Charlton M., Humberston J.W. Positron Physics

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8.3 Antihydrogen 379

As outlined by Wolf (1993) and Holzscheiter et al. (1996), similar con-

siderations to those described above also apply to recombination in traps,

and in particular the nested Penning-trap scheme (see below), again with

appropriate assumptions regarding the speed distributions of the trapped

positrons and antiprotons and their degree of spatial overlap. As an

example, Holzscheiter et al. (1996) argued that the recombination rates

are of the order of one per second (though dependent upon E

) for 10

positrons and 10

antiprotons trapped in a volume of 1 cm

Use of the charge-exchange mechanism, reaction (8.13), to produce

antihydrogen was ﬁrst proposed by Deutch et al. (1986), and subsequently

it was shown that the cross section for this process could be obtained by

applying the charge conjugation and time reversal operators to the process

of positronium formation in positron–hydrogen collisions (Humberston

et al., 1987, and see section 4.2). Under time reversal, the positronium

formation process equation (4.5) becomes

Ps+p→ H+e

, (8.22)

with a cross section σ

. The further application of charge conjugation

then yields reaction (8.13), for which the cross section is also σ

, assum-

ing charge conjugation invariance. Time reversal invariance implies the

symmetry of the S-matrix, from which the relationship between σ

and

can be derived. Consider a positron colliding with a hydrogen atom, in

a state with principal and orbital angular momentum quantum numbers

and l

respectively, and producing positronium with corresponding

quantum numbers n

and l

. If the cross section for this process is

; n

), then the cross section for the time-reversed process is

; n

), and

; n

)=σ

; n

)

(2l

+1)

(2l

+1)

; n

), (8.23)

where k and κ are the wave numbers of the positron and positronium

respectively. Energy conservation gives the relationship between them as

E =

−

. (8.24)

If the initial positronium and the residual antihydrogen are both in the

ground state (n

= 1 and n

= 1) the relationship between the cross

sections reduces to

= σ

. (8.25)

380 8 Exotic species involving positrons

The simple conversion factor k

/κ

was applied by Humberston et al.

(1987) to the positronium formation cross sections of Brown and Humber-

ston (1984) within the Ore gap, the distorted-wave Born approximation

results of Shakeshaft and Wadehra (1980) from the top of the Ore gap to

200 eV, and the Born results of Omidvar (unpublished) beyond 200 eV,

to yield the ﬁrst estimates of the antihydrogen formation cross section.

These results relate only to the formation of ground state antihydrogen

in collisions of antiprotons with ground state positronium, but Darewych

(1987) and Nahar and Wadehra (1988), using the ﬁrst Born approxima-

tion, calculated the cross sections for the formation of antihydrogen in

various excited states with n

≤ 3. The latter authors also attempted

to include contributions to the total formation cross section from antihy-

drogen states with n

> 3: they exploited the fact that, at suﬃciently

high energies, the Born approximation predicts the cross section for the

formation of antihydrogen in a state with principal quantum number n

to be σ

) ∝ 1/n

. However, this scaling law is almost certainly not

valid at the low energies of greatest experimental interest here. Nahar

and Wadehra (1988) also used the Born approximation to calculate the

cross sections for ground state antihydrogen formation in collisions of

antiprotons with positronium in the n

= 2 state. They then used the

above form of scaling law to estimate the contributions to the total ground

state antihydrogen formation cross section from antiproton collisions with

positronium in higher excited states.

All these contributions add up to a total antihydrogen formation cross

section of approximately 2 × 10

−15

in the antiproton energy range

2–10 keV where the charge-exchange production mechanism is likely to be

most eﬀective. This value is consistent with results obtained by Ermolaev,

Bransden and Mandal (1987), who used the classical trajectory Monte

Carlo method, and also with the results of a recent experiment (Merrison

et al., 1997) which measured the hydrogen atom formation cross section

via reaction (8.22).

Further investigations have been made of antihydrogen formation in

collisions of protons with excited state positronium. Igarashi, Toshima

and Shirai (1994), using a hyperspherical coupled-channel method, found

that at an energy of approximately 0.04 ryd the formation cross sections

for antiprotons in collision with positronium in the 2S and 2P states were

each approximately 300πa

, more than ten times the magnitude of the

formation cross section from ground state positronium. They surmised

that the formation cross sections would be even larger for positronium in

higher excited states. Mitroy and Stelbovics (1994), using the unitarized

Born approximation, obtained results that were qualitatively similar but

rather smaller in magnitude.

8.3 Antihydrogen 381

Fig. 8.10. Cross sections for antihydrogen formation in collisions of stationary

antiprotons with positronium atoms (from Igarashi, Toshima and Shirai, 1994).

(a) is for 1S positronium and (b) is for the 2P state (note the changes in scale).

Key (same for both ﬁgures): dotted curve with crosses, formation into the n

state; short-broken line plus squares, formation into the n

= 2 state; long-broken

line plus triangles, formation into the n

= 3 state; very-long-broken line plus

inverted triangles, formation into the n

= 4 states. The solid curve with circles

is the total cross section summed over all n

states and the double chain curve

is this quantity as calculated by Mitroy and Stelbovics (1994).

The possibility that total antihydrogen production (i.e. integrated over

all states) might be markedly enhanced by antiproton collisions with

excited state positronium was ﬁrst suggested by Charlton (1990), who

argued from a classical standpoint that σ

should increase as the area

of the positronium atom, giving an n

scaling. He also argued that as

was increased, the relative speed at which the cross section would

be a maximum would fall as n

−1

. The justiﬁcation for this assump-

tion was based on the Massey criterion, which states that the capture

probability is highest when the speed of the projectile is matched to the

quasi-classical orbital speed of the positron in the positronium. Both these

features are demonstrated qualitatively by the theoretical results shown

in Figure 8.10. Mitroy (1995) investigated antihydrogen formation from

positronium states up to n

= 4, using the coupled-state method, and

found that the scaling for σ

is actually closer to n

than to the fourth

382 8 Exotic species involving positrons

power suggested above. Other work of relevance to this topic has been per-

formed by Mitroy and Ratnavelu (1995) and Mitroy and Ryzhikh (1997).

An interesting method of realizing excited state positronium–antiproton

collisions has been suggested by Hessels, Homan and Cavagnero (1998).

It is clear from the above discussion that the antiproton–positronium

reaction can be a useful means of producing antihydrogen in low-lying

atomic states over a wide range of kinetic energies. Deutch et al. (1988)

suggested a possible method for antihydrogen production via reaction

(8.13), using antiprotons stored in a circular radio-frequency quadrupole

trap. However, given the advances in trapping, cooling and holding

antiprotons in Penning traps described above, it is natural to consider the

feasibility of experiments using such a device.

It is possible to imagine

producing positronium in a Penning trap using a target material such as

SiO

(see subsection 1.5.3), which has been shown to emit positronium in

a cryogenic environment (Mills et al., 1989a). Here, the crucial parameters

are the distance of the antiprotons from the shaped electrode housing the

positronium converter and the kinetic energy of the positronium (which

does not have to correspond to the same temp

erature as that of the

target). At present only conceptual designs of such an arrangement exist

(Deutch et al., 1993; Surko, Greaves and Charlton 1997).

Deutch et al. (1993) also noted that the eﬀective recoil temperature, T

of the antihydrogen is given from conservation of energy and momentum,

ignoring the initial energies of the two reactants, as

2∆Em

3kM

=28.6



−



(K), (8.26)

where m and M are the positron and antiproton masses respectively and

∆E is the diﬀerence between the binding energies of the antihydrogen

and positronium states, governed predominantly by n

and n

. The

most favourable condition for capture, so-called resonant charge transfer,

occurs (formally) when ∆E =0(n

√

) and thus has T

=0. As

an extension to the analysis of Deutch et al. (1993) and in an eﬀort to

simulate a real experimental situation, Cassidy et al. (1999) performed a

Monte Carlo simulation of the antiproton–positronium reaction. This was

done by making assumptions concerning the temperature distribution of

the trapped antiprotons and the energy distribution of the positronium

atoms, taken to be that typically resulting from positron impact on a sur-

face; they used as input the cross sections, diﬀerential and total, of Mitroy

and Ryzhikh (1997) and Mitroy (1997, private communication) for the

antiproton–positronium reaction. This analysis found the antihydrogen

formation rate in the ground state to be rather low, of the order of 100

per hour for 10

trapped antiprotons and with a positron ﬂux of 3 × 10

−1

. (These ﬁgures are close to the maximum which can presently be

8.3 Antihydrogen 383

expected from a laboratory-based positron beam and for antiprotons held

in a Penning trap.) Of these anti-atoms, only approximately 1% had a

temperature below 1 K, which is around the maximum well depth that can

be achieved using magnetic-gradient traps (see e.g. Hess et al., 1987, and

Gomer et al., 1997, for discussions of this type of device) with currently

available technology. Further discussion of the details of antihydrogen

formation by the antiproton–positronium reaction have been given by

Charlton (1996, 1997).

We now turn to reaction (8.14), the trapped plasmas combination

reaction, in which the excess energy is removed by an extra positron

as shown in Figure 8.9(c). The use of this ternary reaction for anti-

hydrogen formation was ﬁrst suggested by Gabrielse et al. (1988), who

noted that its matter equivalent had been studied, mainly in relation to

high temperature plasmas, for some time. These authors were therefore

able to write the antihydrogen production rate, which corresponds to the

recombination rate in conventional plasma physics, as

=6× 10

−12



4.2



9/2

. (8.27)

This formula for the rate per trapped antiproton

is notable for its very

strong positron-temperature dependence, T

−9/2

(in degrees K), and the

presence of the positron density, n

(in cm

−3

), to the second power. The

latter dependence arises because two positrons are needed in the reaction.

In their analysis, Gabrielse et al. (1988) assumed that a plasma of 10

positrons per cm

would be produced at 4.2 K, which yielded R

∼ 600

−1

per trapped antiproton. Glinsky and O’Neil (1991) re-examined this

problem from a plasma physics viewpoint, and found that the combination

rate given by equation (8.27), which is actually that pertaining to zero

ﬁeld, B = 0, is reduced by an order of magnitude in a highly magnetized

plasma (B ≈∞). In essence, this is caused by the constraint imposed on

the positron orbits, since they cannot cross the magnetic ﬁeld lines.

In order to promote reaction (8.14) it is necessary to trap both antipro-

tons and positrons and merge the two swarms. A schematic illustration

of a sequence of traps to achieve this, the so-called nested Penning trap

arrangement (Gabrielse et al., 1988), is shown in Figure 8.11. Again the

radial conﬁnement of the two clouds is provided by large axial magnetic

ﬁelds with appropriate voltage elements that produce potential wells to

trap the opposite charges. The positron well is nested within empty

wells into which the antiprotons can be moved, and the basic idea is to

adjust the depth of the wells so that the antiprotons just pass through the

positron cloud. The nested-trap scheme was developed further by Quint

et al. (1993), who reported the results of preliminary measurements on

384 8 Exotic species involving positrons

Fig. 8.11. Illustration of the principle of nested Penning traps for holding

antiprotons and positrons in close proximity, in order to promote antihydrogen

formation.

merged, trapped swarms of protons and electrons. Hall and Gabrielse

(1996) demonstrated the electron cooling of trapped protons using a

nested-well apparatus. Haarsma, Abdullah and Gabrielse (1995) reported

progress in accumulating positrons under ultra-high vacuum conditions

typical of those in antiproton traps in quantities suﬃcient to promote

reaction (8.14). Experiments carried out at CERN by Gabrielse and

coworkers (reported by Gabrielse et al., 1999) during the last days of

the operation of LEAR managed to conﬁne antiprotons and positrons

together in a multi-electrode apparatus for the ﬁrst time, though the

eﬀects observed when the clouds were merged could not be attributed to

antihydrogen formation.

One of the potential drawbacks of reaction (8.14) is that the antihy-

drogen will initially be produced in a Rydberg state with a high value

of n

, since the energy removed by the extra positron is of the order

of k

T . This has been conﬁrmed by the analysis of Pajek and Schuch

(1997), who studied the time-reversed process (impact ionization) to de-

rive a recombination coeﬃcient for reaction (8.14). Their numerical values

are in accord with equation (8.27), but they found that the coeﬃcient

was proportional to n

, thus dramatically favouring initial capture into

high-lying Ryberg states. Such atoms have long radiative lifetimes, and

Glinsky and O’Niel (1991) and Fedichev (1997) have pointed out mecha-

nisms which can lead to collisional stabilization. These involve so-called

replacement collisions at higher values of n

, whereby once the atom

has been formed, another positron travelling on a path closer to the

antiproton takes the place of the ﬁrst positron, which leaves, carrying

the excess energy. At lower values of n

this process becomes ineﬃ-

cient at the plasma densities envisaged, but another process, transverse

collisional drift, occurs. Here the bound positron drifts across the ﬁeld

8.3 Antihydrogen 385

lines towards the antiproton under the inﬂuence of distant positron–atom

collisions.

Both of these analyses rely on the fact that, to produce stable antihy-

drogen, the relaxation processes must occur before the atom leaves the

plasma, otherwise it will be ﬁeld ionized. Alternatively, as discussed by

Wolf (1993), it may be possible to laser-stimulate transitions to more

tightly bound levels to avoid the ﬁeld-ionization problem.

We now consider the ﬁnal method listed above for antihydrogen pro-

duction. Reactions (8.15) were ﬁrst suggested by Yamazaki (1992), with

further details provided by Ito, Widmann and Yamazaki (1993). The

method is based upon the discovery of Iwasaki et al. (1991) that up to 3%

of antiprotons stopping in helium gas can form a metastable antiprotonic

helium atom (αe¯p) with a lifetime of the order of 3 µs. The basic idea that

such an eﬀect could occur was proposed by Condo (1964). The antiproton,

on stopping in the medium before annihilation, forms this exotic atom

when its kinetic energy falls below the binding energy of the electrons

in the helium (around 25 eV). The captured antiproton then falls

into a

state with principal quantum number, n

, close to (M

∗

/m)

1/2

, where M

∗

is the reduced mass of the antiproton–nucleus system. In this case n

≈

38, and the energy level spacing is ≈ 2 eV. If the antiproton is captured

into a state with large orbital angular momentum, the transition to the

lower levels will most likely occur through a series of radiative transitions,

with a concomitantly long period (hence the appellation ‘metastable’)

before annihilation. The atomic nature of these states was conﬁrmed

by Morita et al. (1994) in a study involving a laser-induced resonant

transition between two of them. Further discussion of the αe¯p ‘atomcule’

has been given by Kartavstev (1996) and Eades and Hartmann (1999).

The proposed scenario for antihydrogen production involves stopping

a bunch of approximately 10

antiprotons in dense helium; this is fol-

lowed shortly afterwards by the injection of a bunch of approximately

positrons into the same medium. A detailed discussion was given

by Ito et al. (1993); this included a treatment of the diﬀerences between

the positrons and the positronium atoms which result from the positron

injection, in terms of the known behaviour of these species in helium

gas; see Chapters 6 and 7. Under optimum conditions, the number of

antihydrogen atoms formed could be as great as 10

–10

per antiproton

bunch.

Ito, Widmann and Yamazaki (1993) also addressed the problem of the

detection of antihydrogen under these conditions. They estimated that

most of the antihydrogen will be formed tens of nanoseconds after the

positron injection into the 37 atmospheres of helium gas, envisaged as

the stopping medium, and will probably be destroyed within 1 ns after

formation. Thus the antihydrogen signature will be a small spike on the

386 8 Exotic species involving positrons

Fig. 8.12. A possible scenario for trapped antihydrogen spectroscopy. The

microwa

ves quench the 2S

1/2

antihydrogen via the 2P

3/2

state, which sponta-

neously decays by emission of a Lyman-α photon.

αe¯p atomcule annihilation spectrum, at a time governed mainly by the

time diﬀerence between the injection of the positrons and the antiprotons

and the diﬀusion coeﬃcients of the lighter species. Ito, Widmann and

Yamazaki (1993) noted that such an observation would, strictly speaking,

indicate only that some of the antiprotons had annihilated more rapidly

with the positrons present and not necessarily that antihydrogen had

been formed. To prove the latter they envisaged using lasers to ionize

the antihydrogen and to observe the associated antiproton annihilation.

For this purpose they asserted that the antihydrogen has to survive for

around 10 ns in the gas, which suggests that the pressure will have to be

lower than the value assumed above.

It should be noted that even if antihydrogen were to be formed by

this technique, the prospects for precision studies of the kind outlined

in subsection 8.3.1, which might oﬀer a challenge to some of the basic

postulates of physics, are not likely to be easy.

4 Prospects for antihydrogen physics

Advances in trapping and cooling antiprotons make it likely that antihy-

drogen will soon be produced with kinetic energies suitable for trapping

and also, possibly, in beams. Almost all the production scenarios outlined

above rely on using a source of low energy antiprotons, such as LEAR

and associated machinery and, although the former unique facility closed

at the end of 1996, continuation of a low energy antiproton programme

at CERN in 1999 and beyond has been assured by the approval of the

Antiproton Decelerator project (Maury, 1997).

A possible glimpse of the future is provided by Figure 8.12, which

shows a schematic illustration of a magnetic-gradient trap for anti-atom

spectroscopy. The scenario described below is due to H¨ansch and Zimmer-

8.3 Antihydrogen 387

mann (1993), who assumed that a sample of 10

antihydrogen atoms could

be held in such a trap at a temperature of 0.2 K. A uniform magnetic ﬁeld

of around 0.1 T (for antiproton and positron conﬁnement) is superimposed

upon the ﬁeld produced by the quadrupole coils, which ensures that a local

ﬁeld minimum exists in the trap centre. Such a ﬁeld conﬁguration is well

known to produce a shallow trap for neutral species with an appropriate

magnetic moment. The schematic shows the anti-atoms, initially in the

1S state, raised into the 2S state by a Doppler-free two photon (243 nm)

transition and further pumped by microwaves into the 2P state, which

then decays back to the ground state with the emission of a Lyman-α

photon. This resonance ﬂuorescence is used as the signature of the tran-

sition. H¨ansch and Zimmermann (1993) also described how this cycle

can be operated on trapped anti-atoms to prevent spin ﬂips and keep

them stored for longer periods. Holzscheiter and Charlton (1999) have

described an alternative scheme of observing the 1S–2S transition which

does not rely on the detection of the Lyman-α photon.

As described brieﬂy in subsection 8.3.1, the ultimate goal for a hydrogen–

antihydrogen comparison for this transition is a fractional frequency

diﬀerence in the region of 10

−18

and as such would provide a stringent test

of CPT invariance and of the WEP. The achievement of such precision

is, however, a distant goal, most probably to be realized through a long

series of steps, with much development work still to be done with ordinary

matter reactions.