Journal of Nuclear and Radiochemical Sciences, Vol. 3, No. 2, pp. R1-R7, 2002

Reviews
Positron Age-Momentum-Correlation (AMOC) Measurements on Organic Liquids
Petra Castellaz, Andreas Siegle,† and Hermann Stoll*
Max-Planck-Institut für MetallforschungHeisenbergstr.1, D-70569 Stuttgart, Germany
Received: November 18, 2002; In Final Form: August 7, 2002
By means of correlated measurements of the individual lifetimes of positrons (called positron ages) and of the
Doppler shifts of their 2γ-annihilation radiation (Age-Momentum Correlation, acronym AMOC) the room-temperature behaviour of positrons and positronium in n-hexane, tetramethylsilane, benzene, methanol, ethanol, nbutanol, n-hexanol, n-octanol, and water has been investigated. The annihilation parameters of “free” positronsand of para- and orthopositronium (lifetimes and Doppler-broadened lines) have been deduced from a two-dimensional analysis of the AMOC data. The lifetimes of positrons are found to be related to the electronic polarizability of the investigated liquid. The behaviour of the Doppler-broadened free-positron and orthopositronium
linewidths is explained in terms of structural and dipolar properties of the liquid molecules.

1. Introduction
Positrons (e+), the positively charged anti-particles of theelectrons (e − ), are widely used as non-destructive mobile
probes in condensed matter, e.g., for the study of vacancy-like
defects in crystals. In many isolating materials positronium(Ps = e+e−), the bound state of a positron and an electron, is at
our disposal as an additional, electrically neutral probe. With
some justification, Ps can be regarded as the lightest isotope ofhydrogen. It may undergo chemical reactions analogous tothose of hydrogen (e.g., oxidation or complex formation). In
contrast to most other branches of reaction chemistry, in which
information on the reactions is primarily derived from studying
educts and products, positronium chemistry offers the possibility to observe the behaviour of one of the reaction partners,
viz. the positronium, rather directly on times scales between
10−11 and 10−8 s.
In the past, positronium chemists studied primarily liquid
solutions that were based on a variety of organic and inorganic
solvents. 1−3 This was particularly true of one of the key
subjects of positronium chemistry, viz. the formation of
positronium in liquids. Here, experiments on scavenger solutions provided insight into the processes involved. 4 Morerecently, this “chemical” approach, based on investigating e+
and Ps reactions with different reactants, has been supplemented by comparing Monte Carlo simulations of the processes
envisaged in Mogensen's spur model of Ps formation 3, 5 withpositron annihilation and electron-radiolysis experiments.6, 7
Clearly, reliable knowledge of the behaviour of e+ and Ps in
the pure solvents is a prerequisite for a deeper understandingof e+ and Ps reactions in solutions. The present work investigates the behaviour of e+ and Ps in various pure organic andinorganic liquids in a “physico-chemical” approach, in whichmaterial parameters such as surface tension, polarizability, or
electron mobility are systematically varied. It is based on the
AMOC (= Age Momentum Correlation) technique, which has
already proved to be an excellent tool for studying e+ and Ps
reactions in liquid solutions.8-10
Section 2 of this publication reviews the experimental
evidences for positron states occurring in organic liquids. The*Corresponding author. E-mail: stoll@mf.mpg.de. FAX: +49711-1932.
† Present address: Robert Bosch GmbH, D-70442 Stuttgart,
Germany.

AMOC set-up used for the measurements and the sample
preparation are briefly described in Section 3. The experimental results will be presented in Section 4 and discussed indetail in Section 5, followed by conclusions in Section 6.
2. Basic Processes
The conventional techniques for studying positron annihilation provide information either on the spectrum of positron
lifetimes or on the distribution function of the momenta ofannihilating positron-electron pairs. The positron lifetime
spectrum is deduced from the distribution function of thepositron ages, i.e., the times spent by the positrons in thesample before they annihilate with electrons. The reciprocal
positron lifetimes associated with the distinct positron states,
called annihilation rates, are proportional to the electron densities at the sites at which the positrons annihilate. The information on the momentum distribution of the annihilating e+e−
pairs is obtained from the 2γ-annihilation radiation. In the rest
system of an annihilating e+e− pair the two γ-quanta are emitted
collinearly, each of them with an energy of mec2 = 511 keV,
where me is the electron mass and c the velocity of light invacuum. In the laboratory system, the e+e− pairs in general
have finite momenta. The momentum components parallel tothe emission axis of the annihilation photons, the so-calledlongitudinal momenta, give rise to Doppler shifts of the photon
energies. For an ensemble of positrons annihilating from acertain positron state these shifts result in a Doppler-broadened
annihilation line characteristic for this distinct positron state.
The AMOC technique, to be described in detail in Section 3.1,
combines lifetime and Doppler-broadening spectrometry bycorrelated measurements of the positron age and the Doppler
shift of one of the annihilation quanta. Therefore, it allows us
not only to separate the momentum distribution associatedwith the annihilation processes with the different lifetimes
from each other but also to follow the behaviour of the e+ as afunction of the positron age.
In most Ps-forming liquids, at least three distinct positronstates may be detected, viz. (i) the singlet Ps ground state,called parapositronium (p-Ps), in which the e+ annihilate with
their “own” electrons, (ii) free positrons (e+ ), annihilating
predominantly with valence electrons of the surrounding molecules, and (iii) orthopositronium (o-Ps), the triplet Ps groundstate. The self-annihilation of isolated o-Ps would lead to the© 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 12/28/2002

Castellaz

J. Nucl. Radiochem. Sci., Vol. 3, No. 2, 2002

emission of three γ-quanta with the total energy 2m e c . In
condensed matter, however, it is much more probable that thee+ of o-Ps annihilates with an electron with opposite spin of
one of the surrounding molecules. This so-called pick-off
process results in the emission of two 511 keV photons.
In the case of “intrinsic” annihilation of p-Ps (= annihilationof the positrons with its “own” electrons) the Doppler broadening provides information on the residual kinetic energy of pPs. In the case of annihilation of “free positrons” or “pick-off”
annihilation of the positrons of o-Ps with “foreign” electrons
the Doppler broadening reflects the momentum distribution of
the annihilating electrons since the momenta of the electrons
involved are generally higher compared with those of thepositrons.
The characteristic annihilation features of the three positronstates described above are summarized in Table 1. The very
weak Doppler broadening associated with the annihilation ofthermalized p-Ps is due to the fact that here both annihilation
partners have very low momenta. Non-thermalized p-Ps,
2

TABLE 1: Annihilation Characteristics of the Positrons in
Ps-Forming Materials
State

Annihilation mode

Lifetime

Doppler broadening

p-Ps

self-annihilation≈ 0.12 nsvery weakfree e+with valence e

−

o-Ps

pick-offo-Ps

3γ-annihilation

0.4 – 0.5 nsstrong0.5 – 5 ns

strong140 nsstrong, asymmetricalhowever, may give rise to appreciable Doppler broadening of
the 2γ-annihilation line.11, 12
However, Doppler-broadening measurements do not show
p-Ps annihilation lines as narrow as they ought to be. This isdue to the limited resolution of the energy detectors used in theexperiment. It leads to a convolution of the annihilation lineshape with the energy resolution function of the detector. In
order to achieve the “true” annihilation linewidths it is necessary to deconvolute the measured annihilation lines after themeasurement.
Lifetime measurements are very suitable for determining the
lifetime of o-Ps and the intensity of the o-Ps state, which is therelative number of positrons annihilating from this state, sincein time regions above 1 ns, o-Ps is the only surviving species,
due to its long lifetime. Therefore, a “pure” o-Ps lifetime spectrum is observed in this time region. However, the limited
resolution of the lifetime spectrometers often makes the separation of the p-Ps and free e+ states difficult, since they annihipositron
beam (4 MeV)

start detector(efficiency 100%)
TAC

plastic
scintillatorl N2

energydetectorγhigh-purity 511
Ge
keV

γ
BaF 2

stop
detectorlate predominantly in the same time region (0 to 0.6 ns). The
AMOC method overcomes this difficulty by allowing us todetermine the Doppler-broadened lineshapes and the annihilation rates of the various positron states rather accurately.14, 15
In contrast to age-integrated Doppler-broadening measurements, AMOC measures the Doppler-broadening associatedwith the various positron states in their characteristic lifetime
ranges. For positron-chemistry studies of positronium-forming
materials such as organic liquids AMOC is the ideal tool since
it combines the o-Ps-sensitive lifetime measurement with the
Doppler measurement, which permits easy identification of pPs with its characteristic narrow momentum distribution.
3. Experimental
3.1. AMOC Spectrometer and Visualization. Figure 1 isa schematic of the AMOC set-up installed at the Stuttgart MeV
positron beam. 16-18 Positrons from a 4 MeV beam passing
through a plastic scintillator with a high detection efficiency
generate the start signal for the age measurement with a timeto-amplitude converter (TAC) before being implanted into thesample. The stop signal is derived from the detection of one of
the annihilation quanta by a BaF 2 scintillator. The second
annihilation quantum is used for the correlated measurement of
the Doppler shift by means of a high-purity Germanium
detector installed opposite to the stop detector. The signals
from the age and the energy measurement are correlated by a
two-parameter multi-channel analyser (MCA). For the data
presented here the time resolution was about 250 ps (FWHM),
the energy resolution (monitored by a simultaneous measurement of the 103Ru γ-photo line) about 1.4 keV (FWHM), andthe count rate typically about 500 triple coincidences per
second.
By means of a two-parameter multichannel analyser theannihilation events are arranged in a two-dimensional
histogram. This histogram may be visualized by an AMOC
relief (Figure 2) where the number of counts is plotted logarithmically versus the positron age and the energy of the annihilation radiation. Summing all counts along the age axis
results in the time-integrated Doppler-broadened lineshape of
the photon energy, summing along the energy axis in the usual
lifetime spectrum.
For the visualization of the correlation between Doppler
broadening and positron age it is instructive to compute the10
counts

R2

10

10

6

4

2

sample2-dim.
MCA

Figure 1. Schematic of the AMOC set-up installed at the Stuttgart
MeV positron beam. The start signal for the positron-age measurement is generated by the passage of a positron through a plastic scintillator, the stop signal by the detection of an annihilation photon in a
BaF 2 scintillator. The second annihilation photon is used for themeasurement of the Doppler shift by means of a liquid-nitrogencooled high-purity-germanium detector.

520
0
pos
10
itro
na
20
ge
[ns
]

515
505
500

[
rgy

ne

ne

to
ho

]

keV

510
p

Figure 2. Room temperature AMOC relief of benzene. The counts
are plotted vs. the individual positron lifetime (= positron age) and theenergy of the annihilation quanta.

Positron Age-Momentum-Correlation (AMOC)

J. Nucl. Radiochem. Sci., Vol. 3, No. 2, 2002

R3

n-hexanol (C6H13OH), n-octanol (C8H17OH), and water (H2O).
All samples were degassed by the freeze-pump-thaw method20
and filled into special glass vessels with thin windows thatallowed implantation of the MeV positrons into the samples.

0.60

4. Results

St

0.55

0.50

0.45
0

2

4
positron age [ns]68

Figure 3. Lineshape function S t(t) of benzene at room temperature,
calculated from the AMOC relief shown in Figure 2.

age-dependent lineshape function S t (t) (Figure 3) by determining separately for each time channel of the AMOC relief(see Figure 2) the lineshape parameter S (defined as the ratio of
the number of counts in the centre of the annihilation line to
the number of counts in the entire line19) and plotting it versus
the positron age. A large S parameter corresponds to a narrow
Doppler-broadening of the annihilation line, a small one to abroad line.*
The characteristic lineshape function S t (t) of a material
forming positronium (see Figure 3) shows large S t (t) values at
short positron ages, resulting from the self-annihilation of thermalized or nearly thermalized p-Ps. This is followed by asteep decrease of S t (t). At larger t the level corresponding to
the rather broad momentum distribution of the electrons whichannihilate with the positrons of o-Ps via the pick-off process isreached (see also Table 1). In cases where the momentum
distribution of the electrons annihilating with free positrons isdifferent from the momentum distribution of the electrons
annihilating with the positrons of o-Ps via a pick-off process,the annihilation of the free positrons shows up as a change in
S t (t) in a time range around 1 ns (see, e.g., Figure 7). Clearly,
this is not the case with benzene (cf. Figure 3).
3.2. Sample Preparation. AMOC measurements were
performed at room temperature on benzene (C6H6), n-hexane(C 6 H 14 ), tetramethylsilane ((CH 3 ) 4 Si, abbreviated TMS),
methanol (CH3OH), ethanol (C2H5OH), n-butanol (C4H9OH),

The AMOC reliefs were analysed using a two-dimensional
fitting procedure.21 In order to make full use of the measured
AMOC data a two-dimensional model function representing
the number of counts as a function of positron age and energyof the annihilation quanta is fitted to the AMOC relief without
prior data reduction. On the age axis, each positron state, i.e.
free positrons, p-Ps and o-Ps, is represented by an exponential
decay function convoluted with the time resolution function of
the spectrometer. On the energy axis, one Doppler-broadened
Gaussian lineshape corresponds to each positron state. The
annihilation with core electrons is taken into account by abroad Gaussian line of low amplitude.
For all liquids investigated, three positron states (p-Ps, e +,and o-Ps) sufficed for describing the data adequately. There
was no need to take reactions among these positron states into
consideration. In particular, no evidence of a fourth positronstate in n-hexane was found as proposed earlier in the literature.3, 22 Moreover, in contrast to Reference 23, no significant
deviation from the statistical ratio 1:3 of the formation probabilities of p-Ps and o-Ps was observed; hence, it suffices toreport the fraction IPs of positrons annihilating from positronium states, the so-called positronium yield. The fraction of
positrons not forming Ps is thus given by 1-IPs.
In all Ps-forming materials examined so far by the AMOC
technique, juvenile broadening arising from the delayedslowing-down of positronium11, 12, 18 was observed: in contrast
to the charged positrons, the electrically neutral positronium
which is formed at an early stage of the slowing-down process
cannot loose its kinetic energy via ionization of molecules orcreation of excitons but only in a less effective way via phonon
excitation. Therefore, a considerable amount of positronium is
not thermalized when it undergoes annihilation. Its kinetic
energy results in a large Doppler broadening visible at young
positron ages. This juvenile broadening was taken into accountby a two-state rate-equation model.12 This model approximates
the Ps energy loss by allowing for transitions between epithermal and slowed-down Ps. An improved analysis of AMOC
data on Ps slowing-down is suggested by Seeger.13
The annihilation parameters of the three positron statesobtained by the two-dimensional analysis program for theliquids examined in the present work are listed in Table 2
together with the typical statistical errors.

TABLE 2: Annihilation Parameters as Obtained from the Two-Dimensional Data Analysis. The last line of the table gives
the typical statistical errors for the various quantities. Since all data analyses were based on similar statistics, the statisticalerrors are about the same for all samples. The Gaussian linewidths (standard deviations) i of the annihilation lines are
given after deconvolution with the energy resolution function
Sampleτ p-Ps [ps]

τ e+ [ps]τ o-Ps [ps]IPs

σ p-Ps [keV]

σ e+ [keV]

σ o-Ps [keV]

benzenen-hexaneTMS
methanolethanol
n-butanoln-hexanol
n-octanol
water

118
127
134
139
132
126
123
122
131

414
452
506
442
435
420
417
412
400

3059
3724
4619
2997
3584
3323
3166
3087
1848

0.563
0.466
0.698
0.293
0.294
0.311
0.318
0.330
0.365

0.33
0.27
0.22
0.26
0.27
0.27
0.27
0.28
0.43

0.89
0.92
0.87
1.05
1.03
1.04
1.03
1.03
1.09

0.89
0.89
0.84
0.96
0.92
0.93
0.93
0.93
1.07

stat.error

±2

±1

±3

± 0.005± 0.02± 0.01± 0.01*The absolute value of S has no immediate physical significance since the width of the centre region of the annihilation lines is deliberately
chosen to give S parameters near 0.5 in order to achieve maximum sensitivity against small variations of the linewidth.

R4Castellaz

J. Nucl. Radiochem. Sci., Vol. 3, No. 2, 2002

5. Discussion
5.1. Orthopositronium Lifetime. It is generally acceptedthat the pick-off lifetime of the o-Ps state in liquids can be
described by the bubble model proposed by Ferrell.24 The Ps
bubble arises from the “quantum” pressure of the Ps motion
which is in equilibrium with the surface tension of thesurrounding liquid. Tao25 showed that the bubble model yields
the following semi-empirical relation between the pick-off
annihilation rate λo-Ps and the macroscopic surface tension γ:
λo-Ps = 1.9 × 10−9 (s−1m1/2N−1/2) × γ1/2.

(1)This relation was shown to be valid for a large variety oforganic and inorganic liquids.
Figure 4 shows the o-Ps annihilation rate as a function of thesurface tension for various liquids. The good agreement of theresults from the present AMOC measurements with eq 1 (solidline in Figure 4), which was derived earlier,20, 25 is obvious.
More recent publications discuss modifications of the bubble
model concerning a modified surface tension of the Ps bubble26(cf. Section 5.3) and a more elegant calculation of the bubble
energy.27
5.2. Free-Positron Lifetime. In 1968, Gray et al.20 showedthat “partial” annihilation rates λ * partial may be derived for
different organic functional molecular groups and that the
specific pick-off annihilation rate λ*po = λpo /VM (λpo denotes the
pick-off annihilation rate and V M the molar volume) of anorganic liquid may be determined by summing up these partial
rates according to the given molecular structure:
λ*po = Σ λ*partial .

lifetime spectrometers which prevented an accurate determination of the free-positron lifetime in the presence of anothershort-lived component such as p-Ps.
A positron implanted into matter polarizes its surroundings
due to its electric charge, attracting the electrons of the molecules. The larger the polarizability, the more the electron
density increases at the positron site and the faster the positronannihilates. Figure 5 shows the specific annihilation rate of thefree positrons λe+* determined from AMOC measurements as afunction of the polarizability α of the liquids determined fromrefraction measurements.28 As expected, the annihilation rateincreases with increasing polarizability of the surrounding
molecules. A fit of a straight line
λ e+*(α) − λ e+*(0) = b × α

to the data points in Figure 5 leads to a proportionality factor
b = 0.555 × 10−3JmC−2s−1 and a constant λe+*(0) = 2.4×10−20
m3s−1.
5.3. Doppler Broadening of the Annihilation Lines. As
already mentioned in Section 4, the linewidths of the Dopplerbroadened lines of the different positron states may be determined rather accurately by the AMOC method. The
two-dimensional analysis uses Gaussian lines with standard
deviations σ i for the modelling of the Doppler-broadened lineshapes for all occurring positron states i. If we assume that thetrue velocity distribution for the motion of positronium in the
bubble is not far away from a Maxwell velocity distribution,
this will lead to a Gaussian Doppler broadening of the annihilation line.17 From this assumption, the mean kinetic energy ofp-Ps can be derived as(2)
E kinp-Ps = 3

It was known that the electronic polarizability does also
have this property. From this coincidence, Gray et al. derived
a correlation between electronic polarizability and pick-off
annihilation rate. It was Tao25 who pointed out that the pickoff annihilation rate may be related to any quantity which
represents the electronic interactions between the molecules.
Both surface tension γ and polarizability are, in fact, such
quantities; since it is not to be expected that the behaviour ofelectrically neutral positronium is influenced by the polarizability of the surrounding molecules the surface tension issurely better suited as a parameter of the bubble model.
However, a correlation between the polarizability of themedium and the annihilation rate would be more straightforward for the free positron since it is a charged particle. To ourknowledge, this correlation has not been discussed in the literature yet. This may be due to the limited time resolution of the(3)

σ 2p-Psme c 2(4)

where me is the electron mass and c the velocity of light. If wegeneralize this relation for the annihilation of free positronsand o-Ps, we have to consider that in both cases themomentum of the annihilating e+e− pair is mainly determinedby the momentum of the electron (which generally is bound toa molecule) whereas the positron can be regarded ascompletely thermalized. As only one particle, the electron,
contributes to the momentum distribution, we have to calculate
its mean kinetic energy according to Reference 17
E kine− (e+) = 6

σ 2e+
σ2
and E kine− (o-Ps) = 6 o-Ps2
2
me c
me c

(5)

from the deconvoluted Gaussian linewidths σ i obtained fromthe two-dimensional data analysis.

0.8

0.8

0.6

0.6
λ e+ [10−18m3/s]

H 2O
benzene0.4

methanolethanol
n-butanoln-hexanol
n-octanoln-hexane
0.2

TMSn-hexanen-hexanol
TMS

0.4

n-butanolbenzene∗

λ 0-Ps [109/s]

n-octanol

ethanol0.2

methanol
H2O

0.0

0.0
0

2

4
γ

1/2

6
−3/2

[10

8

10

1/2

(N/m) ]

Figure 4. o-Ps pick-off annihilation rate λo-Ps vs. surface tension γ(Ref. 29) for benzene, n-hexane, tetramethylsilane (TMS), water, andvarious normal alcohols. The solid line was calculated from eq 1.

0.0

0.2

0.4

0.6
−15

α [10

2

0.8

1.0

2

C m /J]

Figure 5. Specific free-positron annihilation rate λ*e+ = λe+ /VM vs.
polarizability α (Ref. 28) for benzene, n-hexane, tetramethylsilane(TMS), water, and various normal alcohols.

Positron Age-Momentum-Correlation (AMOC)

J. Nucl. Radiochem. Sci., Vol. 3, No. 2, 2002kin

E p - P s was calculated from the Doppler-broadened
linewidths of p-Ps given in Table 2, and the results wereplotted versus the surface tension γ of the measured liquids in
Figure 6. The mean kinetic energy of p-Ps is well above thethermal energy of kBT = 0.025 eV at room temperature. It
increases with increasing surface tension, i.e., with decreasing
bubble radius, as predicted from the bubble model.25 Thismodel can be used for a calculation of the Ps kinetic energy inthe bubble using a simple square potential in which the Ps is
caged. 24 Under the assumption of infinitely high potential
walls Mogensen3 gives
EkinPs = d × γ1/2, with d = 7.68 eVN −1/2m1/2.

(6)Equation 6 gives a qualitative description of our experimental results (cf. full lines in Figure 6) but the calculated
energies are overestimated by about a factor of two. This isthought to be expected when a model with infinitely highpotential walls is used. A comparison of our results with thosefrom the refined bubble model of Byakov et al. 26 shows arather good agreement for the Ps kinetic energies in n-hexane,
benzene, TMS, ethanol, octanol, and water.
In Figure 7 the lineshape functions S t (t) obtained for nhexane and n-hexanol are compared. Both curves show thecharacteristic features of positronium formation. The lineshape functions at old positron ages, which can be attributed tothe pick-off annihilation of o-Ps, are nearly the same. In
contrast to the lineshape function of n-hexanol showing apronounced minimum in the region of 0.5 to 2 ns, S t (t) of nethanol
n-butanoln-hexanol
n-octanolkin

Ep-Ps [eV]

2.0

R5

hexane passes through a quite shallow minimum. Since theminima occur in the time range where the annihilation of thefree positrons takes place, it is straightforward to suppose thatthe electrons which annihilate with the free positrons in nhexanol have a broader momentum distribution than those inn-hexane. The fact that the lineshape function in n-hexane
shows only a shallow minimum seems to indicate that the free
positrons and the positrons of o-Ps in n-hexane annihilate withelectrons of essentially the same momentum distribution.
In order to test whether this finding may be generalized, themean kinetic energies of the electrons annihilating with free
positrons E kine− (e+) and those annihilating with the positrons ofo-Ps E kine− (o-Ps) were calculated from eq 5 and are shown in
Figure 8 for all liquids measured. The two energies are almost
equal for the nonpolar liquids benzene, n-hexane, and TMS,
but different by more than 3 eV for the alcohols which arepolar molecules owing to the large electronegativity of theoxygen of the hydroxyl group. In polar liquids, free positrons
are expected to be solvated by the molecular dipoles. Only thewater molecule, which has the largest dipole moment of all
liquids examined, does not fit into this pattern. This exception
will be discussed in Section 5.5. Figure 9 shows the schematic
molecular structure of methanol with the permanent partial
charges δ+ and δ−.
Free positrons are attracted by the negative partial charge on
the hydroxyl group resulting in a solvation sphere of solvent
molecules around the positrons. They can, therefore, annihilate mainly with the p-electrons of the oxygen since theseelectrons are nearest. As shown schematically by the example
of OH− in Figure 10, the non-binding p-electrons have a larger
energy than the other electrons of the molecules and hencecause a larger Doppler broadening on annihilation.
Positronium as a neutral particle does, in contrast to the free
positron, not interact with the dipole moment of the molecules.

14

water

1.0

13

hexanebenzene

0.0
0

20

40

60

80

100

−3

γ [10 N/m]

E kine− [eV]12methanol

TMS

11
10

Figure 6. p-Ps mean kinetic energy E kinp-Ps vs. surface tension γ (Ref.
29) for various liquids. The solid line was calculated from eq 6.

e+
o-Ps

9

0.60
8

ne ane MS ater anol anol anol anol anolTw eth eth but
exex octn-h
mnn-h n-ze

nbeFigure 8. Mean kinetic energies E kine− of the electrons annihilatingwith free positrons (e+, full symbols) and with o-Ps (open symbols) forsome polar and nonpolar liquids.

0.55
St

n-hexane0.50

Hn-hexanol

δ

+

H

H

C

O
−
δ

H

0.45
0

2

4
positron age [ns]6p-electrons

8

Figure 7. Lineshape functions S t (t) obtained for n-hexane and nhexanol calculated from the AMOC reliefs.

Figure 9. Molecular structure of methanol showing the partial
charges δ+ and δ− and the p-electrons (ellipses). The black triangle
denotes the polarization of the binding electrons towards the electronegative oxygen atom.

Castellaz

J. Nucl. Radiochem. Sci., Vol. 3, No. 2, 2002−

O orbitalσ*energy

1s

15

0.4

10

2p
σ
2s

2s

Figure 10. Schematic diagram for the energy levels of a heteropolarchemical bond on the example of OH−. σ and σ * denote the binding
and non-binding molecular orbitals, respectively.

14.0

H2O
13.5
E kine− (e+) [eV]E kine− (o-Ps) [eV]−OH orbitals

H orbital

rel. p-e−

R6

13.0

methanolethanol
n-butanoln-hexanol
n-octanol

12.5

12.0

0

20

40
60
dielectric constant ε80

Figure 11. Mean kinetic energy E kin e− of the e + e − pair for freepositron (e+) annihilation vs. static dielectric constant ε (Ref. 28) for
CnH2n+1OH (n = 0, 1, 2, 4, 6, and 8).

The pick-off process is, therefore, not selective and its
Doppler-broadening linewidth is a measure for the momentum
distribution of all outer electrons of the molecules. In the caseof non-polar molecules there is no dipole moment, free
positrons are not solvated and, therefore, the free positrons andthe o-Ps annihilate with the same type of electrons.
Our findings agree with the work of Byakov et al.30 who
focus on the orientation of liquid molecules perpendicular to
the Ps bubble surface: their experimental results suggest that
the OH-group of methanol points away from the Ps bubble.
The positron of o-Ps, therefore, annihilates predominantly with“low energy” electrons on the methyl group.
When mean kinetic energies are discussed we should notforget that o-Ps is also caged in the positronium bubble. Thiseffect also contributes to the o-Ps linewidth. As shown above,
the bubble model (eq 6) gives a rather good qualitative description for the kinetic energy of p-Ps in the bubble. The kinetic
energy of o-Ps must be approximately the same. This cannot
be tested by experiment, since o-Ps annihilates via pick-off
process. As the momentum distribution of the bound electrons
annihilating in the pick-off process is much broader than the
momentum distribution due to the motion of the Ps in the
bubble, it gives only a small contribution to the measured annihilation linewidth. It is, therefore, neglected in this discussion.
5.4. Positron Annihilation in Polar Liquids: Normal
Alcohols. In order to test the influence of the strength of themolecular dipole, the results obtained on a series of normal
alcohols with increasing carbon chain length, and, therefore,
decreasing dipole moment can be compared. Figure 11 shows
the mean kinetic energy of the electrons annihilating with free
positrons plotted versus the static dielectric constant ε of the

0.2
0.0
0

2
4
6
carbon chain lenght n

8Figure 12. Upper part: Mean kinetic energy **Ekinp-Ps of the e+e− pairs
for the annihilation of o-Ps vs. carbon chain length n for CnH2n+1OH (n= 0, 1, 2, 4, 6, and 8). Lower part: Fraction of valence electrons being
p-electrons as a function of n.

liquid. The energy E kine− (e+) increases with increasing ε and is
the highest in water which, from this point of view, may be
regarded as an alcohol of chain length n = 0. It may beconcluded from this result that with increasing dipole moment
the probability for the free positron to annihilate with one ofthe p-electrons at the oxygen is also rising.
Finally the seemingly exceptional behaviour of water has tobe discussed where the mean kinetic energies of the electrons
annihilating with free positrons E kine− (e+) and with o-Ps E kine−(o-Ps) are very high and almost equal as shown in Figure 8. In
the previous section we gave an explanation why the energy ofelectrons annihilating with free positrons is high, but what isthe cause for the high energy in the pick-off process (upper
part of Figure 12)? The lower part of Figure 12 gives apossible explanation for this finding. Since positronium is anelectrically neutral particle there should be no preferences of
its position respective to the molecular structure and its
positron can annihilate via pick-off with the same probability
with any valence electron of the molecules. In water, fifty percent of the valence electrons are p-electrons. This number
decreases with increasing carbon chain length, as shown in
Figure 12. The more valence electrons are p-electrons thelarger the Doppler broadening in the pick-off process.
6. Conclusions
The present AMOC measurements performed on benzene,n-hexane, tetramethylsilane, water, methanol, ethanol, nbutanol, n-hexanol, and n-octanol have shown that the annihilation parameters of free positrons and of para- andorthopositronium such as positron annihilation rates andDoppler-broadened linewidths are correlated with macroscopic
and structural features of the liquids. The pick-off annihilation
rate of o-Ps can be well described by the bubble model where
the intramolecular electrical forces are expressed in terms of
the surface tension of the liquid. A clear correlation between
the electronic polarizability of the molecules and the lifetimeof the free positrons could be established for the first time.
The detailed information on the Doppler-broadened linewidthof the distinct positron states obtained by AMOC allowed us todemonstrate that in polar liquids free positrons are solvatedand, therefore, annihilate with electrons with a momentum
distribution which is different to that of the electrons responsible for pick-off annihilation.
The experiments performed so far indicate that the fractionof free positrons annihilating from a solvated state increases
with increasing dipole moment.

Positron Age-Momentum-Correlation (AMOC)

Acknowledgements. The authors would like to thank all of
those who were involved in the Stuttgart MeV positron beam.
This paper would not have been possible without the work of
all of them. In particular we would like to thank Prof. Dr. A.
Seeger for his continuous support and Dr. I. Billard, Strasbourg,
for many fruitful discussions.
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