blow up at the origin.
At the origin there should be no polari-
zation force since, in this limit, spherical symmetry prevails and no
multipole moments can be induced.
For this reason the proper polari-
zation potential should have the property that
_d
Limit
r
-
dr
P
(
vw
m
When the scattering electron comes "inside" the atom, the atomic electrons still tend to hide behind the nucleus.
asymmetry gives rise to a repulsive force.
However in this case the
In this classical picture,
the polarization potential should also have zero slope at a value of r
which is characteristic of the size of the atom.
A function which satisfies these conditions and which will be used
in this calculation is
V (r) p
-
•£$
1
-
(
-12-
exp [-
(r/r
q
Q
)
]
.
]
(7)
5
This function, with q equal either five or eight, was first used by
Biermann to account for polarization in the calculation of oscillator
Garrett and Mann have used this polarization potential,
line strengths. [2]
with q
calculations of electron scattering from alkali atoms. [5,6]
8 in
They obtained good agreement with experimental scattering data by treating
a and
r
Graphs of equation
as parameters.
7
are given in Figures 3 and 4.
The exponential in equation (7) may be expanded so that for, small
r
we have
—
dr
V (r)
K J
?
oc
r-0
r^"
5
From this we see that the requirement that the polarization force vanish
at the origin restricts q to be greater than 5.
In this calculation we
The result with q = 5 represents a
use values of 5, 6, and 8 for q.
limit since any value of q larger than five satisfies the condition
mentioned above.
is taken to be the
The value of r
of the valance electron shell.
expectation value of the radius
Its value is calculated from the Hartree
wave function for the outer shell as follows
00
=
:
°
This yields
r
o
/
r«
Jo
=1.2 atomic
P^(r') dr'.
21
units,
No direct experimental values of the polarizability are available.
Several calculations based on indirect experiments have been made.
Klein
and Brueckner found a to be 5.589 by adjusting it to give the correct
binding energy for
.
[9]
Using a similar procedure, Cooper and Martin
calculate a value of 5.499. [3]
Both these methods do not attempt to
provide the true polarizability of the atom, but yield a number which
-13-
includes exchange and other effects.
Dalgarno and Parkinson found a
value of 6.005 based on experimental determinations of the dielectric
constant. [41J
In our treatment r
o
is fixed and
a
is
taken to be either
5.5 or 6.0, and for both of these values the cross section is computed
for q = 5,
6,
and 8.
14-
4.
Numerical Methods.
Before the wave equation can be integrated it is necessary to know
V (r)
H
This function was calculated from equation (6) using a trapizodial
.
found that the resulting function could be
It was
summation technique.
fitted closely by
2
1
V (r) =
H
-
^~
IVf-^— +Y 2
8
where
+ Y
3
-A r
r
-A.
+ Y
4
e
v
= 1.2806
y
k
= 2.2376
X
e
,
= -0.2806
2
= 18.263
Y 3 = 3.0715
y 4 = -1.9710
X- = 6.803
X,
4
3
]
= 11.548.
This functional form was much easier to enter into the computer than
tabulated data points, and the fitted function matches the computed
*»
function the fifth significant figure.
To solve equation (1) the substitution Y.(r) = r R.(r) was made,
yielding the equation
d
\
dr
(r)
\
=
M
1(1
* +
[
1)
l)
|
+ 2V
r
R
+ 2V
(r)
p
(r)
-
2E
]
Y (r).
Equation (8) was integrated using a Runge-Kutta approximation.
to use such a
(8)
In order
method it is necessary to know an initial value of both Y
and dY/dr at some starting point.
These values were found by assuming
a solution
00
Y
(r)
s
- v
a
S
C r
_
n=0
n
-15-
n
,
=
C
'
o
1.
Since this solution will be used at a small value of r, a series approxi-
mation for V (r) and V
may be used in equation (8).
(r)
Neglecting terms
or smaller, we have
in r
V (r)
R
+ b
+ b
b /r
Q
<*
x
r,
2
where
= 8
b
Q
b
(
Yl + y 2 )
=8;
y^
= 8 (-
b
x
-
Y 2^ 2 + Y 3 + Y4 )
!
= 8
2
Also for small
r, V
«a
—o/—
r
q-4
Only for q =
.
5
will this make a con-
tribution to the total potential in the above approximation, changing the
—
a
+
given above to b
value of b
1
Equation (8) for small
,
r,
becomes
2r
o
r
-
T*
dr
i(X +
1)
+ 2(b
+
r
indical equation
1
c
So
=
+ 2E
b r
into equation
= Z(Z + 1).
1)
2
r
)
=0.
Y
2T
b C
n-l
l
~
8
-
we get the
The only acceptable root is
+(b l +E)C n-2 +b 2 C n-.
n(n +
+
(8'),
The recursion relation is
n
A
„/%
Y (r)^r
s
o/(a -
since Y(0) = 0.
C
+
s
Substituting the series solution for Y
a = i +
2
b r
r
U
4+2
l
16-
+
1
1)
—
dY (r)
,
and
s
m
,
,_>
f
(l+l)r
8U+2)
+ Sff*
X.
t
A+l
(8')
One could, of course, evaluate these equations at
=
r
and start the
integration from there, however it was found that the accuracy of the
Runge-Kutta integration approximation is very sensitive to the step
For this reason we evaluate Y
size chosen near the origin.
and dY /dr
s
s
at r = .01 and use these values to start the iteration procedure.
The
step size was determined by choosing an arbitrary value to do a trial
calculation, reducing the step size and repeating the calculation until
there was no significant difference between two successive results.
A
Although this value is smaller than needed
step size of .002 was used.
for large r, no attempt was made to change the step size as the iteration
progressed
Equation (8) was integrated to the final point
r
= a which was
determined by the condition
V
—
—
+ V
(a)
(a)
^
E
For r > a
(2)
_
4
= 10
we have a region of almost zero potential, so from equation
the solution of the wave equation is
V
r)
=
V
t
cos
b
ih^ kr)
"
sin
6
n (kr)
x 4
1
•
The derivative is
dY>(r)
:
= A„r
r
i
2iVr
/
[Xj
x-i
(kr)
-
-[in^Ckr)
w
-
+
(i
u
W
+ l)n
i+1
kr)
i
cos 6
«
r)
(kr)
]sin
6
\ +
Letting Y^(a) and Y^ (a) denote the computed values of Y, (r) and
17-
V
—
at r = a, we equate logarithmetic derivatives at r = a to get
dr
Y^'(a)
T
Y^(a)
k
=
(22+l)[cos 6^ ^(ka^sin
^(ka)]
6
(
-[jen _
x 1
(ka)-(X+l)n
(ka)]sin6
j6+1
i
l4
Solving for 6. gives
tf'Ca)
'(ufr
V
6
.
*W">
w+d^+iO")
-
)
-777
Y„(a)
V
+
ka)
]
/
V
ka)
= arctan
(9)
Y,(a)
a
(2^ ^.l (ka)
"
Ci+Dn1+l (k.)
]
C
n
-J—
Y,(a)
<ka)]
+ n (ka)
)
From equation (9), for a given I and E, a principle value of 6. is found,
For any E,
value of a
this process is repeated for I
is
found from equation (5)
.
=0,
1,
2,
Then the
and 3.
As expected the contribution to
the cross section from the Z = 3 term is negligible in the energy range
considered.
The entire calculation was done for energies between
steps of .05 for q
=5,
6,
and .50 in
and 8; and for & = 0, 5.5 and 6.0.
putations were done on a C.D.C. 1604 computer using Fortran-60.
All comThe com-
putation time for each run (a given q and a) was about four hours.
This
large amount of computer time could be reduced by increasing the step
size at large values of r.
18-
Previous calculations and experimental data.
5.
The theoretical treatments of the low-energy scattering problem
One approach is similar to the one
may be divided into two groups.
used here, but
different forms of the polarization potential are used.
The other treatment is much more elaborate.
It attempts to solve the
Hartree-Fock equations of the system of scattered electron plus atom.
Figure
7
shows
some
results of previous calculations.
Robinson uses no polarization potential. [12]
is calculated from
His potential function
Hartree-Fock wave functions, and the cross sections
calculated by the partial-wave method for I =
and
1.
He also evaluated
the cross sections using the W.K.B. approximation and showed that it leads
to very large cross sections at low energies.
Klein and Brueckner also treat the problem using the partial-wave
They account for polarization with the potential
method. [9]
V„(r) =
P
2(r
2
o
+
2
x
V2
This potential function has been used often in the treatment of polari-
zation in atomic wave function calculations.
It gives a zero polarization
force at the origin, but differs from the function used here (equation 7)
in that it produces an attractive force for all r.
As mentioned before,
they determined the polarizability from experimental data on the binding
energy of
1)
.
.
Their curve in Figure
7
includes
s
and p waves (i =
and
Cooper and Martin [3] found a mistake in the paper by Klein and
Brueckner.
They use the same procedure and arrive at cross sections which
show a decrease at low energies
-19-
Hammerling, Shine, and Kivel treat polarization in the same way as
Klein and Brueckner.
meter. [7]
They take
o/
= 5.4 and r
as an adjustable para-
They also include a term in the total potential to represent
the exchange energy.
This term varies as 1/r
the Slater approximation.
3/2
and is derived from
'
They show results obtained by neglecting both
polarization and exchange, and neglecting polarization only.
The cross
sections obtained by using just the Hartree potential are different from
any obtained previously.
The cross sections with exchange and polari-
zation are plotted in Figure
7
.
It is not clear why these should be so
different from the other calculations since they reported using the same
method.
Temkin includes polarization effects within the Hartree-Fock formalism. [16]
He associates with each atomic electron's wave function
a polarized part which depends on the coordinate of the scattered elec-
The wave function of the scattered electron is made antisymmetric
tron.
with respect to the total atomic wave function.
This gives rise to the
kind of polarization potential considered here, called direct polarization, and another term, called exchange polarization, which contributes
about 15% to the cross section.
shown in Figure 7.
His results for s-wave scattering are
Bauer and Browne have modified Temkin 's method.
[
l]
They made more drastic simplifications in the Hartree-Fock equations and
are able to calculate s, p, and d-wave scattering.
Their cross sections
are small at low energies and fairly flat at higher energies.
The most accurate experimental data is that of Neynaber, et al. [ll]
They have measured the cross sections in the energy range from about
to
.45
(2.3 to 11.6 eV)
.
.1
They find the cross section to be almost constant
•20-
at
19.5 + 1.3 throughout the energy range.
half of their 64 experimental points fall.
This represents a band in which
Sunshine, Aubrey, and Bederson
measured the cross sections between about .03 and .45.
[15]
There is
more scatter in their data, but there is general agreement between the
two experiments.
Lin and Kivel obtained a low energy point at about .02
energy units using a shock tube. [10]
If this point is accurate it in-
dicates that the cross section decreases sharply at low energies.
Al-
though these experiments are extremely difficult and have low precision,
they do indicate the magnitude of the cross section and show its general
dependence on energy
-21-
Results.
6.
The numerical results of our calculation are given in Tables
and
and summarized in Figures
3
and 6.
5
The curve labeled a =
1,
2,
repre-
sents the cross sections obtained by ignoring the polarization potential.
This curve corresponds almost exactly to that of Robinson.
that the differences between Hartree
-
This indicates
Fock wave functions used by him
and Hartree wave functions used here have an insignificant effect on the
cross section.
The introduction of the polarization potential increases the phase
shift angles but decreases the cross section.
The smaller values of q
produce potentials which give rise to smaller cross sections, and also
straighten the curve out.
The remarkable feature indicated by Neynaber's
experimental results is the almost constant value of the cross section
between
.1
and
.45 energy units,
in this energy range.
value
[ll]
Our results give a similar constant
Further, the measurement of Lin and Kivel
gives evidence that the cross section has a very small value at the
lowest energies.
[
10]
This is also a property of our results.
Of all the theoretical calculations that have been discussed, that
of Bauer and Brown is the most pure (not dependent of experimental para-
meters) which extends of the entire energy range, [l]
with
<y
= 6 and q = 6 almost duplicates their results.
Our calculation
Our results
are also consistent with those of Temkin at the lowest energies. [16]
22-
7.
Conclusions and acknowledgements.
No attempt has been made to justify the fundamental validity of
the form of the polarization potential except in a general way.
Experi-
ments indicate that consideration of polarization in any scattering calculation is important.
Although the "correct" way to treat the effect
has been outlined by Temkin, it is desirable to have a more tractable
Our results indicate that the potential
semi-phenomenological method.
function used here (equations
6
and 7) with
<x
= 6 and q = 5 gives cross
sections which have the energy dependence indicated by experiment and
by the more exact calculations.
Our results seem much more realistic
than previous treatments using other forms of the polarization potential,
The author would like to thank Professor William B. Zeleny for
suggesting this subject as a thesis topic.
The help provided by the
staff of the Computer Facility of the Naval Postgraduate School is
greatly appreciated.
-23-
BIBLIOGRAPHY
1.
Bauer, E. and H.N. Browne. Elastic scattering of electrons by the
many-electron atom. Proceedings of the Third International Conference
on the Physics of Electronic and Atomic Collisions
1964: 16 - 27.
,
2.
Biermann, L. Die oszillatorenstarken einiger Linien in den Spektren
des Nal, KI, und Mgll
Zeitschrift fur Astrophysik v. 22, 1943,
.
157
3.
-
,
163.
Cooper J.W. and J.B. Martin. Electron photodetachment from ions
and elastic collision cross sections for 0, C, CI, and F.
Physical
Review v. 126, 15 May 1962: 1482 - 1488.
,
4.
Dalgarno, A. and D. Parkinson.
Polarizability of atoms from boron
to neon.
Proceedings of the Royal Society of London v. A250, 1959:
,
422.
5.
Garrett, W.R. and R.A. Mann. Model of the scattering of slow
electrons by cesium atoms. Physical Review v. 130, 15 April 1963:
658 - 660.
,
6.
Garrett, W.R. and R.A. Mann.
Elastic scattering of slow electrons
Physical Review v. 135, 3 August 1964:
from alkali atoms.
A580 - A581.
,
7.
Low-energy elastic
Hammerling, P., W .W Shine, and B. Kivel.
scattering of electrons by oxygen and nitrogen.
Journal of
Applied Physics v. 28, July 1957: 760 - 764.
.
,
8.
Self-consistent field,
Hartree, D.R., W. Hartree, and B. Swirles.
including exchange and superposition of configurations, with some
Philosophical Transactions of the Royal Society
results for oxygen.
of London v. A238, 1939: 229 - 246.
,
9.
and K.A. Brueckner Interaction of slow electrons with
Klein, M.M.
Physical Review v. Ill,
atomic oxygen and atomic nitrogen.
15 August 1958: 1115 - 1120.
,
,
,
10.
Lin, S.C. and B. Kivel.
Physical Review
11.
,
v.
114,
Slow electron scattering by atomic oxygen.
15 May 1959: 1026 - 1027.
LowNeynaber, R.H., L.L. Marino, E.W. Rothe, and S.M. Trujillo.
Physical Review
energy electron scattering from atomic oxygen.
v. 123, 1 July 1961: 148 - 152.
,
12.
Robinson, L.B. Elastic scattering of low-energy electrons by atomic
Physical Review v. 105, 1 February
nitrogen and atomic oxygen.
,
1957:
922
-
927.
24-
McGraw Hill, 1955.
13.
Schiff, L.I. Quantum Mechanics
14.
Scott, W.T. The Physics of Electricity and Magnetism
and Sons, 1959.
15.
Low-energy electron
Sunshine, G., B.B. Aubrey, and B. Bederson.
scattering by atomic oxygen. Proceedings of the Third International
Conference on the Physics of Electronic and Atomic Collisions 1965
.
.
John Wiley
,
130
16.
-
133.
Temkin, A. Polarization and exchange effects in the scattering of
Physical Review
electrons from atoms with application to oxygen.
v. 107, 15 August 1957: 1004 - 1012.
-25-
,
(
b/"[
jo srj-mn) £JT SU3 Q a8.iBti3 "[BfpB^
26-
1
300
1
J
250
200
•
150
1
100
_i
CM
0)
>4-l
o
CO
4-1
•r-l
c
3
S 50
V
\
s.
v
•
r
v
4
S"~
(units of a
o
)
b
-»
BEHAVIOR OF THE CORE POTENTIAL, V„(r)
H
Figure 2
-27-
|H
A
.8
•
1
•
.6
•
J,
o
w
i
1
:
fed
U &
C
3
\
'
\
1\
f
,
i
i.
z
r
(units of a
L^~™
-i
)
POLARIZATION POTENTIAL WITH q =
Figure 3
-28j-
5
A
.8
[j
.6
i
\
CM
!
!
1
CO
•1-1
C
3
1
.2
>
T
z>
!
r (units of a
)
'
i
-*
POLARIZATION POTENTIAL WITH q -
Figure 4
-29-
8
".-
«_
v go S3T un ) uof303s sso.13 1*301
(
6
30-
» Vffi''
&3&
lift
a
I:
i
H
m
m
to
z
o
H
O
l-l
CO
§
«_
C
%
jo s^iun) UOT3D3S sscio "[«30i
-31-
vO
4)
H
00
*o r
50
O
<N
.Hammerling,. et al
*4-l
o
CO
40
Klein and Brueckner
Energy (units of e /a Q )
THEORETICAL CROSS SECTIONS
Figure 7
-32-
4<r
O
<M
«d
4-1
O
30
Sunshine, et al
o
CO
4-1
•H
d
3
£
O
2C
a
o
CO
CO
o
u
i
Neynaber, et al
O
cd
o 10
X
— Lin & Kivel
rr
TT
Energy (units of e /a
EXPERIMENTAL CROSS SECTIONS
Figure 8
-33-
7*T
)
^
TABLE
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
1
ALPHA
DELTAO
DELTA1
DELTA2
DELTA3
9770
8872
8269
7813
7440
7122
6843
6594
6371
6167
1.5979
.5018
.5104
.5338
.5561
.5759
.5935
.6091
.6230
.6356
.6471
.6575
.0000
.0055
.0117
.0188
.0267
.0354
.0448
.0550
.0658
.0771
.0888
.0000
.0017
.0035
.0053
.0072
.0091
.0112
.0132
.0154
.0177
.0200
DELTAO
DELTA1
DELTA2
DELFA3
9709
8719
8078
7598
7209
6878
6591
6335
6105
5897
5706
.5006
.5274
.5611
.5889
.6121
.6318
.6488
.6638
.6771
.6890
.6998
.0000
.0055
.0117
.0188
.0266
.0353
.0447
.0547
.0653
.0765
.0880
.0000
.0017
.0035
.0053
.0072
.0091
.0112
.0133
.0154
.0177
.0201
-
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
5.5
SIGMA
6678
7451
4121
5178
7739
5334
0109
3357
24.5858
24.8064
6
15
19
21
22
23
24
24
25.0224
SIGMA
10.4912
22.2733
27.5908
30.0143
30.9885
32.2318
31.1230
30.8598
30.5507
30.2516
29.9895
8
DELTAO
DELTA1
DELTA2
DELTA3
9644
8554
7871
7366
6960
6617
6320
6057
5822
5610
5416
.5028
.5582
.6009
.6318
.6562
.6762
.6934
.7083
.7214
.7332
.7438
.0000
.0055
.0117
.0187
.0265
.0351
.0444
.0543
.0648
.0757
.0869
.0000
.0017
.0035
.0053
.0072
.0091
.0112
.0133
.0154
.0177
.0201
SIGMA
15 ,9588
36, ,8699
42 ,9553
43 ,8549
43, ,0616
41, .7041
40, ,2216
38, ,7979
37, ,5062
36, ,3703
35, 3891
ALL PHASE SHITT ANQbES AP6 QIV6N in MybTiPbes er pi RADIANS
-34-
TABLE
2
ALPHA
=
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
=
6.0
5
DELI A0
DELTA1
DELTA2
DELTA3
9825
8977
8388
7939
7569
7253
6976
6729
6505
6301
6114
.5023
.5040
.5234
.5434
.5619
.5786
.5937
.6074
.6198
.6312
.6416
.0000
.0060
.0129
.0207
.0294
.0390
.0496
.0609
.0729
.0856
.0987
.0000
.0019
.0038
.0058
.0079
.0100
.0122
.0145
.0184
.0193
.0218
DELTAO
DELTA1
DELTA2
DELTA3
9756
8810
8181
7707
7320
6992
9670
1.6450
1.6220
1.6012
1.5820
.5014
.5186
.5483
.5745
.5970
.6165
.6336
.6487
.6622
.6744
.6854
.0000
0060
.0128
.0206
.0293
.0389
.0493
.0605
.0724
.0848
.0976
.0000
.0019
.0038
.0058
.0079
.0100
.0122
.0145
.0168
.0193
.0219
DELTAO
DELTA1
DELTA2
DELTA3
9685
8632
7959
7458
7054
6712
6415
6152
1.5917
1.5703
1.5509
.5006
.5426
.5830
.6140
.6390
.6599
.6778
.6934
.7072
.7196
.7307
.0000
.0060
.0128
.0206
.0292
.0387
.0490
.0600
.0717
.0838
.0963
.0000
.0019
.0038
.0058
.0079
.0100
.0122
.0145
.0168
.0193
.0219
SIGMA
4, ,0100
12, ,8560
16, ,3795
18, ,5432
20, ,0096
21, ,0657
21, ,8826
22, ,5658
23, ,1803
23 ,7635
24, ,3341
SIGMA
•
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
.
7.4262
18.2930
23.2639
25.9655
27.4278
28.2089
28.6273
28.8643
29.0231
29.1604
29.3036
SIGMA
12, ,2868
28, ,7581
35, ,5747
37, ,9080
38, ,3704
38, ,0360
37, ,3904
36, ,6634
35, ,9643
35, ,3414
34, .8102
ALL PHASE SHIFT ANGLES ARE GIVEN IN MULTIPLES OF PI RADIANS
-35-
TABLE
.0050
.0500
.1000
.1500
.2000
.2500
.3000
.3500
.4000
.4500
.5000
3
ALPHA
=
0.0
DELT-AO
DELTA1
DELTA2
DELTA3
1.9154
1.7502
6672
6121
5705
5370
5091
4850
4638
4450
4280
.5004
.5157
.5493
.6000
.6664
.7415
.8139
.8750
.9223
.9575
.9836
.0000
.0000
.0004
.0011
.0020
.0032
.0047
.0064
.0084
.0105
.0128
.0000
.0000
.0000
.0000
.0000
.0002
.0004
.0005
.0008
.0011
.0014
SIGMA
86.6421
63.6804
51.5073
48.9037
53.3931
60.4892
64.6529
63.9188
59.9304
54.7727
49.6356
ALL PHASE SHIFT ANGLES ARE GIVEN IN MULTIPLES OF PI RADIANS
1
-36-
INITIAL DISTRIBUTION LIST
No. Copies
Defense Documentation Center
Cameron Station
Alexandria, Virginia 22314
20
Library
U.S. Naval Postgraduate School, Monterey, California
2
Prof. William B. Zeleny
Department of Physics
US Naval Postgraduate School, Monterey, California
1
LT William L. Johnson
Dept of Physics
US Naval Postgraduate School, Monterey, California
1
.
37-
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26 CROUP
REPORT TITLE
3.
Elastic Scattering of Low-Energy Electrons from Oxygen
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ABSTRACT
The cross sections for the elastic scattering of electrons from atomic
and 13.6 electron volts.
oxygen are calculated in the energy range between
Polarization of the atom is included by using a term in the potential function
which has the correct asymptotic behavior and which vanishes at the origin.
Previous theoretical and experimental results are discussed.
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ROLE
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Scattering
Atomic Scattering
Cross section
Electron scattering
Oxygen
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