PHYSICAL REVIEW A, VOLUME 61, 022719 Positron annihilation on large molecules Koji Iwata,1,* G. F. Gribakin,2,† R. G. Greaves,1,‡ C. Kurz,1,§ and C. M. Surko1,储 1 Physics Department, University of California, San Diego, La Jolla, California 92093-0319 2 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 I. INTRODUCTION 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. 储 Author to whom correspondence should be addressed. Electronic address: [email protected] 1050-2947/2000/61共2兲/022719共17兲/$15.00 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 , 共1兲 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 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 (dir) ⬀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) in 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 (res) 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 7 8 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. II. PREVIOUS EXPERIMENTS 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 022719-2 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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 complex. 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. III. THEORETICAL CONSIDERATIONS 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 , 共2兲 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 . 共3兲 Comparing this expression with the definition of Z eff 关Eq. 共1兲兴, we have Z eff⫽ v r 20 c 共 1⫺e ⫺ / a 兲 . 共4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 022719-3 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 as (dir) ⬃ R a ep , Z eff (dir) 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: 冉 (dir) ⯝F R 2a ⫹ Z eff 共6兲 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 ⫽⫺ 共5兲 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 (dir) ⬃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 冊 ⫹2R a Re f 0 , 4 1 , ⫹ik ⫽ 4 ⫹k 2 2 共7兲 . 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 , 冋 tan ␦ 0 ⫽⫺ak 1⫺ 冉 冑␣ k ␣k 4␣k2 ⫺ ln C 3a 3 4 冊册 ⫺1 , 共8兲 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. 022719-4 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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 ⫽ 冕 ⬁ 0 冉 Z eff共 k 兲 exp ⫺ 冊 4 k dk k . 2k B T 共 2 k B T 兲 3/2 2 ( v ⫽k in atomic units兲. Using Eq. 共2兲 and the definition of Z eff , we obtain (res) ⫽ Z eff 2 共9兲 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. (res) ⫽ 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 共10兲 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 , k 2 D ⌫ a ⫹⌫ c k 共11兲 共12兲 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 22 2 2 ep a ⬅ . k D 共 ⌫ a ⫹⌫ c 兲 k D共 a⫹ c 兲 2 2 ep . k D 共13兲 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 (res) ⬃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 022719-5 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 共14兲 For this potential the s-wave scattering amplitude is f ⫽⫺ 1 , 0 ⫹ik 共15兲 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 : 冏 det ␦ i j 共 i ⫺ 兲 ⫹ 冏 exp共 ⫺ R i j 兲 共 1⫺ ␦ i j 兲 ⫽0, Rij 共16兲 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 0 1 2 3 4 a b 0.111 0.0452 0.053 0.005 ⫺0.014 ⫺0.031 ⫺0.103 ⫺0.071 ⫺0.217 ⫺0.112 a H⫽⫺0.5, F⫽⫺2.0. H⫽⫺0.72, F⫽⫺1.275. b 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 . R 共17兲 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 022719-6 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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. IV. EXPERIMENT 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 关40,41兴. 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. V. RESULTS In Sec. V A, we present experimental measurements of positron annihilation rates of deuterated alkanes and the cor- 022719-7 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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. Molecule j C j H2 j⫹2 C j D2 j⫹2 Ratio 214 116 000 341 000 408 000 641 000 0.96 1.10 0.96 0.70 0.96 Z eff Methane Hexane Heptane Octane Nonane 1 6 7 8 9 222 105 000 355 000 585 000 666 000 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 022719-8 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 TABLE III. Values of Z eff for partially fluorinated hydrocarbons. Molecule Formula Z Z eff CH4 CH3 F CH2 F2 CHF3 CF4 10 18 26 34 42 308 1 390 799 247 73.5 Ethane Fluoroethane 1,1,1-Trifluoroethane 1,1,2-Trifluoroethane 1,1,1,2-Tetrafluoroethane 1,1,2,2-Tetrafluoroethane Hexafluoroethane C2 H6 C2 H5 F CF3 CH3 CHF2 CH2 F CF3 CH2 F CHF2 CHF2 C2 F6 18 26 42 42 50 50 66 1 780 3 030 1 600 1 510 1 110 467 149 Propane 2,2-Difluoropropane 1,1,1-Trifluoropropane Perfluoropropane C3 H8 CH3 CF2 CH3 CF3 C2 H5 C3 F8 26 42 50 90 2 350 8 130 3 350 317 Hexane 1-Fluorohexane Perfluorohexane C6 H14 CH2 FC5 H11 C6 F14 50 58 162 151 000 269 000 630 C6 H6 C6 H5 F C6 H4 F2 C6 H4 F2 C6 H4 F2 C6 H3 F3 C6 H2 F4 C6 HF5 C6 F6 42 50 58 58 58 66 74 82 90 20 300 45 100 32 800 13 100 13 500 10 100 2 760 1 930 499 Methane Methyl fluoride Difluoromethane Trifluoromethane Carbon tetrafluoride Benzene Fluorobenzene 1,2-Difluorobenzene 1,3-Difluorobenzene 1,4-Difluorobenzene 1,2,4-Trifluorobenzene 1,2,4,5-Tetrafluorobenzene Pentafluorobenzene Hexafluorobenzene (dir) 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 (dir) (T) at room temperature on the positions of the virtual Z eff and bound states, as represented by the parameter . This has 022719-9 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 022719-10 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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 022719-11 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 targets. 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, 022719-12 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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 ⌬t⫽ ប , 兩 E⫺E i ⫹E Ps兩 共19兲 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 , 共18兲 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. v 兵 ␥ 关 1⫺ exp„⫺ 兲兴 ⫹ 共 1⫺ ␥ 兲 r 20 c ⫻ 关 1⫺ exp„⫺⌬t 共 sa⫹ po兲 …兴 其 , 共20兲 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 共21兲 , 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 . 共22兲 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 , 共23兲 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 , 共24兲 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 022719-13 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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 molecules. 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. 022719-14 POSITRON ANNIHILATION ON LARGE MOLECULES PHYSICAL REVIEW A 61 022719 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 molecules. 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 process. VI. CONCLUDING REMARKS 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 methane. 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 022719-15 IWATA, GRIBAKIN, GREAVES, KURZ, AND SURKO PHYSICAL REVIEW A 61 022719 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. 关1兴 K. Iwata, R. G. Greaves, T. J. Murphy, M. D. 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