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On positronium formation in crystalline and amorphous ice at low positron energy
Volume 118, number 7
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3 November 1986
O N P O S I T R O N I U M F O R M A T I O N I N C R Y S T A L L I N E A N D A M O R P H O U S ICE AT L O W P O S I T R O N E N E R G Y O.E. M O G E N S E N Chemistry Department, Riso National Laboratory, DK-4000 Roskilde, Denmark
Received 8 July 1986; accepted for publication 12 September 1986
The positronium (Ps) yield for ice, measured by Eldrup et al. using a low-energypositron beam, is discussed in terms of the spur model of Ps formation. The pronounced maxima in the Ps yield for crystalline ice at positron energies below 65 eV are well explained by effects due to energy conservation in the spur processes. Parts of the amorphous ice results are well explained by the spur but not by the Ore model. Important processes influencing the Ps formation are not included in the Ore model.
1. Introduction
Eldrup et al. [ 1 ] recently studied ice using a lowenergy positron beam. A letter [2] and a c o m m e n t [3] with respond [4] concerning these measurements have also been published. In this letter I shall discuss the interpretation [ 1 - 4 ] of the measured positronium (Ps) yields [1,2] at positron energies below ~ 6 5 eV. Eldrup et al. show in fig. 2 ofref. [ 1 ] the measured fraction of 3y decays, f versus energy o f the positrons hitting the ice surface, E. f i s normalized to mainly represent the 3"{ ortho-Ps decay, and hence 0 ~
ing-down processes of the positron in the terminal spur of the positron track. The Ore model predicts that Ps is formed by epithermal positrons in the positron energy interval I-B<~E<~I (the Ore gap), where I is the ionization energy of the molecules and B the formation energy o f P s ( B = 6.8 eV).
2. Spur processes at low energies close to a surface
Before we can explain the two peaks in the Ps yield versus positron energy curve in terms of the spur model, it i~ necessary to discuss the physical process of Ps formation [5 ] and the shortcomings of the Ore model in some detail. It is useful to start by a short discussion of the simple spur picture as it applies to liquids [ 5 ]. The positron is thermalized within the "positron spur", which, apart from the positron, consists of the radiation damage created by the positron itself in its slowing down from some 100 eV to thermal energy. The spur species: electrons, positive ions, free radicals, excited states, etc., and the positron react with each other by nonhomogeneous kinetics. In particular, the positron competes with the other species - mainly the positive ion - for the electrons to form Ps. Only if the positron and one of the electrons happen to thermalize far away from the other charged species, is the Ps formation reaction independent of the presence of the other species. It is important to note that 357
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the light particles will have a short coherence length ( m e a n free p a t h ) c o m p a r e d to the slowing down range and the range o f the C o u l o m b force (the Onsager radius) in most liquids. Hence, the spur reaction problem can be followed by use o f classical diffusion theory in a good a p p r o x i m a t i o n . The equilibrium Ps state is normally the Ps bubble state. In few liquids (e.g. CS2 just above melting p o i n t ) Ps in the bubble state is of higher energy than a state in which the positron and electron are strongly b o u n d to the molecules but kept together by their Coulomb attraction [5,6] (the " f o u r t h " positron state in liquids and solids). As the next step in the argumentation let us replace the liquid by a solid, such as ice. The main change will be that the "excess" electrons and the positron will now be more delocalized, i.e. they will have longer coherence lengths ( m e a n free paths) if the solid is rigid, like ice. The electrons and the positron will still thermalize within the range o f C o u l o m b forces of the other charged particles in most cases. To understand the light particle states better it might be useful to recall that one electron and a positive ion within their C o u l o m b range, and where the electron is delocalized (i.e. with coherence length much longer than the size o f the o r b i t ) , is called an exciton. The dielectric constant of ice at times below 1 ns is 3 or lower, and, o f course, d e p e n d e n t on the very high electric fields ( m e g a v o l t s ) close to the charged species (at ~ 2 n m ) . Hence, the Onsager radius is 2 0 - 5 0 nm depending on t e m p e r a t u r e and effective dielectric constant. The slowing down distances are probably 2 0 - 3 0 nm [ 1 ]. The measured electron mobility, /,=20_+ 10 cm2/V s at 150 K [7], indicates that the electron mean free path is not very large c o m p a r e d with the distance between H 2 0 molecules and, in particular, its wave length at kinetic energy kT, at low electric fields. In summary, the " p o s i t r o n spur" in ice m a y have some "exciton-like" properties over distances of about 2 - 3 nm, but will be "classicalphysics-like" over larger distances, and the normally used radiation-chemistry diffusion-theory language is probably a rather good a p p r o x i m a t i o n after all. Let us now discuss the changes in the spur processes, which are expected when the initial positron energy is decreased from high energies to the lowest energy o f the Ore gap. We designate the upper and lower boundaries o f the Ore gap by Eu and E~, 358
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respectively (see ref. [ 1 ]). At high energies the positron can form Ps with one o f several excess electrons created in its own track. It is normally assumed (but certainly not well known e x p e r i m e n t a l l y ) that the electron, and hence also the positron, loses roughly 20 eV/nm at energies between ~ 103 and ~ 10 eV in condensed m a t t e r o f density ~ 1 g/cm 3, and that one ionization occurs for an average energy loss o f ~ 25 eV [8]. If the positron forms Ps with excess electrons created before the last ionization, it is easy to accept that the e l e c t r o n - p o s i t r o n pair did not just form Ps in a one-step q u a n t u m transition process. The positron takes part in several processes between the creation o f the excess electron forming Ps and the actual Ps formation. At energies below roughly 2 E u - 6 . 8 eV (and above E~,) the positron does not have the possibility to make Ps with more than one electron created in its own track. This is the energy interval where the difference between two types o f description is accentuated. In the simplest "'isolated-positron-molecule description" the positron is supposed either to make Ps or slow down, and the Ps f o r m a t i o n process is considered to be a one-step q u a n t u m transition. Above the lowest excited state energy the positron is also supposed to be able to excite molecules in a q u a n t u m transition, which cannot be followed. Above El, it is supposed to make ionizations as q u a n t u m transitions. In the simplest "diffusion-of-classical-particles description" the positron and electron are supposed to behave like classical particles, which slow down and diffuse under the influence of C o u l o m b fields of the oppositely charged particle and the positive ion. It is supposed that Ps formation occurs as a result of many processes during the diffusion, and not just as one q u a n t u m j u m p . The former description is the Ore model [ 1 ], while the latter is the extreme "classical" model, sometimes preferred by radiation chemists. The former model applies if the interaction o f " t h i r d b o d i e s " is negligible, as in molecular gases at densities below approximately 0.01 amagat. It is my opinion thai: (a) Ps is formed by the latter model m o d i f i e d by quantum mechanics, i.e. the positron spur might be "exciton-like" or "classical" depending upon the structure o f the sample, and (b) that it is possible to influence the Ps formation, by e.g. scavenging one o f the light particles, during the Ps formation processes [ 5 ].
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If a surface "cuts" the positron spur, i.e. that part of the positron track from which electrons can react with the positron, several new processes involving surfaces and vacuum states will occur. For example, one of the particles can thermalize in a surface state and then attract the other particle to form Ps, which is at the same time emitted from the surface (negative Ps affinity of ice). Ps may also form inside the solid and diffuse out of the surface, either as a fairly localized (say within 1 n m ) , or delocalized, as in ice, (i.e. coherence length ~ 5-20 n m ) particle. An extreme case is Ps formation in the energy interval E~ ~
3. The approximations of the Ore model The Ore model was initially formulated for lowdensity gases, where the positron and one of the atoms or molecules can be considered to be the only species present during Ps formation. It was therefore possible to imagine the incoming positron in a wave-packet state, and, after the reaction has taken place, the outgoing positron or Ps in wave-packet states too. At sufficiently long times the atom or molecule, unchanged, excited, or ionized, could be completely spatially separated from the positron or Ps. Hence, it was possible to separate clearly excitation, ionization, and Ps formation at long times after the posit r o n - a t o m or -molecule collision. In the Ore model, as it is used in ref. [ 1 ], the three cross sections for excitation, ionization, and Ps formation, ai, and the three associated rules for calculating the energy loss, AE,, determine the Ps yield, and excited states, excitons, and/or excess electrons, which later form Ps, are not included. Actually, Ps formation in high density matter is
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more complicated. In condensed matter and high density gases an initially ionized electron ion pair can form Ps and excited states after ionization, because of the small electron thermalization distance. Also an excited state can form Ps. In many cases the electron in the "excited molecule" will be influenced by the Coulomb field of the positron roughly as strongly as it is influenced by that of its parent ion even if the initial positron energy is in the Ore gap, because the positron thermalizes normally over a distance which is smaller than the Onsager radius, and comparable to the thermalized electron ion distances. At higher positron initial energies several electrons, positive ions, and the positron will be in each others Coulomb field before and after the light particle thermalizations. As ionized electrons, even o f initial energies much higher than the ionization energy, thermalize over a distance smaller than the Onsager radius in most cases, they will, after thermalization, be in a physical state which is essentially identical to that of a large fraction of the so-called "excited states" [9 ]. Moreover, the "excited states" in condensed matter can also be very different from excited states in v a c u u m because the electron in the excited state may overlap many molecules. Only those excited states where the electron is mainly inside the normal boundary of the molecule, e.g. the lowest excited states on aromatic molecules, have properties fairly similar to the vacuum excited states. The detailed structure of most "excited states" in condensed matter seems not to have been sorted out yet, and the nomenclature is very different from one field of science to another. Photochemists seem to prefer the names: valence states, Rydberg states, and Wannier excitons; and at the highest energies excess electron ion pairs. Radiation chemists speak mainly in terms of excess electrons and excited states. The latter name seems to refer to the very lowest excitation energies. In ionic crystal research mainly excitons are studied and referred to, e.g. the simplest spur recombination in ionic crystals go through the formation of " b o u n d excitons". Hence, the simplest "positron spur" in NaC1 may be an initially ionized or excited electron and a positron, both more or less delocalized and under the influence of their mutual Coulomb field and that of a bound positive ion (the Cls ). In the extreme cases of polar liquids, the excess electrons are known to be solvated with a solvation 359
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energy of 1-2 eV, and localized very strongly even close to (e.g. 1-2 nm) the positive ions [10]. Furthermore, the recombination takes several nanoseconds in polar liquids. For example, flash photolysis experiments seem to indicate that electrons in certain molecules in water are ionized directly into preexisting traps in water to give the absorption of light due to the hydrated electron after only 0.3 ps [ 11 ]. Hence the quantum j u m p processes produces a localized electron at a distance of ~ 1 nm from the positive ion and'not just an excited state as would be the case in vacuum. The positron slowing down processes create also these hydrated electrons close to the positive ion, and the positron can form Ps with the hydrated electrons after further slowing down. Clearly, the Ore model does not include such processes. It is, of course, possible to use the Ore model (as in ref. [1 ]) as an interpolation rule with the six parameters (a,, AE,) for excitation, ionization, and Ps formation as adjustable parameters. The parameters for all the initial excitation and ionization processes which later lead to Ps formation, apart from the "real" Ps formation. This is a rather artificial way of describing the Ps formation process, and the claim that either excitation, ionization, or Ps formation take place is obviously incorrect. Furthermore, the possibility of using such an interpolation rule does not disprove that spur processes give explanations of the fitting parameters. Until now we have discussed the physical processes leading to Ps formation and why the Ore model is too simple compared with the "real world". Let us now discuss specific experimental results which apparently cannot be explained in the Ore model. The Ps yield varies strongly with the structure and chemical composition of condensed matter samples at high initial energy of the positron. For example, it is zero in solid CCL [12], CBr4 [12], and in crystalline anthracene, but not in liquid and defected solid anthracene. The ortho-Ps yield is 28% in H20 and 23% in D 2 0 water, while it is ~ 55% in both H 2 0 and D20 ice, and ~ 3 4 % in amorphous H 2 0 ice [1]. These solids and liquids have roughly identical Ore gap energies, and it seems to be very difficult indeed to argue for parameters (ai, AEi, i = 1-3 ) versus positron energy which explain the measured Ps yields. Clearly, predictions of the used Ore model can be 360
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changed only through a change in these parameters. Hence, the Ore model is insufficient! We have only discussed the Ps yield for some solids. However, we can reach the same conclusion by referring to the yields measured for many other solids. Actually, it has been known for many years that the Ore model cannot explain the Ps yield in condensed matter in general. The disputed problem has always been whether the spur or the "hot-Ps-reaction" model is the correct mechanism. Exceptions are the very few cases where the "excluded-volume" model was advocated. Generally speaking, the main difference between the Ore model, as used in ref. [ 1 ], and the spur model is that in the latter the processes which occur during the positron collisions, slowing down, diffusion, etc., are described in great detail in agreement with the concepts and ideas used in radiation physics and chemistry and in photochemistry. In the Ore model many processes and configurations are not included, and only the possibilities of getting final states of a collision, which are either excitation, ionization, or Ps formation, are taken into account. The actual processes are much more complicated than what can be accounted for by the Ore model. Consequently, many processes influencing the Ps yield in the spur model, and in reality, are totally missing in the Ore model. The spur model is accepted today mainly because it can explain and predict many measured Ps yields in terms of those spur processes which are missing in the Ore model [5]. For example, the spur model explains why the Ps yield decreases strongly if electron scavengers are added to a sample in concentrations of less than one scavenger per thousand solvent molecules [5]. A good illustration is that the addition of roughly 100 ppm radicals to ice by irradiation reduces drastically the Ps yield [13]. Such small changes cannot change the Ore model predictions significantly except if very, unrealistic cross sections are chosen.
4. The spur model explanation
The last topic to discuss is how the spur model can explain the Ps yields in ref. [ 1 ]. Generally, it is fairly easy to show that the results agree with the spur model. One just has to realize that the energy conser-
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vation rules used in the Ore model, apply as well in the spur model. Of course, Ps cannot be formed at positron energies below E~. E~ and Eu are the lower and upper boundaries of the Ore gap for the substance studied. They are generally different from the vacuum values [11. In the Ore gap the electron cannot escape the Coulomb attraction of the positive ion except if it is bound to the positron. The penetration distance of the positron before the electron is ejected is small compared with the Onsager radius, and hence many surface processes contribute to the Ps formation, as discussed above. Intuitively, it seems to be reasonable to propose that the Ps yield is very high in the Ore gap in ice close to the surface because Ps is delocalized in ice, and hence it can be emitted from the surface very rapidly. This is not the case in amorphous ice [ 1 ], where the strong Ps formation in the Ore gap is missing (see below). The strong decrease in the Ps yield at the upper boundary of the Ore gap Eu can be explained in the following way. On increasing the positron energy from just below Eu to just above Eu, a new process becomes possible. Above Eu the electron can escape the positive ion without necessarily being bound to the positron. In the bulk of ice most of the electrons and positrons will slow down at a distance small compared to the range of the Coulomb force anyway, and a strong change in Ps yield is not expected. However, in ice close to the surface a large number of the electrons will be ejected through the surface, and many electrons which just below Eu would have formed Ps can escape the positron just above Eu. This means that Ps formation must compete with a new process, and hence, the Ps yield decreases strongly just above E~. In the same way the maximum at around 25 eV and the small minimum at roughly 35 eV can be explained as caused by the Ore gap associated with the electron band at around 30 eV. It is important to note that the arguments in terms of the spur model concern many intermediate steps leading to eventual ionization and Ps formation, and hence these processes can be intercepted at an intermediate stage, e.g. by trapping the electron. In the Ore model the processes are single step processes. The ortho-Ps yield in amorphous ice is probably
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close to the 37 decay intensity of 34%, given in fig. 12 of ref. [1]. This agrees with 34_+ 2% for 10 M NaOH glassy ice at 85 K, extracted by a very rough analysis in ref. [ 14]. In alcohols the ortho-Ps yield of the amorphous state seems to be close to, but somewhat larger than, that of the liquid, and much less than that of the crystalline solid [ 15 ]. These results, which seem to be the only ones published for the amorphous state of polar liquids, are of course for annihilation in the bulk. They have been considered to be important tests of the spur model for the following reason: In nonpolar liquids the electron-ion recombination takes 10 ps or less, and Ps formation is fast because the electron and the positron are moving rapidly as they are either not, or only partly, localized. In polar liquids the electron and positron are solvated. Hence, recombination takes 1-10 ns [10] which also must be roughly the Ps formation time for solvated particles far enough apart. However, the positron lifetime is only ~ 400 ps in polar liquids, and Ps is therefore formed either if both particles are not yet solvated, or if they solvate close to each other. This is the spur explanation of the approximately two times lower Ps yield in polar compared with nonpolar liquids. The difference between the Ps yields in crystalline and amorphous polar solids is explained similarly in the spur model [5]. In crystalline polar solid the electron and positron are rapidly moving, and Ps is expected to be formed within picoseconds. In amorphous solids the particles are trapped in preexisting traps. Hence, they can move only by a very slow trapto-trap hopping mechanism, tunnelling, etc., and Ps formation will be limited by the finite positron lifetime of 400-500 ps, i.e. the Ps yield is lowerin amorphous solids. In summary, an ortho-Ps yield of 34% in the bulk of amorphous ice is in full agreement with the spur model and previously obtained yields for other polar amorphous polar solids, while it is difficult to explain in the Ore, the "hot-Ps-reaction" and the "excludedvolume" models. It is a good illustration of the main difference between the Ore and spur models. In the latter the Ps yield is influenced by a reaction of the light particles, here trapping, during the complex spur processes of Ps formation. In the former model such reactions are not included at all.
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A discussion o f Ps f o r m a t i o n in the o t h e r substances, m e n t i o n e d a b o v e , can be f o u n d in ref. [6] and references therein.
5. Conclusions In conclusion, the Ps yields r e p o r t e d in ref. [ 1 ] can be well e x p l a i n e d in t e r m s o f the s p u r m o d e l for all energies o f the i n c o m i n g positron, m a i n l y because the energy c o n s e r v a t i o n rules, used in the O r e m o d e l in ref. [ 1 ], apply as well in the spur m o d e l . T h e Ps yield for a m o r p h o u s ice, p r e s e n t e d in fig. 12 o f ref. [ 1 ], is the yield e x p e c t e d fo~ a spur process in the bulk o f a m o r p h o u s ice. M a n y solids, w i t h O r e m o d e l p a r a m eters roughly i d e n t i c a l to t h o s e o f ice, give no Ps in the bulk (e.g. CC14) c o m p a r e d to 55% o r t h o - P s in the bulk o f ice, in full a g r e e m e n t with the spur model. T h e s e results c a n n o t be e x p l a i n e d by the O r e m o d e l .
Acknowledgement M a n y s t i m u l a t i n g discussions w i t h M. E l d r u p and F.M. J a c o b s e n are a c k n o w l e d g e d .
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References [ 1] M. Eldrup, A. Vehanen, P.J. Schultz and K.G. Lynn. Phys. Rev. B32 (1985) 7048. [2] M. Eldrup, A. Vehanen, P.J. Schultz and K.G. Lynn, Ph}s. Rev. Lett. 51 (1983) 2007. [3] J. van House, A. Rich and P.W. Zitzewilz. Phys. Rev. Left. 53 (1984) 953. [4] M. Eldrup. A. Vehanen, P.J. Schultz and K.G. Lynn. Phys. Rev. Lett. 53 (1984) 954. [5] O.E. Mogensen, in: Positron annihilation, eds. P.G. Coleman, S.C. Sharma and L.M. Diana (North-Holland, Amsterdam, 1982) p. 763. [6] O.E. Mogensen, N.J. Pedersen and F.M. Jacobsen, in: Proc. 7th Intern. Cone on Positron annihilation (Delhi, January 1985), eds. P.C. Jain, R.M. Singru and K.P. Gopinathan (World Scientific, Singapore, 1985) p. 205. [7] J.M. Warman, M.P. DeHaas and J.B. Verberne, J. l)hys. Chem. 84 (1980) 1240. [8] A. Mozumder. Adv. Radiat. Chem. I (1969) 1. [9] K. Funabashi, Adv. Radiat. Chem. 4 (1974) 103. [10] J.W. Hunt, Adv. Radiat. Chem. 5 (1976) 185. [II]J.M. Wiesenfeld and E.P. Ippen, Chem. Phys. Lett. 73 (1980) 47. [12] W. Brandt, private communication. [ 13] M. Eldrup, O.E. Mogensen and J.H. Bilgram. J. Glaciol. 2 I (85) (1978) 101. [ 14] M. Eldrup, O.E. Mogensen and L. Kevan. Chem. Phys. Lett. 10(1971) 379. [15] O.A. Anisimov and Yu. N. Molin, Khim. Vys. Energ. 9 (1975) 376 [High Energ. Chem. 9 (1976) 331].
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O N P O S I T R O N I U M F O R M A T I O N I N C R Y S T A L L I N E A N D A M O R P H O U S ICE AT L O W P O S I T R O N E N E R G Y O.E. M O G E N S E N Chemistry Department, Riso National Laboratory, DK-4000 Roskilde, Denmark
Received 8 July 1986; accepted for publication 12 September 1986
The positronium (Ps) yield for ice, measured by Eldrup et al. using a low-energypositron beam, is discussed in terms of the spur model of Ps formation. The pronounced maxima in the Ps yield for crystalline ice at positron energies below 65 eV are well explained by effects due to energy conservation in the spur processes. Parts of the amorphous ice results are well explained by the spur but not by the Ore model. Important processes influencing the Ps formation are not included in the Ore model.
1. Introduction
Eldrup et al. [ 1 ] recently studied ice using a lowenergy positron beam. A letter [2] and a c o m m e n t [3] with respond [4] concerning these measurements have also been published. In this letter I shall discuss the interpretation [ 1 - 4 ] of the measured positronium (Ps) yields [1,2] at positron energies below ~ 6 5 eV. Eldrup et al. show in fig. 2 ofref. [ 1 ] the measured fraction of 3y decays, f versus energy o f the positrons hitting the ice surface, E. f i s normalized to mainly represent the 3"{ ortho-Ps decay, and hence 0 ~
ing-down processes of the positron in the terminal spur of the positron track. The Ore model predicts that Ps is formed by epithermal positrons in the positron energy interval I-B<~E<~I (the Ore gap), where I is the ionization energy of the molecules and B the formation energy o f P s ( B = 6.8 eV).
2. Spur processes at low energies close to a surface
Before we can explain the two peaks in the Ps yield versus positron energy curve in terms of the spur model, it i~ necessary to discuss the physical process of Ps formation [5 ] and the shortcomings of the Ore model in some detail. It is useful to start by a short discussion of the simple spur picture as it applies to liquids [ 5 ]. The positron is thermalized within the "positron spur", which, apart from the positron, consists of the radiation damage created by the positron itself in its slowing down from some 100 eV to thermal energy. The spur species: electrons, positive ions, free radicals, excited states, etc., and the positron react with each other by nonhomogeneous kinetics. In particular, the positron competes with the other species - mainly the positive ion - for the electrons to form Ps. Only if the positron and one of the electrons happen to thermalize far away from the other charged species, is the Ps formation reaction independent of the presence of the other species. It is important to note that 357
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the light particles will have a short coherence length ( m e a n free p a t h ) c o m p a r e d to the slowing down range and the range o f the C o u l o m b force (the Onsager radius) in most liquids. Hence, the spur reaction problem can be followed by use o f classical diffusion theory in a good a p p r o x i m a t i o n . The equilibrium Ps state is normally the Ps bubble state. In few liquids (e.g. CS2 just above melting p o i n t ) Ps in the bubble state is of higher energy than a state in which the positron and electron are strongly b o u n d to the molecules but kept together by their Coulomb attraction [5,6] (the " f o u r t h " positron state in liquids and solids). As the next step in the argumentation let us replace the liquid by a solid, such as ice. The main change will be that the "excess" electrons and the positron will now be more delocalized, i.e. they will have longer coherence lengths ( m e a n free paths) if the solid is rigid, like ice. The electrons and the positron will still thermalize within the range o f C o u l o m b forces of the other charged particles in most cases. To understand the light particle states better it might be useful to recall that one electron and a positive ion within their C o u l o m b range, and where the electron is delocalized (i.e. with coherence length much longer than the size o f the o r b i t ) , is called an exciton. The dielectric constant of ice at times below 1 ns is 3 or lower, and, o f course, d e p e n d e n t on the very high electric fields ( m e g a v o l t s ) close to the charged species (at ~ 2 n m ) . Hence, the Onsager radius is 2 0 - 5 0 nm depending on t e m p e r a t u r e and effective dielectric constant. The slowing down distances are probably 2 0 - 3 0 nm [ 1 ]. The measured electron mobility, /,=20_+ 10 cm2/V s at 150 K [7], indicates that the electron mean free path is not very large c o m p a r e d with the distance between H 2 0 molecules and, in particular, its wave length at kinetic energy kT, at low electric fields. In summary, the " p o s i t r o n spur" in ice m a y have some "exciton-like" properties over distances of about 2 - 3 nm, but will be "classicalphysics-like" over larger distances, and the normally used radiation-chemistry diffusion-theory language is probably a rather good a p p r o x i m a t i o n after all. Let us now discuss the changes in the spur processes, which are expected when the initial positron energy is decreased from high energies to the lowest energy o f the Ore gap. We designate the upper and lower boundaries o f the Ore gap by Eu and E~, 358
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respectively (see ref. [ 1 ]). At high energies the positron can form Ps with one o f several excess electrons created in its own track. It is normally assumed (but certainly not well known e x p e r i m e n t a l l y ) that the electron, and hence also the positron, loses roughly 20 eV/nm at energies between ~ 103 and ~ 10 eV in condensed m a t t e r o f density ~ 1 g/cm 3, and that one ionization occurs for an average energy loss o f ~ 25 eV [8]. If the positron forms Ps with excess electrons created before the last ionization, it is easy to accept that the e l e c t r o n - p o s i t r o n pair did not just form Ps in a one-step q u a n t u m transition process. The positron takes part in several processes between the creation o f the excess electron forming Ps and the actual Ps formation. At energies below roughly 2 E u - 6 . 8 eV (and above E~,) the positron does not have the possibility to make Ps with more than one electron created in its own track. This is the energy interval where the difference between two types o f description is accentuated. In the simplest "'isolated-positron-molecule description" the positron is supposed either to make Ps or slow down, and the Ps f o r m a t i o n process is considered to be a one-step q u a n t u m transition. Above the lowest excited state energy the positron is also supposed to be able to excite molecules in a q u a n t u m transition, which cannot be followed. Above El, it is supposed to make ionizations as q u a n t u m transitions. In the simplest "diffusion-of-classical-particles description" the positron and electron are supposed to behave like classical particles, which slow down and diffuse under the influence of C o u l o m b fields of the oppositely charged particle and the positive ion. It is supposed that Ps formation occurs as a result of many processes during the diffusion, and not just as one q u a n t u m j u m p . The former description is the Ore model [ 1 ], while the latter is the extreme "classical" model, sometimes preferred by radiation chemists. The former model applies if the interaction o f " t h i r d b o d i e s " is negligible, as in molecular gases at densities below approximately 0.01 amagat. It is my opinion thai: (a) Ps is formed by the latter model m o d i f i e d by quantum mechanics, i.e. the positron spur might be "exciton-like" or "classical" depending upon the structure o f the sample, and (b) that it is possible to influence the Ps formation, by e.g. scavenging one o f the light particles, during the Ps formation processes [ 5 ].
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If a surface "cuts" the positron spur, i.e. that part of the positron track from which electrons can react with the positron, several new processes involving surfaces and vacuum states will occur. For example, one of the particles can thermalize in a surface state and then attract the other particle to form Ps, which is at the same time emitted from the surface (negative Ps affinity of ice). Ps may also form inside the solid and diffuse out of the surface, either as a fairly localized (say within 1 n m ) , or delocalized, as in ice, (i.e. coherence length ~ 5-20 n m ) particle. An extreme case is Ps formation in the energy interval E~ ~
3. The approximations of the Ore model The Ore model was initially formulated for lowdensity gases, where the positron and one of the atoms or molecules can be considered to be the only species present during Ps formation. It was therefore possible to imagine the incoming positron in a wave-packet state, and, after the reaction has taken place, the outgoing positron or Ps in wave-packet states too. At sufficiently long times the atom or molecule, unchanged, excited, or ionized, could be completely spatially separated from the positron or Ps. Hence, it was possible to separate clearly excitation, ionization, and Ps formation at long times after the posit r o n - a t o m or -molecule collision. In the Ore model, as it is used in ref. [ 1 ], the three cross sections for excitation, ionization, and Ps formation, ai, and the three associated rules for calculating the energy loss, AE,, determine the Ps yield, and excited states, excitons, and/or excess electrons, which later form Ps, are not included. Actually, Ps formation in high density matter is
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more complicated. In condensed matter and high density gases an initially ionized electron ion pair can form Ps and excited states after ionization, because of the small electron thermalization distance. Also an excited state can form Ps. In many cases the electron in the "excited molecule" will be influenced by the Coulomb field of the positron roughly as strongly as it is influenced by that of its parent ion even if the initial positron energy is in the Ore gap, because the positron thermalizes normally over a distance which is smaller than the Onsager radius, and comparable to the thermalized electron ion distances. At higher positron initial energies several electrons, positive ions, and the positron will be in each others Coulomb field before and after the light particle thermalizations. As ionized electrons, even o f initial energies much higher than the ionization energy, thermalize over a distance smaller than the Onsager radius in most cases, they will, after thermalization, be in a physical state which is essentially identical to that of a large fraction of the so-called "excited states" [9 ]. Moreover, the "excited states" in condensed matter can also be very different from excited states in v a c u u m because the electron in the excited state may overlap many molecules. Only those excited states where the electron is mainly inside the normal boundary of the molecule, e.g. the lowest excited states on aromatic molecules, have properties fairly similar to the vacuum excited states. The detailed structure of most "excited states" in condensed matter seems not to have been sorted out yet, and the nomenclature is very different from one field of science to another. Photochemists seem to prefer the names: valence states, Rydberg states, and Wannier excitons; and at the highest energies excess electron ion pairs. Radiation chemists speak mainly in terms of excess electrons and excited states. The latter name seems to refer to the very lowest excitation energies. In ionic crystal research mainly excitons are studied and referred to, e.g. the simplest spur recombination in ionic crystals go through the formation of " b o u n d excitons". Hence, the simplest "positron spur" in NaC1 may be an initially ionized or excited electron and a positron, both more or less delocalized and under the influence of their mutual Coulomb field and that of a bound positive ion (the Cls ). In the extreme cases of polar liquids, the excess electrons are known to be solvated with a solvation 359
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energy of 1-2 eV, and localized very strongly even close to (e.g. 1-2 nm) the positive ions [10]. Furthermore, the recombination takes several nanoseconds in polar liquids. For example, flash photolysis experiments seem to indicate that electrons in certain molecules in water are ionized directly into preexisting traps in water to give the absorption of light due to the hydrated electron after only 0.3 ps [ 11 ]. Hence the quantum j u m p processes produces a localized electron at a distance of ~ 1 nm from the positive ion and'not just an excited state as would be the case in vacuum. The positron slowing down processes create also these hydrated electrons close to the positive ion, and the positron can form Ps with the hydrated electrons after further slowing down. Clearly, the Ore model does not include such processes. It is, of course, possible to use the Ore model (as in ref. [1 ]) as an interpolation rule with the six parameters (a,, AE,) for excitation, ionization, and Ps formation as adjustable parameters. The parameters for all the initial excitation and ionization processes which later lead to Ps formation, apart from the "real" Ps formation. This is a rather artificial way of describing the Ps formation process, and the claim that either excitation, ionization, or Ps formation take place is obviously incorrect. Furthermore, the possibility of using such an interpolation rule does not disprove that spur processes give explanations of the fitting parameters. Until now we have discussed the physical processes leading to Ps formation and why the Ore model is too simple compared with the "real world". Let us now discuss specific experimental results which apparently cannot be explained in the Ore model. The Ps yield varies strongly with the structure and chemical composition of condensed matter samples at high initial energy of the positron. For example, it is zero in solid CCL [12], CBr4 [12], and in crystalline anthracene, but not in liquid and defected solid anthracene. The ortho-Ps yield is 28% in H20 and 23% in D 2 0 water, while it is ~ 55% in both H 2 0 and D20 ice, and ~ 3 4 % in amorphous H 2 0 ice [1]. These solids and liquids have roughly identical Ore gap energies, and it seems to be very difficult indeed to argue for parameters (ai, AEi, i = 1-3 ) versus positron energy which explain the measured Ps yields. Clearly, predictions of the used Ore model can be 360
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changed only through a change in these parameters. Hence, the Ore model is insufficient! We have only discussed the Ps yield for some solids. However, we can reach the same conclusion by referring to the yields measured for many other solids. Actually, it has been known for many years that the Ore model cannot explain the Ps yield in condensed matter in general. The disputed problem has always been whether the spur or the "hot-Ps-reaction" model is the correct mechanism. Exceptions are the very few cases where the "excluded-volume" model was advocated. Generally speaking, the main difference between the Ore model, as used in ref. [ 1 ], and the spur model is that in the latter the processes which occur during the positron collisions, slowing down, diffusion, etc., are described in great detail in agreement with the concepts and ideas used in radiation physics and chemistry and in photochemistry. In the Ore model many processes and configurations are not included, and only the possibilities of getting final states of a collision, which are either excitation, ionization, or Ps formation, are taken into account. The actual processes are much more complicated than what can be accounted for by the Ore model. Consequently, many processes influencing the Ps yield in the spur model, and in reality, are totally missing in the Ore model. The spur model is accepted today mainly because it can explain and predict many measured Ps yields in terms of those spur processes which are missing in the Ore model [5]. For example, the spur model explains why the Ps yield decreases strongly if electron scavengers are added to a sample in concentrations of less than one scavenger per thousand solvent molecules [5]. A good illustration is that the addition of roughly 100 ppm radicals to ice by irradiation reduces drastically the Ps yield [13]. Such small changes cannot change the Ore model predictions significantly except if very, unrealistic cross sections are chosen.
4. The spur model explanation
The last topic to discuss is how the spur model can explain the Ps yields in ref. [ 1 ]. Generally, it is fairly easy to show that the results agree with the spur model. One just has to realize that the energy conser-
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vation rules used in the Ore model, apply as well in the spur model. Of course, Ps cannot be formed at positron energies below E~. E~ and Eu are the lower and upper boundaries of the Ore gap for the substance studied. They are generally different from the vacuum values [11. In the Ore gap the electron cannot escape the Coulomb attraction of the positive ion except if it is bound to the positron. The penetration distance of the positron before the electron is ejected is small compared with the Onsager radius, and hence many surface processes contribute to the Ps formation, as discussed above. Intuitively, it seems to be reasonable to propose that the Ps yield is very high in the Ore gap in ice close to the surface because Ps is delocalized in ice, and hence it can be emitted from the surface very rapidly. This is not the case in amorphous ice [ 1 ], where the strong Ps formation in the Ore gap is missing (see below). The strong decrease in the Ps yield at the upper boundary of the Ore gap Eu can be explained in the following way. On increasing the positron energy from just below Eu to just above Eu, a new process becomes possible. Above Eu the electron can escape the positive ion without necessarily being bound to the positron. In the bulk of ice most of the electrons and positrons will slow down at a distance small compared to the range of the Coulomb force anyway, and a strong change in Ps yield is not expected. However, in ice close to the surface a large number of the electrons will be ejected through the surface, and many electrons which just below Eu would have formed Ps can escape the positron just above Eu. This means that Ps formation must compete with a new process, and hence, the Ps yield decreases strongly just above E~. In the same way the maximum at around 25 eV and the small minimum at roughly 35 eV can be explained as caused by the Ore gap associated with the electron band at around 30 eV. It is important to note that the arguments in terms of the spur model concern many intermediate steps leading to eventual ionization and Ps formation, and hence these processes can be intercepted at an intermediate stage, e.g. by trapping the electron. In the Ore model the processes are single step processes. The ortho-Ps yield in amorphous ice is probably
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close to the 37 decay intensity of 34%, given in fig. 12 of ref. [1]. This agrees with 34_+ 2% for 10 M NaOH glassy ice at 85 K, extracted by a very rough analysis in ref. [ 14]. In alcohols the ortho-Ps yield of the amorphous state seems to be close to, but somewhat larger than, that of the liquid, and much less than that of the crystalline solid [ 15 ]. These results, which seem to be the only ones published for the amorphous state of polar liquids, are of course for annihilation in the bulk. They have been considered to be important tests of the spur model for the following reason: In nonpolar liquids the electron-ion recombination takes 10 ps or less, and Ps formation is fast because the electron and the positron are moving rapidly as they are either not, or only partly, localized. In polar liquids the electron and positron are solvated. Hence, recombination takes 1-10 ns [10] which also must be roughly the Ps formation time for solvated particles far enough apart. However, the positron lifetime is only ~ 400 ps in polar liquids, and Ps is therefore formed either if both particles are not yet solvated, or if they solvate close to each other. This is the spur explanation of the approximately two times lower Ps yield in polar compared with nonpolar liquids. The difference between the Ps yields in crystalline and amorphous polar solids is explained similarly in the spur model [5]. In crystalline polar solid the electron and positron are rapidly moving, and Ps is expected to be formed within picoseconds. In amorphous solids the particles are trapped in preexisting traps. Hence, they can move only by a very slow trapto-trap hopping mechanism, tunnelling, etc., and Ps formation will be limited by the finite positron lifetime of 400-500 ps, i.e. the Ps yield is lowerin amorphous solids. In summary, an ortho-Ps yield of 34% in the bulk of amorphous ice is in full agreement with the spur model and previously obtained yields for other polar amorphous polar solids, while it is difficult to explain in the Ore, the "hot-Ps-reaction" and the "excludedvolume" models. It is a good illustration of the main difference between the Ore and spur models. In the latter the Ps yield is influenced by a reaction of the light particles, here trapping, during the complex spur processes of Ps formation. In the former model such reactions are not included at all.
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A discussion o f Ps f o r m a t i o n in the o t h e r substances, m e n t i o n e d a b o v e , can be f o u n d in ref. [6] and references therein.
5. Conclusions In conclusion, the Ps yields r e p o r t e d in ref. [ 1 ] can be well e x p l a i n e d in t e r m s o f the s p u r m o d e l for all energies o f the i n c o m i n g positron, m a i n l y because the energy c o n s e r v a t i o n rules, used in the O r e m o d e l in ref. [ 1 ], apply as well in the spur m o d e l . T h e Ps yield for a m o r p h o u s ice, p r e s e n t e d in fig. 12 o f ref. [ 1 ], is the yield e x p e c t e d fo~ a spur process in the bulk o f a m o r p h o u s ice. M a n y solids, w i t h O r e m o d e l p a r a m eters roughly i d e n t i c a l to t h o s e o f ice, give no Ps in the bulk (e.g. CC14) c o m p a r e d to 55% o r t h o - P s in the bulk o f ice, in full a g r e e m e n t with the spur model. T h e s e results c a n n o t be e x p l a i n e d by the O r e m o d e l .
Acknowledgement M a n y s t i m u l a t i n g discussions w i t h M. E l d r u p and F.M. J a c o b s e n are a c k n o w l e d g e d .
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References [ 1] M. Eldrup, A. Vehanen, P.J. Schultz and K.G. Lynn. Phys. Rev. B32 (1985) 7048. [2] M. Eldrup, A. Vehanen, P.J. Schultz and K.G. Lynn, Ph}s. Rev. Lett. 51 (1983) 2007. [3] J. van House, A. Rich and P.W. Zitzewilz. Phys. Rev. Left. 53 (1984) 953. [4] M. Eldrup. A. Vehanen, P.J. Schultz and K.G. Lynn. Phys. Rev. Lett. 53 (1984) 954. [5] O.E. Mogensen, in: Positron annihilation, eds. P.G. Coleman, S.C. Sharma and L.M. Diana (North-Holland, Amsterdam, 1982) p. 763. [6] O.E. Mogensen, N.J. Pedersen and F.M. Jacobsen, in: Proc. 7th Intern. Cone on Positron annihilation (Delhi, January 1985), eds. P.C. Jain, R.M. Singru and K.P. Gopinathan (World Scientific, Singapore, 1985) p. 205. [7] J.M. Warman, M.P. DeHaas and J.B. Verberne, J. l)hys. Chem. 84 (1980) 1240. [8] A. Mozumder. Adv. Radiat. Chem. I (1969) 1. [9] K. Funabashi, Adv. Radiat. Chem. 4 (1974) 103. [10] J.W. Hunt, Adv. Radiat. Chem. 5 (1976) 185. [II]J.M. Wiesenfeld and E.P. Ippen, Chem. Phys. Lett. 73 (1980) 47. [12] W. Brandt, private communication. [ 13] M. Eldrup, O.E. Mogensen and J.H. Bilgram. J. Glaciol. 2 I (85) (1978) 101. [ 14] M. Eldrup, O.E. Mogensen and L. Kevan. Chem. Phys. Lett. 10(1971) 379. [15] O.A. Anisimov and Yu. N. Molin, Khim. Vys. Energ. 9 (1975) 376 [High Energ. Chem. 9 (1976) 331].