The L=0 and 1 partial wave amplitudes obtained by a two-state coupled static approximation with correlation with the L greater than or equal to 2 Born amplitudes were combined to obtain the differential cross section for positronium formation in electron-atomic hydrogen collisions. For positron energies of 0.64 and 0.75 ryd, minima at the scattering angles of 57 deg and 51 deg are found. Total cross sections for positronium formation for low and intermediate impact energies are given. Measurement of the differential cross section for the process positron + helium yields positronium + helium ion for the detection of possible minima is suggested

Drachman, R. J.

Mcguire, J. H.

Omidvar, K.

English

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PREPR I NT
NASA TU X- M/ J'Z
DIFFERENTIAL CROSS-SECTION FOR
POSITRONIUM FORMATION IN
ELECTRON-ATOMIC HYDROGEN
COLLISIONS
(NASA-TM-X-71082)
	 DIFFERENTIAL	 N76-19923
CROSS-SECTION FOR POSITRONIUM FORHATICN IN
ELECTRON-ATOMIC HYDROGFN COLLISIONS (NASA)
15 p HC $3.50	 CSCL 20H	 Unclas
63/73 22159
R. J. DRACHMAN
K. OMIDVAR
J. H. McGUIRE
MARCH 1976 NA Sit FA001INPUT bRANCH
1
---^ GODDARD SPACE FLIGHT. CENTER
GREENBELT, MARYLAND
1
p
0	 t
c
DIFFERENTIAL CROSS-SECTION FOR POSITRONIUM FORMATION
IN ELECTRON-ATOMIC HYDROGEN COLLISIONS
Richard J. Drachman
Theoretical Studies Grou,)
Goddard Space Flight Center
Greenbelt, Maryland 20771
Kazem Omidvar
Theoretical Studies Group
Goddard Space Flight Center
Greenbelt, Maryland 20771
J. H. McGuire
Department of Physics
Kansas State University
Manhattan, Kansas 66505
We have combined the L=o and 1 partial wave amplitudes obtained
by a two-state coupled static approximation with correlation with the
L > 2 Born amplitudes to obtain the differential cross section for
positronium formation in electron-atomic hydrogen collisions. For
positron energies of 0.64 and 0.75 ryd minima at the scattirinn
angles of 57 0 and 51 0
 are found. Total cross sections for positronium
formation for low and intermediate impact energies are given. Measure-
ment of the differential cross section for the process a ++ He a Ps + He+
for the detection of possible minima is suggested.
1
0
c
I. INTRODUCTION
We consider the formation of a positronium atom in its ground
state due to collision of a positron and a hydrogen atom. Aside
from purely theoretical interest, the formation cross section is of
interest in a number of astrophysical problems. A case to be mentioned
is the formation and annihilation of positronium in the sun following
an energetic solar flare. The emitted gamma rays are expected to pro-
vide sensitive probes of conditions in the annihilation region.l
The f{	 calculation of the positronium formation cross section
was done by Massey and Mohr, 2 using the Born approximation. This
calculation was repeated later, and errors in the numerical values of
the cross section were corrected. 3 Chen and Kr,amer4	have calculated
the differential cross section for positronium formation in both the
Born and the Faddeev-Watson multi ple-scattering approximations. Similarly,
Sural and Barman 5 have performed an approximate form of the second
Born approximation calculation.
For low impact energies a number of elaborate calculations have been
performed by several authors. 6,7 ' 8
 These include both the hydrogen and
positronium ground states, as in the coupled-static approximation, while
representing polarization and distortion either by the addition of
correlation terms or by the inclusion of effective potentials.
To obtain a reliable low energy dif ferential cross section, partial
wave amplitudes for all values of the angular momentum quantum number L
are needed. In the absence of any better calculation for the differential
2
O
f
(,
cross section we have combined the seemingly Lccurate L=o and 1
partial wave amplitudes of Refs. 6 and 7, which are based on two-state
coupled static approximation with correlation, with the L > 2 Born
amplitudes to obtain the differential cross section. This cross section
for a number of low impact energies will be presented. In addition,
the total gross section for low and intermediate impact energies will
be given.
The first Born approximation meets with some difficulties for the
similar process of proton-atomic hydrogen electron transfer. It has
been shown that in the range of relatively high impact energies of a
few MeV, th.e second Born terms are comparable in magnitude to the
first Born terms. g As the energy increases the contribution to the
cross section from the second order terms dominates the cross section.
For impact energies below MeV region the contributions of the second and
higher order Born terms are negligible.
The (p,H) process has two characteristics which are absent in the
(e ,H) process: (1) The (p,H) process is a resonance charge exchange
collision, with zero energy transfer. Due to this effect the collision
amplitude has a different analytic form compared to the non-resonance
charge-exchange amplitude. (2) In the (p,H) process due to their
heavy masses the motion of the projectile and the nucleus can be
treated classically. It then can be shown that in the limit of high
impact energies the role of the projectile-nucleus interaction is to
introduce only a phase, which depends on the relative velocity of the two
particles,in the total wave function. This leads to the fact that this
interaction has a negligible effect on the exact transition probability. 10
3
't
1
IO
Due to these differences it is difficult to draw any conclusion for
the (e+ , H) process. The second Born calculation of Ref. 5 does not show
the domination of the second Born terms, although the calculation is
approximate, and it is not extended to high enough incident energies.
In this paper we have concerned ourselves with the low energy positro-
ilium formation cross section. We have made the assumption that as L in-
creases, the partial wove first Born amplitudes approach the true values.
A test for t"e validity of this assumption is that as the more accurate
partial wave amplitudes become available in the future, they should converge
to the first Born amplitudes.
II.	 _TTHOD OF CALCULATION AND RESULTS
Using the R-matrix of Chan and Fraser 6 for L=o and the R-matrix of Chan
and McEachran 7 for L=1, the corresponding T-matrix can be calculated. Similar-
ly, a partial wave expansion is made of the total T-matrix according to the
Born approximation given analytically by Omidvar and Puget. 11 The T-matrices
for different L are then combined to obtain the differential cross section.
The Justification for using the Born approximation for L > 2 is that in
going from L=0 to L= 1 the discrepancy between the Born approximation and the
more accurate approximation of Refs. 6 and 7 is substantially reduced. In
going from L=1 to L=2 it is hoped that a similar reduction will take place.
More explicitly, for an incident energy of 0.64 ryd, for which a calcula-
tion of the differential cross-section has be y.., carried out here, the T O value
for the matrix element connecting the incident channel to the Ps-formation
channel according to the 2-channel, 26-correlation term calculation of Ref. 5
is 0.042 - 0.017i, while this value according to the Born approximation is
4
f
i
i
0
(
1
f
0.851, fifty times larger than the imagi.ary part of the more accurate
calculation. Fo L al and the same incident energy the corresponding value
according to Ref. 7, which is obtained as in Ref. 6 but with 56 correlation
terms, is -0.00613 -0.3231, while the Born approximation value is -0.5351.
The ratio of the imaginary parts in the two calculations has decreased to 1.7.
Furthermore, it should be noted that while the magnitudes of the real and
imaginary parts for L =0 in the more accurate calculation are comparable,
for L=1 the real part is two orders of magnitude smaller than the imaginary
part, indicating the increased accuracy of the Born approximation.
These considerations suggest that for L > 2 the Born approximation
values which are purely imaginary, may not be too far from the true values.
In Fig. 1 we show the differentia" cross-section for an incident energy
of 0.64 ryd. The more accurate cross section represented by the solid line
shows a deep and narrow minimum at 570 . The vane of the cross section drops
at this minimum by two orders of magnitudes. The width of the minimum where
the cross section has drop ped by one order of magnitude is approximately 70.
Angles from zero up to' the minimum angle contribute 80% to the total cross-section.
In Fig. 2 the similar differential cross section for 0.75 ryd incident
energy is shown. The minimum in the more accurate solid line occurs at 510
and is shallower then that in Fig. 1. However, a second broad minimum occurs
here at 1470
As the incident energy increases, the angle at which the minimum occurs
becomes smaller, and the percentage of the cross-section arising from angles
smaller than the minimum angle increases.
In Fig. 3 the differential cross-section for an incident energy of 20
ryd is shown. Here we have used the Born approximation to calculate all
5
o	,,
	
I
c
partial wave amplitudes. This is ,justified by noticing that the partial
cross-sections due to L-0 and L=1 at this energy in the Born approximation
are only 1.46% and 1.47% of the tote 	 iss-section. The main contribution
comes from the higher partial waves, 	 the Born approximation is apparently
reliable. In this case, instead of a minimum in the differential cross-section
a zero occurs. This is due to the fact that the Born amplitude is pure
imaginary, and the amplitudes due to the attractive and repulsive parts V `1e
potential are equal but opposite in sign. The zero in Fig. 3 occurs at at
angle of 23.30 , and 90.9% of the total cross-section comes from angles smaller
than the minimum angle. The results given in this figure are consistent with
the results given in Ref. 4.
In order to see how the cross-section peaks in the forward direction for
different incident energies, we consider the ratio 4w Cdv/d p le=o//o, which for
an isotropic scattering is equal to unity. do/dQ and o are the differential
cross section per unit solid angle and the total cross section. For energies
of 0.64, 20, and 10 3 ryL' this ratio is 16.0, 171 and 197, respectively.
Similar to the calculation of the differential cross section, we have
computec the total cross section for a number of impact energies b y combining
the more accurate L=O, 1 partial wave cross sections of Refs. 6 and 7 with the
Born cross section for higher waves. The results are shown in Table I.
By studying the cross sections for L=0 and 1 given in Rafs. 6 and 7
a prescription for approximate computation of the total cross section for
energies not given in Refs. 6 and 7 can be found. This study shows that at
all impact energies except the lowest one,the L=0 cross section is less
than 1% of the L =l cross section. With acceptable accuracy we then can
neglect the L= 0 contribution. Furthermore, by taking the ratio of the
cross section given in Ref. 7 to the Born cross section for L=1, we see
6
1_— 4
ithat this ratio for impact energies of 0.504, 0.563, 0.64, 0.723, and 0.750
ryd are 0.69, 0.48, 0.38, 9.40, and 0.42, respectively. It may not be a
Lad approximation if we take this ratio to be 0.4 for higher energies. The
following formula for the cross section then follows:
a ti OT  
_ 00 8 - 0.6 al 
	
(1)
where oT6, 006 and a16 are the total, L=O, and L=1 Born cross sections.
The total cross sections calculated using (1) are indicated by an asterisk
in Table I.
The results of Tabl p
 I up to 3 ryd energy are shown graphically in
Fig. 4.
As a test of the consistency of our results, the total cross sections
are obtained in two ways, one by summing over all partial wave cross sections,
and the other by integrating the differential cross section with respect to
the scattering angles. The discrepancy is about or less than one percent.
III. CONCLUSIONS
With the approximations presented in the text we have found minima in
the differential cross sections for o.64 and o.75 ryd impact energies.
According to the first Born a pproximation zeroes are found in the differential
cross section for all impact energies. The approximate form of the second
Born approximation of Ref. 5 shows two shallow minima at 12 eV, and no
minimum at 100 eV impact energies, while the Faddeev-Watson multiple-
scattering approximation of Ref. 4 shows no minima in the differential
cross section for 200 and 500 eV impact energies.
Zeroes or minima were found previously in the differential cross-
section for the similar (p,H) exchange process in different approximations.
The zero appears in the first Born 12 and distorted-wave approximations, 13
7
0
0
,
0while the minimum appears in the second Born approximation. 9 Since the
zeroes and minima occur at small scattering angles of the order of tenths
of a degree, it has not been possible to substantiate these findings
experimeitally. For the e+ - H system, however, the minima occur at large
scattering angles, and the difficulty of the (p,H) experiment does not arise
here, although the present status of low-energy po r 'tron beam technology
makes this experiment extremely difficult. Minima similar to those obtained
here may also occur in the differential cross-section for the process e + +
He + Ps + He although at smaller angles due to the increased nuclear
charge. A measurement of the differential cross section for this process
is easier to be realized, and the results will be an aid in discriminating
between different theories.
ACKNOWLEDGMENTS
We would like to thank H. M. Eiserike and V. A. Turner for their
programming assistances, and Y. F. Chan and R. P. McEachran for
providing us with their data before their publication.
t
j` 8
1t^
1, C. J. Crannell, R. Ramaty, and C. Werntz, 14th International Cosmic
Ray Conference, Munchen, Germany, 1975, Conference Papers, p. 1656;
C. J. Crannell, G. Joyce, R. Ramaty, and C. Werntz, submitted to
Astrophysical Journal.
2. H. S. W. Massey and C. B. 0. Mohr, Proc. Ph ys. Soc. A 67, 695 (1954).
3. I. M. Cheshire, Proc. Phys. Soc. 83, 227 (19F4).
4. J. C. Y. Chen and P. J. Kramer, Phys. Rev. A 5, 1207 (1972).
5. D. P. Surat and S. Barman, Phys. Rev. 10, 1162 (1974)
6. Y. F. Chan and P. A. Fraser, J. Phys. 8 6, 2504 (1973).
7. Y. F. Chan and R. P. McEachran, J. Phys. B, to be published.
8. B. H. Bransden and Z. Jundi, Proc. Phys. Soc. 92, 880 (1967); M: F.
Fels and M. H. Mittleman, Phys. Rev. 182, 77 (1969); Y. Hahn and
J. F. Dirks, Phys. Rev. A 3, 1513 (1971); S.E.A. Wakid and R. W.
LaBahn, Phys: Rev. A 6, 2039 (1972); J. Stein and R. Sternlicht,
Phys. Rev. A 6 9 2165 (1972).
9. P. J. Kramer, Phys. Rev. A 6. 2 1.29 (1972).
10. I. M. Cheshire, Proc. Phys. Soc. 84, 89 (1964).
11. K. Omidvar and J - L. Pugit, Astrophysical Journal, to be published.
12. J. D. Jackson and H. Schiff, Phys. Rev. 89, 359 (1953).
13. R. H. Bassel and E. Gerjuoy, Phys. Rev. 117, 749 (1960).
9
0
TPJLE I
Values of the total cross section for positronium formation as a
function of the incident energy. a is obtained by combining the L=0,1
partial wave cross sections of Refs. 6 and 7 with L > 2 Born partial
cross sections. aB is the Born cross section given for comparison.
The numbers marked by asterisks are obtained using Eq. (1).
E(ryd) a B (na 0 2 ) a(na0 1 ) E(ryd) aB(nao2) a(nao2)
0.5041 2.96-1 1.63-2 3 7.85-1 7.09 -1*
0.5625 1.87 5.45-1 4 3.51-1 3.24-1*
0.64 3.34 1.39 5 1.73-1 1.62-1*
0.7225 4.28 2.25 6 9.21-2 8,70 -2k
0.75 4.47 2.50 8 3.13-2 2.99-2*
1 4.74 3.31* 10 1.26-2 1.21 -2*
2 1.97 1.69* 20 5.90-4 5.76-4*
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FIGURE CAPTIONS
Fig. 1. Differential cross-section for positronium formation in
positron-hydrogen atom collision as a function of the scattering
angle e, and for an incident energy of 0.64 ryd. Th e, dashed
line contains the L=0 amplitude of Ref. 6, and the Born
amplitudes for higher L. The solid line contains the L =0, 1
amplitudes of Refs. 6 and 7, and the Born amplitudes for
higher L.
Fig. 2. The same as Fig. 1, but for an incident energy of 0.75 ryd.
Fig. 3. The same as Fig. 1, but for an incident energy of 20 ryd.
Here all amplitudes are according to the Born approximation.
Fig. 4. Total cross-section for positronium formation in positron-
hydrogen atom collision as a function of the incident enermy. 1
represents the Born approximation. 2 is obtained by combining
the L=0 partial wave cross section of Ref. 6 with higher
partial wave Born cross sections, 3 is obtained by combining
the L=0 and 1 partial wave cross sections of Refs. 6 and 7 with
the higher partial wave Born cross sections. The dashed
line in 3 is obtained using Eq. (1).
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Differential cross-section for positronium formation in electron-atomic hydrogen collisions

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