Journal of Nuclear and Radiochemical Sciences, Vol. 7, No.1, pp. R37-R52, 2006

Positronium Atom in Solids –Peculiarities of Formation and Interconnection with Free
Volume Nanostructure–
V. P. Shantarovich*
N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Kosygin st. 4, 119991 Moscow, Russia
Received: March 10, 2006
The recent several decades have seen the spectacular advances in a new area at the junction of elementary particle
physics and chemistry, the so called chemistry of new atoms in which either electron is replaced by another negative
particle or a proton by another positive particle. The area of research does not confine itself to chemical studies
alone. External influences on the properties of new atoms provide useful, sometimes unique information about thestructure of solids. In this review, we survey briefly properties of the new atoms and illustrate the last statement on
example of positronium, the simplest hydrogen-like atom, where pick-off annihilation process gives valuable information on the free volume nanostructure of solids.

l. Introduction to new atoms
One feature of the vehement growth of physics in the 20th
century is, undoubtedly, the correlation of this science withother fields of human endeavor, their mutual enrichment. The
recent several decades have seen the spectacular advances, forexample, in a new area at the junction of elementary-particle
physics and chemistry, the so-called chemistry of new atoms,
in which either an electron is replaced by another negative particle or a proton by another positive particle. This area of
research does not confine itself to chemical studies alone.
External influences on the properties of new atoms, and also
those of mesons and positrons, provide useful, sometimes
unique, information about the structure of solids. Therefore,
unlike other reviews on positron annihilation, this paper
includes a special part, describing intrinsic properties of positronium in comparison with properties and composition ofother new atoms, so that this part can be considered as a general introduction into the field of new atoms. On example ofpositronium, the main part of this paper demonstrates external
influences of the surrounding matter on properties and characteristics of one of the new atoms and the use of this effect forstructural studies in solids. Correspondingly, a newcomer willfind in this paper a little more than one of the special applications of the positron annihilation method.
Atoms with an electron replaced by a negative meson are
called mesoatoms. When chemically bound to conventional
molecules and radicals they form mesomolecules. Typicalexamples are µ- and π-mesoatoms.
Also possible in principle are E-mesoatoms, in which an electron is replaced by a negative hyperon, antiproton, or antihyperon.
Atoms with a proton replaced by a positive particle arecalled by the name of that particle with an ending ‘ium’. Sincea penetration of any positive particle into a nucleus is hinderedby Coulomb repulsion (which can be compensated for by the
nuclear forces alone), for light positive particles not involved innuclear interactions, such as positrons e + and µ + -mesons, theatoms of this type cannot have nuclei consisting of many nucleons. These atoms only form when a positive lepton replaces asingle nucleon, namely, a proton. In that case, an atom of apositron and an electron is called positronium (Ps). A boundsystem of µ+ -meson (positive muon) and an electron is known
as muonium (Mu). Both atoms can be treated as light isotopesof hydrogen.
*Corresponding author. E-mail: Shant@chph.ras.ru

What is more, positronium is the lightest of all known newatoms, the only atom without a nucleus. In fact, since both constituent particles have the same mass, the centre of mass does notcoincide with any of them (otherwise such a particle could be
regarded as a nucleus), but lies strictly in between them.
A nuclear proton can also be replaced by a positively charged
hyperon in much the same way as a neutron by a neutral
hyperon. The result is the so-called hypernuclei, and the corresponding atoms are referred to as hyperatoms. The structureof their electron shells and their chemical properties are, however, essentially the same as in conventional atoms with thesame nuclear charge. Therefore, hyperatoms are generally notincluded into the list of new atoms.
Interest in new atoms chiefly comes from the fact that theircharacteristics are appreciably influenced by the chemical
properties and the structure of their environment. We will belooking at them in more detail in the next section. Examinationof the characteristics of new atoms can supply unique evidence
concerning the properties of the environment. One importantcharacteristic of new atoms is their lifetime (all of them unstable). The decay characteristics of new atoms are studied using
the apparatus of nuclear physics.
The first stage of new atom research is the study of the lawsby which the environment affects the characteristics and fate of
new atoms. The second stage, more important for otherresearch fields, consists in obtaining more complete and new
structural and chemical data or data on the kinetics of certain
processes based on observations of characteristics of formation, transformation, and disintegration of various new atoms.
At the present time both the investigations into annihilation(death) of positrons and positronium and the mesochemical
studies of µ- and π-mesoatoms and muonium have attained thesecond stage and started to yield information of general chemical significance. The field is very extensively promising. The
aim of this review is not to embrace the whole problem of“new atoms” or the most of directions of positron annihilation
studies. Instead, after general introduction into physics andmethodology, detailed discussion is given of so called positronium pick-off annihilation and application of this effect for thestudies of the free volume nanostructure in solids, in polymers
particularly.
2. Composition and properties of positronium and othernew atoms.
Table 1 is a summary of some of the properties of elementary particles constituting the new atoms. Particle masses are© 2006 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 06/30/2006

R38 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Shantarovich

TABLE 1: Some properties of the elementary particles of which new atoms are made up
Name

Symbolm, rel. units

m, MeV/c2

ChargeLifetime, sElectronPositronµ-mesonπ-mesonProton−1
1
206.77273.151836.1

0.511
0.511
105.66139.58
938.2

-1
+1
±1
±1
+1

StableStable2.26×10 -6
2.56×10 -8
StableComposition

µ

r, cm

e
e+
µ±
π±
p

TABLE 2: Properties of hydrogen-like atoms
Name

SymbolHydrogen
Positronium
Muoniumµ-Mesohydrogen
π-Mesohydrogen

H
Ps
Mu
-

−

pe
e + eµ+ e pµpπ-

1
½
0.995
185.8
237.8

Vion, eV-80.529×10
1.05×10 -8
0.532×10 -8
2.85×10 -11
2.23×10 -11

13.595
6.80
13.539-given in relative (in relation to the electron mass) units and in“energy” units that correspond to Einstein equation. Unstablemesons decay in the following way:µ± → e ± + νe (ν~e)+ ν~µ (νµ)(1)
π ± → µ±+ ν (ν~ ),
(2)
µ

µ

where ν and ν~ are the corresponding neutrino and antineutrino.
The positron as such is a stable particle and it only disappears
as a result of its interaction with the environment. In the previous section we have already mentioned that positronium andmuonium are hydrogen-like. Replacement of the electron in
the hydrogen atom by π – - or µ– -meson yields mesohydrogen.
Characteristics of hydrogen-like new atoms are given in Table
2. The reduced mass – a parameter that is fairly important for
determining other characteristics of new atoms– is calculated
from the given masses of the constituent particles m1 and m2 : µ= m1 m2 /(m1 + m2).
In terms of reduced masses positronium has only half themass of the atomic hydrogen, while in terms of absolute masses
the latter is 920 times heavier. Because of the difference in
reduced masses the ionization potential Vion (i.e. the binding
energy) of positronium is half of that for the hydrogen atom,and the distance r between the electron and the positron is
twice that between the proton and the electron (twice the Bohrradius). Although positronium consists of stable particles (theelectron and positron are particle and antiparticle), they cankill each other (annihilate) and give off γ -quanta. This process
is responsible for the instability of the positronium, and for its
extremely short lifetimes (~10 -7 and 10 -10 s in a vacuum).
Although muonium and atomic hydrogen differ markedly in
mass, their reduced masses and hence some other characteristics
are similar.
The main feature of mesohydrogen is that the radius of themeson orbit is about 1/200 of the Bohr radius. Therefore, thepositive charge of the proton is highly screened and in many
respects mesohydrogen behaves as a neutral particle similar toneutron.
3. Positron annihilation studies
At present the bibliography of this expanding area of
research1 includes more than 2000 publications. The details of
the theories and experimental techniques are discussed inbooks2-6 and in a number of review articles,7-15 including thosedevoted to the general overview of new atoms.10,13 Recently the
13th and 14th International Conferences on Positron Annihilation(ICPA13, 14 ) and 8th International Workshop on Positron andPositronium Chemistry PPC-8 were held.16, 17 Experimental
data are summarized in review tables.18, 19

Figure 1. Scheme of positron annihilation processes

This review does not claim to be a profound analyses of thecurrent work. Its objective is to give a general idea of positronannihilation research and application of these investigations
mostly in the free volume studies. In some way, this review
continues a discussion of the possibilities of positron annihilation experiments, started several years ago by publication inthis journal20 and in the other Journal.21
3.1. Annihilation and interaction of positrons with matter. As
a positron and an electron interact, their kinetic energies andrest masses transform into γ -quanta. This process is calledannihilation and is governed by general physical laws of conservation (of energy, momentum, parity, etc.).
By the law of parity conservation in annihilation two or
three γ -quanta are produced depending on the relative orientation of the spins of the electron and the positron (for parallel
spins three quanta, and for antiparallel spins two quanta).
In the case of a two-photon annihilation of an electron-positronpair at rest the γ -quanta fly off in strictly opposite directions(at 180˚) in accordance with the law of momentum conservation.
The energy of each quantum is equivalent to the electron rest
mass (0.511 MeV). Slight deviations from 0.511 MeV of thephoton energies or from collinearity of the emission directions
give valuable information about the properties of matter andare the core of a number of experimental techniques.
The fate of energetic positrons emitted by radioactive sources(e.g. 22Na) is annihilation. But before this really happens someprocesses may occur, which are shown in Figure 1. In a condensed
phase positrons are slowed down in only several picoseconds(mainly by ionization and excitation of surrounding molecules).

Positronium Atom in Solids –Peculiarities of Formation and Interconnection

TABLE 3: Properties of the positronium atom
Type of state
Property
SymbolSpin state
Quantum number
Share, %Proper lifetime, s
Annihilation type

Singlet (p-Ps)
S

Ps
e-↑↓e +
J=025
1.25×10 -10
2γ

Triplet (o-Ps)tPse-↑↑e +
J=175
1.4×10 -7
3γ

A fraction of positrons undergo free annihilation with surrounding electrons. In liquids this process may be preceded bysolvation. Some positrons may get involved in chemical reactions and further annihilate from the bound state Me + with amolecule or ion M. In the case of free annihilation a two-photon
event is 372 times more probable than a three-photon event. And
for the free annihilation of positrons the lifetime is about 0.4 ns.
The remaining positrons form a bound positron-electron
systems called positronium (Ps). Depending on the relativealignment of spins of the electron and the positron, positronium
can have two states: singlet and triplet. The properties of thepositronium atoms are tabulated in Table 3. It is seen that theproper lifetime (in a vacuum) of a triplet positronium (3γ -annihilation) is three orders of magnitude longer than for a singlet
positronium. But a variety of interactions shown in Figure 1
shorten the lifetime of positronium in a condensed phase down
to several nanoseconds. In positronium-forming systems thefree annihilation of some of triplet positronium increases theshare of 3γ -annihilation as compared with the case of freeannihilation of positrons.
Measurements of lifetimes of triplet positronium and theangular distribution of the annihilation radiation of singlet andtriplet positronium, as well as Doppler broadening of annihilation line, provide an insight into positronium reactions and thechemical properties of the environment. These studies are themain part of positronium chemistry.
3.2. Positronium formation. As is seen in Figure 1, not all
positrons (and not in all substances) form positronium.
Experimentally, this share can be estimated, for example, by
measuring the intensity of the positronium-related long-lived
component. Now there are two views of the process of positronium formation, although their combination seems to give themost complete picture of the phenomenon.
By the model of the Swedish physicist Ore22 positronium canform within a narrow energy interval between the ionization
potential of the surrounding molecules, Vion and (Vion −6.8) eV,
where 6.8 eV is the binding energy for positronium (half a rydberg). If we take into account the excitation of the first electron level E1 of surrounding molecules, a process that competes
with positronium formation, the interval will become narrower;
and eventually for a positron that effectively forms a positronium, the energy interval will be E1<Ee+ <Vion −6.8, and the proportion of positrons that form positronium, P, will be given by
eq 3:

6.8
E − (Vion −6.8)
<P< 1
Vion
E1

(3)

Positronium here can have a kinetic energy from zero to Vion or
to E1. It seems obvious now that the model only gives a correct
prediction of positronium formation in inert gases though someattempts were made to use the same considerations in liquids23,24 and participation of unthermalized positrons in positronium formation can not be ruled out.
Another, so-called spur, model was first suggested by
Mogensen25 and developed by other authors26-29 as a diffusion-

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R39

recombination model. This model better explains the evidence
for positronium formation in liquids. The underlying idea ofthe spur model is the assumption that positronium forms whena positron at the end of its moderation combines with one of
the electrons of its own spur or, more exactly, with one of the“blob” electrons. By “blob” is here meant the unstable clusterof electrons, positive ions and radicals that emerge as a positron
is moderated in a liquids and perhaps in some solids.30-33
The probability of positronium formation in the positron spurdepends mainly on the rates of the competing processes. In thegeneral form the process can be represented as follows:
e + + e−e− + M +e− + S1
M + + S2
e + + S3

k1

> Ps
> M*
k3
> products
k4> products
k5
> S3+
k2

(4)
(5)
(6)
(7)
(8)

In pure solvent, obviously, only reactions (4) and (5) should beconsidered. In the presence of active impurities S1, S2, S3 theprocesses (6)–(8) also play a part. They lead either to inhibition or an increase in the percentage of positrons that form
positronium. The rates of the processes (5)–(7), and hence theprobability of positronium formation, are quite sensitive to thephysico-chemical properties of the environment and they aremeasured in radiation chemistry. No wonder, therefore, that a
number of interesting correlations have been found between
positronium chemistry and radiation chemistry.34- 40 Much
work is being done currently in this direction to study thedetails of the spur mechanism.41-43 A new technique of singlephoton count with radioactive (including positron) sources has
appeared, which a11owed observations of the correlation of
inhibition of the recombination fluorescence intensity and positronium formation by the same acceptors.44 Recently, similar
behavior was observed for low-temperature positronium formation and thermo-stimulated luminescence (TSL) in some polymers.45 The presence of these correlations is evidence of the
predominance of spur processes in positronium formation not
only in liquids but also in solids. We have to stress howeverthat in solids positronium observations are influenced also by
the presence and characteristics of the so-called free or unoccupied volume.46,47
3.3. Positron annihilation experiments. Information about
the fate of positrons and positronium and about the propertiesof their environment can be obtained from examining the characteristics of the annihilation radiation emitted in positron
decay. Each of the techniques available is aimed at investigating certain characteristics:(1) The number of γ -quanta emitted in annihilation. The
corresponding method of double or triple coincidences is theleast informative.
(2) The angular distribution of annihilation radiation. Hereinformation is derived about the momentum distribution ofelectrons involved in annihilations. If positrons or positronium
take part in chemical reactions, the shape of the distributionchanges.
(3) The energy spectrum of annihilation γ -quanta (deviationfrom 0.511 MeV registered using a Ge(Li) -semiconductor
detector of the Doppler broadening) also represents the energy
distribution of electrons involved in annihilation. Especiallyinformative is a work with the coincidences of signals from thetwo Doppler broadening detectors (CDB technique), where thismethod becomes sensitive to positron annihilation on the coreelectrons of different elements.
(4) The time distribution of annihilation radiation. It isstrongly dependent on electron density at the location of a positron or positronium and on the rate at which positronium enters
into chemical reactions.
Various radioactive isotopes can be used as positron sources.
Of these 22Na is the commonest both due to its rather long half-R40 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Figure 2. The decay scheme of the positron-active 22Na.

Figure 3. Experimental arrangements: (a) triple coincidence circuit,(b) observation of lifetime distribution of positron annihilation, (c)measurements of angular correlation of positron annihilation.

life (2.58 years) and to its applicability in various experimental
arrangements. The decay scheme for 22Na is shown in Figure 2.
Depending on the technique used, sources can have different
activities which vary from tens of µCi for positron lifetime measurements and studies of the Doppler broadening of the annihilation line to a hundred mCi in angular correlation measurements.
In the first case activities are limited by the growth of the random coincidence background. A source is prepared by evaporating a drop of 22NaC1 radioactive solution on a thin (1–2 mg/cm2)
metal foil or synthetic film that pass positrons well, the source isthen covered with the same material (foil or film). When working with liquids it is also convenient to use sources prepared bydiffusing at an elevated temperature of 22NaC1 into 2–4 mg/cm2
glass foil.48 Another positron source can be a positron beam ofcertain energy produced at accelerators.49 The last technique iscommonly used in studies of the chemical state and physicalproperties of surfaces of solid specimens, and also in studies ofthe depth distribution of defects.
The common experimental arrangements are depicted in
Figure 3. Shown in Figure 3a is the triple coincidence arrangement to record ortho-positronium (o-Ps) annihilation events withthe emission of three γ -quanta. They are recorded by counters
C1, C2 , and C3. The technique has the disadvantage that thecounting rate for triple γ -coincidences (C3γ ) is proportional tothe product of two variables — the percentage of positrons, P,
that form positronium and the probability of an annihilation
with the ejection of three γ -quanta W3γ = τ 3 / τ T0 (τ 3 is the real
lifetime of positronium in matter, and τ T0 is the lifetime of triple
positronium in a vacuum (see Table 3)). Therefore, C3γ can go

Shantarovich
down both because of shorter positronium lifetime (smallerW3γ ) and because of positronium formation inhibition (smaller
P). It is desirable, however, to distinguish the contributions ofthese two interactions.
The most variegated information is supplied by observationsof the time spectrum of positron annihilation (see Figure 3b)
using coincidence circuits that record delayed coincidences
N (t) between a nuclear γ -quantum with energy 1.3 MeV (counter C1) emitted essentially simultaneously with the ejection of apositron from a 22Na nucleus, and a 0.5 MeV γ -quantum (counterC2) produced by a 2γ -annihilation of that positron. Thus, thetimes of creation and decay of that positron are recorded. The
short-lived components of the time spectrum (lifetimes τ1, τ 2
with intensities I1, I2) are due to the annihilation of positrons inpara-positronium (SPs, see Table 3) and in free collisions withelectrons; whereas the long lived component (lifetime τ 3, intensity I3 =3/4P) is due to the annihilation of the positrons thathave formed triplet positronium. The time resolution (thewidth of the prompt coincidence curve at half maxima FWHM)
for present-day apparatus is about 0.2–0.3 ns.
This technique distinctly distinguishes positronium quenching and positronium formation inhibition. In the simplest caseof pure inhibition the intensity I3 of the long-lived component
decreases without decreasing the slope of the “tail” of thiscomponent that characterizes its lifetime τ 3. But with purequenching, τ 3 decreases while I3 remains unchanged. In positron annihilation experiments we will encounter examples ofchanges in time spectra of annihilation due to inhibition andquenching. Strictly speaking, when the lifetime of positronium
in a solution becomes shorter (quenching) due to the use ofactive additives, kinetic equations50 determine the relation (seeeq 9) between the percentage of positrons P that form positronium in the solution and the observed intensity of the long-lived
component I3:
I3 = (3P/4)(λ B − λ P)/{(λ B − λ P) − k [Ac]},(9)

where λP is the rate of annihilation at solvent molecules, the socalled pick-off annihilation; λB is the rate of annihilation of positronium in bound state with active additive (λB = 1/τ B, τ B ≤ 0.4
ns, see Figure 1); k[Ac] is the positronium quenching rate.
The single-photon count technique with positron sources,44
mentioned in sec. 3.2, was in fact a modification of apparatus
designed to study positron lifetimes in which the annihilation
γ -quantum detector is replaced by a single-photon one.
Figure 3c represents a third major technique — the observation of correlation curves, i.e. dependences of coincidence
counting rate of two annihilation γ -quanta (C2γ ) with energy
0.5 MeV (C1 and C2) on the angle θ, i.e. on the deviation from180˚ of the angle between two ejected quanta. The detectors
are located at about 2 m from the sample behind lead collimators. Differentiating dC2γ /dθ gives the momentum distributionfunction for electrons in matter ρ (PZ ). The narrowest component of the complicated correlation curve corresponds to singlet
positronium annihilation. Therefore, by separating in a large
experimental run the correlation curves shown schematicallyin Figure 3c into the narrow I N and wide components we canobtain important additional information about positronium
chemical reactions, namely we can distinguish the various positronium quenching forms. Actually, the annihilation of o-Ps
on “foreign” electrons in collisions, just as the formation ofchemical compounds PsM in substitution and addition reactions, and the annihilation of o-Ps oxidation products — free
positrons — yield correlation curves several milliradians wide.
Special approaches were suggested to distinguish and identify
both the case of addition and the case of oxidation. 51, 52
Positronium conversion from a triplet state into a singlet one
with subsequent 2γ -annihilation increases in the correlation
curve the intensity of the narrow component and can also bePositronium Atom in Solids –Peculiarities of Formation and Interconnection

easily identified.
The coming of Ge (Li)-semiconductor detectors (SD) withhigh energy resolutions (1 keV at the energies about 500 keV)
opened up new possibilities to do without angle correlation set
up. In fact, if X is the direction from a positron source to a
detector, then since ∆Eγ =PX c/2, the electron energy E (positronis assumed to be at rest), the momentum component PX and theannihilation line broadening ∆Eγ will be related by eq 10:
E = PX2 / 2me = 2(∆Eγ )2 / me c2 .

(10)Thus, the Doppler shift of a γ -quantum by 1 keV, i.e. only by
0.2%, corresponds to an electron energy of 3.9 eV. Since underexperimental conditions
sinθ = PZ / me c ≅ θ(11)(Z is the direction normal to the plane passing through the collimator slit of the stationary detector and through the sample— see Figure 3c), then in isotropic case

θ = 2∆Eγ / me c2(12)Here a 1-keV broadening is equivalent to a deviation of 3.9
mrad from collinearity in quantum ejection. In other words, adetector with a high (in modern terms) energy resolution isequivalent to an angular installation with a resolution of about
4 mrad, a resolution four times worse than in a conventional
angular experiment. Nevertheless special computer programs
take into account resolution functions and make it possible toextract the information required.53 It was mentioned already,that a set up with coincidence of the two detectors (CDB technique) essentially diminishes a background and gives an elemental sensitive instruments for solid state structure studies
and for identification of the mechanisms of the positron trapping.54-56
3.4. Positronium quenching. The lifetime of o-Ps in normal liquids and solids (0.6–5 ns) appears to be much shorter
than its proper lifetime (140 ns, see Table 3). This is due topick-off annihilation, i.e. the annihilation of positronium’s positron on electrons of surrounding molecules. The lifetime of
positronium becomes yet more shorter in the presence of small
amounts of positronium-active molecules (quenchers).
Positronium is quenched through its chemical transformations.
They were discussed previously.6,20,21 Here, we will look at
pick-off annihilation.
Pick-off annihilation. Bubble formation. “Free volume” model.
This subsection is concerned with two main hypotheses of thepick-off annihilation of positronium. For liquids this is the socalled bubble model which has first been formulated for condensed gases;57-59 it is also of importance for an understandingof the mechanism of positronium chemical reactions in liquids.60-69 For solids, e.g. polymers, the “free volume” model issuggested in Reference 70.
The bubble model assumes that in the majority of liquids
around positronium a bubble is formed. This model approximates the bubble around a Ps atom as a square well with spherical geometry. This potential well has a radius R a depth U andthe energy of positronium E, measured from the bottom of thewell. According to References 57–59, the bubble radius isdetermined by minimizing with respect to R the total energy ofthe bubble Etot = Ecm (R, U) + (4/3) π R 3 P + σ 4π R 2 , where Ecm isthe zero point energy of the trapped positronium, the second
term is the pressure-volume work, and the last term is the surface energy due to the bubble-fluid interface with specific surface energy σ . The minimization of Etot gives eq 13:

π h 2γsin4(kR),=−
m eU 2kR(tan(kR)− kR)(13)

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R41

where k = (4me E / 2)1/ 2 , and me is the mass of an electron.
Existance of the solution of eq 13 is a condition of the bubble
formation.60
The specific rate of pick-off annihilation is given by eq 14:

λ P /N = π r02 cnZ eff P(kR) = π r02 cNA

Z eff
P(kR),
V(14)

where P(kR) = sin 2 (kR)[1 − kR × cot(kR)] -1 and expresses thequantum mechanical probability for Ps atom coming out of thewell; r0 is classical radius of an electron, c is the velocity of light,
n is the number of molecules in unit volume, NA is Avogadro’s
number and V is the molecular volume, Z eff is the effective number of annihilating electrons per molecule, which was proved tobe equal to the number of valence electrons per molecule.57-59
Experimental lifetimes (τp = λ P-1) in majority of liquids give thefollowing bubble parameters: R =3– 4.5 Å, U ≈ 1 eV.
Growth and shrinking of the positronium bubble influence
the rate of positronium pick-off annihilation and chemical reactions. In the work,71 the effect of the bubble on chemical reaction is seen from the good correlation of the o-Ps quenchingrate with surface tension of various solvents and with energy ofthe system “Ps in a bubble”. Obviously, bubble formation influences PsAc complex formation and the complex decomposition. We touched here the idea of the positronium bubble
formation since this peculiarity of Ps interaction with a substance may be important for positronium annihilation in elastic
polymers at high temperatures. The role of this effect must be
found from comparison of the rate of the bubble growth andthe rate of positronium annihilation, as it was done for a number of liquids in paper.72 It was shown that in such viscous liquids as glycerol the bubble has no time to reach equilibrium
size before o-Ps annihilation. Experiments73 show that one of
the signs of equilibrium bubble formation can be found inappearing of additional intermediate momentum component in
the ratio curves of CDB spectra of liquids relative to those in
their solid state for n-hexane and cyclohexane (Figure 4). In
contrast, such components were not found at low temperatures insome liquids with low freezing points and in rubbery polymers

Figure 4. Ratio curves of CDB spectra for n-hexane (a), and cyclohexane (b) in liquid state relative to their solid state. The dashed lines
as well as the values shown at the right side of each graph illustrate
the expected level or ratios due to I3 difference. The dotted lines demonstrate the expected variation tendency of the ratio curves due to I3
difference.73

R42 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Shantarovich(above Tg), which may show absence of equilibrium bubble inthese cases. Detailed discussion of the free volume factor insupercooled liquid dynamics can be found in Reference 74.
When positronium is formed in solids, e.g. glassy polymers
or ice, it becomes impossible to push molecules aside to makeroom for a bubble. Under these conditions positronium canlocalize in microvoids, called “free volumes”. Annihilation
with possible captures of positronium in free volumes was firstconsidered by Brandt.70,75,76 The method offers new possibilities for investigation into the structure of a number of materials, including polymers.77-82 According to the results83 thelifetime distributions of positron annihilation in solid molecular systems have more than two components and for amorphous
structures they have the following nature: τ 1 is the component
associated with annihilation of singlet Ps, partly with free positron annihilation and annihilation of nonlocalized positronium,τ 2 is the component of annihilation of free and trapped positrons and non-localized o-Ps, τ 3 is the component of annihilation of localized (captured in microvoids) atoms of triplet
positronium. In amorphous-crystalline systems, a spectrum
structure becomes more complicated.
Figure 5. Distribution of the free volume hole sizes in polymeric sorbent LPS200X found from the sorption experiments.87

4. Pick-off annihilation and polymer structure
According to mentioned above circumstances, this process
is determined by the positron spur (Ps formation) and also by
the number and size of elementary free volumes (Ps trapping).
Consider now a correlation of annihilation characteristics withthese parameters.
4.1. Effective size of elementary free volume and o-Ps
lifetime. In some approximation,62,84-86 a pore can be represented by a spherical potential well with effective radius R 3.
For this case, well calibrated semi-empirical eq 15 represents a
dependence of o-Ps lifetime τ3 on the pore effective radius:{ [τ 3 = λ 0T + 2 1-

(

2πR3
R3
1
+
sin
R3 + ∆R
R3 + ∆R
2π

-1

)]}

(ns),

(15)

Parameter λ 0T stands here for the intrinsic o-Ps annihilation rate
(0.7×109 s-1), ∆R = 0.166 nm, determined by calibration on substances with well known free volume elements and denotes
penetration depth of positronium wave function inside of the“walls” of the elementary free volume. The problem of correctness of such estimations, especially in the cases of large(more than 1nm) pores, typical for recently synthesized sorbents and membrane materials still exists. Paper87 describes
such results for the cross-linked polystyrenes and gives achance to check contemporary models. The names of the sorbents are explained elswhere.88 The largest free volume holes
in the polymer sorbents (in LPS200X at RT, τ 5 =73–75 ns, and,according to Tao-Eldrup eq 15, R5 =20 Å) were big enough forcomparison with the data of nitrogen sorption experiments.
Sorption experiments treated using Brunauer-Emmett-Teller
theory.89 Results for LPS200X are given in Figure 5,87 and they
do not contradict to the positron annihilation lifetime (PAL)
estimations given above, though R5 =20 Å is a little bit too large.
This comparison along with a discussion of temperature dependence (TD) of annihilation characteristics (Figure 6) were useful for checking possibility to apply contemporary theoreticalmodels, connecting EFV size and positronium lifetimes, to ourexperimental results. We mean here extended Tao-Eldrup(ETE) model suggested by Goworek et al.90,91 and so called“classical regime”, considering Ps atom as a classical particle
in a large hole.92,93 Analysis of TD-data87 shows that the classical free path concept for Ps lifetime in a rectangular pore, as wellas its modification for spherical or cylindrical pore in CPS150X,94
LPS200X87 and also in silica-based glasses (see Figure 2 inReference 95), though predicting a decrease of the longest
leaved o-Ps lifetime on heating, does not give good enough fitting to the experimental points over the entire temperature

Figure 6. Temperature dependence of the long-lived o-Ps componentτ5 , I5 for LPS200X (PATFIT data). Solid lines illustrate different
attempts to describe temperature dependence in terms of the classicalparticle model87,92,93

range between 50 and 350 K. The slope of experimental
dependence is too flat compared to the theory. For example(LPS200X, Figure 6), if the model curve is normalized onhigh-temperature data, increase of experimental lifetime at low
temperature is not as strong as it is expected from the theory.
For silica-based glasses and polystyrens, an explanation of thisdiscrepancy was found88,95,96 in combination of the “large hall”
effect with chemical interaction of positronium with radicals
induced by irradiation and stored in a sample during the experiment with positron source at low temperature or with atmospheric oxygen strongly trapped between aromatic rings.
However, another discrepancy comes out when the estimationsof the pore size found on the bases of the ETE model and sorption experiments are compared. The ETE model91 considers
Tao-Eldrup eq 15 as a limiting case of zero temperature. If
this idea is followed and we apply this equation to LT data5 (τ 5
= 100 ns in LPS200X at 50 K) the pore radius larger than 25 Å
is obtained. This size is poorly represented in the distribution
in Figure 5 and demonstrate the discrepancy between PAL andsorption data or, more exactly, between the sorption data andresults of theoretical estimations made on the bases of the PAL
data.
The results obtained show that the effects of Ps annihilation

Positronium Atom in Solids –Peculiarities of Formation and Interconnectionin the large free volume holes are too complicated and combined PAL and sorption experiments can be very promising for
elucidation of this problem.
4.2. Concentration of elementary free volumes. This isvery important characteristic of various materials determining
their mechanical and physico-chemical properties, such aselasticity, aging, permeation, selectivity to different gases,
sorption etc. A consideration given here is a simple representation of the idea of o-Ps localization in elementary volume of a
substance. The simplification is in the assumption that mobility of non-localized positronium before the final trapping is
described by some effective diffusion coefficient D Ps, and annihilation rate before final localization is taken equal to the free
positron annihilation rate: λ ƒ+ = λ ƒPs = λ ƒ .88 In principle, theprocess can be more complicated: distribution of the spur electrons can be inhomogeneous. Localization can be a process ofthe consecutive positronium transitions to deeper and deeper
traps with longer and longer o-Ps lifetimes and, correspondingly, with non-exponential distribution of positronium annihilation. Importance of this effect has to be estimated in eachgiven case from the correctness of multi-exponential fitting.
Theoretical considerations of the complicated process can befound in References 97–99.
Generally speaking, a possibility to estimate concentrationof elementary free volumes depends on the conditions of formation of localized o-Ps. Analytical dependence of the longlived component intensities on the number of elementary fee
volumes is important for interpretation of the PAL results.
Fraction of positrons, which formed o-Ps and is observed as
long-lived component in the PAL spectrum, is dependent not
only on the number density of EFV (possibility of localization)
but also on the density of electrons in positron the spur (possibility of Ps formation). Positrons produce spur electrons during their slowing down. The validity of the spur model of Ps
formation in polymers was strongly confirmed recently by
experiments on additional Ps formation in irradiated polymers
at low temperatures.45,100 Some results will be discussed later.
The model infers that in the case of lack of the spur electrons,
no Ps will be formed, though the number of EFV is very high.
In addition, precursors of the trapped o-Ps can be either nonlocalized o-Ps, or non-localized e+, which forms Ps on the spurelectron inside or in close vicinity to EFV. This situation canbe described by a simple system of kinetic equations (16) where
the relation among positronium formation rate ν form , positronium trapping rate ν it , and annihilation rate of non-localized
positronium λ f determines the choice of the precursor for
localized o-Ps:
14243

dPf+ /dt = − (λ +f + ν form)Pf+ ,Pf+ (0) = 1

Ps
Pst
dP fPs /dt = (3/4) ν form Pf+ −(λ Ps
f + ∑ ν i )P f , P f (0) = 0

dP /dt = ν P − γ i P ,
t
i

Ps
f

Ps
t,i

{

}

I i = (3/4)Q ν it /( λ f − γ i + ∑ ν it ) ,i(17)

where Q = ν form /( λ f − γ i + ν form ) ≈ ν form /( λ f + ν form ) and is afraction of positrons forming Ps in a system.
The trapping rateν it = 4π DPsRi Ni , (i = 3,4, etc.)(18)

where D Ps means diffusion coefficient of nonlocalized Ps, Ri is
a radius for interaction, which we suppose equal to the EFV
effective radius, Ni is the number density of trapping centers ofa given type.
The condition

ν it ≤ λ f ,(19)when some part of nonlocalized o-Ps have time to annihilate
before trapping, implies nonlocalized o-Ps as a precursor of
localized o-Ps, when intensities of the long-lived positronium
components Ii are dependent on the concentration Ni and radius
Ri of the holes of corresponding type (see eq 17). This is thefirst case in terms of the previous qualitative discussion. For
the PAL spectrum with one long-lived component (τ 3, I3), ν t
can be expressed also in terms of annihilation characteristicsasν t=4π DPsR3N3 =(4I3 /3)(λ f -γ 3)/(Q-4I3 /3) ≈(4I3 /3)(λ f -γ 3)/(I1-I3 /3)(20)

The accurate values of D Ps in a given polymer are not known.
However, Jean et al.101 had found D Ps = 0.2×10 -4 cm2 /s for positronium in the bulk of the porous resins. An attempt was madeto estimate diffusion coefficient of non-trapped Ps in a polymer
bul k using posit ron a n n i h ilation data for the porouspoly(phenylene oxide) with the known specific surface area,
and it was found that 0.5×10 -4 < D Ps <1.5×10 -4 cm 2 /s 102-104 We
suppose that scattering of the data may cover the difference inthe mobility of non-localized Ps atom in the most of polymer
materials, though this uncertainty decreases precision of theresults. In principle, the data from this range and annihilation
characteristics τ3, I3 can be used in eq 20 for estimation of N3 invarious polymers, if the relation (19) is fulfilled. Obviously,
the diffusion coefficient for nonlocalized positronium D Ps isdifferent from that for localized positronium. The last parameter was estimated as (2.6–3.2)×10 -6 cm 2 /s by Hirata et al,105
from the results on positronium quenching by dinitrobiphenyl
in amorphous polysulfone, polycarbonate and polystyrene.
The second case (see discussion above) implies thatν it >> λ f .

(21)

(16)

i

Ps
t,i

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R43

P t,iPs (0) = 0

with summing up over the numbers of the long-lived components or over the types of holes (i = 3, 4, etc), and also with
Pf+ , P fPs, P t,iPs standing for the probabilities to find free positron,
nonlocalized or trapped positronium, correspondingly; γ i = 1/τ iis the rate of annihilation in the i-hole (elementary free volume
EFV) and depends on the hole size (see eq 15).
For convenience of discussion, we assign the same label i forthe lifetime component and for the type of a hole in a sample.
In the simplest case of three-component analyses of the PAL
spectrum, when only the third component is related with o-Ps,
i =3. Transitions between the EFV of different types are
neglected. Factor 3/4 in eq 16 is coming because we consider
behavior of the long-lived triplet positronium. From eq 16, theintensities of the long-lived components (coefficients before theexponents in the PAL distribution:

In fact, validity of this condition means that free positron is aprecursor of localized o-Ps.
From eq 17
I i = (3Q /4) ν it / ∑ ν it = (3Q /4)R i N i / ∑ R i N i .
i

(22)

i

For two types of the holes k and m, for example,
I k /I m = Rk N k /Rm N m .

(23)

When the specific surface area S of the sample is known, and it
is determined mostly by the largest EFV (S ≈Sm ≈ 4π R m2 N m ),eq 5 can be used for calculations of Nk.88
However, if there is only one long-lived component (one typeof elementary volumes), intensity of the long-lived component
depends only on Q (I3 = 3Q/4), and it can not be related directly
to concentration of EFV. Sometimes it is not obvious what
condition, eq 19 or 21, is valid and, furthermore, it is knownR44 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Shantarovichthat I3, the only long-lived component in a system, can be influenced by the positron trapping with an acceptor, like polar carbonyl group. To avoid this uncertainty and to calculate thenumber density of free volume holes in this case, temperaturedependence of o-Ps lifetime τ 3 below and above the glass-transition temperature Tg might be used.106,107
Let us consider the effect of temperature on the free volume
characteristics of polymers. The coefficient of thermal expansion of the total volume α can be written asα = α h h + α b (1 − h),(24)where h is fractional free or hole volume (Vh /V), α i are thermal
expansion coefficients of the holes (subscript “h”) and of thebulk (subscript “b”). Hence,h = (α − α b )/(α h − α b )(25)

Hristov et al.108 had shown that eq 25 can be applied both in
the temperature ranges T< Tg and T > Tg. Assuming also that hbehaves continuously at Tg and α b (T < Tg) = α b (T > Tg), thevalue α b can be eliminated, and for T → Tg

Figure 7. Variation of annihilation characteristics I3, τ 3 in PI189 asfunction of the time of measurements (temperature) for the twocycles of measurements (20 ˚C→200 ˚C→20 ˚C). Each point corresponds to two hours of measurements and to variation of sample temperature by 5 ˚C. Non-regularity is seen between 125 and 165 ˚C.

Parameters α r and α g (with subscripts “r” for rubber and “g”
for the glass phases) are related to macroscopic volume while
α h ,r and α h ,g describe behavior of the hole volumes.
Now, in the case when α r and α g for a given polymer are notknown, the Simha-Boyer value ( α r − α g )Tg ≅ 0.113 may be
used,109hg = 0.113/ {Tg (α h ,r − α h ,g )},o-Ps Lifetime τ 3 (ns)

(26)

τ3
I3

20

2

10

1.8

o-Ps Intensity I3(%)

2.2

hg = (α r − α g)/(α h ,r − α h ,g).

(27)

α h ,r = (1/νh ,r)dνh ,r / dT = 95.5 ×10 K ,α h ,g = (1/νh ,g )dνh ,g / dT = 3.9×10 -4 K -1.
-4

-1

(28)

Naturally, α h ,g << α h ,r. According to positron annihilation data,
Tg for PI189 is readily found from the crossing point of theslopes for the glass and rubber phases. It comes equal to 160
˚C, which is in agreement with literature. Combination of eqs
27 and 28 gives hg = 3.35%, and at the glass transition tempera-

1.6

50

100

150

200

Temperature (˚C)
Figure 8. Dependence of I3, τ 3 on temperature for PI189. Heatingstage of the second cycle (20 ˚C→200 ˚C).
120
110
100

Vh (Å3)and the fractional free volume hg at T=Tg can be found from eq
27 using only positron annihilation data. From hg = νg Ng weestimate the number density of holes Ng =hg /vg at this temperature. It seems to have been demonstrated now106 that in theamorphous phase the hole volume fraction h(T) is controlled
mainly by the temperature dependence of the mean hole volume νa , and the proportional constant, that is the number N ofholes per cm 3 , does not depend much on temperature, i.e.
h (T) = Nνa (T) so that N ≅ N r ≅ Ng . Further testing of thisassumption may be in so called pressure-volume-temperature(PVT) experiments.110
4.3. Typical examples. Calculations using eq 27 can beillustrated on example of some polyimides, particularly PI189
and PI304.111 This example is interesting since in many polymers, as it will be seen below, initial structure is not stable andirreversibly changing under influence of the temperature. Thisis shown in Figure 7 for PI189. Only on the second cycle ofheating above glass transition temperature Tg=165 ˚C temperature dependence of annihilation characteristics becomes, what
we shall call, “regular” (Figure 8) and corresponding to linear
variations of the elementary volumes of the free volume elements below (ν h ,g )and above (ν h ,r ) Tg (Figure 9). These volumes were found from the hole radius R 3 (o-Ps lifetimes τ 3).
The dependence shown in Figure 9 makes it possible to determine thermal expansion coefficients of holes occupied by Ps in
the rubbery (r) and glassy (g) phases.

90
80
70
60

50

100

150

200

Temperature (˚C)
Figure 9. Effect of temperature on the hole volume νh in PI189 calculated using the data from Figure 7 and equation eq 17.87

ture Ng = hg /vg = 0.44 ×1027 m-3. For comparison, according toReference 106, similar approach gives for PE and PTFE
0.73×1027 m-3 and 0.36×1027 m-3, respectively.
The case represented by eq 23 can be illustrated on example
of polymers with highly developed free volume and specificsurface, cross-linked polystyrenes96 (Table 4, see also sec. 4.1).
Effective radius of the voids is readily found from eq 15, andfor LPS-150X, for example, R3 =2–3 Å, R4 =5 Å, R5 =15 Å. Since
R5 is essentially larger than R3 and R4, density ρ , specific sur-

Positronium Atom in Solids –Peculiarities of Formation and Interconnection

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R45

TABLE 4: Annihilation characteristics τ i, Ii at room temperature, specific surface SL and SBET, density ρ and the largest
radius of nanopores R5 sensed by positronium in some of the cross-linked polystyrenes
substance

τ 3, ns
I3, %τ 4, ns
I4, %τ 5, ns
I5, %

R5, Å

ρ, g/cm3

SL, m2 /g

SBET, m2 /g

CPS(0.3)-100E, vac

1.23±0.05
6.76±0.24

5.58±0.11
7.67±0.10

39.49±0.28
15.00±0.06

13

0.76

1000

718

CPS(0.3)-150E, vac

1.69±0.15
7.27±0.37

5.31±0.32
8.33±0.33

16.56±0.33
9.52±0.33

8

0.71

1200

972

CPS(0.3)-150E, nitr

1.84±0.15
5.69±0.27

6.81±0.34
8.25±0.22

21.58±0.42
10.33±0.32

10LPS150X, vac

1.23±0.06
5.47±0.22

6.12±0.17
5.20±0.09

52.13±0.24
29.29±0.06

15

0.60

1600

965

face S, and fractional free volume of the porous samples Vf isdetermined mostly by the largest voids (R5).
Vf = (ρ Ps − ρ150)/ρ Ps = 0.429 ; N5 = 3Vf /4π R53 = 3×1019 cm-3

(29)

Correspondingly, from eq 23 for LPS150X N4 =1.45×1019 cm-3
and N3 =3.36 ×1019 cm-3. Along with results of five-term analyses of the PAL spectra, largest hole radius R 5 and density ρ,
Table 4 contains also specific surface area determined fromthermal desorption of argon and using Langmuir formula (SL)and also from nitrogen sorption measurements (SBET) combinedwith BET theory.89 Estimation of N5 from the equation N5 = SL/
4 π R 52 gives about the same value as in eq 29. On the other
hand, BET theory and Kelvin equation,89 applied to data on
nitrogen sorption makes it possible to obtain size distributionof EFV in samples from Table 4 shown in Figures 10 and 11 at
R > 10 Å. Remarkably, PAL estimation R 5 = 15 Å is just in a
middle of this distribution. Integration of the distribution for
LPS150X in the vicinity of R 5 =15 Å gives concentration ofvoids 2.5×1019 cm-3 which corresponds to the estimation given
above. Thus, combined application of PAL and BET techniques mutually confirms obtained results and extends theranges of the pore diagnostics in polymers.
4.4. Effect of low-temperature Ps formation. Low-temperature
positronium formation in polymer is known since nineties.45,112-117
Not referring to all publication, we wanted to mention someideas important for understanding of this effect, since it cameout to be useful in the polymer structure investigations. The
authors114 suggested a qualitative explanation of additional Ps
formation by interaction of positrons with weakly bound electrons, produced by preliminary γ -irradiation or by irradiation
during the positron annihilation experiment and localized at
low temperature. Simple quantitative consideration of thiseffect for PE and PMMA polymers was given in Reference 116
on the bases of the same system of kinetic equations, eq 16.
The only difference is in the zero moment condition in theequation for positrons: Pf+ (0) = 1− Q, where Q is a fraction of
the positronium formed in a polymer without γ -irradiation.
The rest of the positrons (1− Q) escape from the spur and have achance to form Ps on the trapped electrons. The solution of thecorrected system of eq 16 gives an increase in I3, ∆I3 due toadditional Ps formation on the trapped electrons:

Figure 10. Size-distribution of pores in cross-linked polystyrenes
from Table 4 obtained from incremental pore area data (BET).96∆I3 = (3/4)(1− Q)ν form(λ f +ν form− γ 3 ) -1ν T (λ f +ν T − γ 3 ) -1 = (3/4)(1− Q)AB,(30)
where B = ν T (λ f +ν T − γ 3 ) -1 and meaning of A is obvious from eq
30; ν form = 4π D+f R Ne = kform Ne and ν T = 4π DfPs R3 N3. Parameters
Df+ and DfPs are diffusion coefficients of free e + and Ps, correspondingly; R is effective radius for interaction of e + andtrapped electron, and R3 describes effective size of the trapping
center.

Figure 11. Distance between pores, calculated from the data shownin Figure 9.96

R46 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006
It becomes obvious that fitting of eq 30 to experimental
points for PE114 is possible only with k form (PE) = ν form /Ne =10 -7
cm3/s and ν T >> λ f , i.e. B = 1 (Figure 12). A description of the
PMMA results for the highest Ne = 2×1017 cm-3 114 is possible bysuggesting the two following alternatives:(a) ν T >> λ f , B =1, k form (PMMA) = (1/50)k form (PE) = 0.24×10 -8 cm3/s.
The last ratio probably means that Dƒ+(PMMA) = (1/50)Df+ (PE).
The diffusion coefficient for PE at 30 K is known to be about
0.1 cm 2 /s.118 Therefore, the positron diffusion coefficient at
30 K in PMMA is about 0.002 cm2 /s. This result corresponds
to recently published data.119

additional I3, %(b) ν T and λ f are comparable. Fitting to the highest value of∆I3 gives the probability of Ps trapping B = 0.2, if the rate constant of Ps formation on the trapped electrons in PE andPMMA are roughly the same: k form (PMMA) = k form (PE)=10 -7
cm3/s.

concentration Ne, 1015 cm-3
Figure 12. Additional intensity ∆I3 of the longest lifetime component
in PE as a function of the trapped electron density.114 The dependence
is described (solid line) by eq 30 with kform (PE)=1.2×10 -7 cm3/s and B=1. The dashed line here and in Figure 13 shows linear dependence
for comparison.

Shantarovich
A comparison of calculated dependences for the two cases
mentioned above is given in Figure 13.
We have to conclude that for PMMA, just as for PE, thealternative (a) gives a much better description of the whole setof points114 than the alternative (b). And this is the first direct
argument in favor of the point of view that in such polymers as
PE and PMMA all o-Ps atoms are probably trapped before
annihilation (B = 1). It means that in such cases I3 is dependent
mostly on the probability of Ps formation in a substance Q but
not on the number of the trapping sites N3 , since the number ofPs traps is too high (ν T >> λ f ). Further experiments of this type
could be useful. In the scope of the further study of the mechanism of Ps formation on the trapped electrons, a comparison of
the observed effect with results on thermo-stimulated luminescence (TSL) for the same samples at low temperatures was suggested in Reference 116. In agreement with the described
results, integral TSL intensity (Figure 14) for PE is much higher
than for PMMA, since both effects, additional Ps formation
and TSL are determined by the amount and stability of the
weakly bound electrons.
4.5. Some cases when free Ps trapping rate and annihilation rate are comparable. It follows from the previous sectionthat in PE and in PMMA most of o-Ps atoms are trapped before
annihilation (relation 21 from sec. 4.2 is fulfilled). How typical is this situation for solid substances, and are there any cases
when some of o-Ps atoms have chance to annihilate before
final localization (relation 19 in sec. 4.2)? The problem withpolymers will be discussed below. However some examples
can be found now among solid organic substances. In solidbiphenyl for example, o-Ps intensity I3 decreases to zero atpressure about 108 Pa.120,121 Probably, due to compression, elementary free volume size is reduced below the value, when thequantum level for positronium in the well can exist. In otherwords, o-Ps is not localized before annihilation.
Another example is found among highly dispersed polycrystalline oxides and salts of metals.122 Hydroxyapatite
Ca10 (PO 4) 6 (OH) 2 , vaterite CaCO3 and also CaSO 4 and Al2O3
are the materials where the intensity of extremely long (20–60ns) lifetime component of relatively low intensity depends not
only on the probability of o-Ps formation inside of the microcrystallite, but also on the probability of this atom to diffuse to
micro-crystallite surface and to localize inside of the intercrystallite space. It was seen from the fact that increase of theeffective size of crystallites after annealing was found to inhibit
long-lived o-Ps component because this enlargement prevents
positronium localization. Experiments122 are interesting also
by comparison of positron annihilation and sorption data. As

additional I3, %

8000

A

B

Intensity, arb. units

6000

x5
4000

2000

concentration Ne, 1017 cm-3
Figure 13. Additional long-lived component intensity ∆I3 as a function of the trapped electron density114 in PMMA. Solid curves correspond to different sets of parameters in eq 30.
kform (PMMA)=kform (PE)=1.2×10 -7cm3/s, B=0.21 or kform (PMMA)=(1/50)kform (PE)=0.24×10 -8cm3/s and B=1.

0
80

130

180

230

280

Temperature, K

Figure 14. Integral intensity of the TSL curves for PE (A) and
PMMA (B). Both curves are represented in the same relative scale.116

Positronium Atom in Solids –Peculiarities of Formation and Interconnection

it was discussed above for cross-linked polystyrenes, data on
low-temperature nitrogen sorption enable calculation of sizedistribution of elementary free volumes in a system. This distribution for vaterite is shown in Figure 15. Lifetime
measurements indicate that positronium is sensitive only to theleft part of this distribution: experimental lifetime of localized
positronium τ 4 = 22.87 ± 2.36 ns corresponds to pore radiusR≈10 Å and number density of free volume holes (intercrystillite spaces) N4 =(2–3)×1016 cm-3. This concentration gives aver3
age distance between inter-crystallite spaces √1/N attainable
for positronium atom (see also Figure 11) and, correspondingly,
positronium diffusion length L. In the result, diffusion coeffi2
cient D Ps
cryst =L / τ of non-localized Ps in a crystallite can be-1
found 0.4×10 cm2 /s. Amorphization leads to strong decreaseof positronium mobility. Comparison of the results of sorption
experiments in polymer glasses (Figures 5 and 10) and crystallites (Figure 15) show that in disordered structures positronium
is three orders of magnitude less sensitive. It is trapped bypores having three orders of magnitude higher concentration(about 3×1019 cm-3 ). This means three orders of magnitude-42
lower mobility (DPsgrass = 0.4×10 cm /s). These values and eq 20
give, for example, in polycrystalline α-Al2O3 N4 =1.14×1016 cm-3
for the number density of inter-crystallite spaces andN3 =9.7×1019 cm-3 in amorphous Al2O3 for the number density ofdefects.
5. Polymer structure as seen by positron annihilation
5.1. Structural transitions. A study of temperature dependence of annihilation characteristics (and low-temperature Ps
formation) became a useful instrument for investigations ofstructural transitions in polymers. The intermolecular spaces
available for localizing positronium are situated mostly in theamorphous regions and crystalline-amorphous interfaces.
Generally, a transition of a polymer from one state to another isaccompanied by changes in thermal expansion coefficient of
the material, which influences directly the thermal expansion
coefficient of the free volume holes. Consequently, by investigating the lifetime changes as a function of temperature, thetransitions could be observed. As soon as polymer segmental
motion, which is frozen at low temperature, commences byheating polymer samples above the relaxation temperature, theshallow potentials are smeared out. Then the trapped electronsin shallow potentials are excited, resulting in the reduction of
the amount of electrons available for Ps formation. Hence, thiscontributes to the decrease in the Ps formation. In such a way,

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R47

by monitoring the o-Ps intensity, an onset of a new motion ofsome parts of the polymer-chain below the glass-transition
temperature could be detected in polypropylene (see Figure 16
95). When the temperature of measurements is higher than Tg,
the polymer is in rubbery state, and large molecules in theamorphous regions can move like in viscous liquid, and Ps
probe manifests its nature simultaneously to be a seeker and a
digger of free volume holes, and could expand additionally thefree-volume hole where it is localized. As the temperature
increases, and, due to thermal expansion, more and more freevolume holes become available for localization of Ps, sharp
increase of the o-Ps intensity I3 is observed (Figure 16). In thisway, structural transformations in PP were observed. The identification of transition and relaxation temperatures can be done
according Krevelen and Hoftyzer123: Tγ : γ relaxation, rotationof the methyl group, Tβ : β relaxatrion, motion of very small
section of polymer chain, Tg : glass transition (lower), main
chain local motionwhich arise from purely amorphous material, Tgu : second (upper) glass transition, main chain local
motion which arises from amorphous material which is underrestraint due to the vicinity of crystallites, Tα c : premelting transition, hindered rotation of polymer chains inside the folded
crystals, and Tm : melting point. It is very difficult to estimate
precisely the error of determination of the corresponding values, but it is probably not higher than 5 K.
5.2. Heterogeneity of polymeric systems. Elastic andglassy polymers are normally considered as homogeneous disordered materials with unimodal (statistical) distribution ofelementary free volumes. In this paragraph, we discuss resultsof PAL measurements for a number of elastic and glassy polymers, where observations of the two long-lived o-Ps components in the PAL spectra and irregular (nonlinear, see sec. 4.3)
variations of annihilation characteristics in the vicinity of glasstransition temperature reveal structural heterogeneity of thesesystems. The conclusion is confirmed by measurements of
TSL, thermo-mechanics and mobility of penetranrs.
Elementary free volumes, the carriers of the free volume inmaterials, are very important characteristic of the matter,
determining such properties as permeability, swelling ability,
selectivity, mechanical strength, etc. It is generally accepted toconsider polymers in glassy and elastic states as homogeneous
disordered materials with unimodal, statistical distribution ofelementary free volumes. Positron annihilation lifetime measurements give a chance to check this point of view. If size
distribution of elementary free volumes is one-centered (onlyone average effective radius), PAL distribution, analyzed withthe known mathematical programs PATFIT or MELT, will
include only one long-lived (several nanoseconds) o-Ps component. Observation of two or more long-lived components in ao-Ps Lifetime (ns)

10

1

Tα c

D = 0 kGy
Lifetime
Intensity

40

Tgu

3

Tγ

35

Tg

30

Tβ

2

45

Tm 25

o-Ps Intensity (%)

N, 1017 g-1

100

20
1
0.00

20.00

40.00

60.00

80.00

100.00120.00pore radius R, Å
Figure 15. Size-distribution of elementary free volumes (inter-crystallite sites) in vaterite (CaCO3) polycrystal obtained from low-temperature sorption isothermof nitrogen gas (Brunauer-Emmett-Teller
BET method).122

100

200

300

400

500

Temperature (K)Figure 16. o-Ps lifetime and intensity versus temperature for polypropylene.95 Tγ : γ -relaxation temperature, Tβ : β -relaxation, Tg : glasstransition (lower), Tgu : glass transition (upper), Tα c : pre-melting transition, Tm : melting point.

R48 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Shantarovich

rubber or polymer glass would mean inhomogeneity of corresponding material. However the number of the long-lived Ps
components is not obvious sometimes. Thus, it was demonstrated recently124,125 that, in the cases of small (about 2 ns) difference between two lifetimes or between two peaks in theprobability density function of positron annihilation with agiven lifetime (obtained from the MELT program), the splitting can be an “artefact”, i.e. a result of incorrect mathematicaltreatment (deconvolution) of experimental non-exponential
lifetime distribution of positron annihilation. Nevertheless, weare interested in revealing heterogeneity of amorphous polymers in principle. Therefore, we concentrate here on the PALS
data for a number of polymers, such as polyisobutylene, polybutadiene, synthetic rubbers, cross-linked polystyrene, whereobservation of the two long-lived positronium components inthe spectrum (bi-central distribution) corresponds to the signsof heterogeneity of these compounds, obtained by supplementary techniques, such as TSL, X-ray scattering, mobility of themarker, thermo-mechanics and demonstrates heterogeneity ofthese materials. Being considered together, the experimental
results corroborate the idea of heterogeneity of some of thestudied structures.
Cross-linked polystyrene. Typical probability density function of different positronium lifetimes in the cross-linked networks CPS200E and LPS200X (References 96 and 126, seealso sec. 4.1) is given in Figure 17. Three groups of lifetimes(three effective sizes of EFV) are clearly resolved: τ 3 =1.5 ns
with the intensity I 3 =2.3%; τ 4 =7.4 ns and I 4 =2.8%; τ 5 =73 ns
and I 5 =32%. According to eq 15, these lifetimes correspond to
the following effective radius of the free volumes: R3 = 2–3 Å,
R4 = 6 Å and R 5 = 20 Å. On the other hand, mechanical tests(under compression) were performed for such super-crosslinked polystyrenes (CPS(0.3)100E and CPS(0.3)200E) fullyswollen in water and kept at different temperatures.127 The
derivative (dh/dT) of the spherical bead diameter h at different
temperatures and given pressure is shown in Figure 18. Curves
1 and 2 are related to mentioned swollen polystyrenes andcurve 3 for the dry CPS(0.3)100E is reproduced for comparison. Curve 4 related to the pure ice, obtained by crystallizationof water onto a quartz plate at -150–100 °C. At least two peaks
dh/dT are clearly seen for each sample at the same temperatures. It seems reasonable that these peaks correspond todifferent plastification effects of water in free volumes, characterized by inhomogeneous size-distribution and measured forthe first time by positron annihilation technique.
Rubbers. We suggest to consider reliable measurement of
the two long-lived o-Ps components as an argument in favor ofheterogeneity of amorphous polymers. This point is illustrated
on example of polymer rubbers. Table 5 contains the longestlived positronium components obtained from the PATFIT
unconstrained four-component analyses of PAL spectra for therubbers. Left part of the Table 5 includes materials described
well in terms of the two long-lived o-Ps components. This
description gives better fitting and characteristics of the components have a good precision. In the right part of the Table 5,
there are materials for which description in terms of the twolong lifetimes is not necessary: uncertainty of the longest lifetime intensity is too high. Therefore, in the frames of the previous discussion, positron annihilation method considers thefollowing group of substances as heterogeneous systems: cis1,4-polybutadiene (cis-1,4-PB); polyisobutylene (PIB), i.e. a
copolymer of isobutylene and isoprene; polybutadiene (SKB);
water

4

0.5

(dh/dT)

2

1

3
heat

3

0.0
-100
Figure 17. Probability density function of positron annihilation witha given lifetime (MELT results) for hyper-cross-linked polystyrene
network96,125 at different temperatures (RT and 35 K).

ice

-50

0
T, ˚C

50

100

Figure 18. Derivative thermo-mechanical curves dh/dT for sphericalbeads of the hyper-cross-linked polystyrene CPS(0.3)100E (1) andalso for this polymer (2) and CPS(0.3)200E (3) fully swollen inwater, and (4) ice on a quartz plate, crystallization at -150–100˚C.127

TABLE 5: Longest-lived o-Ps components obtained from the PATFIT unconstrained four-component analyses for a numberof rubbers
Inhomogeneous structure
τ3, nsI3, %Homogeneous structure
τ3, nsI3, %τ 4, ns
I4, %Substanceτ 4, ns
I4, %

PIB
ϕ=50%

1.59±0.11
18.66±2.67

2.68±0.12
13.67±2.97

BNKS-40ϕ=31%1.66±0.60
8.33±6.68

2.51±0.29
11.0±8.09

PIB, 180C
annealed

1.51±0.22
11.90±2.47

2.55±0.15
11.42±3.41

BNKS-40
Ann. 90C

1.95±0.10
14.17±0.52

3.95±1.0
0.93±0.87

cis 1,4 PB
ϕ=13%

1.29±0.59
4.54±1.50

2.66±0.05
24.50±1.85

PI
ϕ=25%

2.28±1.06
10.09±25.02

3.01±0.39
23.28±25.62

SKB
ϕ=69%

1.59±0.48
6.80±1.88

2.95±0.11
17.43±2.99

BNKS-28ϕ=47%2.02±0.16
13.8±1.6

3.54±0.70
2.24±2.05

BNKS-18ϕ=37%1.65±0.39
7.00±2.27

2.82±0.17
10.25±3.09

SubstancePositronium Atom in Solids –Peculiarities of Formation and Interconnection(τ 3 and I3) were found, so that glass transition temperature evencould not be determined from positron annihilation data.
Obviously, this uncertainty infers inhomogeneity of the last
polymer too. Both conclusions were confirmed using X-ray
scattering.
TFEMVE copolymer and poly(norbornen). A dependenceof the long positronium lifetime on temperature in the elastic
copolymer of tetra-fluoro-ethylene and perfluoro-methyl-vinylether (TFEMVE) (Figure 22) demonstrates smeared glasstransition (slow transformation of the structure) between −20
and +60 ˚C. Similarly, in metathesis poly (norbornen) of a composition 2(-SiMe3)-PNB annihilation characteristics show complex variations of the structure between 50 and 170 ˚C (Figure23) while listed Tg = 167 ˚C. We consider these results as an evidence of in-homogeneity of these two polymers too.

2.5

τ3
I3

P145htPS5mw-t

Lifetime (ns)

2.4

31
30

2.3

29

2.2

28

2.1

27

2

50

100

150

Intensity (%)

a copolymer of acrylonitrile (x) and butadiene (1-x) (BNKS)
with x=18%. This conclusion is confirmed using TSL technique. According to this method, materials from the left sideof the Table 5 have two glass-transition temperatures: 174/177
K for cis-1,4-PB; 230/233 K for PIB; 210/225 K for SKB, and226/246 K for BNKS (18%). Figure 19 gives a typical exampleof Tg splitting in BNKS (18%) compared to BNKS(28%) andBNKS(40%). In addition, heterogeneity of cis-1,4-PB was
revealed with so called mobility (penetration) technique by
measuring variations of permeation activation energy E D ofpenetrant with temperature. Thus, for the penetrant diphenylguanidine (DPhG), this variation (from 32.3 to 57.4 kJ/M) was
found above Tg at 65 °C. This increase of E D with heating is
not expected in the frames of simple free volume model forhomogeneous structure and can be related with the increase of
the volume content ϕ (Table 5) of the ordered sites of smaller
permeation, i.e. in fact, by phase transformation of the inhomogeneous rubber structure.
Polystyrene. Examples of temperature dependence of positronium annihilation characteristics in polystyrene (PS) samples with different molecular weights MW=3.6×103 and 8×105
are shown in Figures 20 and 21. Irregular variation of intensity
I3 of the long-lived Ps component in PS (MW=3.6x103, Figure20) on heating above Tg (well defined by the typical bending of
τ3 at 75 ˚C) correspond to structural transformations of the rubber in this temperature range. For PS with MW= 8×105 (Figure21), irregular variations of the both annihilation characteristics

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R49

26

Temperature, ˚CFigure 21. The same as in Figure 20, but for PS with MW=8×105.127

Figure 19. Results of TSL studies of BNKS.Integral intensity of TSL
spectra as function of sample temperature. Typical example of Tg
splitting (two peaks in the intensity curve) for BNKS(18%), compared to BNKS(28%) and BNKS(40%), illustrating in-homogeneityof BNKS(18%) structure.127

31

2.2
2.2

3

RT 12

2

100
200
300
Measurement temperature, K

3.2

τ3
I3

L177Si2Me3PNB

35

3

34

2.8

Intensity (%)

2.4

Intensity (%)

32

14

Figure 22. Temperature dependence of annihilation characteristics in
TFEMVE.

33

τ3
I3

2.6

16

10

Lifetime (ns)

PS MW=3600

Lifetime (ns)2.8

4

L178TFEMVEIntensity (%)

Lifetime (ns)

18
τ3
I3

30

50

100

150

Sample temperature, ˚CFigure 20. Variation of the positronium annihilation characteristics
in polystyrene (MW=3.6×103, Tg =75 ˚C) on heating from RT to 180
˚C. Irregular (nolinear, compare with Figure 8) variations of o-Ps
intensity manifest structural transformations of the polymer above Tg.

2.6

50

100

150

33

Measurement temperature, ˚CFigure 23. Temperature dependence of annihilation characteristics
for 2(-SiMe3)-PNB.127

R50 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006
5.3. Relaxation effects in polymer glasses. As we haveseen from the previous section, PAL investigations show thatmany polymer glasses and rubbers may be inhomogeneous.
This point of view is in agreement with one of the modern concepts of the glass structure, according to which amorphous
polymers are characterized by long-lived fluctuations of density of a size about 300–2000 Å.128 These fluctuations are
responsible for super-slow λ-relaxation processes observed bydynamic mechanical spectroscopy technique above Tg. Figure24128 illustrates this point of view by the temperature dependence of internal friction, i.e. peaks on the curve of mechanicallosses (logarithmic decrement ∆) at relaxation transition temperatures above Tg. Obviously, such situation infers slow relaxation of amorphous material, and such effects have to be seen
in PAL measurements. In positron annihilation experiments,
effects of thermal expansion (without structural transformations) are normally seen as linear (regular) variations of orthopositronium annihilation characteristics (lifetime τ 3 andFigure 24. Spectrum of internal friction in PMMA (MW=5×105,
Tg=122 ˚C). ∆ is logarithmic decrement of mechanical losses.128

Shantarovich
intensity I3) and free volume parameters(see Figures 8 and 9).
However, recent temperature-controlled experiments in polystyrene129 revealed non-regular variations of these characteristics (minium in variations of I 3 at 220 –340 K), showing
structural changes (Figure 25). The variations could not be
reproduced in full scale in the repeating cycles of measurements with the same sample kept for some time above glasstransition temperature. Therefore such effects can be called“vanishing” effects130 and are explained probably by slow
relaxation processes in a polymer glass described above.
Particularly, differential scanning calorimetry (DSC) measurements131 demonstrated the role of intramolecular relaxation
associated with mobility of phenyl groups ( β -transition).
Obtained results illustrate complex, dependent on thermal history, inhomogeneous character of the glass structure. In fact,
structure of some polymer glasses is changing continuously.
6. Conclusion
On example of positron annihilation studies, we have got a
glimpse of a quite interesting area of research, which came into
being at the junction of several disciplines: elementary-particle
physics, solid-state physics, and chemical kinetics. Referring
to a number of examples we have seen how the employment ofmodern facilities of nuclear physics makes it possible, on the
one hand, to identify a specific formation, one of the newatoms, called positronium, to study its transformations; on theother hand, to obtain valuable information about the structure
and chemical properties of substances from the influence ofthe medium on the transformations of positronium. Analysis
of intrinsic characteristics of positronium and bound states of
positrons and comparison of these characteristics with predictions are also of great interest as a check on several model concepts and calculating techniques.132
We were not able to go into details of experimental arrangements, results of measurements and theoretical interpretations
for other new atoms. This could be a topic of another review.
It seems however that with section “introduction to new atoms”
this paper is showing to a newcomer a little more than one of
the specific applications of the positron annihilation method.
In principle, the data on each of the new atoms and their comparison make physical information more complete and enableobtaining new interesting conclusions. The bibliography contains references to special reviews and original works to which
the reader is referred for more profound treatments of the issues
covered. We hope, it is seen from the review that studies of thephysics and chemistry of the positronium at this stage not only
made it possible to formulate the principal underlying concepts,
but also resulted in a quite promising method of probing into
physical and chemical properties of matter. And much of theinformation is either inaccessible with other methods or is
obtained much simpler.
Beyond doubt, further development of experimental techniques and the joint efforts of physicists and chemists will
expand these possibilities.
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