"Positron Annihilation on Large Molecules" Phys. Rev. A , 61 (2000), 022719 K. Iwata, G. F. Gribakin, R. G. Greaves, C. Kurz and C. M. Surko

"Positron Annihilation on Large Molecules" Phys. Rev. A , 61 (2000), 022719 K. Iwata, G. F. Gribakin, R. G. Greaves, C. Kurz and C. M. Surko
Positron annihilation on large molecules
Koji Iwata,1,* G. F. Gribakin,2,† R. G. Greaves,1,‡ C. Kurz,1,§ and C. M. Surko1,储
Physics Department, University of California, San Diego, La Jolla, California 92093-0319
School of Physics, University of New South Wales, Sydney 2052, Australia
共Received 6 May 1999; published 18 January 2000兲
Positron annihilation on molecules is known to depend sensitively on molecular structure. For example, in
the case of hydrocarbon molecules, modest changes in molecular size produce orders of magnitude changes in
the observed annihilation rates. Although this process has been studied for more than three decades, many open
questions remain. Experimental studies are described which are designed to test specific features of the annihilation process. Two possible mechanisms of the annihilation are considered theoretically: direct annihilation
of the positron with one of the molecular electrons, including possible enhancement of this process when
low-lying virtual or bound positron-molecule states are present, and resonant annihilation through positron
capture into vibrationally excited states of the positron-molecule complex. The dependence of annihilation
rates, ␭, on positron temperature T p is studied for the first time for molecules, and at low values of T p the
dependence follows a power law ␭⬀T ⫺ ␰ , with ␰ ⬇0.5. These data are used to test the predictions of direct
numerical calculations and theories of the virtual-level enhancement. Partially fluorinated hydrocarbons are
studied in order to understand the rapid changes in annihilation rate produced in hydrocarbons as a result of
fluorine substitution. These data are compared with the behavior expected due to direct annihilation when there
is virtual or bound level enhancement. Measurements of positron annihilation on deuterated hydrocarbons are
described which test the dependence of the annihilation on the nature of the molecular vibrations. The relationship of the presently available experimental data for annihilation in molecules to current theories of the
annihilation process is discussed.
PACS number共s兲: 34.85.⫹x, 34.50.⫺s, 78.70.Bj, 71.60.⫹z
The annihilation of low-energy positrons on atoms and
molecules is a fundamental phenomenon in the field of
atomic and molecular physics 关1,2兴. Experimental studies of
this subject have been conducted for more than four decades
关3,4兴. The introduction of a modified Penning-Malmberg trap
a decade ago to accumulate large numbers of roomtemperature positrons has expanded experimental capabilities for these studies 关5,6兴. The quality of the data was further improved by subsequent increases in the number of
positrons available for experimentation 关2,7兴. The variety of
substances studied has also expanded due to improvements
in the low-pressure operation of the positron accumulator
关1,2兴. Stored positrons can now be manipulated for other
kinds of experiments, including heating the positrons for
temperature dependence studies 关8,9兴, and the creation of
positron beams with very narrow energy spreads for a new
generation of scattering experiments 关10兴. While these ad*Present address: University of California, San Francisco, Physics
Research Laboratory, 389 Oyster Point Blvd., Suite #1, South San
Francisco, CA 94080.
Present address: Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast
BT7 1lNN, United Kingdom.
Present address: First Point Scientific Inc., 5330 Derry Avenue,
Suite J, Agoura Hills, CA 91301.
Present address: Kasernenstrasse 8 A-7000 Eisenstadt, Austria.
Author to whom correspondence should be addressed. Electronic
address: [email protected]
vances and complementary theoretical work have illuminated many facets of the interaction of positrons with atoms
and molecules leading to annihilation, a detailed understanding of the phenomenon has yet to be achieved.
Historically, the annihilation rates of positrons with atoms
or molecules have been expressed in terms of the dimensionless parameter
Z eff⬅
␲ r 20 cn
where ␭ is the observed annihilation rate, r 0 is the classical
radius of an electron, c is the speed of light, and n is the
number density of atoms or molecules 关1兴. Measured values
of Z eff for a variety of substances are summarized in Ref. 关1兴.
The parameter Z eff is a modification of the Dirac annihilation
rate for a positron in an uncorrelated electron gas. For small
atoms and molecules, Z eff is typically regarded as the effective number of electrons contributing to the annihilation process. For these species, values of Z eff are similar to the number of electrons in the atom or molecule, Z. However this
approximation is crude; for example, even for atomic hydrogen, which has only one electron, Z eff is 8.0 at low energies
关11兴. There is extensive evidence that annihilation occurs
only on outer-shell electrons 关2兴. Thus, in the case of large
atoms, one should consider that it is not all the electrons but
only the valence electrons 共e.g., 8 for noble gases heavier
than helium兲 that participate in the annihilation process, yet
Z eff⫽400 for Xe. Annihilation rates as much as two orders of
magnitude larger than Z were observed for molecules such as
butane by Paul and Saint-Pierre in 1963 关3兴. Surko et al.,
taking advantage of the low-pressure capabilities of the pos-
61 022719-1
©2000 The American Physical Society
itron trap 关5兴, were able to extend these studies to larger
organic molecules, including alkanes as large as hexadecane
(C16H34) and a variety of aromatic molecules and annihilation rates Z eff up to five orders of magnitude larger than Z
were observed. Thus the data clearly indicate that a model of
the annihilation process based upon Eq. 共1兲 and uncorrelated
dynamics of the positron and bound electrons is inadequate.
While a detailed explanation of the experimental data is
still lacking, we believe it is useful to relate the experimental
results to two possible mechanisms of the annihilation process. Here we consider annihilation in the case where there is
a thermal distribution of low-energy positrons interacting
with atoms or molecules. The simplest mechanism is direct
annihilation of the incident positron with one of the atomic
or molecular electrons. The contribution of this mechanism
to the annihilation rate is proportional to the number of valence electrons available for annihilation. It will be enhanced
by the attractive positron-electron interaction, which tends to
increase the overlap of the positron and electron densities on
the atom or molecule. For example, this is the case when a
low-lying virtual level at energy ␧ 0 ⬎0 or a shallow bound s
state (␧ 0 ⬍0) exists for the positron 关12兴. It is known that, in
⬀1/( 兩 ␧ 0 兩 ⫹␧) for small positron kinetic enerthis case, Z eff
gies ␧ⱗ 兩 ␧ 0 兩 关13–15兴. It has been predicted that this effect is
responsible for the large Z eff values observed in the heavier
noble gases (Z eff⫽33.8, 90.1, and 401 for Ar, Kr, and Xe,
respectively 关1,16兴兲.
In the case of annihilation on molecules, which have vibrational and rotational degrees of freedom, a second potentially important mechanism is resonant annihilation. In this
process, the positron annihilates with a valence electron after
being captured into a Feshbach-type resonance in which the
positron is bound to a vibrationally excited molecule. In
analogy with a mechanism frequently used to explain electron attachment to molecules, this mechanism was advanced
关5兴 to explain the high annihilation rates observed in alkane
molecules, and the strong dependence of annihilation rates
on molecular size. This model assumes that the positron can
form bound states with the neutral molecules 共i.e., that the
positron affinity of the molecule is positive, ⑀ A ⬎0). Capture
is then possible if the positron energy is in resonance with
one of the vibrationally excited states of the positronmolecule complex. Such resonances have been observed in
electron scattering from some simple molecules, e.g., NO
关17兴, that have positive electron affinities.
The density of states ␳ (E) due to the vibrational excitation spectrum of the complex can be high, even if the available energy E⫽ ⑀ A ⫹␧ is only a few tenths of an eV 共making
the plausible assumption that the presence of the positron
does not alter significantly the molecular vibrational spectrum兲. For a thermal 共i.e., Maxwellian兲 distribution of posi(res)
tron energies, the observed resonant contribution Z eff
large molecules is an average over many resonances located
at specific positron energies. Accordingly, the magnitude of
Z eff
is proportional to ␳ (E). This density of states increases
rapidly with the size of the molecule, ␳ (E)⬀(N v ) n v , where
N v is the number of vibrational modes, n v ⬃ ⑀ A / ␻ is the
effective number of vibrational quanta excited in positron
capture, and ␻ is a typical molecular vibrational frequency.
Thus the resonant annihilation mechanism provides a possible explanation for the rapid increase in Z eff that is observed when the size of the molecule is increased. For ther(res)
as large
mal positrons, we have estimated that values of Z eff
as 10 –10 might be expected as a result of this process.
These values are comparable with the largest values of Z eff
observed so far: 4.33⫻106 for anthracene 关18兴, and 7.56
⫻106 for sebacic acid dimethyl ester 关19兴.
One necessary condition for resonant annihilation is the
existence of a positron-molecule bound state. Indirect evidence for the existence of such states comes from the experimental results and their interpretation by Surko et al. 关5兴.
Many-body theory calculations by Dzuba et al. 关20兴 predicted that positrons can be bound to metal atoms such as
Mg, Zn, Cd, and Hg. Variational calculations by Ryzhikh
and Mitroy proved rigorously that positrons form bound
states with Li atoms, and showed that bound states also exist
for Na, Be, Mg, Zn, and Cu 关21兴. It is likely that molecules
have essentially much larger long-range ‘‘potential wells’’
for the positron, and therefore many molecules are likely to
be capable of binding positrons.
The objective of the present study was to try to investigate
specific features of the annihilation process by studying the
dependence of annihilation rates on such parameters as positron temperature, the electronic structure of the molecules,
and the frequency spectrum of molecular vibrational modes.
As discussed below, we have not been entirely successful in
this objective. Nonetheless, the studies described here can
provide important benchmarks with which to test refined
models of the annihilation process.
This paper is organized in the following way. In Sec. II,
previous experimental results are reviewed. Theoretical considerations regarding the annihilation process are described
briefly in Sec. III. The positron trap and the experimental
procedure for measuring annihilation rates are described in
Sec. IV. The results of a new series of experiments and the
relationship of these studies and other available data to current theoretical work are discussed in Sec. V. We also test a
recently proposed phenomenological model of the annihilation process in Sec. V D. Finally, our current understanding
of the physics involved in the positron annihilation processes
is summarized in Sec. VI, together with a discussion of open
questions in this area.
The existence of very high annihilation rates on large
molecules was discovered in the early 1960s in the seminal
work of Paul and Saint Pierre 关3兴, and complementary experiments were later carried out by Heyland et al. 关4兴. Later,
Surko et al. used a positron accumulator to extend these
studies to much larger molecules 关5兴. Murphy and Surko
discovered very strong dependences of the rates of positron
annihilation on the chemical composition of the molecules.
For example, they found that perfluorinated molecules have
much smaller annihilation rates than those of the analogous
hydrocarbons 关18兴. They also discovered an empirical linear
scaling of ln(Zeff) with (E i ⫺E Ps) ⫺1 , where E i is the atomic
or molecular ionization potential, and E Ps⫽6.8 eV is the
binding energy of a positronium atom 共Ps兲.
This scaling was found to be valid 共to better than an order
of magnitude in Z eff) for all noble-gas atoms and nonpolar
molecules studied thus far 共i.e., species in which E i ⬎E Ps),
that do not contain double or triple bonds. While this scaling
has not been understood theoretically, it has been conjectured that it provides evidence for a model in which a highly
correlated electron-positron pair moves in the field of the
resulting positive ion, and that this dominates the physics of
the annihilation process 关18兴.
Recent theoretical work on positron annihilation with
noble gas atoms 关15兴 and ethylene 关22兴 confirms that virtual
Ps formation makes a large contribution to positron-atom
and positron-molecule attraction, and is crucial for determining the low-lying virtual levels for the positron that give rise
to large Z eff values. However, if the ionization energy of the
system is greater than E Ps by one or a few eV, the Psformation process is strongly virtual 共i.e., far off the energy
shell兲, and consequently the lifetime of this temporary ‘‘ion
⫹Ps’’ state, ␶ ⬃ប/(E i ⫺E Ps) is not large enough to produce
any direct effect on the positron-atom or positron-molecule
In a separate set of experiments, the spectra of 511-keV ␥
rays from positrons annihilating on various atoms and molecules were studied in a positron trap 关2兴. The observed spectra are Doppler broadened due to the momentum distribution
of annihilating electron-positron pairs which, for the case of
room-temperature positrons, is dominated by the momentum
distribution of the bound electrons 关23兴. Thus the Doppler
broadening measurements provide information about the
quantum states of the annihilating electrons. The results obtained in Ref. 关2兴 are consistent with a model in which the
positrons annihilate with equal probability on any valence
electron 共i.e., a model in which the positron density is distributed evenly around the molecule兲. These measurements
indicate that the large annihilation rates that are observed
depend on global properties of the molecule as opposed to
共localized兲 positron affinity to a particular atomic site.
In the following paper Gribakin discusses two basic
mechanisms of positron annihilation, direct and resonance
annihilation, that are potentially relevant to the interaction of
low-energy positrons with molecules 关24兴. Here we briefly
summarize the key results of this analysis.
The physical processes responsible for the observed large
values of Z eff can be understood qualitatively in the following way. The interaction rate ␭ i of a positron with an atom or
a molecule can be expressed as ␭ i ⫽n ␴ v , where ␴ is the
interaction cross section and v is the velocity of the positron
relative to the atom or molecule. If the positron-atom or
positron-molecule interaction time 共or the ‘‘dwell time’’兲 is
denoted by ␶ , the probability of the positron annihilating
during an interaction can be written heuristically as (1
⫺e ⫺ ␶ / ␶ a ), where 1/␶ a ⬅⌫ a is the annihilation rate for the
positron localized near the atom or molecule during the in-
teraction. It is obtained from the two-photon spin-averaged
annihilation cross section as
⌫ a ⫽ ␲ r 20 c ␳ ep ,
where ␳ ep is the positron density on the atomic or molecular
electrons 关25兴. If we use ␳ ep ⫽1/(8 ␲ a 30 ) for the ground-state
Ps atom as an estimate, then ␶ a ⬇5⫻10⫺10 s is the familiar
spin-averaged Ps lifetime. Thus the annihilation rate ␭ in
positron-atom or positron-molecule interactions is given by
␭⫽n ␴ v共 1⫺e ⫺ ␶ / ␶ a 兲 .
Comparing this expression with the definition of Z eff 关Eq.
共1兲兴, we have
Z eff⫽
␲ r 20 c
共 1⫺e ⫺ ␶ / ␶ a 兲 .
Therefore, enhanced values of Z eff can be achieved by either
having a large interaction cross section ␴ , or by making the
interaction time ␶ large.
In this section, we discuss cases in which the interaction
of positrons with atoms and molecules can result in relatively
large value of ␴ or ␶ . We first discuss direct annihilation in
atoms and molecules. We then discuss resonant annihilation
in molecules that possess vibrational and rotational degrees
of freedom. Finally, we discuss the circumstances by which
molecules with several atoms are likely to have virtual or
weakly bound levels, which, in turn, can have an important
effect on the annihilation process.
We have omitted from discussion two other possible
mechanisms which lead to the formation of quasibound 共or
bound兲 positron-atom or positron-molecule states 共i.e., states
that would produce large values of ␶ ). Formation of a bound
state is energetically prohibited in a two-body collision, and
so another particle is necessary. Below we discuss the case
where vibrational excitations 共i.e., phonons兲 play the role of
the third particle. Other possible mechanisms involve another atom or molecule in the collision 共i.e., three-body collision兲 or a photon. We do not discuss the possibility of
three-body collisions involving the positron and two atoms
or molecules because our experiments are performed at low
pressures of the test gas, and the annihilation rates are observed to depend linearly on test-gas pressure 关1兴. This indicates that the annihilation process is due to a two-body interaction of a positron and an atom or a molecule.
The positron-atom or positron-molecule quasibound state
formed by positron capture could be stabilized by the emission of a photon. However, the radiative lifetime for infrared
emission is much larger than typical atomic radiative lifetimes, and so it is also much larger than the positron annihilation lifetime in the atom or molecule. The annihilation
event has a much greater probability than radiative stabilization. The positron could also be captured into a true bound
state in a binary collision with the atom or molecule by the
emission of a photon 共i.e., ‘‘radiative recombination’’兲. In
this case, ␴ in Eq. 共4兲 would be the radiative recombination
cross section, and ␶ in Eq. 共4兲 would be infinite. However, it
can be shown that the probability of this process has the
same order in inverse powers of c as direct positron annihilation. Numerically, this gives a contribution to Z eff , which
is less than 1. Since this effect does not increase rapidly with
the size of the molecule, it also appears to be negligible.
One feature of the available data runs counter to the idea
that different annihilation mechanisms are operative for different classes of atomic and molecular species. As we have
reported previously and discuss in Sec. V D 1, there is an
empirical scaling of the form ln(Zeff)⫽A(E i ⫺E Ps) ⫺1 , which
fits all of the data for atoms and single-bonded molecules
reasonably well, with only one fit parameter A. This scaling
could be interpreted as evidence that one mechanism describes annihilation in both atoms and molecules. In this picture, the major differences in annihilation rates are due only
to differences in the electronic structure of the atoms and
molecules 共i.e., in contrast to the resonant vibrational mode
model discussed above兲. Thus one mechanism would be responsible for both small and large values of Z eff . However,
we are not aware of any existing theoretical picture that
could explain the large observed values of annihilation rates
on the basis of electronic structure alone. Consequently, here
we present a theoretical framework in which different annihilation mechanisms are dominant for different classes of
atomic and molecular species, but we encourage further investigation of this issue.
A. Direct annihilation
Suppose first that positron-atom or positron-molecule interaction is a simple elastic collision, and that annihilation
takes place directly between the incident positron and one of
the bound electrons. The dwell time ␶ ⬃R a / v , where R a is
the atomic or molecular radius, is small compared to the
annihilation time ␶ a . Hence the annihilation probability is
just ␶ / ␶ a Ⰶ1, and the rate of direct annihilation is estimated
⬃ ␴ R a ␳ ep ,
Z eff
cross section ␴ ⯝4 ␲ k ⫺2 corresponds to Z eff
⬃103 . This
value of Z eff is still much smaller than the values observed
for large molecules. We conclude that the positron dwell
time near the molecule, ␶ , must be much larger than that of
the simple direct annihilation process.
Equation 共5兲 is too simple to describe direct annihilation
quantitatively. However, it is possible to derive a more ac(dir)
to the scattering properties
curate formula that relates Z eff
of the system at low positron energies 关24兴:
⯝F R 2a ⫹
Z eff
where ␴ is the elastic cross section, f 0 is the s-wave scattering amplitude, R a is the average positron-atom or positronmolecule separation at which the annihilation occurs, and F
is a factor that takes into account the overlap of the positron
and electron densities. Note that unlike Eq. 共5兲, the above
expression does not vanish even when the scattering cross
section is very small. Indeed, the positron wave function is
always a sum of the incident and scattered waves, and, even
if the scattering amplitude is very small, the incident wave
contributes to the annihilation rate. Formula 共6兲 contains
contributions of both, as well as the interference term. When
the scattering cross section is anomalously large, 兩 a 兩 ⰇR a ,
Eq. 共6兲 coincides with Eq. 共5兲. Comparison of theoretical
cross sections and Z eff for noble gases 关27兴 and C2 H4 关22兴
shows that Eq. 共6兲 works well at energies of up to 0.5 eV, if
R a and F are used as fitting parameters (R a ⬃4 and F⬃1
a.u. are the typical values兲. When low-energy scattering is
dominated by the presence of a virtual level or a weakly
bound s state, both ␴ and Z eff become large. They also show
a similar rapid dependence on the positron momentum. For a
short-range potential, this dependence is determined by the
standard formulas 关28兴
f 0 ⫽⫺
where we used Eq. 共2兲 to estimate ␶ a . If we consider a
typical low-energy positron-atom or positron-molecule cross
section in the range ␴ ⫽10⫺15 –10⫺14 cm2 , R a ⫽5a 0 , and the
⬃10–100 is obtained.
Ps value of ␳ ep , then Z eff
The long-range positron-atom or positron-molecule interaction is attractive due to dipole polarization of the electron
cloud by the positron. At low incident energies this interaction may increase the collision cross section ␴ above the
value determined by the geometric size of the atom or molecule, if a virtual ( ␬ ⬍0) or a shallow bound ( ␬ ⬎0) s state
exists for the positron-atom or positron-molecule system at
␧ 0 ⫽⫾ប 2 ␬ 2 /2m. In this situation the scattering length a
⫽ ␬ ⫺1 and the cross section at zero energy ␴ ⫽4 ␲ a 2
⫽4 ␲ ␬ ⫺2 can be much greater than the size of the atom or
molecule 关13,26兴. This effect can explain the rapid increase
and large values of Z eff in Ar, Kr, and Xe 关14,15兴. The enhancement, due to this mechanism, is limited by the size of
the positron wavelength. For room-temperature positrons the
wave number is k⬃0.045a ⫺1
0 , and the maximal possible
⫹2R a Re f 0 ,
␬ ⫹ik
␬ ⫹k 2
Since the target has a nonzero dipole polarizability ␣ , these
formulas must be modified to account for the long-range
⫺ ␣ e 2 /2r 4 positron-target interaction. This can be done by
using the modified effective-range expression for the s-wave
phase shift ␦ 0 ,
tan ␦ 0 ⫽⫺ak 1⫺
冑␣ k
␲␣k 4␣k2
ln C
together with the usual relations ␴ ⫽4 ␲ sin2␦0 /k2 and Re f 0
⫽ sin 2␦0/2k 共atomic units ប⫽m⫽e⫽1 are used hereafter兲
关29兴. If ␣ is known, Eq. 共8兲 contains basically one free parameter, the scattering length a, since the dependence of ␦ 0
on the positive constant C is rather weak. For ␣ ⫽0 Eq. 共7兲
with ␬ ⫽a ⫺1 are recovered. The polarization potential qualitatively changes the behavior of ␴ and Z eff at small momenta. They now contain terms linear in positron momentum
k, and ␴ ⫽4 ␲ (a⫹ ␲ ␣ k/3) 2 follows from Eq. 共8兲 in the case
where k→0 and 兩 a 兩 Ⰷ ␲ ␣ k/3.
To describe annihilation of thermal positrons, one must
fold Z eff(k) with the Maxwellian distribution at temperature
T, and the result is
Z eff共 T 兲 ⫽
Z eff共 k 兲 exp ⫺
4 ␲ k dk
2k B T 共 2 ␲ k B T 兲 3/2
( v ⫽k in atomic units兲. Using Eq. 共2兲 and the definition of
Z eff , we obtain
Z eff
At room temperature T⫽293 K the typical positron energies
are k 2 /2⬃k B T⫽9.3⫻10⫺4 a.u., which corresponds to thermal positron momenta k⬇0.045.
Z eff
The interaction time ␶ can be made much greater if the
low-energy positron is captured by the molecule in a process
involving the excitation of a narrow resonance in the
positron-molecule system. Enhancement of annihilation due
to the excitation of a single resonance was considered theoretically in Refs. 关26兴 and 关30兴. The possibility of forming
such resonances by excitation of the vibrational degrees of
freedom of molecules was discussed by Surko et al. 关5兴. Suppose that the positron affinity ⑀ A of the molecule is positive
共e.g., ⑀ A is a fraction of an eV兲. Vibrationally excited states
of the positron-molecule complex would then manifest as
resonances in the positron continuum, and provide a path for
resonant annihilation. In this process the positron is first
trapped temporarily by the molecule. In this case, there are
two possibilities. The positron can annihilate with one of the
molecular electrons, or it can undergo detachment and return
to the continuum. As a result, the resonant annihilation rate
␭ (res) is proportional to the probability of positron capture
multiplied by the probability of its annihilation in the quasibound state.
A positron-molecule resonance is characterized by its total linewidth ⌫⫽⌫ a ⫹⌫ c , where ⌫ a and ⌫ c are the rates 共or
partial widths兲 for annihilation and capture, respectively.
These quantities are directly related to the lifetime of the
resonant state against annihilation, ␶ a ⫽1/⌫ a , and positron
detachment, ␶ c ⫽1/⌫ c , and ␶ in Eq. 共4兲 is 1/⌫. The probability of annihilation in the resonant state is determined by the
competition of these two processes: P a ⫽⌫ a /(⌫ a ⫹⌫ c ). The
resonant annihilation rate is given by
where ␴ c is the capture cross section. If the molecule absorbed all incoming positrons, ␴ c would be given by ␴ max
⫽␲⑄2⫽␲k⫺2. This cross section corresponds to the s-wave
capture, which dominates at low positron energies. The true
capture cross section is smaller than ␴ max , because the capture takes place only when the positron energy matches the
energy of the resonance. For positrons with a finite-energy
spread 共e.g., thermal兲, the capture cross section is then ␴ c
⬃(⌫ c /D) ␴ max , where D is the mean energy spacing between the resonances. More accurately, ␴ c ⫽(2 ␲ ⌫ c /
D) ␴ max 关28兴, and Eq. 共10兲 yields 关24兴
␭ (res) ⫽n
␲ 2␲⌫c
⌫ a ⫹⌫ c
This expression estimates the average contribution of resonant capture to the positron-molecule annihilation. It becomes especially simple if the capture width is greater than
the annihilation width, ⌫ c Ⰷ⌫ a ⬃1 ␮eV, or ␶ a Ⰷ ␶ c :
B. Resonant annihilation
␭ (res) ⫽n ␴ c v P a ,
␳ ep ⌫ c
2 ␲ 2 ␳ ep ␶ a
k D 共 ⌫ a ⫹⌫ c 兲
k D共 ␶ a⫹ ␶ c 兲
2 ␲ 2 ␳ ep
k D
Therefore, the contribution of resonances to the annihilation
rate is proportional to the density of positron-molecule resonances ␳ (E)⫽D ⫺1 , evaluated at the energy released when
the positron binds to the molecule, E⬇ ⑀ A ⫹k 2 /2.
Suppose the resonances correspond to vibrationally excited states of the positron-molecule complex, and a single
vibrational mode with frequency ␻ is excited. Then we estimate D⫽ ␻ ⬃0.1 eV⬃4⫻10⫺3 a.u., and for thermal posi(res)
⬇4⫻103 . In larger
trons, k⫽0.045 a.u., Eq. 共13兲 gives Z eff
molecules several vibrational modes can be excited, and the
resonance spectrum density D ⫺1 is much higher. Thus resonant annihilation can lead to very large values of Z eff . However, they cannot be arbitrarily large. The theoretical maximum is achieved in Eq. 共12兲 at D⬃⌫ c ⬃⌫ a , and it yields
⬃108 for room-temperature positrons. Of course, some
Z eff
of the modes may not be excited in the positron capture due
to symmetry constraints, and others may have very small
coupling to the positron-molecule channel 关small ⌫ c in Eq.
共12兲兴. In the latter case the positron-molecule resonant states
will have very large lifetimes against positron detachment.
However, this does not mean that they contribute much to
Z res ; if ␶ c →⬁ their contribution is very small, since they are
effectively decoupled from the positron-molecule continuum, meaning ␴ c →0.
Another interesting property of resonant annihilation is an
apparent violation of the 1/v law that governs the cross sections of inelastic processes at vanishing projectile energies
关28兴. This law means that the corresponding rate should be
constant at low k, whereas Eqs. 共12兲 and 共13兲 indicate a 1/k
increase of the rate toward zero positron momenta 共and a
E ⫺1 dependence of the annihilation cross section兲. This apparent contradiction is resolved if we recall that the capture
width ⌫ c is also a function of the projectile energy. For
s-wave capture, ⌫ c ⬀kR a . Hence Eq. 共13兲 becomes invalid at
very small positron momenta, while the complete expression
共12兲 approaches a constant value. The contribution of partial
waves with higher orbital momenta l to the resonant annihilation have the structure of Eq. 共12兲 times a 2l⫹1 factor.
However, the corresponding capture widths behave as ⌫ c
⬀(kR a ) 2l⫹1 . Hence at low positron energies the s-wave contribution dominates, and the contribution of l⭓1 become
noticeable only at higher positron energies—first the p wave,
then the d wave, etc.
The s-wave resonant annihilation’s behavior of 1/k means
a T ⫺1/2 temperature dependence. This law breaks down for
very small k 共or T), where Z eff becomes constant. Higher
partial waves contributions (p, d, . . . 兲 emerge as T, T 2 , etc.
at small T. The latter statement is valid for the direct contribution to Z eff as well.
Qualitatively resonant annihilation is similar to electronmolecule attachment. The treatment of Christophorou and
co-workers 关31,32兴 for electron-molecule collisions assumes
that the light particle 共in their case, the electron兲 distributes
its kinetic energy statistically over the vibrational modes of
the molecule. Their treatment provides a way to estimate the
capture lifetime in the limit of complete mixing of the vibrational modes. However, a complete quantum-mechanical expression for the positron annihilation rate averaged over the
resonances has the form of Eq. 共12兲, and depends on the
density of the resonant spectrum D ⫺1 , as well as on the
relation between the widths of the competing processes,
which for positron annihilation, are ⌫ c and ⌫ a .
C. Virtual and weakly bound positron-molecule states
As we discussed in Sec. III A, the existence of virtual or
weakly bound states leads to enhanced direct annihilation
rates for both atoms and molecules. Positron-molecule binding is also a necessary condition for resonant annihilation
which can result in very high values of Z eff . In this section
we consider a simple model of a positron interacting with a
molecule composed of several atoms. This model illustrates
how the chemical composition of the molecule can influence
the binding, thereby changing the molecular annihilation rate
significantly. We specifically discuss the case of methane
and its fluorosubstitutes.
Let us approximate the interaction between a low-energy
positron and an atom by the zero-range potential 关33兴. This
potential is characterized by a single parameter ␬ 0 , which
determines the behavior of the positron wave function at
small distances,
1 d共 r␺ 兲
⯝⫺ ␬ 0 .
r ␺ dr
For this potential the s-wave scattering amplitude is
f ⫽⫺
␬ 0 ⫹ik
where k is the positron momentum, and the scattering length
is given as a⫽1/␬ 0 . If ␬ 0 ⬎0, there is a bound state at E⫽
⫺ ␬ 20 /2 共atomic units are used throughout兲, and ␬ 0 ⬍0 corresponds to a virtual level.
When we consider low-energy scattering or a weakly
bound state for n scattering centers 共atoms兲, each scattering
center can be approximated by a zero-range potential with
␬ i ⫽1/a i , where a i is the scattering length of the ith atom
(i⫽1, . . . ,n). For this system, the eigenvalue problem is
reduced to the following algebraic equation for ␬ :
det ␦ i j 共 ␬ i ⫺ ␬ 兲 ⫹
exp共 ⫺ ␬ R i j 兲
共 1⫺ ␦ i j 兲 ⫽0,
TABLE I. Effect of fluorination on the parameter ␬ of the
bound or virtual levels for positrons with CH4⫺x Fx molecules.
No. of F atoms
␬ H⫽⫺0.5, ␬ F⫽⫺2.0.
␬ H⫽⫺0.72, ␬ F⫽⫺1.275.
where R i j is the distance between atoms i and j. Depending
on the sign of ␬ , Eq. 共16兲 can yield either a true bound state,
E⫽⫺ ␬ 2 /2 ( ␬ ⬎0), or a virtual level, E⫽ ␬ 2 /2 ( ␬ ⬍0). The
case n⫽2 was considered in detail in Ref. 关34兴. This model
was also used to investigate positron binding to small xenon
clusters 关14兴.
If the atoms form a symmetric configuration, Eq. 共16兲 can
be simplified. For example, for n⫽2, 3, or 4 identical atoms
( ␬ i ⬅ ␬ 0 ) separated by equal distances R 共a diatomic molecule, triangle or tetrahedron configuration兲, the lowest
eigenstate is found from the simple transcendental equation
␬ ⫺ 共 n⫺1 兲
e ⫺␬R
⫽␬0 .
We note that even if none of the individual atoms possesses
a bound state ( ␬ i ⬍0 for all i), the system of several atoms
may well support a bound state. One can easily see this from
Eq. 共17兲, which has a positive ␬ solution for (n⫺1)/R⬎
⫺ ␬ 0.
Let us use the zero-range potential model to consider positron binding to the methane molecule and its fluorinated
counterparts (CH4 to CF4 ). The positron cannot penetrate
very deeply into the molecule because of the repulsion from
atomic nuclei, and we neglect the effect of the central carbon
atom in these compact, rounded-shape molecules. The ␬ 0
parameters of the zero-range potentials for hydrogen and
fluorine can be taken from positron-atom calculations. For
hydrogen ␬ H⫽⫺0.5 is derived from the positron scattering
length a⫽⫺2.1 关35兴. The value for fluorine can be roughly
estimated as ␬ F⫽⫺2 by using the positron scattering length
for Ne, a⫽⫺0.43 关15,27兴. As shown by calculations for
heavier halogens 关36兴, their scattering lengths are close to
those of the neighboring noble-gas atoms. The interatomic
distances R i j are derived from the geometrical parameters
given in Ref. 关37兴. Using these values, Eq. 共16兲 is solved
numerically for ␬ . In the two simplest cases, CH4 and CF4 ,
Eq. 共17兲 can be used with n⫽4. For CH4 we take ␬ 0 ⫽
⫺0.5, R⫽3.38 a.u. and obtain ␬ ⫽0.111, and for CF4 we use
␬ 0 ⫽⫺2, R⫽4.07 a.u., and the result is ␬ ⫽⫺0.217. Thus a
tetrahedral configuration of four hydrogen atoms provides a
bound state for the positron, whereas that of fluorine atoms
does not.
The calculated values of ␬ for all five CH4⫺x Fx molecules
are given in Table I. We see that only the two first members
of the series have bound states, whereas for the molecules
with two, three, and four fluorine atoms the binding does not
take place, because the fluorine atoms are less attractive for
the positron than hydrogen. In all cases the corresponding
FIG. 1. Final stage of the positron trap showing schematically
an accumulated positron cloud and the ␥ -ray detector.
scattering lengths a⫽1/␬ are large, which justifies the use of
the zero-range potential model. If we use the simple estimate
of the direct annihilation rate, Eq. 共5兲 combined with Eq. 共7兲,
we conclude that Z eff should peak ‘‘between’’ CH3 F and
CH2 F2 , in accord with the experimental results 共i.e., see Sec.
V B兲. This is an indication that larger alkane molecules are
likely to be able to form bound states with positrons, whereas
their perfluorinated analogues are probably not capable of
positron binding. The implication of this result is that the
model predicts that annihilation rates of large alkanes could
be determined by resonant annihilation. If so, the annihilation rates for these species are expected to be orders of magnitude greater than those of the perfluorinated alkanes, since
only direct annihilation is possible for perfluorinated alkanes
because their positron affinities are negative.
The experiments were performed using a technique similar to previous studies 关1,5,18兴. However, ongoing refinements in the trapping techniques have substantially enhanced
the quality of the data. A schematic diagram of the experiment is shown in Fig. 1. Positrons, emitted at high energies
from a 60-m Ci 22Na radioactive source, are moderated to a
few eV by a solid neon moderator 关38,39兴. They are then
guided magnetically into a modified three-stage PenningMalmberg trap. A magnetic field (⬃1 kG兲 produced by a
solenoid provides positron confinement in the radial direction, and an electrostatic potential well imposed by an electrode structure provides confinement in the axial direction.
The positrons experience inelastic collisions with nitrogen
buffer gas molecules introduced into the electrode structure
and become trapped in the electrostatic potential well. In a
time of the order of 1s, the trapped positrons cool to room
temperature through vibrational and rotational excitation of
nitrogen molecules. The trap is designed to accumulate an
optimal number of positrons with minimal losses from annihilation on the buffer gas molecules. More detailed accounts
of the operation of the positron trap are given elsewhere
The positrons end up in the final stage of the trap, which
is shown in Fig. 1. A cold surface in the vacuum system is
chilled with a water-ethanol mixture to ⫺7 °C in order to
reduce impurities. The base pressure of our system is typically 5⫻10⫺10 torr, and the positron lifetime with the buffer
gas turned off is typically 180 s. The cold surface can be
cooled with liquid nitrogen, resulting in positron lifetimes
exceeding 1 h. However, this is not useful for the experiments described here, since most of the gases under study
condense on surfaces at liquid nitrogen temperature.
For annihilation-rate measurements, the test substances
are introduced into the final stage of the trap as gases at
pressures less than 10⫺6 torr. Substances that exist as liquids
at room temperature are introduced as low-pressure vapors.
Use of low-pressure test gases ensures that the process studied here is dominated by binary encounters of the positrons
and atomos or molecules. Annihilation rates are measured by
the following procedure. Positrons are accumulated for a
fixed time, and then the positron beam is shut off. The positrons are stored in the positron trap for a few seconds in the
presence of the test atoms or molecules and then dumped
onto a collector plate 共Fig. 1兲. The intensity of the ␥ -ray
pulse from the annihilating positrons is measured. The annihilation lifetime is measured by repeating this procedure for
various values of the positron storage time in the presence of
the gas. The measurements are performed for various testgas pressures. The slope of the plot of annihilation time versus pressure is proportional to the 共normalized兲 annihilation
rate of the test atoms or molecules. A more detailed account
of this technique can be found in Ref. 关1兴.
The dependence of annihilation rate on positron temperature was measured with the technique described in Ref. 关9兴.
This experiment consists of repeated cycles of positron filling, heating the positrons by applying rf noise, and monitoring the subsequent annihilation. After positron filling, the
positron beam is switched off, and the trapped positrons cool
down to room temperature. The buffer gas is then switched
off and pumped out. After a delay time to ensure that the
buffer gas density is negligible, the test gas is admitted to the
trap. Following an appropriate time delay 共to allow the pressure to stabilize兲, the positrons are heated by applying a
pulse of broadband rf noise to one of the confining electrodes. The positrons are heated to temperatures in the range
0.1–0.5 eV for atomic test gases and 0.1–0.3 eV for molecular test gases 共where the maximum temperature is limited by
vibrational excitation of the gas molecules兲. The positrons
then cool by collisions with the test gas atoms or molecules
after the rf noise is off. Concurrent with the cooling, the
positrons annihilate on the test gas while the annihilation is
measured using a Na I (Tl) detector to count the ␥ rays.
Before and after each run, the positron temperature is measured as a function of elapsed time since the end of the
heating pulse. This is accomplished by reducing the depth of
the confining well to zero and analyzing the number of positrons escaping the trap as the function of well depth. A more
detailed account of this type of measurement was presented
in Ref. 关9兴.
In Sec. V A, we present experimental measurements of
positron annihilation rates of deuterated alkanes and the cor-
TABLE II. Measured values of Z eff for protonated and deuterated alkanes with number of carbon atoms j. All values are measured in the positron trap. The last column is the ratio of Z eff for
deuterated alkanes to those for protonated alkanes.
C j H2 j⫹2
C j D2 j⫹2
116 000
341 000
408 000
641 000
Z eff
105 000
355 000
585 000
666 000
responding protonated alkanes. The annihilation rates of alkanes and benzenes with varying degrees of fluorination are
presented in Sec. V B. The dependence of annihilation rates
of noble gases, hydrocarbons, and fluorinated methanes on
positron temperature is described in Sec. V C.
The data presented here differ in certain instances from
those reported previously 关1兴. The values of Z eff reported
here are larger than the previous measurements by as much
as 50%, due to a faulty ion gauge. However, the same gauge
is used for all the data sets presented here, so the relative
error is expected to be of the order of 10%. Since the models
discussed in this paper are compared with the relative values
of Z eff measured with the same ion gauge, the conclusions
reached remain valid in spite of the uncertainties in the absolute values of Z eff . Where two values of Z eff are reported,
those in Ref. 关1兴 are more accurate.
A. Comparison of annihilation rates for deuterated
and protonated hydrocarbons
The annihilation rates of deuterated and protonated alkanes were measured systematically, and the results are
listed in Table II. The ratio of Z eff for deuterated alkanes to
those for protonated alkanes is listed in the last column of the
table, and is plotted in Fig. 2. As can be seen from the figure,
the annihilation rates for the deuterated and protonated alkanes are very similar if not identical. A factor of 2–3
change in annihilation rate was observed previously for deuterated benzenes 关1兴. However, in contrast to data for the
benzenes, the systematic study of alkanes presented here
does not provide support for a mechanism in which the positron forms long-lived vibrationally excited resonant states
with molecules.
This result would be natural if the annihilation process
involved only electron-positron degrees of freedom and proceeded by direct annihilation as described in Sec. III A. This
mechanism is likely to dominate for smaller molecules with
moderate Z eff and relatively high vibrational frequencies, and
for those with negative positron affinities 共like perfluorocarbons兲. Thus, the agreement between Z eff for CH4 and CD4 is
consistent with the direct annihilation mechanism. However,
the measurements show that Z eff values are quite similar for
protonated and deuterated forms of larger alkanes. Based on
the estimates given above, these large values of Z eff cannot
be explained by direct annihilation.
In the context of the theory of resonant annihilation 共Sec.
III B兲, the corresponding annihilation rate should be proportional to the density of vibrational excitations. The substitution of deuterons for protons in the molecules studied here
lowers the frequencies of the high-frequency vibrational
modes significantly. Consequently, it increases ␳ (E), and
one could anticipate that the resonant mechanism would predict significantly larger values of Z eff for deuterated alkanes,
which was not observed.
One explanation for these observations is that the coupling between the electron-positron degrees of freedom and
nuclear motion is weak, effectively either reducing or completely shutting off the process of resonance formation. This
coupling might also be smaller for the deuterated alkanes
compared with protonated ones. In this case the capture
width ⌫ c might become very small, and if ⌫ c ⬍⌫ a , the regime described by Eq. 共13兲 does not take place. Another
possibility is that only lower-frequency vibrational modes
take part in the resonance process, and, thus, contribute to
the density factor D ⫺1 in Eq. 共13兲, although these are more
difficult for the relatively light positron to excite. Deuteration
will not have a large effect on the frequency of these modes,
which are dominated by the masses of the carbon atoms.
Therefore, the effective mean vibrational spacing D could be
roughly the same for protonated and deuterated alkanes.
Thus far we have not succeeded in devising a way to test the
possible effect of these low-frequency modes on the annihilation process.
B. Annihilation rates for partially fluorinated hydrocarbons
FIG. 2. The ratios of Z eff for deuterated alkanes to those for
protonated alkanes plotted against the number of carbon atoms, j.
As reported previously 关1,3,5兴, large alkane molecules
have very large annihilation rates Z eff compared with the
number of electrons Z. In contrast, the analogous perfluorinated alkanes have annihilation rates that are orders of magnitude smaller 关18兴. Besides this, Z eff increases very rapidly
with the size of the molecule, approximately as Z eff⬀Z 5 , for
alkanes with 3–9 carbon atoms, whereas for perfluorocarbons it follows a much slower Z eff⬀Z 1.7. This large difference in annihilation rates between hydrocarbons and fluorocarbons can potentially provide insights into the physical
processes responsible for the annihilation. In order to pursue
this issue, we studied annihilation in molecules in which the
TABLE III. Values of Z eff for partially fluorinated hydrocarbons.
Z eff
CH2 F2
1 390
C2 H6
C2 H5 F
C2 F6
1 780
3 030
1 600
1 510
1 110
C3 H8
CF3 C2 H5
C3 F8
2 350
8 130
3 350
C6 H14
CH2 FC5 H11
C6 F14
151 000
269 000
C6 H6
C6 H5 F
C6 H4 F2
C6 H4 F2
C6 H4 F2
C6 H3 F3
C6 H2 F4
C6 HF5
C6 F6
20 300
45 100
32 800
13 100
13 500
10 100
2 760
1 930
Methyl fluoride
Carbon tetrafluoride
FIG. 3. The dependence of Z eff
(T) at room temperature on the
parameter ␬ of the virtual and bound s state, calculated from Eqs.
共6兲, 共8兲, and 共9兲 using F⫽0.93, R a ⫽4, ␣ ⫽17.6 共polarizability of
CH4 ), and C⫽1. The solid circles are values of Z eff for CF4 ,
CHF3 , CH2 F2 , CH3 F, and CH4 共from left to right兲 of the present
experiment, normalized to Z eff⫽158.5 共for CH4 ), plotted as a function of ␬ values obtained from the zero-range positron-molecule
binding model 共Table I, second line兲.
hydrogen atoms in hydrocarbons have been selectively replaced with fluorine atoms to form partially fluorinated hydrocarbons.
The measured annihilation rates for a selection of partially
fluorinated hydrocarbons are listed in Table III. It is interesting that, within a given series, the molecule with a single
fluorine atom has the highest annihilation rate. Further fluorination decreases the annihilation rate gradually, with the
perfluorinated molecule having the lowest annihilation rate.
We note that molecules with one fluorine atom are highly
dipolar. Although the effect of a permanent dipole moment
on the annihilation rate is not understood, empirical evidence
关1兴 indicates that this does not account for the large increases
in annihilation rates that are observed for the monofluorinated molecules. In particular, partially fluorinated molecules containing more than one fluorine have dipole moments comparable in magnitude to or larger than that of the
monofluorinated compound, but significantly smaller annihilation rates.
For larger alkanes, the high values of Z eff and their strong
dependence on the size of the molecule are consistent with
the resonant annihilation mechanism with a positron affinity
⑀ A ⬇5 ␻ , where ␻ is the typical frequency of molecular vibrations excited in the positron capture 共see estimates in
Secs. I and III B兲. Fluorination reduces the vibrational frequencies and increases the vibrational spectrum density at a
given energy. This, together with the loss of symmetry of the
molecule, could be the reason for the increase in Z eff with the
first fluorine substitution. However, the rapid decrease of Z eff
observed when several H atoms are replaced with fluorines
can be interpreted as a ‘‘switching off’’ of the resonant
mechanism due to the fact that the positron-molecule binding
becomes weaker and then disappears with the addition of
fluorine atoms. Note that for heavier halogen-substituted alkanes the annihilation rates are much larger 关1兴. Both Cl and
Br are much more attractive for positrons than F. Thus, in
this case, the resonant annihilation model predicts that there
will be a softening of the vibrational spectrum, but no loss of
positron binding.
For the smallest of the alkanes, methane, the annihilation
rate is relatively small, Z eff⬃102 , although much larger than
the number of valence electrons. Combined with the sparse
vibrational spectrum of the molecule, this can be interpreted
as evidence that 共i兲 for room-temperature positrons annihilation proceeds via the direct mechanism, and 共ii兲 the direct
annihilation rate is enhanced by the presence of a virtual
level, or a weakly bound state, cf. Sec. III A. In the context
of the zero-range potential model in Sec. III C, the variation
of Z eff is then consistent with the change in the position of
this level, when hydrogen atoms are substituted by fluorines.
To test this hypothesis, we plot in Fig. 3 the dependence of
(T) at room temperature on the positions of the virtual
Z eff
and bound states, as represented by the parameter ␬ . This has
been calculated using Eqs. 共6兲, 共8兲, and 共9兲. Solid dots show
measured values of Z eff as a function of ␬ calculated in the
zero-range potential model 共Table I, second line兲. These values of ␬ for the five CH4⫺x Fx molecules are determined by
the parameters, ␬ H and ␬ F that describe the interaction of the
positron with isolated H and F atoms. In the second line of
Table I we use ␬ H and ␬ F as free parameters, and find that
␬ H⫽⫺0.72 and ␬ F⫽⫺1.275 give the best fits to the experimental data shown in Fig. 3.
The main feature in Fig. 3 is the maximum in the dependence of Z eff on ␬ . It corresponds to the ␬ ⫽0 point, where
the virtual level ( ␬ ⬍0) turns into a bound state ( ␬ ⬎0), and
where the scattering length becomes infinite. The annihilation rate remains finite at ␬ ⫽0 because we consider finitetemperature positrons 关cf. Eq. 共7兲 with k⬎0]. Therefore, in
the context of the model, the dependence of Z eff on the degree of fluorination can be understood as a gradual change in
the position of the level, from a bound state in CH4 共maximal
binding energy ⑀ A ⫽ ␬ 2 /2⬇28 meV兲 and CH3 F, to the virtual
levels in difluoromethane, trifluoromethane, and tetrafluoromethane. The small binding energy of methane explains
why the vibrational resonances do not contribute to the annihilation rate. We note that there is a discrepancy between
measured Z eff and the calculation for larger negative values
of ␬ . This may be a result of the assumptions used that
individual hydrogen and fluorine atoms contribute equally to
Z eff . Also, for larger 兩 ␬ 兩 , the zero-range potential model becomes less accurate. The main result of this study of annihilation in methane and its fluorosubstitutes is evidence that
the bound level disappears as the number of fluorines is increased. This effect could explain the difference between
very large Z eff in larger alkanes, due to resonant annihilation,
and orders of magnitude smaller Z eff for perfluoroalkanes,
where the resonant mechanism would be switched off by the
absence of binding.
C. Dependence of annihilation rates on positron temperature
1. Noble-gas atoms
Annihilation rates as a function of positron temperature
for noble-gas atoms were measured previously 关9兴. These
data for the temperature dependence of Z eff were found to be
in good agreement 关9兴 with calculation by Van Reeth et al.
for He 关7兴 and calculation by McEachran et al. for Ne, Ar,
Kr, and Xe 关27兴. The data are plotted in Fig. 4 on a log-log
scale. We relate the observed temperature dependences for
these atoms to that expected for direct annihilation 共cf. Sec.
III A兲. We find that we are able to fit the data using Eqs. 共6兲,
共8兲, and 共9兲 using the known dipole polarizabilities ␣
⫽2.377, 11.08, 16.74, and 27.06 a.u. for Ne through Xe,
respectively. The values of the scattering length a and the
constant C are taken from the scattering calculations of
McEachran et al. 关27兴 for the s wave: a⫽⫺0.61, ⫺5.3,
⫺10.4, and ⫺45.3 a.u., and C⫽0.001, 0.60, 0.35, and 0.005
for Ne, Ar, Kr, and Xe, respectively. The only free parameter
in the fits is R a , and we determine it by comparison with
experimental data in the range of positron temperatures T
⫽0.025–0.1 eV, where Eq. 共8兲 is valid. The fits shown by
FIG. 4. Dependence of annihilation rates on positron temperature for noble gas atoms 共data are from Ref. 关9兴兲: (䊊) He, (䊉) Ne,
共solid square兲 Ar, 共solid triangle兲 Kr, and 共solid diamond兲 Xe. The
annihilation rates are normalized to their room-temperature values.
The experimental data are fit with the direct annihilation formulas
关Eqs. 共6兲, 共8兲, and 共9兲兴 共solid curves兲. Power-law fits to the lowtemperature parts of the data are also shown, corresponding to exponents of ⫺0.036 共He兲 共dash-dotted line兲, ⫺0.039 共Ne兲, ⫺0.23
共Ar兲, ⫺0.32 共Kr兲, and ⫺0.67 共Xe兲 共dashed lines兲.
solid curves in Fig. 4 correspond to R a ⫽3.2, 3.2, 4.2, and 4.2
a.u. for Ne, Ar, Kr, and Xe, respectively. We see that the
direct annihilation mechanism gives an accurate description
of the measured temperature dependences at low positron
energies. The stronger temperature dependence observed for
heavier noble-gas atoms is caused by the increasing magnitudes of the scattering length from Ne to Xe. As seen from
Eq. 共8兲 this causes more rapid variation of the phase shift,
and hence the cross section ␴ , which, for heavier noble-gas
atoms, gives a dominant contribution to Z eff in Eq. 共6兲. Large
negative scattering lengths 共i.e., small negative ␬ parameters兲 correspond to the existence of low-lying virtual s levels for positrons on Ar, Kr, and Xe. This in turn enhances the
absolute values of the annihilation rates at low positron energies 共cf. Z eff⫽33.8, 90.1, and 401 for Ar, Kr, and Xe,
respectively, at room temperatures 关1,16兴兲. The data can also
be fit accurately, over almost the entire energy range, by a
power law Z eff(T)⬀T ⫺ ␰ 共dash-dotted and dashed lines in
Fig. 4兲, with ␰ ⫽⫺0.036, ⫺0.039, ⫺0.23, ⫺0.32, and ⫺0.67
for He, Ne, Ar, Kr, and Xe, respectively.
2. Partially fluorinated hydrocarbons
Annihilation rates were measured as a function of positron temperature in an attempt to test the hypothesis that a
FIG. 5. Dependence of annihilation rates on positron temperature: (䊉) methane (CH4), and (䊊) fluoromethane (CH3 F). The
annihilation rates are normalized to their room-temperature values.
The dotted line (•••) is a fit to the lower-temperature data with the
coefficient of ⫺0.53.
large s-wave scattering cross section 共small ␬ ) due to weakly
bound or virtual positron states can explain the trend of Z eff
in the partially fluorinated hydrocarbons. A smaller value of
␬ for CH3 F as compared with that for CH4 would result in a
larger value of Z eff , and one would expect that Z eff for CH3 F
would have a more rapid temperature dependence at low
temperatures, since its value of ␬ is smaller. Measurements
for these molecules are presented in Fig. 5. As can be seen
from the figure, the dependence of the annihilation rate on
positron temperature is similar for CH3 F and CH4 at low
temperatures. The dotted line shown in the figure is a fit to
the low-temperature part of the data with the coefficient of
⫺0.53, which is between those of Kr and Xe 共Fig. 4兲. This
indicates that the absolute value of positron scattering length
for these molecules is probably between those of Kr and Xe.
In Fig. 6, the data are plotted on an absolute scale and
compared with the analytical direct annihilation fits from
Eqs. 共6兲, 共8兲, and 共9兲, based on a⫽1/␬ values from Table I.
In this comparison, the data and theory are in reasonable
agreement at low positron temperatures 共i.e., energies兲. In
spite of a large difference in ␬ values for CH4 and CH3 F, the
slopes of their temperature dependences are rather similar.
The key point appears to be that due to the terms containing
the dipole polarizability in Eq. 共8兲, the temperature dependence of Z eff increases, and this effect is more pronounced
for methane which has a larger value of ␬ 共which would
otherwise, for ␣ ⫽0, give a rather flat temperature dependence兲. Thus the data and model are in reasonable
agreement—the model predicts similar positron temperature
dependences of Z eff for both species, even though they have
FIG. 6. Temperature dependence of Z eff for CH4 共solid circles—
experiment, solid lines—theory兲 and CH3 F 共open circles—
experiment, dashed lines—theory兲. The theoretical curves are obtained using R a ⫽4, ␣ ⫽17.6, and C⫽1, and the following
parameters. For CH4 : F⫽1, ␬ ⫽0.045 共upper curve兲, F⫽0.93, ␬
⫽0.0452 共lower curve兲; for CH3 F: F⫽0.93, ␬ ⫽0.005 共upper
curve兲, F⫽1, ␬ ⫽0.01 共lower curve兲.
different values of ␬ . The fact that the temperature dependences are so similar 共i.e., as shown in Fig. 5兲 might have led
to the conclusion that very similar parameters were responsible for this. However, in the context of the model presented
here, this does not appear to be the case.
The fit in Fig. 6 gives ␬ ⫽0.045 for methane, and the
scattering length a⫽22 a.u. is comparable in magnitude to
those of Kr (a⫽⫺10) and Xe (a⫽⫺45 关27兴, or a⫽⫺100
关15兴兲. The positive sign of a implies that the positron has a
weakly bound state with CH4 . As for CH3 F, the fit gives ␬
⫽0.01 or so (a⬃100), which has a large uncertainty, because for ␬ 2 /2Ⰶk B T the temperature dependence becomes
insensitive to the precise value of ␬ . We should also point
out that CH3 F is a polar molecule, and that the dipole force
changes the description of low-energy scattering.
3. Hydrocarbons and deuterated hydrocarbons
The annihilation rate Z eff has recently been predicted for
ethylene, C2 H4 , by da Silva et al., using a large-scale numerical calculation which included short-range correlation of
the positron and the molecular electrons 关22兴. In order to test
this prediction, we measured the dependence of Z eff on positron temperature, which is shown in Fig. 7. The experimental data are scaled with the room-temperature value of Z eff
⫽1 200, measured in a previous experiment, which has the
uncertainty of 20% 关1兴. The theoretical calculation 关22兴 is
shown in Fig. 7 as a solid line, and it underestimates the data.
The calculated values are also shown by the dashed line,
which is obtained by multiplying the theory by a scale factor
FIG. 7. Temperature dependence of the annihilation rate for
ethylene; experiment (䊉) and calculation 共—兲 关22兴. The dashed
line 共- - -兲 is the calculation fit to the experimental data, which
requires a scale factor of 1.3.
of 1.3. The data and calculation are in reasonable agreement.
As pointed out in Ref. 关22兴, the calculated value of Z eff for
C2 H4 is sensitive to the inclusion of electron-positron correlations. Thus the agreement between theory and experiment
provides evidence that such correlations are important in determining the annihilation rate.
The calculations of da Silva et al. demonstrate a strong
dependence of both the elastic cross section and Z eff on the
positron energy. We note that, in the framework of the model
for direct annihilation presented above, this behavior can be
interpreted as evidence for the existence of a virtual level for
the positron on C2 H4 with ␬ ⫽⫺0.05, and can be fitted using
the formulas of Sec. III A. This value of ␬ is in agreement
with the scattering length a⫽⫺18.5 a.u. determined from
the zero-energy limit of the elastic scattering cross section
␴ ⫽4 ␲ a 2 presented in Ref. 关22兴. Thus it appears that the
large value of Z eff for C2 H4 at low temperatures is due to the
large scattering cross section ␴ . In relation to this, it is interesting to note that the increase of the annihilation rates for
the molecules C2 H6 , C2 H4 , and C2 H2 (Z eff⫽660, 1200, and
3160, respectively 关1,2兴兲 correlates with the increase in the
total scattering cross sections for low-energy positrons on
these molecules, which were measured down to 0.7 eV by
Sueoka and Mori 关42兴. This is consistent with the predictions
of Eq. 共6兲 for direct annihilation, as the elastic cross section
␴ dominates in the total scattering cross section at low positron energies. The term with ␴ also dominates in Eq. 共6兲,
since the scattering lengths are expected to be large for these
We have measured the dependence of annihilation rate on
positron temperature for the deuterated methane CD4 , and
butane C4 H10 , and these data are compared with those for
methane in Fig. 8. Z eff for CD4 is quite similar to that of
CH4 ; see Sec. V C 2. The dependence for butane is similar as
well, but with much greater absolute values of Z eff . The
dotted line shown in the figure is a fit to the low-temperature
part of the data with the slope ⫺0.55. At low temperatures,
FIG. 8. Temperature dependence of the annihilation rates for
methane (䊊), deuterated methane (䉭), and butane (䊉). The dotted
line (•••) is a fit to the lower-temperature data with the coefficient
of ⫺0.55. The theoretical fit for methane is shown in Fig. 7. The
data for butane show 1/冑T behavior at small temperatures, which is
characteristic of the resonant annihilation.
the dependence can be derived from Eq. 共13兲 to follow 1/冑T
law for the resonant annihilation (s wave兲. The origin of the
plateau in Z eff that is observed at larger values of positron
temperature is unclear. It could be due to higher partial-wave
contributions to the resonant annihilation which emerge as T,
T 2 , etc. for p, d, etc. partial waves, respectively. However, if
these contributions were present, the exponent in the powerlaw dependence of Z eff on temperature would appear to be
less than 0.5, and this is not observed. In smaller molecules
where direct annihilation is expected to dominate at low positron temperatures, the plateau could result from both the
direct contribution of the higher partial waves and from excitation of vibrational resonances by the positron. We note,
however, that this interpretation does not provide an obvious
explanation for the fact that the temperature dependences of
Z eff for CH4 , CD4 , and C4 H10 are all so similar, and so
several unanswered questions remain.
We had hoped that this study of the dependence of annihilation on positron energy would aid in distinguishing the
two annihilation mechanisms considered here. At present,
this is not the case. Whether there is a more universal picture
that describes the self-similar temperature dependences that
are observed remains to be seen. One interesting facet of the
data is that no plateau has been seen in Z eff for CH3 F, suggesting that further studies of the temperature dependence of
Z eff for a wider variety of molecules might be useful in determining the origin of the physical phenomena responsible
for this feature of the data.
D. Phenomenological models
As discussed in Sec. II, phenomenological models have
been proposed in the past. We discuss two of these models,
2. Larrichia-Wilkin model
Laricchia and Wilkin modeled the annihilation rate as follows 关43兴. They began by arguing that energy conservation
can be violated for a time interval, ⌬t, given by the uncertainty principle, and concluded that virtual positronium can
be formed for a time
兩 E⫺E i ⫹E Ps兩
where E is the kinetic energy of the positron. They consider
the total annihilation rate to be the sum of direct annihilation
and the annihilation of virtual positronium due to ‘‘self’’ and
‘‘pickoff’’ annihilation. This is formulated as
Z eff⫽
FIG. 9. Scaling of Z eff with (E i ⫺E Ps) ⫺1 . The data plotted are
all the atoms and molecules for which physical parameters are
available for calculation of the predictions of the other models discussed in Sec. V D: (䊉) noble gases, (䉮) H2 , 共solid triangle,
down兲 SF6 , (䊊) alkanes, (䉭) perfluorinated alkanes, 共solid square兲
perchlorinated alkanes, and 共solid diamond兲 CBr4 .
including one proposed by Laricchia and Wilkin 关43,44兴, by
testing their predictive values in comparison with our experimental data 关1兴.
1. Scaling relation of Murphy and Surko
Murphy and Surko observed a scaling relation between
the logarithm of Z eff and the quantity (E i ⫺E Ps), where E i is
the ionization energy of the atom or molecule and E Ps is the
binding energy of a positronium atom. This scaling is valid
for all the atoms and single-bonded nonpolar molecules 关18兴.
In particular,
ln共 Z eff兲 ⫽A 共 E i ⫺E Ps兲 ⫺1 ,
where A is a positive constant. This scaling is illustrated in
Fig. 9 for comparison with other models. The peak-to-peak
spread in measured Z eff values is generally better than one
order of magnitude. There is no apparent distinction between
atoms and molecules or any change in the scaling at values
of Z eff⬃103 . To the extent that this simple relation matches
the data, this scaling indicates that it is the electronic structure of the atom and the molecule that determines the annihilation rate, and other aspects of atomic and molecular
structure, such as the character of the vibrational modes, play
a relatively minor role in determining the annihilation rate.
Murphy and Surko 关18兴 found that this scaling was not
applicable to other molecules, such as polar molecules and
those containing double and triple bonds. For these species,
there are different ionization potentials for different bonds.
While the authors found that using other than the lowest
ionization potential improved the correlation of Z eff with
(E i ⫺E Ps) ⫺1 , they considered such a model to have too much
ambiguity to be useful.
兵 ␥ 关 1⫺ exp„⫺␭ ␶ 兲兴 ⫹ 共 1⫺ ␥ 兲
␲ r 20 c
⫻ 关 1⫺ exp„⫺⌬t 共 ␭ sa⫹␭ po兲 …兴 其 ,
where ␥ is the fraction of direct annihilation, ␭ is the direct
annihilation rate, ␶ is the positron-atom or positron-molecule
interaction time, ␭ sa⫽2⫻109 s⫺1 is the self-annihilation
rate, and ␭ po is the pickoff annihilation rate. It can be noted
that the first term 共direct annihilation contribution兲 in Eq.
共20兲 is identical to Eq. 共4兲 with the factor of ␥ . The direct
annihilation rate can be calculated from the spin-averaged
Dirac rate of ␭⫽ ␲ r 20 cn e , where the n e is the electron density. They chose to estimate the electron density by putting
all of valence electrons Z v in a sphere of the size given by
the Bohr radius, a 0 . Thus
n e⫽
3Z v
4 ␲ a 30
and ␭⫽3r 20 cZ v /(4a 30 ). In their model, they consider pickoff
annihilation to mean that the positron in the positronium
atom annihilates with an atomic or molecular electron other
than the electron forming the positronium atom. Laricchia
and Wilkin assumed that this rate is enhanced by the atomic
or molecular polarizability ␣ :
␭ po⫽ ␣ ␭⫽
3r 20 cZ v ␣
4a 30
The value of ␥ is estimated as ␥ ⫽ exp(⫺⌬t/␶), where the
interaction time is taken as ␶ ⫽a 0 / v for this approximation.
The collision cross section is approximated by
␴ ⫽ 共 10⫺15␣ 兲 cm2 ,
with ␣ in units of Å3 in Ref. 关43兴. In Ref. 关44兴, Laricchia and
Wilkin chose to modify the assumed cross section by an
additional factor,
␴ ⫽ 关 10⫺16␣ 共 1⫹ ␣ 兲兴 cm2 ,
arguing the collision cross section will scale as ␴ ⬀ sin2(␦0),
where ␦ 0 is the phase shift 关45兴. We note that the factor (1
FIG. 10. Scaling of Z eff with values calculated using the model
of Ref. 关43兴. 共—兲 is the line y⫽x. The same symbols are used as in
Fig. 9.
⫹␣) introduces another numerical constant for the relative
weight of the two terms 共which the authors choose to be 1兲.
Figure 10 shows the correlation of experimental Z eff with
the quantity calculated with Eq. 共20兲 using the cross section
关Eq. 共23兲兴 for the same atoms and molecules plotted in Fig.
9. The predicted values of Z eff of noble gases correlate reasonably well. The model underestimates the observed values
for alkane molecules by an order of magnitude, while it overestimates those for perfluorinated molecules by as much or
more. Figure 11 shows the predicted values calculated using
Eq. 共24兲 for the same atoms and molecules. While this scaling improves the agreement for the alkanes, it results in
poorer agreement for the perfluorinated compounds. Comparing Figs. 9, 10, and 11, we conclude that the scaling proposed by Murphy and Surko, although not perfect, is a better
FIG. 11. Scaling of Z eff with values calculated from the model
of Ref. 关44兴. 共—兲 is the line y⫽x. The same symbols are used as in
Fig. 9.
FIG. 12. Scaling of Z eff with values calculated from the model
of Ref. 关43兴: (䊉) noble gases, (䉮) inorganic molecules, (䊊) alkanes, 共solid triangle, down兲 alkenes and acethylene, 共solid triangle,
up兲 aromatic hydrocarbons, (䉭) perfluorinated alkanes, 共solid
square兲 perchlorinated alkanes, CBr4 , CH3 Cl, and CCl2 F2 , (〫)
alchohols, carboxylic acids, ketones, 共solid diamond兲 substituted
benzenes, and (䊐) partially fluorinated hydrocarbons. 共—兲 is the
line y⫽x.
predictive parameter for atoms and single-bonded nonpolar
Murphy and Surko observed that the scaling they proposed in Ref. 关18兴 is not valid for polar molecules and molecules with double and/or triple bonds 共see Ref. 关1兴 for further analysis兲. Figures 12 and 13 show the predicted values
calculated for the Laricchia-Wilkin model, using Eq. 共20兲
and the cross section of Eqs. 共23兲 and 共24兲, respectively, for
all available data. The values calculated from these two models correlate as well to all of the data as they do to the data
for atoms and single-bonded molecules. The largest discrep-
FIG. 13. Scaling of Z eff with values calculated from the model
of Ref. 关44兴. The same symbols are used as in Fig. 12.
ancies are underestimates of the alkanes and overestimates of
the values for the perfluorinated molecules. In this more general comparison, the predictions for the partially fluorinated
hydrocarbons fall naturally in between these two groups of
3. Remarks and one more scaling relation
The model by Laricchia and Wilkin appears to us to include questionable assumptions. One such assumption is that
all of the valence electrons are concentrated in a sphere of
radius a 0 关i.e., Eq. 共21兲兴, which is much smaller than the size
of the molecule. This clearly overestimates the electron density. Yet the high annihilation rates predicted by this model
are due in large part to this assumption. The enhancement of
pickoff annihilation by the polarizability factor 关Eq. 共22兲兴
might also be questioned, since ␥ -ray spectral measurements
indicate that the positron wave function is distributed rather
evenly over molecular species 关2兴. Finally, the form of the
cross section given by Eq. 共24兲 introduces one additional
parameter, and does not appear to improve substantially the
agreement with the available data.
The model of Laricchia and Wilkin predicted a divergence of annihilation rate at the positronium formation
threshold, where the positron energy E⫽E i ⫺E Ps . An ab
initio calculation by Humberston and Van Reeth also predicted a divergence of annihilation rate at the positronium
formation threshold 关46,47兴. The divergence found by Humberston and Van Reeth can also be derived from the diagrammatic expansion of the annihilation rate; see Eq. 共14兲 and
Fig. 10 of Ref. 关15兴. However, the singular behavior of annihilation rate near the positronium formation threshold in
the latter two calculations is of the form Z eff⬀ 兩 E⫺E i
⫹E Ps兩 ⫺1/2. It is qualitatively different than the singular behavior predicted by the Laricchia-Wilkin model, which is of
the form Z eff⬀ 兩 E⫺E i ⫹E Ps兩 ⫺1 . We note that it is now possible that positron annihilation in this energy range can be
investigated experimentally in a precise manner using the
intense, cold positron beam recently developed by Gilbert
et al. 关10兴. These experiments are now in preparation.
Finally, we considered whether we might obtain agreement similar to that for the Laricchia-Wilkin model 共i.e.,
Figs. 12 and 13兲, for all the available atomic and molecular
data using a purely empirical model with fewer parameters.
Plotted in Fig. 14 is Z eff against ␣ /(E i ⫺E Ps). We note that,
while the correlation is not linear on a log-linear scale, it is
as good as those shown in Figs. 12 and 13, and the model
uses only one parameter 共i.e., the polarizability兲 besides the
quantity E i ⫺E Ps . The fact that inclusion of ␣ in the scaling
improves the correlation over (E i ⫺E Ps) ⫺1 may reflect the
importance of the collision cross section in the annihilation
We have conducted experimental studies of positron annihilation on molecules. We have also theoretically considered two mechanisms which could contribute to the large
annihilation rates that are observed. Our estimates indicate
that the direct annihilation mechanism is capable of giving
FIG. 14. Scaling of Z eff with ␣ /(E i ⫺E Ps). The same symbols
are used as in Fig. 12.
Z eff⬃103 . The resonant annihilation mechanism, which involves positron capture into the vibrationally excited states
of the positron-molecule complex, appears, at least in principle, to be able to produce values of Z eff as large as 108 .
This mechanism is analogous to the electron-molecule capture mechanism thought to be responsible for very large dissociative attachment rates in some molecules.
In the case of direct annihilation, enhanced rates can be
observed if there are weakly bound states or low-lying virtual levels. The annihilation rates for hydrocarbons with various degrees of fluorination were measured in order to test the
predictions of this model. It was found that molecules with
one fluorine have the largest annihilation rates, and successive fluorination monotonically decreases the rates. This
trend was explored in detail for methane and its fluoroderivatives, and appears to be consistent with the simple zero-range
potential calculations presented here. The model suggests
that the first two members of the CH4⫺x Fx series form
weakly bound states with the positron, whereas for x⫽2 –4
the molecules have only a virtual level for the positron. The
dependence on temperature of the measured annihilation
rates for methane and fluoromethane were found to be rather
similar at low positron temperatures. Within the context of
the direct annihilation mechanism, this is interpreted as a
competition between the effect of a low value of ␬ for fluoromethane and a larger effect of the dipole polarizability for
For larger molecules that possess a broad spectrum of
vibrational resonances, we conjectured that the resonant annihilation mechanism is dominant. In this case, the absence
of positron binding in the perfluorinated alkanes can explain
the large difference in Z eff values for these compounds as
compared with alkanes which, according to the estimates discussed here, appear to be able to bind positrons. This resonant annihilation mechanism involves the formation of longlived positron-molecule compounds through transfer of the
positron’s energy to the molecular vibrational modes. To test
this model, measurements of annihilation rates of deuterated
alkanes were made and compared to those of protonated
ones. It was found that the deuterated alkanes have similar
annihilation rates to the protonated ones. Thus this test did
not confirm the predictions of the simplest interpretation of
this model for the alkanes. We note that deuteration of benzene molecules did produce some changes in Z eff . Thus the
overall result of these tests is inconclusive.
Data were presented for the dependence of annihilation
rates on positron temperature. Empirically, we noted similarities in the data for methane, deuterated methane, and butane, over a relatively wide range of positron temperatures,
and for methane and fluoromethane at low positron temperatures. The dependence of annihilation rates on positron temperature follows power law with the coefficients of ⫺0.53
for the combined data of methane and fluoromethane, and
⫺0.55 for those of methane, deuterated methane, and butane.
We find that we are able to explain these data within the
context of simple models of direct and resonant annihilation
described above. However this explanation required using
共specific values of兲 a number of parameters, and did not provide universal explanations for these trends. Whether there is
a more general theoretical framework to explain these dependences appears to us to be an open question which might
benefit from further scrutiny.
The two possible annihilation mechanisms that are considered theoretically in this paper do not involve Ps formation in a direct way, since it is forbidden by energy considerations for low-energy positrons and atoms or molecules
with E i ⬎E Ps . In addition, one of the two mechanisms directly involves the molecular vibrations. In contrast, the empirical scaling described by Eq. 共18兲 seems to indicate that
the dominant mechanism for enhanced annihilation rates involves only the electronic structure of the atom or molecule
共i.e., not the molecular vibrational modes兲. We are not aware
of any theoretical framework that has these characteristics,
and so we can offer only a couple of vague suggestions. If
there were low-lying electronic excitations of a positronatom or molecule complex, then a resonance model, such as
that described above, might be possible, with the resonant
modes now electronic, as opposed to vibrational, in nature.
To our knowledge, there is no analogous phenomenon involving low-lying electronic excitations in electron-atom or
electron-molecule interactions, and so the positron would
have to play a fundamental role in these modes. We have
speculated previously that the states might be thought of as a
Ps atom moving in the field of the positively charged atomic
or molecular ion 关18兴.
The positron annihilation rate is proportional to the overlap of positron and electron wave functions. Thus the shortrange correlation between the positron and an electron is
important. It poses a challenge to theory to include shortrange correlation into the scattering problem. As discussed
above, recent advances in computational approaches have
enabled large-scale calculations of positron-molecule interactions to be carried out for small molecules such as ethylene. The agreement between theory and experiment for ethylene, as illustrated in Fig. 7, is encouraging 关22兴. This
comparison provides support for the importance of shortrange electron-positron correlations in determining annihilation rates. Vibrational motion is not included in these calculations, and the estimates presented above indicate that these
vibrational excitations are crucial in obtaining Z eff values
larger than about 103 . If the numerical calculations could be
done for larger molecules, one could test this prediction.
Phenomenological models, including the model proposed
by Laricchia and Wilkin 关43,44兴, were analyzed using our
experimental data. Their model describes the observed annihilation rates reasonably well. However, the annihilation
rates predicted by this model appear to us to arise from questionable assumptions. In Sec. V D 3, we proposed a scaling
with the parameter ␣ /(E i ⫺E Ps). This scaling exhibits a
somewhat better correlation with measured values of Z eff
than the model by Laricchia and Wilkin. Nevertheless, we
note that this new scaling is purely empirical, and its physical meaning is unclear. It was conjectured previously that the
strong dependence of Z eff on E i ⫺E Ps might indicate that the
positron interacting with an atom or a molecule could be
thought of as a highly correlated electron-positron pair moving in the field of the resulting positive ion 关18兴. The inclusion of the factor ␣ could mean that the collision cross section is also an important parameter in determining the
annihilation rate.
In conclusion, we do not find a ready and universal explanation for the anomalously large positron annihilation
rates of organic molecules that have been observed in many
experiments and for a wide range of molecules. Nevertheless, advances in the experimental measurements and formulating a theoretical framework for this problem have provided new insights. They place new constraints on
theoretical models of this phenomenon.
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We thank E. A. Jerzewski for expert technical assistance.
The work at the University of California, San Diego, was
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