Variational approach to calculation of charge exchange cross sections for adiabatic collisio

Alzofon, F. E.

Witteborn, F. C.

English

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NASA TN D-1654
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TECHNICAL
D-1654
NOTE
VARIATIONAL APPROACH TO THE CALCULATION OF CHARGE
EXCHANGE CROSS SECTIONS FOR
ADIABATIC COLLISIONS
By Frederick E. Alzofon and Fred C. Witteborn
Ames Research Center
Moffett Field, Calif.
NATIONAL AERONAUTICS
WASHINGTON
AND SPACE ADMINISTRATION
May 1963
https://ntrs.nasa.gov/search.jsp?R=19630006515 2020-03-24T05:58:23+00:00Z

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
TECHNICAL NOTE D-1654
VARIATIONAL APPROACH TO THE CALCULATION OF CHARGE
EXCHANGE CROSS SECTIONS FOR
ADIABATIC COLLISIONS
By Frederick E. Alzofon and Fred C. Witteborn
SUMMARY
Cross sections for the reaction H+ + H(is) _ H(is) + H+ are calculated
for incident particle energies between i0 ev and 2250 ev using an impact
parameter method. The required electronic energy levels were found by varying
one parameter in the molecular ion wave function at every value of proton
separation. The resulting cross sections are compared with those obtained
using more accurate wave functions and with experimental data.
I}YfRODUCTION
Interest in charge exchange cross sections has increased in recent years
because of their usefulness in explaining the presence of certain ions in the
upper atmosphere and because of their application to :ion beam techniques.
Methods for calculating cross sections were described by Bates, Y_ssey, and
Stewart (ref. i). Their "perturbed stationary state" (PSS) method was used by
Dalgarno and Yadav (ref. 2) to calculate cross sections for the resonant reaction
H+ + H(is) _ H(is) + H + at energies too low for use of the Born approximation.
Exact molecular ion wave functions were used in the calculation. For analogous
reactions among larger atoms, exact wave functions are not available. Approxi-
mate wave functions with three variable parameters were used by Moiseiwitsch
(ref. 3) to calculate cross sections for the reaction He + + He _ He + He + ,
but the parameters were optimized only for infinite separation of the nuclei.
In order to examine the errors introduced by the use of approximate wave
functions, calculations of cross sections for the reaction H++ H(is) _ H(is) + H+
were performed at the Ames Research Center by means of the impact parameter
method of reference i, using hydrogen-like wave functions with a variable param-
eter. In this method as in the PSS method, the adiabatic approximation is used;
that is, it is assumed that the nuclei move much more slowly than the electron,
so that the latter can be described by stationary state wave functions which
must be determined for each internuclear distance. For a hydrogen molecular ion
only one parameter is needed to obtain a fairly good approximation to the wave
function (ref. 4). An algebraic expression for this parameter in terms of the
particle separation has been obtained by using the variational principle, and
the resulting energy eigenvalues were used in conjunction with the equations of
reference i to obtain cross sections for the reaction H+ + H(is) _ H(is) + H+
for incident particle energies between i0 ev and 2250 ev. The calculations were
programmedfor the IBM709 computer by Miss Marilyn A. Myers.
SYMBOLS
a
b
e
e_
E
Eo
H
i
m
Q
ra
rb
R
Rc( )
t
v(R)
X+
X-
proton moving in the laboratory frame of reference
proton stationary in the laboratory frame of reference
base of natural logarithms
charge on an electron
initial kinetic energy of proton a
ground state energy of the electron in a hydrogen atom
Planck's constant divided by 2_
Hamilt onian operator
mass of the electron
charge exchange cross section
distance from electron to proton a
distance from electron to proton b
distance between protons a and b
distance of closest approach of protons a and b
time
relative velocity of protons
relative velocity of protons
symmetric wave function
antisymmetric wave function
a and b
a and b at infinite separation
c+
c-
P
Ii
variational parameter in equation (4)
energy eigenvalue of the symmetric state
energy eigenvalue of the antisymmetric state
phase shift defined in equation (3)
distance of closest approach of protons a ancL b
paths are undeviated in the encounter
wave function of the electron
assuming that their
ANALYSIS AND DISCUSSION
Consider a proton a approaching a second proton b at constant speed,
initially directed along a straight line whose closest distance to proton b is
p as shown in the sketch. An electron e_ is originally in the is state of
system a. Its distance from proton a is ra, from proton b, rb. Since
Q_
protons a and b are identical, the electronic wave function must be symmetric
or antisymmetric with respect to interchange of the two protons. Let X+ be an
appropriate symmetric wave function, X- an antisymmetric wave function, and let
the initial wave function of the system be (I/H) (X+ + X-)exp[-(i/_)Eot]. After
the collision the electronic wave function is (ref. i):
= _i--- exp _ (e+ - Eo)dt + X-exp -_ (e- - Eo)dt exp -i Eo t
(i)
in the approximation that there are no transitions to higher states, where Eo
is the ground state energy of the electron at infinite separation of the protons,
c+ is the ground state energy of the symmetric state, ¢- is the ground state
energy of the antisymmetric state. The cross section for charge exchange is
(ref. i)
3
Q = p sin (2)
where
v2(X)R2J
(3)
where v(R) is the relative velocity of the two protons, and Rc(p) is the
distance of closest approach. Moiseiwitsch (ref. 3) used the further approxi-
mations v(R) = v(_) and Rc(P) = p which we shall use for the case of hydrogen.
The symmetric and antisymmetric spatial wave functions used in our
calculations are, respectively,
-_rb) and X- 1 (e-_ra -_rb) (4)X + = ___i{/e-_ra + e - - e
where _ is the variational parameter. Inserting these into the expression for
the initial wave function shows that the electron was assumed to be originally
in a is state near proton a. The energy eigenva!ues for the symmetric and
antisymmetric states are, respectively,
c+ = /x+mx+ and c- - /X' X aT (5)
/X+X + dT /X X dT
where
_i 2 e_ 2 e_ 2 e_ 2
H- V 2 +--
2m ra rb R
is the Hamiltonian operator for the electron in a hydrogen molecular ion, m is
the electron mass, e_ is the electron charge, dT is the increment of volume,
and the integrations are carried out over all space. Evaluation of the integrals
is simple except where ra and rb appear together in the integrand. In the
latter case it is convenient to use confocal elliptic coordinates (ref. 4). The
integrals of equation (5) are separately minimized with respect to _ to deter-
mine c+ and c- in terms of R. The integrations for _(p) were carried out
on an IBM 709 computer for many values of p. These values were then used in
the integration for Q at several values of v(_).
A plot of the energy eigenvalues of the symmetric and antisymmetric states
is shown in figure i. Plots of the integrand of equation (2) appear in
figures 2 and 3- The error in assuming Rc(O) = P in equation (3) may be
estimated by noting that the c+ curve determines the classical limit of prox-
imity of the protons. Contributions to Q for smaller values of P should be
omitted. However, figures 2 and 3 show that these are very small anyway. In
fact, the contribution is only i percent at i0 ev and gets smaller for higher
4
energies. At higher energies the effect of assuming v(R) constant is mainly
to ignore the slowing down of the protons when they are very close (p < i).
Figure 3 shows that for small p the resulting change in _(p) would make little
difference, since the integrand is very small. This is not the case at low
energies where the velocities are changing by a significant fraction at several
Bohr radii. Thus the calculation is not made below i0 ev. Ferguson (ref. 5)
finds that consideration of momentum transfer of the electron (neglected in our
approximation) reduces the cross section by 5 percent at 2000 ev and by larger
amounts at higher energies, so our calculation was not extended much above
2000 ev.
The resonant charge exchange cross sections obtained using the variational
wave functions are shown in the following table at various incident particle
energies E:
(ev)
i0
4o
i00
160
2D0
i000
2250
v(_)(lO6cm pe r second)
4.38
8.76
13.84
17._l
21.9
43.8
6> .7
Q
38 ._
30.4
2D .6
23.3
21.3
l> .3
12.2
They are compared with those of reference 2 and with the experimental results of
Stebbings, Smith, and Fite (ref. 6) in figure 4. The good agreement between
these results and those obtained by more exact calculations suggests that the
variational method may be applied with satisfactory rcsults to more complicated
atoms.
Ames Research Center
National Aeronautics and Space Administration
Moffett Field, Calif._ March 4, 1963
REFERENCES
Io
2o
•
4•
•
_6.
Bates, D. R., Massey, H. S. W., and Stewart, A. L.: Inelastic Collisions
Between Atoms. I. General Theoretical Considerations. Proc. Roy.
Soc. A216, 1953, P. 437.
Dalgarno, A., and Yadav, H. N.: Electron Capture II: Resonance Capture
From Hydrogen Atoms by Slow Protons. Proc. Fnys. Soc. A66, 1953, P. 173.
Moiseiwitsch, B. L.: Interaction Energy and Charge Exchange Between Helium
Atoms and lons. Proc. Phys. Soc. A69, 1956, p. 653.
Pauling, L., and Wilson, E. B., Jr.: Introduction to Quantum Mechanics.
McGraw-Hill Book Company, Inc., New York and London, 1935.
Ferguson, A. F.: Resonance Charge Transfer Between H(is) and H+
Calculated by Means of an Approximation Based on an Expansion in Molecular
Eigenfunctions. Proc. Roy. Soc. A264, 1961, p. 540.
Stebbings, R. F., Smith, A. C. H., and Fite, W. L.: Charge Transfer in
Collisions Involving Symmetric and Asymmetric Resonance. Preprint GA-2783,
General Atomic, Div. of General Dynamics, Jan. 1962.
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VARIATIONAL APPROACH TO THE CALCULATION OF CHARGE EXCHANGE CROSS SECTIONS FOR ADIABATIC COLLISIONS

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