Antihydrogen, consisting of a positron orbiting around an antiproton, is the simplest few body system consisting entirely of antimatter and as such is of considerable importance in providing additional tests of the validity of charge conjugation invariance. In addition, the nature of the gravitational interaction between matter and antimatter might more readily be investigated for an electrically neutral system than one which is charged. Before such studies can be undertaken the antihydrogen must, of course, be produced by attachment of a positron to an antipositron. Several production mechanisms have been proposed, the two most favored of which are radiative capture (spontaneous or stimulated) and charge exchange in positronium-antiproton collisions. The cross section for radiative capture is very much less than that for charge exchange, so that it might be thought that the latter process is greatly to be preferred. Various calculations of the cross section for the charge exchange process are briefly reviewed

Humberston, J. W.

English

NASA Technical Reports Server

N90-18988
ANTIHYDROGEN FORMATION IN COLLISIONS OF POSITRONIUM WITH ANTIPROTONS
J.W. Humberston
Department of Physics and Astronomy
University College London, Gower Street, London WCIE 6BT, U.K.
INTRODUCTION
Antihydrogen, consisting of a positron orbiting around
an antiproton, is the simplest few body system consist-
ing entirely of antimatter and as such is of considerable
importance in providing additional tests of the validity
of charge conjugation invariance. In addition, the nature
of the gravitational interaction between matter and an-
timatter might more readily be investigated for an elec-
trically neutral system than one which is charged.
Before such studies can be undertaken the antihy-
drogen must, of course, be produced by attachment of
a positron to an antiproton. Unfortunately, both con-
stituents can only be produced in very small quantities
and the viability of a particular method of antihydrogen
production therefore depends critically on the efficiency
of the attachment process.
Several production mechanisms have been proposed,
the two most favoured of which are radiative capture
(spontaneous or stimulated)
e+ +_--* H+7,
e + +_+ n7 ---. H+ 7 + nT,
(1)
(2)
and charge exchange in positronium-antiproton collisions
Ps + _--* H + e-. (3)
Both methods are being actively investigated.(1-3)
The cross section for radiative capture is very much
less than that for charge exchange, so that it might be
..... thought that the latter process is greatly to be preferred.
-" However, in the proposed experiment using the radia-
tive capture process the positrons will be confined in
a storage ring (as also will be the antiprotons in both
methods) and, therefore, those positrons which do not
undergo capture by antiprotons on the first occasion will
be recycled until capture occurs. In this way nearly all
the antiprotons can be converted into antihydrogen.
The charge exchange process (equation (3)) is a sin-
gle pass method because positronium is neutral, but the
somewhat simpler nature of the proposed apparatus (see
figure 1) and the much larger antihydrogen formation
cross section are advantages which may, nevertheless,
make this an effective method of production also. We
shall, therefore, now consider various calculations of the
cross section for the charge exchange process.
223
PRECEDING PAGE BLANK NOT FILMED
e •
CONSTANT
RF DRIVE
PULSEDRF DRNE
I_JECTION
1 cm
Figure 1. Proposed experimental arrangement for the
formation of antihydrogen(3).
CALCULATIONS OF THE CHARGE
EXCHANGE CROSS-SECTION
Until very recently the prospect of forming antihydro-
gen in positronium-antiproton collisions seemed so re-
mote that no calculations of the cross-section had been
considered. However, the required cross-sectlon is rather
simply related by charge conjugation and time rever-
sal invariance to the cross-section for positronium for-
mation in positron-hydrogen collisions, a process which
has received extensive theoretical attention for several
years.(4-_)
Starting with the process of interest
Ps + _ -_ H + e- ( cross-sectionc_-), (4)
the charge conjugate system is
Ps T p --_ H + e + (cross-section crH) , (5)
and charge conjugation invariance requires (rH = aft.
The time reversed process is then
H + e + --_ Ps + p (cross-sectlon errs). (6)
https://ntrs.nasa.gov/search.jsp?R=19900009672 2020-03-19T22:56:58+00:00Z
Time reversal invariance implies the symmetry of theS-
matrix which provides the following connection between
the cross-sections o_- and ap,. Consider a positron col-
liding with a hydrogen atom in a state with energy and
orbital angular momentum quantum numbers n I and £1
respectively, to produce positronium with corresponding
quantum numbers n 2 and t 2. If the cross-section for this
process is ap° (n2, t2; hi, £1) then the cross-section for the
time reversed process is _rn(nl, £1; n2, £2), and
k2 (2£1+ 1)
= --_ (2£2+ 1)¢:rp,fn2,t2; nl,ex), (8)
where _ and ,_are the wave numbersof the positron and
positronium respectively. They are related by energy
conservation such that (in atomic units)
k2 1 _2 1
Es+e+ = Ep,+p = 2 n_--- 4 2n]" (9)
If the initial positronium and residual antihydrogen are
both in their ground states
42
a_-= a H = _-ap,, (10)
where
1
/c_-1= 1_ 2-_. (11)
This simple rescaling formula may be applied to the
various positronium formation cross sections already cal-
culated.
As we shall see, the peak in _- occurs at low ki-
netic energies of the positronium relative to the antipro-
ton, corresponding to positronium formation in, and just
above, the Ore gap (6.8 - 10.2 eV for hydrogen). In the
Ore gap, where the only two open channels are elastic
scattering and positronium formation, the most accurate
results are probably those of Humberston(4), and Brown
and Humberston(5). They used a two channel version of
the Kohn variational method of the form
K21 K22 -[Kh Kh-[(%,z%) (%,z%)
(12)
where _ = 2( H - E).
The two components of the trial wave function, @l
and ff22, represent positron elastic scattering plus positro-
nium formation and positronium elastic scattering plus
hydrogen formation respectively. Each component con-
tains many short range correlation terms with associated
variational parameters to allow adequately for the strong
electron-positron correlations. Investigations of the vari-
ations of the values of the K-matrix elements with re-
spect to systematic improvements in the trial functions
suggest that the most accurate results obtained in this
manner have converged to within a few per cent of the
exact values.
The partial wave cross-section for positronium forma-
tion is then (in units of Irao2)
4(2£+ 1) I/_K 2
Results were obtained for s-, p- and d-wave scatter-
ing and compared With results from other simpler meth-
ods of calcuiatlon. The accurate variational values are
several orders of magnitude smaller than the results of
the Born approximation for s-wave sCgttering but only
a factor of three smarer for p-wavescattering, and for
the d-wave the differencebetween the two methods is
rathersmall. It is therefore reasonable to suppose that
the Born approximation provides quite accurate values
for the positronium formation cross'sectionfor all_igher
partial waves. With this assumption, the total posjtro-
nium formation cross secii0n' in the Ore gap is a_ given
in figure 2. The Born results were first calculated by
Omidvar (unpublished).
all parti¢/_
k
g ! I i I
05 0.6 0._"
k2 (o o-P)
Figure 2. Partial wave cross sections for ground state
positronium formation in positron collisions with atomic
hydrogen (4'5). The s-wave contribution is goo smalI to
be shown on this scale.
2.0-
1.5
k_ l.O
0.5
0
224
• ::w = 7 _7:
At positron energies somewhat above the top of the
Ore gap, where the Born approximation is still not sufH-
cient]y accurate for the low partial waves, the most reli-
able calculations of the positronium formation cross sec-
tion are probably those of Shakeshaft and Wadehra(e),
who used a distorted wave Born approximation and ob-
tMned results in the positron energy range 13.6 - 200eV.
These results match on reasonably well at the lower end
of the range to the more accurate variational results.
Beyond 200eV the Born results of Omidvar may be
used.
These three sets of positronium formation cross sec-
tions produce, upon rescaling according to equation (10),
the hydrogen (antihydrogen) formation cross-section in
collisions of positronium with protons (antiprotons)
shown in figure 3 (7) . The peak value of the cross-section
is approximately 3.5 x 10-1%m 2 at a positronium energy
of ,,, 3.5eV relative to the antiproton, corresponding to
an antiproton energy relative to the positronium of ,-,
3keV.
energy(keV}
20 40 60 BO 100
0.02
0.01
0 8 12 16
x2,Ps energy (6.8eV)
Figure 3. Theoretical estimates by Humberston et
al(7), of the ground state antihydrogen formation cross
section from ground state positronium.
There is an interesting feature of the antihydzogen
formation cross section at very low positronium energies,
as is shown on the expanded scale in figure 4, with the
cross section tending to infinity as g2 _. 0. The zero
orbital angular momentum contribution to e_-is, from
equation (13),
(14)
and the K-matrix elements have the following behaviour
as the positronium threshold is approached from above:
so that
•_11 _ congt.
Kz2 -' _½ /
J
o (15)
)_x- as g --* 0, (16)
PC
as is to be expected for an exothermic reaction such as
this. However, the rate of production of H is vc,_- oc
Iccr_- --, finite const, as _ --, 0.
energy (keY)
0 1, 8 12 0 0.2 0._, 0.6
.... / ....... /
0 O.S 1.6 0 CtO& 0.08 0.12
xa, Ps energy (6,8 tVJ
Figure 4. Low energy portions of figure 3 on expanded
SCKIeS.
Thus far we have considered the formation of antihy-
drogen in its ground state, using positronium also in its
ground state. It would be equally acceptable to form
antihydrogen in one of its excited states, and several au-
thors have recently considered this process. Darewych (s)
used the first Born approximation to calculate the cross
sections for antihydrogen formation into the ls,2s, 2p
and 3s states and his results are shown in figure 5. As
can be seen, these excited state formation cross sections
225
are comparable to the ground state formation cross sec-
tion at low positronium energies and, therefore, make a
very significant contribution to the total formation cross
section. Almost certainly the Born approximation signif-
icantly overestimates the s-wave contributions to each of
these cross-sections, as it does for the ground state, but
even if this partial wave contribution is totally supressed
the excited formation cross sections remain significant,
as shown in figure 5.
for antihydrogen formation into a state with quantum
numbers n, l is
1
o_ff(n, t) v_ -_-_. (17)
However, such a scaling formula is almost certainly not
valid in the low energy range of interest here.
1
06
01
0 Oil
006
004
1 2 ) _ S 6 7 8
Figure 5. Cross sections for antihydrogen formation
in various states from ground state positronium accord-
ing to Darewych (s). Full curves; Born approximation:
broken curves; s-wave subtracted Born approximation:
chain curve; 'exact' results of Humberston eta[ (r)
Similar investigations, also using the first Born ap-
proximation, have recently been conducted by Nahar and
Wadehra (9) and are shown in figure 6. They obtained
good agreement with Darewych for the s-state but not
quite so good for the p-state, where Darewych had used
an approximate method to evaluate some of the angular
integrals. These authors attempted to include the contri-
bution to the total antihydrogen formation cross section
from even higher energy states by making use of the fact
that, at sufficiently high energy, the Born cross section
226
I0'
10'
0 SO 10@ RiO 10@ IN _ lira 400 _ I_@
AntipfotonImplct En_rgv (koY_
Figure 6. Cross sections for antihydrogen formation in
various states from ground state positronium according
to the Born calculations of Nahar and Wadehra.(9)
Ermolaev e_ aL (1°) used classical and semiclassical
methods to investigate the same formation cross sections,
and their total results for formation of antihydrogen into
an states with n _< 3 are shown in figure 7. Although
the low mass of the incident positronium must cast some
doubt on the validity of such methods for this system,
the results are, nevertheless, in tolerably good agreement
with the quantum mechanical results.
oRIGINAL PAGE IS
OF pOOR QuALiTY'
b
1¢
00'
',,t\
",,! \
"\ \
\
I_ton Energy (keY tab}
\
'i \
% 100
Figure 7. Total antihydrogen formation cross-sections
from ground state positronium according to ErmolaevO°)
......; impact parameter method: e; classicaltrajectory
Monte-Carlo: ._.-; Born approximation (Darewych(S)).
,--.,i .... i .... i .... i .... i .... 1 .... ! .... i .... t'''_
_+ p,ln.o)-_ fills) + s
\
10'
0 liO loll lit _ illO _ NO 40Q 41_ I_0
Antiprolon Imp_:l Enecgy(keY)
Figure 8. Cross sections for ground state antihydrogen
formation from various states of positronium according
to the Born calculations of Nahar and Wadehra. (_)
In all the investigations described so far,the incident
positronium has been assumed to be in its ground state,
but antihydrogen formation from positronium in excited
states should also be considered.
Nahar and Wadehra (9) have used the Born approxi-
mation to investigate the formation of ground state anti-
hydrogen in colllsions of excited state positronium with
antiprotons. They included the positronium states with
n= 1 and 2 explicitly and again used the scaling law given
in equation (17) to estimate the effect of allother states.
Their results are given in figure 8..
These Born calculations were not conducted at very
low energies around the peak in the antihydrogen forma-
tion cross section and, even had they been, they would
probably have suffered again from a similar overestima-
tion of the low partial wave contributions to that men-
tioned previously. Nevertheless, it is very likely that an-
tihydrogen formation from positronium in excited states
will make a further very significant contribution to the
total antihydrogen formation cross section.
Taking all these contributions into account, it would
not seem unduely optimistic to predict a total antihydro-
gen formation cross section of at least 207ra2o. With the
expected currents of posltronium and antiproton, this
cross section suggests a possible rate of production of
antihydrogen atoms of a few per second.
227
I should like to thank Prof. R.P. McEachran and his
colleagues, Profs. J.W. Darewych and A.D. Stauffer for
their hospitality at York University, Toronto during the
writing of this article.
REFERENCES
1. H. Poth, "Atomic Physics with Positrons", edited by
J.W. Humberston and E.A.G. Armour (Plenum, New
York, i987) p. 307.
2. A. Rich, R. Conti, W. Frieze, D.W. Cidley, H. Griffin,
M. SkMsey, T. Steiger, :l. Van House, W. Zheng and
P.W. Zitzewitz, ibid, p. 321.
3. F.M. JacoB_sen_ L.H, Andersen, B.L Deutsch,
P. Hvelpluncl, H. Knudsen, M. Cha_lton, O' Larlcchia
and M. Holzscheiter, ibid, p. 333_ :_
3.W. Humbe_ston, Can. J. Phys. 60, 591 (1982).
...... , J. Phys. B. 1T, 2353 (1984).
....... , Adv. At. Mol. Phys. 22, 1 (1986).
5. C.J. Brown and J.W. Humberston, J. Phys. B. 1T,
L423 (_1984),
.... , :I. Phys. B. 18, L401 (1985).
P_. Shakeshaft and 3.M. Wadehra, Phys. Rev. A22,
968 (1980).
J,W. Humberston, M, Charlton, F.M. Jacobsen and
B.I. Deutch, J. Phys. B20, L25 (1987).
J.W. Darewych, 3. Phys. B20, 5917 (1987).
S.N. Nahar and AM. Wadehra, Phys. Rev. AgT, 4118
(1988). _
A.M. Ermoiaev, B.H. Bransden and C.R. Mandal,
Phys. Lett. A125, 44 (1987).
.
,
7.
8.
9.
I0.
228


Antihydrogen formation in collisions of positronium with antiprotons

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server