Journal of Nuclear and Radiochemical Sciences, Vol. 1, No. 1, pp. 11−13, 2000

11

Positronium as a Probe of Small Free Volumes
Tomasz Goworek*,†
Radiation Science Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
Received: October 13, 1999
The relation between free volume size and the lifetime of ortho-Ps trapped in that volume is usually described in
the framework of simplified model, proposed by Tao and Eldrup et al. In that model the positronium trap has
spherical shape and the potential is rectangular one with infinite depth. In spite of its simplicity in most cases themodel gives proper values of o-Ps lifetime and its zero-point energy in the trap. It can be also easily extended
toward nonspherical and large-size free volumes. In that last case the population of particle states lying above thezero-point energy has to be taken into account.

Positronium is a hydrogen-like structure, formed by an electron and positron. Owing to very low reduced mass the energies of all levels of that structure are scaled down by the factorof two comparing to hydrogen (e.g., the binding energy of theground state is 6.8 eV). Positronium forms when it is energetically favorable, i.e., the energy of bound system is lower thansum of energies of quasi-free electron and positron in themedium. If a pair of electron and positron is on the surface of
medium or close to that surface, it can escape to vacuum taking with it the energy gain in the form of kinetic energy of thebound system. When positronium is ejected not into an open
space, but to a cavity (void) inside the medium, it is trapped,loosing the surplus of energy in the interactions with the bulk.
The kinetic energy of positronium reaches finally the level ofzero-point oscillations (Figure 1). Trapping of positronium has
two important consequences:- the uncertainty principle determines the momentum of particle as dependent on the trap size,- interaction of positronium with surrounding bulk material
gives additional possibility of annihilation and thus increases
the decay rate (shortens the lifetime).

Figure 1. a) Energetic conditions of Ps formation; positronium work
function is negative. b) Rectangular potential steps for Ps at the medium-vacuum boundary; from left to right: open space, broad void, narrow void. The narrower is the well, the higher zero-point energy.
*Corresponding author. E-mail: goworek@tytan.umcs.lublin.pl.
FAX: +48-81-537-6191.
†on leave from Maria Curie Sklodowska University, 20-031
Lublin, Poland

Positronium atom can exist in two hyperfine structure states:with antiparallel electron and positron spins (parapositronium, p-Ps) and with parallel ones (ortho-positronium,o-Ps), distinctly differing as to their annihilation properties. 1
Para-positronium annihilates in vacuum in a very short time
0.125 ns transforming into two gamma quanta, 511 keV each;ortho-positronium in vacuum decays via three quantum
process (continuous energy spectrum) with the lifetime 142 ns.
Due to very short lifetime para-Ps is weakly affected by surrounding medium, while it is very probable that ortho-Ps picks
one of electrons of the medium with appropriate spin orientation (antiparallel one) and annihilates with it in a short time,
via two quantum emission.
Trapped p-Ps annihilates as a whole entity and the width ofits momentum distribution is simply related to the trap radius R
by uncertainty principle2:
Θ1/2 = 16.6/R(1)

where Θ1/2 is half-width of angular distribution of annihilation
quanta (in mrad, for R in angstroms). The deviation from colinearity of the quanta Θ is proportional to the transverse component of the momentum p⊥ of annihilating system: Θ = p⊥/mc.
Longitudinal pII component of the momentum can be observedin Doppler broadening of the 511 keV annihilation gammaline. The Doppler shift in that case is ∆E = pIIc/2. The lifetimeof ortho-Ps annihilating by pick-off process is determined by
the overlap of Ps wave function with bulk electrons of surrounding medium and the o-Ps lifetime becomes dependent on
the trap size. Thus, positron annihilation methods (positron
lifetime measurements PAL, or measurements of momentum
distribution of annihilating pairs) can be used in the study ofsmall free spaces in the structure of solids − defects in the
crystals, free volumes in amorphous media. In the majority ofpractical applications of PAL method the relation between freevolume size and ortho-positronium decay rate λ o-ps is takenfrom Tao-Eldrup model,3,4 assuming Ps as trapped in a spherical void representing for Ps a rectangular potential well. In
order to simplify the calculations, in that model the penetrationof Ps wave function outside the void is substituted by broadening the potential well by ∆R and assuming infinite height ofthat well (Figure 2). The value of ∆R was chosen to give thesame probability to find the Ps atom in the range (R, ∆R) as
outside the well in real finite potential. With these assumptions the model gives

λo-Ps = λ b [1−

R + 1 sin(2π R )]+λt
R+∆R 2πR+∆R(2)where λ b, R, λ t are o-Ps annihilation rate in the bulk, sphere© 2000 The Japan Society of Nuclear and Radiochemical Sciences

Goworek

12 J. Nucl. Radiochem. Sci., Vol. 1, No. 1, 2000

Figure 2. Tao-Eldrup model. Finite depth well with penetration of
the wave function to outside (a) is substituted by infinite well broadened by ∆R (b); the wave function does not extend behind R+∆R.

radius, intrinsic decay rate of o-Ps in vacuum, respectively.
There is to note, that at very small R no energy levels in thewell, thus positronium cannot be bound. The minimal radiusat which the energy level for Ps exists is ≈ 0.216/√V0, whereV0, the depth of potential well, is in eV for R in nanometers.
Thus, probing the distribution of free volumes in the mediumby positron methods, we do not see the smallest objects with R
below ≈ 0.12 nm. The equation (2) was applied to the free volume holes from 0.15 nm (e.g., Ref. 5) to 0.60 nm (zeolites6)
and in that range of R the last term in eq 2 can be neglected.
The equation eq 2 contains in fact a set of empirical parameters. The value of λ b is usually assumed as weighted averageof p-Ps and o-Ps intrinsic decay rates

λb =

1
3
λ+ λ
4 s 4 t(3)

In that approach λ b looks independent on electron density ofthe medium surrounding the free volume hole. In earlier models of the pick-off annihilation7 the λ b was assumed proportional to the medium density and effective number of electrons
per molecule Zeff participating in that process:λ b = πr02cnZeff(4)where r0 is the classic electron radius and n is the number ofmolecules per volume unit. Zeff is also an empirical parameter,
related to some extent to the number of outer electrons in themolecule, but leaving certain freedom in fitting λ b to the experimental data. Thus, λ b vs. electron density dependence in one
model (eq 3) is contrasting to that in other one (eq 4). The
annihilation rate is determined by the electron density at thesite of positron. This density is created partly by o-Ps own
electron, partly by the electrons of surrounding medium.
However, owing to the Pauli principle only one of these electrons can found itself in the nearest vicinity of e+ (that one withthe spin opposite to o-Ps electron), repulsion between o-Ps andother electrons reduces their influence on the annihilation rate.
Thus, the configuration giving maximal electron density issimilar to that in Ps− ion and Ps− decay rate is known to be thesame as that given by eq 38: λ (Ps−) = 2ns−1. The λ b value
equal 2 ns-1 is thus rather the maximal expected rate, the real
ones can be lower and dependent on n. If the Tao-Eldrup
model works (as we observe in polymers, molecular crystals,
liquids) it means that we are in the range of high densities andλ b approaching its maximum; the dependence of λ b on n and
Zeff is thus weakened.
The equation 2 is valid for spherical voids. The real freevolume holes are irregular, the lifetime depends on theirshapes. Figure 3 shows the o-Ps lifetime as a function of void
volume for several simple shapes. It is seen that the positron
lifetime experiments bring no information strictly about thevolume; all what we determine is an equivalent radius of ahypothetical sphere, giving the same lifetime as that observed
experimentally. Note, that in the case of elongated free volume
holes the lifetime stops to be dependent on volume (length) if
the hole cross section remains constant. Depending on the
crystalline structure or polymer chain arrangement we canFigure 3. The o-Ps lifetime as a function of void volume for several
void shapes: a) sphere; b) ellipsoid with the eccentricity ε = 0.2; c)cube; d) cuboid with constant square cross section 0.5 × 0.5 nm. In allcases ∆R is assumed 0.166 nm.

make the choice between the idealized bodies (sphere, cuboid,
ellipsoid, cylinder) most suitable for description of particular
medium, but always we receive equivalent cylinder radii,
equivalent cuboid sides etc. The “equivalent” dimensions
should be compared with these obtained using other methods,
expecting they are identical (or at least close to each other).
The parameter ∆R is introduced as a means to simplify thecalculations. The ∆R value is chosen to receive the probability
to find Ps between R and R+∆R, equal to the probability tofind Ps outside the real well, i.e., between R and ∞ (Figure 2).
The value of ∆R = 0.17 nm was proposed by Eldrup et al. forplastic crystals investigated by them.4 In liquids it was noticed
that ∆R can vary in certain limits.9 Almost all the data, on
which the estimation of ∆R was based, relate to narrow classof media, organic solids and liquids, composed of similar
atoms. It cannot be excluded that for other compounds different values of ∆R would be needed. The ∆R is not an universal
number fitting to all cases, thus, giving too many digits in its
value (e.g., ∆R=0.1656… etc.) has no physical meaning. By
the way, even the value of R is not too precisely defined: theboundaries of such a small hole are not a stepwise function of r
and the electron density changes from the bulk value to zero incertain range of r. For large R this transition region can beneglected, but not for smallest holes.
In some cases, e.g., interpretation of the results of momentum distribution measurements we need to know the zero point
energy of Ps particle in the well. Comparing to real well(finite depth V0, radius R), the transition to infinite well shifts
the zero point energy upwards, while substitution of R by
R+∆R shifts it down. However, for ∆R = 0.166 nm these two
shifts do not compensate exactly. Figure 4 shows the energies
for real void radius R and V0 = 1 eV or 1.5 eV (exact solution
in rectangular potential) and for R+∆R, V0 = ∞ (Tao-Eldrup
model) in the units of zero point energy for the radius R andh2
infinite potential V0, i.e., in the units of 2m
2 . It is seen thatPs R
the application of R+∆R instead of R gives acceptable energy
values for 0.30 nm < R < 0.55 nm if V0 ≈ 1 eV, but for larger R
or deeper potential the shift to lower energies, following from
Tao-Eldrup model, is exaggerated. The misfitting is not toolarge, but should be taken into account. With the rise of R therole of potential depth reduces.
The Tao-Eldrup model gives to us relatively good estimateof free volume size and Ps zero-point energy for definite kindof that volume. It is enough when the structure of free volume
in a solid is simple, like in crystals: one dominant type of
defects (e.g., vacancies), or several well defined ones (divacancies, free spaces in the vicinity of impurities a.s.o.). In theamorphous structures, like polymers, we have rather continuous spectrum of free volume sizes which produces a continu-Positronium as a Probe

J. Nucl. Radiochem. Sci., Vol. 1, No. 1, 2000 13

Figure 4. Zero point energy in spherical well in the units of h 2 π 2 /2mPsR2: a) infinitely deep well, radius R; b) finite potential depth V0 =1.5 eV and radius R; c) infinite potential depth and radius R+∆R (∆R
assumed 0.166 nm); d) finite potential depth V0 = 1.0 eV and radius R.
Cases b) and d) represent real situations, case c) - the Tao-Eldrup
model.

ous spectrum of decay rates S( λ ). Fitting programs like
CONTIN10 or MELT11 allow to determine that S(λ) spectrum.
However, can the λ spectrum be directly “translated” into voidradii spectrum, is still an open question. In the experiment weobserve not the radii distribution V(R), but the product
V(R)T(R), where T(R) describes the dependence of Ps trapping
probability on R. From eq 2 we have12:
V(R)T(R)=λ b ∆R

1-cos (2π

R )R+∆R S(λ)(R+∆R)2(5)

The T(R) function is not well determined yet. Moreover, thetransformation from S(λ) to V(R) as in eq 5 needs to assume
that positronium once trapped does not change its location up
to the decay, i.e., the processes like tunneling from void tovoid, averaging the lifetime (with preference to larger voids),
are neglected.
The Tao-Eldrup model describes the ground state of a particle in the potential well. In deep or broad wells we should
expect the existence of excited levels. However, if the freevolume holes are of the sizes not exceeding (0.3−0.4) nm, likeusually in polymers or crystals, the spacing of second state in
the well from the ground one is much larger than thermal energy at the temperatures usually applied. If the thermal equilibrium is achieved, we can neglect the population of excitedstates.
In spite of all limitations, created by approximate characterof the model, rough agreement of model and experiment isobserved. The model is well tested in the case of plastic crystals,4,13 however the range of free volumes equivalent to vacancies cover a very narrow range, from 0.10 to 0.25 nm3, several
points for zeolites6 also fit well to the expectations from the
Tao-Eldrup model. In principle, the data for liquids cannot be
the test of our model, for the sizes of zero-point bubbles arecalculated ones, basing on the macroscopic surface tension,
which need not be true in micro scale. One can compare thepositron data on free volumes with similar data obtained byother methods; as an example of such a comparison can serve
the Reference 14, where the free volumes were independently
estimated from gas diffusivity measurements.

In our recent papers (e.g., Ref. 15) we have tried to extend
this model to the case of large R and cylindrical free volumes,e.g., to the pores, without introducing too radical changes inorder to retain smooth transition from small to large radii within one model. The only change was the increase of ∆R to 0.19
nm; it has proven necessary to match the annihilation radii tothose determined by classic chemical methods: LN desorption,
thermogravimetry etc.). This extended model (i.e., accounting
the annihilation from excited levels in the well) explains e.g.,
the decrease of o-Ps lifetime with the rise of temperature, commonly observed in porous media.
At low temperature limit, like for small R, only the groundlevel in the well is populated and the o-Ps lifetime is determined by classic eq 2, i.e., at T→0 the o-Ps lifetime should
tend to the constant value. This “saturation” value wasobserved in some cases, however, in others more complex
behavior appears. If we assume, as in the model presented
above, that the overlap of Ps wave function with the bulk isdetermined only by the size and shape of the void and there areno other factors influencing the Ps decay beside pick-off, thedeviation from expected lifetime value needs to accept the possibility of λ b variation.
There is to note that in standard applications of positron lifetime technique to the determination of free volume holes, only
the term λ po relating to pick off process is taken into account,the intrinsic decay rate λ T is usually neglected as a small contribution. In the case of large voids, already at R = 2 nm thepick-off rate is smaller than intrinsic decay one. The pick-off
process, which is the subject of Tao-Eldrup and similar models, is no longer the dominant one. Various factors which inusual case are of secondary importance for annihilation rate
can begin to play substantial role, e.g., chemical effects, sorption processes. The lifetime is reciprocal of λ , thus if λdecreases, the lifetime should lengthen.
The author wishes to thank the Ministry of Education,
Culture, Sport and Science of Japan for the grant enabling him
the stay at KEK Institute in Tsukuba.
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