Positron physics in a new perspective: Low-energy antihydrogen scattering by simple atoms and molecules
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Introduction
The existence of the positron, a particle with the same mass as the
electron but with opposite charge, was predicted in 1930 by Dirac [1],
Weyl [2] and Oppenheimer [3]. It was not long before its existence was
confirmed experimentally by Anderson [4] and Blackett and Occhialini
[5]. The existence of this particle, the antiparticle of the electron, made
it possible to envisage a whole new range of scattering experiments in
atomic and molecular physics in which the incident particle is a positron.
Since then, extensive experimental and theoretical research has been
carried out on positron scattering by atoms and molecules. This work
has been extended recently to include the scattering of positronium (Ps)
by these targets. Ps is a system, similar to hydrogen, but with the proton
replaced by a positron.
Charlton and Humberston have given a detailed description of this
work in a recent book [6]. In an appendix they list references for all
but the most recent conference proceedings (e.g. [7]) on positron and
positronium scattering by atoms and molecules. The most recent work,
together with probable future developments, are described in this volume.
Quantum field theory predicts that, corresponding to every particle,
there is an antiparticle. If a particle is charged, its antiparticle has the
same mass but opposite charge, as in the case of the electron and the
positron. The particles of which the Earth is made, e.g. electrons and
protons, are referred to as matter and the corresponding antiparticles,
e.g. positrons and antiprotons, as antimatter.
Antihydrogen, in which a positron is bound to an antiproton, was
quite recently observed at CERN by Baur et al. [8, 9] and at Fermilab
by Blanford et al. [6, 10]. However, the high speed ( ~ 0.9c) of the
few antihydrogen atoms that were obtained ruled out any possibility of
determining their properties.
Experimentalists want to trap at very low temperatures, i.e. < 1
K, so that the is in essentially its rest frame. This will make possible
tests of the predictions of two fundamental theories of modern physics:
quantum field theory and Einstein’s general theory of relativity. In particular, tests can be made of the charge conjugation, parity interchange
and time-reversal (CPT) symmetry of quantum field theory. See, for
example, Hughes [11] and Schweber [12] (note that Schweber refers to
this symmetry as the TCP theorem).
It follows from the CPT symmetry that a charged particle and its
antiparticle should have equal and opposite charges and equal masses,
lifetimes and gyromagnetic ratios. In addition, the CPT symmetry
Positron physics in a new perspective
55
of quantum electrodynamics predicts that hydrogen and antihydrogen
should have identical spectra [6, 13, 14, 15]. Experimentalists plan to
test, as far as possible, whether H and do have these properties. In
particular, they intend to compare the frequency of the 1s-2s two-photon
transition in H and
The equivalence principle, according to which all bodies fall at the
same rate in a gravitational field, led Einstein to his general theory of
relativity in which gravitation manifests itself as a metric effect, curved
spacetime, as opposed to the flat spacetime of special relativity in the
absence of gravitation [16, 17]. The exact form of the curvature is determined by Einstein’s field equations. It is proposed to test the validity
of the above formulation of the principle of equivalence by carrying out
a null redshift experiment in which the frequency of the two-photon 1s
to 2s transition is observed for H and
as both are moved through the
same gravitational field. Any difference between the two sets of values
would indicate that the two atoms experienced a different gravitational
red shift, which can be shown to be a violation of this principle [13, 17].
A large group of experimentalists is currently working on the preparation of the ATHENA (ApparaTus for High precision Experiments on
Neutral Antimatter) [13, 14, 15]. The size and extent of the collaboration
on this project can be seen from [14], which is entitled ‘Antihydrogen
production and precision experiments’ and has 54 authors from 7 different countries.
What can positron theoreticians do to contribute to this important
and rapidly developing field ? In fact, they have already made an important contribution. One possible way of making antihydrogen is in
collisions between Ps and antiprotons:
The Hamiltonian that is used to describe this process only involves
the Coulombic interaction and is invariant under charge conjugation.
Thus the cross sections for reaction (1) are the same as for:
Humberston [18] had previously carried out calculations using the
Kohn variational method for the reverse reaction:
for incident positron energies below the first excited state of H. Humberston [19] readily adapted his calculation so that it could be applied to
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the calculation, in this energy range, of the cross section for H formation
in reaction (2) and hence for in reaction (1).
There are other ways in which positron theoreticians can assist the
experimentalists working on the ATHENA project. These involve
scattering by simple atoms and molecules. As a prelude to describing
these processes, we shall review the work that has been carried out to
date on
Unless otherwise stated, the units we will use in this article
are Hartree atomic units in which, Planck’s constant e, the charge on
the positron and,
the mass of the electron and the positron, all have
the value one. Note that 1 a.u. of energy corresponds to 27.212 eV.
1.
PREVIOUS WORK ON ANTIHYDROGEN
Some work was carried out on the interaction between H and in the
seventies and early eighties [23, 24, 25, 26]. The most significant paper
amongst these is the paper by Kolos et al. [25]. Kolos and Wolniewicz
had extensive experience of carrying out calculations of very accurate energies and properties of hydrogen-like molecules using the Rayleigh-Ritz
variational method and basis sets containing Hylleraas-type functions,
i.e. functions that depend on an odd power of the distance between the
light particles. See, for example, refs. [27, 28]. Encouraged by Schrader,
they used this experience to carry out calculations of the energy of
system at internuclear distances, R, ranging from
They made the reasonable assumption that the ground state wave
function for the
system is invariant under all the operations of
the
symmetry group. If the nuclei are fixed, it follows that the
wave function is of symmetry and is invariant under the operation of
interchange of the electron and the positron, followed by reflection of
their coordinates in the plane containing the perpendicular bisectors of
the internuclear axis. The spin wave function need not be considered as
interactions involving spin have been neglected.
Using a total of 77 basis functions, each of which contains a power of
the distance between the electron and positron ranging equal to 0, 1 or
2, Kolos et al. were able to show that a maximum in the interatomic
potential at about
found by Junker and Bardsley [23] using the variational method and configuration-interaction (CI) type basis functions,
was an artefact of their calculation. CI-type basis sets do not contain
Hylleraas-type functions. As pointed out, for example, by Lebeda and
Schrader [30, 25], such basis functions are unsuitable for describing the
wave function for the system when the electron is close to the positron.
This is because an infinite number of CI-type basis functions is required
to represent the cusp behaviour [31] of the wave function when the dis-
Positron physics in a new perspective
57
tance between the electron and the positron tends to zero. Hylleraastype functions are needed to represent the wave function in this region.
The importance of describing this region accurately is greater when,
as in this case, the interaction between the light particles is attractive
rather than repulsive, as in molecules, where CI-type basis functions are
known to give quite accurate results for interatomic potentials. See, for
example, [32].
Clearly, if the internuclear distance R = 0, the electron and the
positron are not bound to the nuclei. In fact, there is a certain critical
internuclear distance,
of the order of
below which the electron
and the positron are not bound to the nuclei. This is the basis of a
semiclassical method of calculating the cross section for rearrangement
annihilation [24, 25], using the very accurate potential, V(R), between
the and H atoms calculated by
et al. The largest impact parameter,
for which the classical turning point of the trajectory is
less than
is then determined. The rearrangement channel that leads
to annihilation is
where Pn is protonium, i.e. a proton bound to an antiproton, and both
the Pn and the Ps may be in excited states. If it is assumed that this
rearrangement takes place, if and only if
then the cross section
for rearrangement annihilation of the nuclei (and the light particles), is
given by
Uncertainties arise not only from this assumption but also from the
fact that
is not known exactly and from the breakdown of the classical
treatment at very low energies. The uncertainties about the exact value
of
are not thought to affect the accuracy [25] of the calculation until
far above room temperature, the maximum temperature that we need
to consider. We will have more to say about the breakdown at very low
energies later on in this section.
The problem of calculating
is the next stage on from the problem
of calculating the critical R value below which the dipole made up of the
fixed proton and antiproton is unable to bind an electron (or a positron).
This latter problem aroused a great deal of interest culminating in the
mid-sixties with several proofs by different methods that the critical
value of the internuclear distance is
See the references on this
subject in
et al. [25], and also in Armour and Byers Brown [29].
This problem can be solved exactly. However, this is not possible for the
more complicated problem of calculating the critical value,
below
which the fixed nuclei are unable to bind an electron and a positron.
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We have calculated an upper bound to
using the Rayleigh-Ritz
variational method [33, 34]. Our basis set was similar to that of
et al. in that it contained functions of the form
where
and
are prolate spheroidal coordinates defined by
Particle 1 is the electron and particle two is the positron. A is the proton,
B is the antiproton.
for example, is the distance from the proton
to the point under consideration.
is the usual azimuthal angle. The
is taken to be in the direction of AB.
is of the form
This factor only differs from the corresponding factor in the basis set
used by
et al. when
They take this factor to be
where
is the distance between the electron and the positron. As
this difference is unimportant as it is the presence of our choice of factor
on the right-hand side of this expression that gives rise to a basis function
linearly independent of the basis functions for which
As we wished to calculate the energy of the light particles for R values
for which they are very weakly bound to the nuclei, or not at all, we included an additional basis function, not included in the set used by
et al. Its role is to represent ‘virtual’, i.e. weakly bound, positronium.
It is of the form
where is the distance of the centre of mass of the positronium from the
centre of mass of the nuclei.
is the normalised positronium groundstate wave function.
is a variational parameter.
is a shielding
Positron physics in a new perspective
function to ensure good behaviour at
form
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It was taken to be of the
After some numerical trials, was taken to be 3 and to be 0.5.
Including a function of the form (10) is similar to the procedure used
by Rotenberg and Stein [35] to obtain an upper bound on the mass a
‘positron’ would have to have to form a bound state with an H atom. In
their variational calculation, they included a basis function to represent
a weakly bound ‘positron’. This enabled them to obtain an improved
upper bound.
It is well known that the inclusion of a ‘virtual’ positronium function
in the basis set improves the convergence of the results of Kohn variational calculations of scattering parameters at incident positron energies
just below the threshold for positronium formation [36]. The effect on
the annihilation parameter,
of including such a function at these
energies is discussed by Van Reeth and Humberston [37].
In our calculation, the parameter analogous to the positron energy
is the internuclear distance, R. As it approaches the critical value,
from above, the light particles become more and more weakly bound.
The channel that is close to being open is the ground state positronium
formation channel. It is this channel that is represented by the ‘virtual’
positronium function (10). As R becomes less than
in this function
becomes imaginary and the channel becomes open.
The non-relativistic Hamiltonian for an electron and a positron in the
field of the fixed proton and antiproton a distance R apart, in the form
appropriate for representing the separation of
into
as
is:
The –1/R term in the potential is the energy of the fixed nuclei. Matrix
elements involving only the basis functions (5) can be evaluated straightforwardly using well established procedures. See, for example, references
[33, 38]. Matrix elements involving only the positronium function (10)
can easily be calculated analytically by transforming the kinetic energy
operators in
above to Jacobi coordinates appropriate for separation
into unbound positronium, as in (13) below, and using the symmetry of
the ground-state positronium charge distribution.
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where
is the position vector of the positron with respect to the electron and is the position vector of the centre of mass of the positronium
with respect to the centre of mass of the nuclei.
The problem of evaluating the matrix elements involving both types
of function is the usual one encountered when including a rearrangement
channel: the coordinates that are convenient to use in the rearrangement
channel are very different from those used in the entrance channel. One
coordinate could be eliminated as all the basis functions we use are
functions of
but not of
and
separately [39].
Integration over the remaining five coordinates was carried out using
two different methods, Gaussian quadrature and the Boys boundaryderivative reduction method [40, 41]. The positronium function was
taken to be on the right of the matrix element so that operation on
it with the Hamiltonian,
in the form in equation (13) appropriate
for separation of
into the bare nuclei + positronium, removed the
singularity in the matrix elements due to the
term in
The
results we obtained for the energy of
for R in the range
are given in table 1.
Our results are compared with the results of two other calculations,
the calculation by
et al. [25], referred to earlier, who carried out
calculations for R = 1.0 and
and a recent calculation by Jonsell
et al. [42, 43], who carried out calculations for
using the
Rayleigh-Ritz variational method and a basis set containing 908 basis
functions. These functions are of a more restricted form than those used
by
et al. and in our calculation as they contain two independent
non-linear parameters, rather than four. However, they were able to
include basis functions containing
in addition to basis functions
containing lower powers of
or equivalent, as in the calculation by
et al. and in our calculation.
It can be seen that the size of basis set used by Jonsell et al. [42, 43]
and their inclusion of basis functions containing
more than compensates for the restriction on the form of their basis functions and they
obtain more accurate results than
et al. and ourselves, except for
where our calculation, with the inclusion of the basis function that represents weakly bound positronium, gives a slightly more
accurate value.
We have also carried out calculations for
In this case, the
calculated value of the binding energy was less than 0.001 and great care
had to be taken with evaluation of the inter-channel matrix elements.
Positron physics in a new perspective
61
Results were calculated using the Boys method of numerical integration [40, 41] and 31 points per dimension. No significant change in the
binding energy was found if the number of points was increased to 71
per dimension or if Gaussian quadrature was used with 64 points per
dimension.
We have recently found a way to include factors of
with
in
our basis functions. It seems probable that using 246 basis functions of
this form will make it possible for us to show that the light particles are
bound to the nuclei when
without the inclusion of the basis
function that represents ‘virtual’ positronium (10).
The results obtained for various values of the parameter in the basis
function representing ‘virtual’ positronium, are given in table 2. It can
be seen that the maximum calculated binding energy is obtained for
between 0.06 and 0.07, In an exact calculation, it would satisfy the
relation
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It can be seen that for this relation to be satisfied for our maximum
value of the binding energy, would have to equal 0.051.
Our calculations show that an electron and a positron are bound to a
fixed proton and an antiproton if
It is reasonable to assume
that the critical value,
for binding is greater than the value
the critical value of the internuclear distance below which the proton
and the antiproton are unable to bind an electron or a positron on its
own. This is because, below this value, the
dipole would be unable
to bind the electron and the positron, even if there were no interaction
between them that gives rise to positronium.
So far, all the calculations we have described are within the BornOppenheimer approximation in which coupling between the nuclear and
light particle motion is neglected. The nuclei are regarded as fixed in
space and the energy of the light particles is calculated as a function of R.
This energy, together with the potential for the Coulombic interaction
between the nuclei, is then taken as the potential in the Schrodinger
equation for the relative motion of the nuclei.
The omitted terms in the kinetic energy of the system all contain the
factor
where
is the reduced mass of the nuclei.
Thus they can be expected to be small, in comparison with the light
particle kinetic energy terms. The Born-Oppenheimer approximation is
very widely used in quantum chemistry to calculate energies and wave
functions for molecules. In most cases, the corrections to it are small
enough that they can be neglected, except in calculations that require a
very high level of accuracy.
The situation regarding
is less clear. There are no bound states
of
[44]. This is due to the very high binding energy, 459.04 a.u.,
of protonium and the very small expectation value of R in this state.
Positron physics in a new perspective
63
Also, if
the light particles are not bound to the fixed nuclei.
Prom the point of view of calculating the internuclear potential this
is not a serious difficulty as the potential in this region is effectively
the – 1/R Coulombic potential representing the attraction between the
nuclei. However, it is not clear what form the wave function for
should take in this region. We will have more to say about this when we
come to consider
scattering.
We have seen earlier that in the semi-classical treatment of
scattering used by
et al., rearrangement into protonium + positronium, followed by annihilation, is assumed to take place if the classical
value of the closest distance between the nuclei on a given trajectory is
less than
It is reasonable to expect that there should be a quantum
mechanical analogue of this behaviour.
We have carried out calculations to determine the effect of taking into
account the omitted terms in treatments using the Born-Oppenheimer
approximation. In the first set of calculations, we calculated the changes
to the energy of the light particles, due to the coupling of the light
particle and nuclear motion [45]. We will go on to consider the effect of
this coupling on the calculated scattering parameters for low energy
scattering. The corrections arise from the dependence of the prolate
spheroidal coordinates,
on R and on the spherical polar angles
of the internuclear axis, AB, with respect to a non-rotating axis system,
with origin at the centre of mass of
i.e. with respect to the centre
of mass inertial frame. We take this centre of mass to be the centre
of mass of the nuclei. This is an approximation but the error incurred
should be small as the nuclei are much more massive than the electron
and the positron.
The dependence of and on R and the spherical polar angles can
be obtained from equation (6). In terms of the non-rotating Cartesian
coordinates,
with origin at the nuclear centre of mass,
and
where
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are the coordinates of the antiproton, B, with respect to the non-rotating
axis system. The coordinates of the proton, A, can be obtained by
inverting the coordinates of B. and are the spherical polar angles
of AB, with respect to these coordinates. The third prolate spheroidal
coordinate of light particle
is the azimuthal angle of the
particle with respect to body-fixed axes with axis in the direction of
AB and an arbitrarily chosen axis. (This can be arbitrarily chosen on
account of the symmetry of
about the internuclear axis).
We can see that
and
depend on R from the equations for
and
and also from the inverse power of R in the equations (6) for
these coordinates in terms of
and
This inverse dependence on
R means that the coordinates are not defined if R = 0. We will have
more to say about this when we come to consider
scattering.
The non-relativistic Hamiltonian for the internal motion of
is
obtained by separating out the centre of mass motion and expressing
the kinetic energy operator for the internal nuclear motion in terms of
derivatives with respect to the spherical polar coordinates, (R,
) of
one nucleus, in this case the antiproton, with respect to the other. It is
of the form
where
and
is the reduced mass of the proton-antiproton pair which has
been taken to be 918.08 a.u.,
is the sum of the proton and antiproton
masses and
is the reduced mass of the electron or positron,
The kinetic energy terms for the internal motion of the electron and
the positron are first expressed in terms of the non-rotating coordinates.