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Positron physics in a new perspective: Low-energy antihydrogen scattering by simple atoms and molecules

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



59



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.



60



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.



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