Born Cross Sections for the Excitation of Helium by Fast Protons

Abstract

Cross sections for excitation of helium by proton impact have been calculated in the Born approximation. Transitions from the ground state to the n1P(n = 2 and 3) states have been considered. Highly accurate wave functions of the Hartree–Fock and configuration-interaction type have been used to represent the ground state. Approximate wave functions due to Messmer have been employed for the final states. The results are compared with other calculations and with experiments.

Full text

Born Cross Sections for the Excitation of Helium
by
Fast Protons
S. P. OJHA, P. TIWARI,
AND
D.
K.
RAI
Department
of
Physics, Banaras Hindu Utricersity, Varanasi
5,
Indicl
Received February 14, 1972'
Cross sections for excitation of helium by proton impact have been calculated in the Born approximation.
Transitions from the ground state to the
nlP(n
=
2 and 3) states have been considered. Highly accurate wave
functions of the Hartree-Fock and configuration-interaction type have been used to represent the ground
state. Approximate wave functions due to Messmer have been employed for the final states. The results are
compared with other calculations and with experiments.
Les sections efficdces pour l'excitation de I'helium par impact protonique ont ete calculees I'approximation
de Born. Les transitions du niveau fondamental aux etats
tilP
(11
=
2
et 3) ont ete considerees. Des fonctions
d'onde tres exactes, du type Hartree-Fock et du type "interaction de configurations", ont ete utilisees pour
representer I'etat fondamental. Les fonctions d'onde approximatives de Messmer ont kte
employees
pour Ics
etats finals. Les resultats sont compares
i
d'autres calculs et aux donnees experimentales.
[Traduit par le journal]
Canadian Journal of
Physics,
50,
2253
(1972)
1.
Introduction
Proton impact excitation of helium has been
the subject of extensive studies in recent years.
Thomas and Bent
(1967)
and Thomas
(1967)
have reported new experimental measurements
of the cross sections for excitation of neutral
helium from its ground state to various excited
states. Van den Bos
(1969)
has used the impact
parameter method to calculate the cross sections
for excitation of helium by protons having
energy in the range
10-10 000
KeV. Bell
et
01.
(1968)
have also calculated the cross sections
for proton excitation of helium from its ground
state to
nlP(n
=
2-6)
and
nlD(n
=
3-6)
states.
Significant differences are observed between the
results of various workers. For instance, the
data of Van den Bos
et
01.
(1968)
and that of
Thomas and Bent
(1967)
do not agree, and the
two reported values of the cross sections differ
by almost a factor of
2
for a proton energy of
0.15
MeV.
The.
major source
of
unreliability in the com-
puted cross sections is the use of approximate
wave functions as the cross sections are very
sensitive to errors in the wave functions for the
combining states, specially for the ground state.
It is therefore of interest to compute the excita-
tion cross sections with better wave functions,
specially the ground state. Wave functions of the
configuration-interaction type (Joachain and
Vanderpoorten
1970)
are very suitable for this
'Revision received May 23, 1972.
type of work as they not only include a major
part of the correlation but are also relatively
simple as far as computation is concerned.
In the present paper, cross sections for
excitation of atomic helium from its ground
state to the
2lP
and
3lP
excited states due to
impact of fast protons have been calculated in
the first Born approximation. Since the present
computation has been performed only for high
energy incident protons no account has been
taken of charge transfer, polarization or distor-
tion.
2.
Theory
At high impact energy, the cross section for
proton impact excitation from the ground state
to any excited state is given in the first Born
approximation by the expression
where
Eo
and
El,
are the initial and final state
energies of the target atom,
E
is the energy of the
incident nucleus having charge
Z
and mass
M,,
and
K
is the change in momentum of the
scattered nucleus. The limiting values of the
momentum transfer
Kmin
=
ko
-
k,
and
Kmax
=
ko
+
k1,
where
ko((koJ
=
M,(2E/M,)li2)
and
k,,
are the
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2254
CANADIAN
JOURNAL
OF
PHYSICS.
VOL.
50,
1972
momenta of the incident and outgoing nucleus
respectively, are determined by the principle
of conservation of energy, i.e.
Here,
M,
is the reduced mass of the colliding
system.
The generalized oscillator strength
jb,,
is
defined as
3.
Choice of Wave Functions
A.
Excited State Wave Function
The wave function employed in the present
calculations to remesent the helium excited state
is the one given by Messmer2 (1969). It is written
in the following form
:
4,
and
4,
orbitals are represented as a linear
combination of four basis functions in the
following way:
where
4
(
2
~o,,(K)
=
y,,(rl, r2)1
1
eiK.'j l~~(r~, r2))
4s
=
1
cs,"xs,"
n=l
j=
1
5
YO(rl, r2) and Y,,(rl, r,) are the wave functions
4p
=
C
Cp.nxp."
for the ground and excited state of helium
n=2
respectively. The basil; functions
x,,,
and
x,,,
are
The parameters are taken from the paper due to Messmer (1969).
B.
Ground State Wave Functions
To describe the helium ground state we have used the following wave functions:
(i) An analytical fit to a Hartree-Fock (HF) (Roothaan, Sachs, and Weiss 1939) wave function
using a sum of two exponentials. In this form the wave function is written as
For the parameters we have taken the values given by Byron and Joachain (1966), viz.
a
=
1.41,
p
=
2.61,
q
=
0.799, and NlS
=
1.302525. Results employing this function will be denoted by HF
(BJ).
(ii) A configuration-interaction wave function (Joachain and Vanderpoorten 1970) which takes
account of angular correlation between the two electrons is written as
where
-
-
."
-*
,
-.
F,(,.,
,
,.,I
=
1
~;;f;,.;~;(,.;ll,.;
+
,.n,..l)
e-P(rl
+ rz)/2
12
Both
m
and n take values from 0 to 5 subject to employing this wave function will be labelled
the condition
m
+
n
<
6. Thus there are 15 by JV.
terms in each partial wave. Taking three partial
4.
Results and Discussions
waves
and
putting
1,'
=
3.7, the ground state The integrals in the expressions for the
energy
is
given by
E~
=
-2.9020
a.u.
generalized oscillator strengths could be evalu-
'=his corrected
form
of
the
wave
function
was
communi-
ated analytically. The wave functions described
cated to us in a personal communication.
in the earlier section and the corresponding
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OJHA
ET
AL.:
BORN
CROSS SECTIONS FOR
THE
EXCITATION
OF
HELIUM
2255
FIG.
1.
Crosssections for proton excitation ofthe ground
state of helium to the
2'P
state. The present work: HF (BJ)
result
(---)
and JV result
(-).
Bell
er
al.
(1968) (A);
Van
den Bos
(1969)
(0);
Vriens and Carriere
(1970)
(a);
and
Bell
(1961)
(-
x
-1.
theoretical eigenenergies have been used to
calculate cross sections.
Figure
1
compares our HF (BJ) and JV results
with other theoretical values for
2'P
excitation.
As can be seen from the figure, our results are
in excellent agreement with the data reported by
Van den Bos (1969). The results obtained by
Bell (1961) using the distorted wave approxi-
mation are also shown in Fig.
1.
In Fig.
2
we have compared our results with
different theoretical and experimental data.
Vriens and Carriere (1970) have calculated
proton impact excitation cross sections using
the generalized oscillator strengths reported by
Kim and Inokuti (1968). These values are
supposed to be very accurate and it is indeed
satisfying that our JV results almost coincide
with these accurate values. Our results are also
in excellent agreement with the results of Van
den Bos (1969) and Bell
el
al.
(1968). Also, a
comparison of our HF (BJ) results with the
accurate results due to Vriens and Carriere
(1970) shows that the use of a correlated ground
-,
-.-
state wave function i'sB great improvement. It
is observed that the effect of correlation is more
pronounced at low proton energies.
As regards the validity of the first Born
approximation it is seen that, even at very high
energy, theoretical and experimental data do
not agree too well. These theoretical estimates
are almost the best first Born results and the
above disagreement is indicative of the failure
of the first Born approximation. Bell (1961)
using the distorted wave approximation has
FIG.
2.
Cross sections for proton excitation of theground
state of helium to the
3'P
state. The present work: HF (BJ)
result
(---)
and JV result
(-).
Bell
er
al.
(1968)
(0);
Vriens
and Carriere
(1970)
(a);
and Bell
(1961)
(-
x-).
Experi-
mental cross section: Thomas and Bent
(1967) (A)
and
Van den Bos
er
al.
(1968)
(0).
concluded that the Born approximation is
reliable only for proton energies above 400
KeV. Experimental results due to Thomas and
Bent (1967) and due to Van den Bos
el
al.
(1968) are also presented for comparison in
Fig.
2.
The shape of the experimental cross
section curve is well represented by the first
Born results and the former may be suitably
normalized to bring them into better agreement
with the latter. This normalization would lead
to very good overall agreement over a wide
range of proton energies.
Acknowledgments
We wish to thank Mr. K. P. Srivastava for
computer programming. One of us (S.P.O.) is
thankful to the C.S.I.R., New Delhi, for
financial assistance.
BELL, R. J.
1961.
Proc. Phys. Soc.
78,903.
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L.,
KENNEDY,
D.
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1968.
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BYRON, F.
W.
and JOACHAIN, C. J.
1966.
Phys. Rev.
146,
1.
JOACHAIN, C. J. and VANDERPOORTEN, R.
1970.
Physica,
46,
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