IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL
J. Phys. A: Math. Theor. 40 (2007) 11923–11937 doi:10.1088/1751-8113/40/39/014
Annihilation of the electron–positron pairs in
polyelectrons
Alexei M Frolov and Farrukh A Chishtie
Department of Applied Mathematics, University of Western Ontario, London,
Ontario N6A 5B7, Canada
Received 11 June 2007, in final form 14 August 2007
Published 11 September 2007
Online at stacks.iop.org/JPhysA/40/11923
Abstract
Annihilation of the electron–positron pairs (or (e
−
, e
+
)-pairs, for short) in
various polyelectrons e
+
n
e
−
m
= e
−
m
e
+
n
(where n 1 and m 1) is considered. In
particular, we discuss the three- and four-photon annihilation of the (e
−
, e
+
)-
pairs in the three-body Ps
−
ion and four-body bi-positronium system Ps
2
. It
is shown that the five-body e
+
2
e
−
3
ion is an unbound system. The closed-
form expression is derived for the amplitude-square |M|
2
of the three-photon
annihilation of (e
−
, e
+
)-pair at arbitrary energies of the colliding particles.
Analogous amplitude-square |M|
2
for the four-photon annihilation is reduced
to the form which is convenient for future analytical calculations. A method
which can be used to produce macroscopic polyelectrons is briefly discussed.
PACS numbers: 36.10.−k, 36.10.Dr
1. Introduction
Annihilation of the electron–positron pairs (or (e
−
, e
+
)-pairs, for short) in various polyelectrons
e
+
n
e
−
m
is considered. The polyelectrons discussed in this work include the three-body
positronium ion Ps
−
(
e
+
e
−
2
)
and four-body bi-positronium system Ps
2
(
e
+
2
e
−
2
)
. We also consider
annihilation of the (e
−
, e
+
)-pairs in macroscopic polyelectrons, i.e. in the e
+
n
e
−
m
systems,
where n ≈ N
A
and m ≈ N
A
and N
A
≈ 6.022 × 10
23
is the Avogadro number. Here
and everywhere, the notation e
+
stands for the positron, while the notation e
−
means the
electron. Theory of polyelectrons and analysis of annihilation in such systems are required
in many applications. Note that in small polyelectrons e
+
n
e
−
m
, where n 5 and m 5,
the leading annihilation process is the annihilation of electron–positron pairs from bound
states. In higher polyelectrons, annihilation of the (e
−
, e
+
)-pairs from unbound states also
contribute.
The positronium ion, bi-positronium and higher polyelectrons are of great interest in some
applications to astrophysics [1], solid state physics [2] and other problems [3–5]. Most of
such applications are related to the electron–positron pair annihilation in these polyelectrons.
1751-8113/07/3911923+15$30.00 © 2007 IOP Publishing Ltd Printed in the UK 11923