Measurement of the decay e+e− → 4γ at rest

21 April 1994 PHYSICS LETTERS B

ELSEVIER

Physics Letters B 325 (1994) 300-307

Measurement of the decay e + e -

>47 at rest

H. von Busch 1, p. Thirolf, Ch. Ender, D. Habs, F. Kfck, Th. Schulze 2, D. Schwalm Max-Planck-lnstitut fiir Kernphysik, P.O. Box 10 39 80, D 69029 Heidelberg, Germany and Physikalisches Institut der Universitat Heidelberg, Philosophenweg 12, D 69120 Heidelberg, Germany

Received 18 February 1994 Editor: R.H. Siemssen

Abstract The decay of e+e - at rest into four photons has been measured in a 20-day experiment at the Heidelberg-Darmstadt Crystal Ball. From a total of 4.9 x 10 l° annihilations observed, 406 47 events with an estimated background contamination of less than 5% have been isolated. Energy and angular distributions are found to agree excellently with simulations based on QED. Using a parallel measurement of 3y-annihilation for a reduction of systematic uncertainties, the branching ratio F4r/F2r is obtained to be (1.50 :t: 0.07(stat.) + 0.09(syst.)) x 10-6, in agreement with lowest-order QED.

1. Introduction The e+e - system at rest [ 1 ], called positronium as a bound state, annihilates via the standard electromagnetic interaction into two or more photons. In this process, s-wave initial states overwhelmingly dominate because of their non-vanishing probability density at zero distance. C-parity conservation requires the spin singlet and the spin triplet s-wave states to decay into even and odd numbers of photons, respectively. The QED predictions for the corresponding branching ratios F n ~ y / F 2 v and Fno~:,/F3:, decrease strongly with the photon multiplicity n; for F4y/F2r, a value of 1.48 x 10 -6 has been calculated in lowest order [ 2 - 4 ] . While the standard decays into two and three `/-rays are well-known and have been measured with high

i Current address: Universit~t Kaiserslantern, Fachbereich Physik, Erwin-Schr0dinger-Str. 46, D 67663 Kaiserslautern, Germany. E-mail: [email protected]. 2 Current address: Institut fOr Energiewirtschaft und rationelle Energieverwendung, Hegbriihlstr. 49a, D 70565 Stuttgart, Germany.

precision [5,6], the existence of 4,/-annihilation has only recently been experimentally confirmed; based on 26 four-photon events, a 4T/2,/branching ratio of (1.30 4- 0.26(stat.) 4- 0.16(syst.)) × 10 -6 has been derived [7]. Following a series of exploratory investigations started in our laboratory almost 10 years ago [ 8-10], we report in the present letter on the first detailed measurement of the 4,/-decay of e+e - at rest [ 11,12]. Our results confirm the QED calculations of this high-order process in the most fundamental leptonic two-body system to an accuracy of better than 10%.

2. Experiment and analysis Observing this rare 4 , / d e c a y mode requires high efficiency both in detection and background suppression. Optimal detection efficiency is provided by 4vr multidetector systems of high granularity such as the Heidelberg-Darmstadt Crystal Ball spectrometer [ 13 ], a spherical NaI shell of 25 cm inner radius and

0370-2693/94/$07.00 (g) 1994 Elsevier Science B.V. All rights reserved SSDI 0370-2693 (94) 00264-8

H. von Busch et al. / Physics Letters B 325 (1994) 300-307

20 cm thickness, which is segmented into 162 separate scintillation detectors of equal solid angle and pentagonal or hexagonal shapes. Each module provides energy and timing signals. In the present experiment, four-photon annihilation is to be isolated from considerable background arising from the dominant decays into two and three photons, which by Compton scattering may prompt four detectors to fire within the 4 ns time resolution, thus giving a spurious 49' signature. Suppression of this background is achieved using both active shielding of the detectors by their neighbours and a passive reduction of backscatter through the inner volume of the Crystal Ball by means of a collimator system [ 14]. This compound of hollow lead cells replicates the honeycomb structure formed by the NaI modules and extends from a radius of 24 cm inward to a radius of 7.25 cm. Accordingly, the thickness of the lead wall separating adjacent solid angle cells ranges from 1 cm to 0.3 cm, reducing the average opening angle of the detectors to 15.3 °, equivalent to a shadowing of 28% of the solid angle of 4zr. The collimator system is contained within a spherical carbon fibre chamber of 2.8 mm thickness. Its mechanical support requires the removal of one of the 162 detectors of the complete NaI shell. A 140 kBq 22Na fl+ source, deposited as NaC1 on a 16 /zm polyester foil, is placed at the centre of the Crystal Ball within a spherical aluminium shell of 2.3 mm inner radius and 1.4 mm thickness which serves as the annihilation medium. The 22Ne 1275 keV ),-ray which is emitted in 99.9% of all 22Na decays and is accompanied by e + emission with a probability of 90.5% [ 15] provides a trigger signal. Positrons entering the aluminium (except for a fraction of (0.85 + 0.10)% which annihilate in flight [ 16] ) are decelerated to thermal kinetic energies before annihilating with electrons from the metal. The centre-of-mass system of their annihilation is hence virtually equivalent to the laboratory system, in which the collinearity of the 2y-decay and the co-planarity of the 3),-decay are thus preserved, allowing straightforward suppression of two- and three-photon annihilation via their geometrical signatures in the Crystal Ball. Since the formation of bound states is energetically forbidden in aluminium [ 17], e+e - annihilation in this medium proceeds only from scattering states. The resulting positron lifetime (~- < 0.5 ns) [ 18] is small in comparison to the NaI time resolution (4 ns),

301

making the y-rays from the annihilation appear to coincide promptly with the trigger photon. Moreover, as the singlet annihilation rate markedly exceeds that of triplet states [ 19], the 3y annihilation probability is low; in aluminium, a 39' yield of (2.64-/-0.08) x 10 -3 has been found [20], which coincides with the value of 1/379 expected theoretically for free annihilation [ 19]. However, a high-resolution (FWHM < 0.3 ns) time spectrum of our source, obtained from a conventional yy-coincidence measurement with BaF2 scintillators, revealed a decay component with a lifetime of ( 1.83 d: 0.02) ns and an intensity of (4.6 + 0.7) % [ 21 ], which we attribute to triplet bound-state formation within the source-supporting polyester foil in the presence of strong 'pick-off' triplet-singlet conversion [22]. The corresponding excess in the 3), yield compared to annihilation in aluminium is calculated according to Ref. [20], giving (16.3 4- 2.6)%. In addition, we observed in the main experiment a delayed 3), component with an intensity of (0.11 d: 0.01)% and an average lifetime of ( 7 0 + 10) ns, which is approximately half the triplet ground-state vacuum value of 142 ns [6]; this component is believed to result from positronium drifting through residual air volumes within the source after being formed at the surface of the NaC1 powder or the polyester foil. The increase of the 3), signal within the standard coincidence window due to this long-lived orthopositronium component is found to be ( 4 . 0 + 1.1)%. We thus expect the effective 3), yield of our source to be (3.19-4-0.12) × 10 -3. The positronium populations identified above do not significantly alter the 4), yield from that expected for free annihilation. Read-out of the Crystal Ball data is initiated by the logical ' O R ' of two triggers with common dead-time. While the down-scaled 'reference trigger' merely demands a photopeak detection of the 1275 keV trigger photon, the 'main trigger' additionally requires a detector multiplicity > 5, at the same time rejecting - by means of a dedicated fast-coincidence anti-Compton and anti-2), unit - events with adjacent or opposed detectors firing. Thus, at a full-scale reference trigger rate of about 30 kHz, a hardware-triggered data rate as low as 20 events/s is achieved, which is reduced even further by a parallel processor system ( 'Heidelberg POLYP' [23]) imposing a real-time software filter [24] on the total sum-energy of the trigger-), and the annihilation photons in main trigger events

302

H. yon Busch et al. / Physics Letters B 325 (1994) 300-307

(1500 keV < Er,to~ < 2800 keV). This trigger scheme not only reconciles high effective trigger rates with low read-out dead-time, but also enables complete cancellation of dead-time and source activity by forming ratios between numbers of events recorded in the main and in the reference trigger. Moreover, reference trigger events allow a check of the real-time data reduction in the main trigger. At the same time, they provide the basis for a two-parameter stabilisation of the energy calibration of the individual Crystal Ball modules by means of the trigger line at 1275 keV and the annihilation line at 511 keV. For undisturbed detection by the NaI counters, all events are required to be separated by > 1200 ns from the preceding Crystal Ball response. Further data reduction is carded out off-line: only NaI energies surpassing a threshold of 30 keV are considered. All counts in an event must lie within a coincidence window of nominal width 16 ns; however, as the Crystal Ball is run in a self-trigger mode, imperfections in the timing alignment of the individual detector modules lead to a reduction of the effective coincidence width to about 10 ns. After identification and extraction of the photopeak count of the 1275 keV trigger photon, a sum-energy E~. sum <- 1370 keV is required for suppression of background and random coincidences. While this step already concludes the analysis of reference trigger events, the isolation of 4)'annihilation from the main trigger demands additional cuts: a fold of four is required, and events involving detectors that are first or second neighbours are rejected (extended active anti-Compton shielding). Likewise, events comprising opposed detectors or neighbours thereof are discarded (suppression of 2)'-related backgrounds). In addition, the projections of the photon momentum-sum onto the directions of each of the four firing counters are required to be zero within the energy and angular resolution. At this stage, while the originally overwhelming 2)'-decay background has already been reduced to less than a tenth of the 4)" signal [ 10], 3)'-related background dominates the spectra, which arises from two mechanisms: Compton scattering into a third neighbour through the ~ 1 cm gap between the lead collimator and the NaI shell, and coincidence of a three-photon annihilation with a bremsstrahlung photon emitted by the positron while slowing down. As a measure of an event's deviation from co-planarity, a '3-D parameter' is introduced,

which is given by the sine of the smallest angle in the event between the direction of any one firing detector and the plane containing the centre of the Crystal Ball and any two different other counters involved. For a fold-four event, values range from 0 (in the case of coplanarity of the center and at least three of the firing detectors), to X / ~ ~ 0.82 (for a tetrahedral configuration, which is however forbidden for 4)'-decay from an s-wave state [25] ). By selecting events surpassing an appropriately chosen 3-D parameter threshold, both of the aforementioned 3y-related background contributions can be suppressed, yet at the cost of severely impairing the efficiency for true 4)' events. It has proven advantageous to use instead a relatively modest minimum 3-D parameter condition ( > 0.2) in combination with a separate suppression of the principal background mechanisms: scattering events are virtually eliminated by third-neighbour anti-Compton shielding, while coincidences with bremsstrahlung photons are reduced by rejection of events with individual energies below 120 keV. Finally, a sum-energy cut is applied (944 keV < Er, s u m <-- 1100 keV). In twenty days of data-taking, 4.9 x 101° annihilations (for which the trigger photon was identified) were observed, yielding 406 events passing all 4), filter conditions. In a similar analysis of an equivalent measurement of 3)'-annihilation - the only modifications being the requirement of fold three, in addition to the trigger photon, and the omission of the 3-D parameter condition - 6.1 × 105 events were isolated from 2.6 x 109 annihilations recorded in a one-day run.

3. Simulations

For comparison of the experimental results to the theoretical predictions as well as for background estimates, Monte Carlo detector simulations [ 12] were conducted by means of the code GEANT3 (version 3.15) [ 26], using a detailed representation of the complex geometry of the Crystal Ball, including the collimator system and the annihilation source. The calculated energies deposited in the detector modules are convoluted with Gaussians of widths that are derived individually for each detector module by interpolation between an experimentally obtained set of linewidths at different energies. In order to reproduce the effect of the trigger-y, a 1275 keV photopeak energy is added

H. yon Busch et al. / Physics Letters B 325 (1994) 300-307

to each simulated annihilation. The resulting events are fed into the off-line analysis procedure. The input to the detector simulations consists of photon energies and directions chosen according to the QED predictions [3,27], which give for each decay mode a probability distribution over its phase-space. A reference against which the QED spectra may be compared is the 'phase-space prediction', which simply assumes the invariant decay amplitude to be constant over phase-space. Trivially valid in 27-decay, this assumption yields in the 3y case a triangular energy distribution which still rather closely resembles the QED result [ 27 ], while four-photon decay exhibits a more prominent difference between the simple phase-space prediction and that of QED (Fig. 1). As the majority of the positrons, although emitted centrally, annihilate off-centre in the aluminium shell of the source, the simulations use annihilation vertices distributed randomly on a spherical surface of r~ = 2.38 nun radius, equivalent to an intrusion into the aluminium by one mean penetration depth [ 28 ]. Due to geometrical shadowing by the lead collimator, noncentral emission of a photon leads to an average loss of detection efficiency which, at the radii in question, varies approximately linearly with off-centre distance. In simulations of single monoenergetic (344 keV) 7rays as well as of three- and four-photon decays, the defect in detection efficiency for emission at r a w a s found to be consistently interpretable as an average loss of (12 4- 1)% per photon compared to central emission, in agreement with simple estimates. However, neither the absolute photopeak detection efficiency of the Crystal Ball nor the similarly acting effect of non-central photon emission may be expected - both being raised to a power of four in 4y-decay - to be reproduced by the simulations with sufficient precision to match the statistical uncertainty of only 5% achieved in the total number of 4), events. Hence, in order to fully exploit the significance of the experimental data in the determination of the branching ratio F4~/F2z,, a reduction of these systematic errors is indispensable. Such a correction is made possible in a nearly ideal way by recourse to the equivalent measurement and simulation of three-photon annihilation.

303

47-Decay Predictions 2 mec2 •

lOe

\ c-

i

•

i

•

i

•

i

,

|

•

'

11(~0

'

' 1200

'

Sum-Energy

10'L TO3 10 2

LU

101

10o 700

6 0' 0

'

9 0' 0

'

1000 '

"

' 1300

'

-~'~,s~ [keV] rneC '

/~0000.

:

,

i

•

i

•

~

Ji

Photon Energies •-~ 30111111CO 20000-

O 1011110-

o 0 0

100

200

300

/.00

500

600

E./ [ k e V ] 0)

60000

.

,

E

0 0

,

.

,

.

,

•

¢j')

.Q

.

.

,

.

,

t;jt.r...-"-"

.

,

.......

.

,

.

'

' 160

.

3oooo2oooeIOOOOO" O

'

2 '0

~ '0

"

6 '0

'

8'0

'

" 100 '

'

' 120

' 140 '

" 100

(Pij, 1_<~
i

•

i

•

i

•

i

,

i

•

i

•

i

•

i

,

J

,

3-D Parameter

aoo0o- I

~o .~ 80000"~

4.0000-

,,>,

-

'

~

.

20000-

0

10

20

30

~0

50

[%]

60

70

80

90

100

Fig. l. QED (solid lines) [3] and phase-space (dashed) predictions for 4y-decay of e+e - at rest with respect to the sum-energy, to the single-photon energies, to the distribution of the angles between the directions of photon emission, and to a measure of the 3-dimensionality of the events ('3-D parameter', see Section 2 for explanation). QED can be seen to enhance extreme photon energies and relative angles, as well as co-planar configurations.

304

H. von Busch et al. / Physics Letters B 325 (1994) 300-307

4. Results and discussion

In combination with earlier results based both on numerical and semi-experimental simulations [9,10], the present experimental data and Monte Carlo simulations show that the principal background contributions surviving the 4)' analysis originate from 39' events, viz. from )'-scattering and from coincidences with a bremsstrahlung photon emitted by the positron while slowing down. Background from 2),-annihilations is found to be considerably smaller and hence negligible. Scattering has to elude at least three rings of detector modules between the counters involved in order to pass the analysis. Using the fourth-neighbour scattering probability observed in 2),-decay, the relative background contribution of 3)' scattering events to the final 4)" signal is estimated to be (0.6 4- 0.6)%. By combining simulated 3),-annihilations with the bremsstrahlung spectrum obtained experimentally from a detailed analysis of 2),-decays, coincidences of a 3),-decay and a bremsstrahlung photon are found to give a (1.0 4- 0.3)% contamination. For background resulting from random coincidences or not associated with an annihilation, a fraction of (0.8+0".8)% is obtained. The total estimated background contamination of the final 4), signal is thus (2.4+L.~)%. In a similar way, the background contribution to the final 3)" signal is estimated to be less than 0.5%. Thus, almost pure sets of equivalently re'corded and as similarly as possible analysed 3)'- and 4),-annihilation events are obtained from the presented experiment. They are displayed in Figs. 2 and 3, which show the following distributions: the sum-energy spectrum (with the indicated cut representing the final analysis condition), the spectrum of the individual energies, the distribution of the n ( n - 1)/2 relative angles between the n firing detectors, and the spectrum of the '3-D p a r a m e t e r ' defined above. Identically analysed Monte Carlo detector simulations, normalised to equal number of events, are presented for comparison. As a result of the restrictive cuts, the spectra exhibit distinct deformations compared to the original distributions, which for 4),-decay are displayed in Fig. 1. The high-statistics 3)" data (Fig. 2) show excellent agreement between experiment and QEDbased simulation. Slight deviations in the sum-energy distribution are believed to reveal the limitation of the modelling of the detector lineshapes by Gaussians.

For the 4),-decay (Fig. 3), the experimental data are compared to the QED prediction as well as to a simple phase-space calculation. Experimental and quantumelectrodynamical distributions are found to agree, the only deviation again being a slight difference in the sum-energy lineshape. Phase-space prediction can be seen to retain its characteristic original (Fig. 1) differences from QED throughout the analysis. While in some spectra the statistical uncertainty of the experimental data points precludes discrimination between QED and phase-space calculation, the experimental result for the relative angles shows a clear preference for the QED prediction. The present measurement is thus found, for threeas well as for four-photon annihilation, to agree well with the QED-based simulations in the signatures of the decays as given by their energy and angular distributions, in which experiment and theory can readily be compared, since background contamination of the experimental data is small. By contrast, checking the prediction for the probability of four-photon annihilation requires careful consideration of a variety of overall corrections in the deduction of the experimental result, which is performed as follows: both in the 4)'- and the 3),-measurement the numbers of main trigger events passing the analysis are divided by the corresponding numbers of e+e - annihilations as deduced from the reference trigger, with the identification probability for the trigger photon in reference trigger events ( simulated by 2),-decay) taken into account (0.9395 4-0.0002). The results are then divided by the simulated detection efficiencies for 4),-decay ( (4.28± 0.07) × 10 -3) and 3),-decay ((6.004-0.02) x 10-2), respectively. Corrections are applied for background, for loss by the coincidence window, for annihilation in flight, and for the effect of the sum-energy cut with regard to imperfections in the modelling of the sumenergy lineshape in the simulations. Background and coincidence loss in the reference trigger are also allowed for. These corrections amount to + ( 1.J_2~9 ) ' 2 +S 2 x "/o in four- and + (3.1 +~.~)% in three-photon decay. As discussed in Section 3, it must be anticipated that imperfections in the modelling of the photondetector interaction or an inaccuracy of the assumed off-centre distance of annihilation may lead to significant systematic errors of the simulated detection efficiency. However, since 3),- and 4),-annihilation

305

H. yon Busch et al. / Physics Letters B 325 (1994) 300-307

3y-Decay

47-Decay 2 meC2

2 mac 2 =

=

=

=

=

i

i

> 300011-

v-

'

"~.

. ! .

(I) 20000-

~

.g 20-

10000UJ 0 700

Energ,e/ 800

900

1000

1100

U.l

1200

rrleC

'

~00Q0-

,

'

,

'

,

NaI ..~

30000-

to ~"

•

~

.

,

•

;,oo

1300

Ex. . . . [keY]

J

i

i

11oo

12oo

i3bo

J

. 60-

• m 'tO"

Cut Window

.

i

Sum-Energy

co-

.~

to

i

0oo

900

lOOO

E, sum [keV]

2 150

•

,

i

•

i

•

'

.

'-~ LO ~r-

i

2

•

l

i

mec

•

i i

Nal Energies

i

t

100-

I

'

i ;

-

20O00-

50-

C.)

10000-

Q O 0

0

•

do

'

26o ' 3~o ' E [keV]

,~o

~oo

600

::i: I

..~ 0

,

I00

,

.

,

200

,

~x.w'Z~r!: ' * : ;'

~t00

500

,

300

600

E [keV] 200

-~

.o_ ,~--

30000-

"i~

Relative

20000.

. Relative Angles

""

.

,_,, 1 r- T

h

t

100-

~

.

10000,

C)

0

,

,

,

20

40

g0

80

100

i

o~

10 ~

to

104

,~

i

,

i , ~,

3 ~ . . L _- . n

m

•

i

•~

i

120

11.0

180

160

0

20

,tO

60

[deg]

(Pij, 1
i

1

•

80

100

120

1~0

100

190

g>ii, 1_~i<]_~4[deg] i

•

i

,

i

100-

•

3 D Parameter-

i0s

"

i

,

i

.

i

•

i

•

i

•

i

.

J

•

i

.

i

,

3-D Parameter

,

~ ~.

. 00-

u)

Go

"~ > ILl

"O

.

10 z ,

LLJ

101

i

20-

]

, 10 °

. 10

20

30

/~0

50

60

70

, 60

•

, 90

,

(~ 100

[%] Fig. 2. Experimental (solid lines) and simulated (dotted) energy and angular distributions of three-photon annihilation. The Monte Carlo simulations shown are based on the QED prediction for

3"y-decay [ 27 ] and are normalised to the total number of observed events. Except for slight imperfections attributed to the assumption of Gaussian energy resolution functions, excellent agreement between experiment and theory is observed.

--i :,:1:,:

0

I0

20

e 30

i 40

i 50

60

70

i 80

90

100

[%1 Fig. 3. Experimental (dots) and simulated (QED: solid lines, phase-space: dashed) energy and angular distributions of four-photon annihilation. Simulations are normalised to the total number of observed events. While the pure phase-space calculation does not reproduce well in particular the measured relative angles spectrum, QED is found to be consistent with all experimental spectra.

306

1-1.von Busch et aL / Physics Letters B 325 (1994) 300-307

involve photons of comparable energies, the 39" measurement allows the determination of a correction in terms of the absolute photopeak detection efficiency for a single 9,-ray. Given the small relative difference of the folds in question, such a correction still substantially reduces at the same time any systematic errors common to 39"- and 49,-decay that are independent of photon multiplicity. From our experiment, a 39" yield of (3.79_6.05) +0 o4 × 10 -3 is obtained, as compared to (3.19 4-0.12) x 10 -3 expected for our source. We conclude from this discrepancy that the simulated photopeak detection efficiency for a single 9,-ray is too small by (5.6+_~3)%. This correction is applied to the simulated 49' detection efficiency, with the slight difference of (3.1 4- 1.2) % between the effective 39' and 49' offcentre distances of annihilation (caused by the formation of positronium within the inner volume of the source) taken into account. The resulting additional correction for the 49' yield is - ( 19.2 4- 4.4)%. Since the difference between the 49" annihilation yield and the 49"/29" branching ratio is negligible in the present case, we obtain finally F4r/I'2r = ( 1.50 ± 0.07(stat.) -4- 0.09(syst.) ) x 10 -6.

(1)

Our result is considerably more accurate than the value given in [7] and is in excellent agreement with the lowest-order QED value of 1.48 x 10 -6 [ 2 - 4 ] . Calculating the 49" detection efficiency from the phase-space simulations gives less than half the above 49"/29" ratio, demonstrating the sensitivity of the analysis to the decay signature.

5. Acknowledgements The authors wish to thank Dr. K.-H. Streng for providing the 49' event generator code and Prof. Dr. P.M. Zerwas and Prof. Dr. O. Nachtmann for valuable advice. We are indebted to Dr. R.H. Howell and Prof. Dr. K. Maier for helpful discussions on the source corrections and gratefully acknowledge Dr. R. Wiirschum and Prof. Dr. H.-E. Schaefer for the high-resolution time measurement of our annihilation source and for valuable comments. We wish to thank Dr. J.B. Fitzgerald for helpful suggestions on the draft of this paper. This work was supported by the Bundesministerium

ftir Forschung and Technologie, Germany, under contract No. 06HD133I, and by the Gesellschaft ffir Schwerionenforschung, Darmstadt, Germany.

References [1] A. Rich, Rev. Mod. Phys. 53 (1981) 127. [2] G.S. Adkinsand ER. Brown, Phys. Rev. A 28 (1983) 1164, and references therein. [3] G.P. Lepage, P.B. Mackenzie,K.-H. Strengand P.M. Zerwas, Phys. Rev. A 28 (1983) 3090. [4] S. Adachi, Bachelor's thesis, Tokyo MetropolitanUniversity (1990), unpublished. We obtained this result through Ref. [7]. [5] D.W. Gidley,A. Rich, E. Sweetmanand D. West, Phys. Rev. Lett. 49 (1982) 525, [6] J.S. Nico, D.W. Gidley, A.Rich and EW. Zitzewitz, Phys. Rev. Lett. 65 (1990) 1344, and references therein. [7] S. Nagayama, Y. Nakamitsu, T. Sato and T. Yamada, Phys. Rev. Lett. 65 (1990) 2634. [8] W. Wahl, D. Habs, V. Metag, D. Schwalm and R.S. Simon, Annual Report MPI-H (1984) 104. [9] G. Enders, D. Habs, D. Schwalm, W. Wahl and R.S. Simon, Annual Report MPI-H (1986) 78. [10] Th. Schulze, G. Enders, J. Gerl, D. Habs and D. Schwalm, Annual Report MPI-H (1989) 63. [ 11] H. von Busch, Diplomathesis, MPI-H Report V 30 (1992); H. von Busch et al., Annual Report MPI-H (1992) 83. [12] P. Thirolf et al., Annual Report MPI-H (1992) 85. [13] V. Metag et al., Nucl. Phys. A 409 (1983) 331c. [ 14] M. Mu~'ic,Ch. Ender, D. Habs, J. Kramp, J. Schirmer and D. Schwalm, Annual Report MPI-H (1985) 98. [15] C.M. Lederer and V.S. Shirley, Table of Isotopes, Seventh Edition (John Wiley & Sons, New York, 1978). [16] Obtained (with ZAI = 13 and Emax,~+; 22Na= 0.545 MeV [15]) by means of the nomogram in: J. Kantele and M. Valkonen,Nucl. Instr. Meth. 112 (1973) 501. [17] P. Hautoj~rvi and A. Vehanen, in: Positrons in Solids, Topics in Current Physics 12, ed. P. Hautoj~vi (Springer, Heidelberg, 1979) p. 1. [18] H,-E. Schaefer, Phys. Stat. Sol. (a) 102 (1987) 47; M. Puska and R. M. Nieminen, J. Phys. F: Met. Phys. 13 (1983) 333; C. Hidalgo, S. Linderoth, G. Gonz~ilez-Donceland J. San Juan, in: Positron Annihilation,eds. L. Dorikens-Vanpraet, M. Dorikens and D. Segers (World Scientific, Singapore, 1988) p. 371. [191 A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696; for a discussionof the most recent progress in experimentand theory see also Refs. [5,6]. [20] M. Bertolaccini,C. Bussolati and L. Zappa, Phys. Rev. 139 (1965) A 696, and references therein. [21] R. Wiirschum and H.-E. Schaefer, University of Stuttgart (1993), unpublished. [22] M. Bertolacciniand L. Zappa, NuovoCimento 52 B (1967) 487.

H. yon Busch et al. /Physics Letters B 325 (1994) 300-307

[23] Ch. Eader, R. M/inner and P. Bauer, IEEE Trans. Nucl. Sci. 36 (1989) 1518. [24] E K6ck, Ch. Ender, J. Gerl, D. Habs and D. Schwalm, Annual Report MPI-H (1989) 41. [25] H.S. Mani and A. Rich, Phys. Rev. D 4 (1971) 122.

307

[26] R. Brun, M. Hansroul and J.C. Lassalle, GEANT User's Guide, CERN, DD/EE/82 edition (1982). [27] G.S. Adkins, Ann. Phys. (New York) 146 (1983) 78. [28] T. Baltakmens, Nucl. Instr. Meth. 142 (1977) 535.