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On the anomalous magnetic quenching of positronium in solutions of nitrobenzene in hexane
Volume163,number2,3
CHEMICAL PHYSICS LETTERS
10 November 1989
ON THE ANOMALOUS MAGNETIC QUENCHING OF POSITRONIUM IN SOLUTIONS OF NITROBENZENE IN HEXANE O.E. MOGENSEN Ugiebakken28, DK-4040Jyllinge, Denmark Received 4 July 1989: in final form 22 August 1989
A new explanation is proposed of the anomalous magnetic quenching ofpositronium found by others for solutions of nitrobenzene in hexane, but not benzene and cyclohexane. It is assumed that particular “swollen”-Ps-nitrobenzene states occurring during attachmen&detachment of Ps on nitrobenzene at higher temperatures, give the effect. Depending on temperature and acceptor structure, the effect is expected for certainother solutionsof electronacceptors. The effect is related to a similar effect in solid naphthalene.
1. Introduction Positronium (Ps) is the bound state of a positron and an electron. An external magnetic field mixes the para positronium (p-Ps) and the ortho positronium (o-Ps) (m = 0) states of Ps in vacuum, resulting in a reduction ofthe o-Ps (m=O) lifetime, which can be measured by the positron lifetime technique. In condensed matter the magnetic quenching of Ps may be somewhat stronger (swelling of Ps), or, in rare cases, weaker (compression of Ps), than in vacuum. Recently Rochanakij and Schrader [ I] discovered an anomalous magnetic quenching of o-Ps (m=O) in hexane solutions of nitrobenzene, but not in nitrobenzene/benzene, nitrobenzene/cyclohexane, carbon tetrachloride/hexane, and biphenyl/hexane solutions, and not in the pure solvents. The weighted average, TV, of the o-Ps (m = 0 ) and o-Ps (m = I 1) lifetimes could be measured. For example, for a 0.1 M nitrobenzene/hexane solution 753decreased from 2.6 to 2.33 ns for a magnetic field increase from 0 to 2 kG, while the theoretically expected decrease in 73 would be only about 0.02-0.04 ns. At higher magnetic fields the difference between the expected and measured 7, values decreased, and at 15 kG it was only just detectable. The anomalous magnetic quenching was found also for a 0.05 M, but it was not significant for a 0.01 M, nitrobenzene/ hexane solution. A similar strong effect had not been found
for any liquid before; however, only a few organic solutions have been studied. The anomalous effect has been reproduced by Billard et al. [ 2,4] in both lifetime and Doppler broadening of the annihilation radiation measurements. Three attempts to explain the anomalous magnetic quenching of Ps in nitrobenzene/hexane have been published [2-51. However, these attempts do not explain why, in particular, the hexane,but not the benzene and cyclohexane, solutions of nitrobenzene give the anomalous effect. The purpose of the present paper is to propose another explanation of the anomalous magnetic field effect. It will be shown that the proposed model explains why only the nitrobenzenejhexane solution gives the effect. Furthermore, the model predicts that the effect is present in some other solute/solvent combinations as well, and that it depends on the temperature.
2. Positronium-molecule
reactions
The proposed explanation of the anomalous magnetic quenching of Ps is based on the available knowledge of the reaction of Ps with diamagnetic and paramagnetic molecules in organic liquids [ 6-9 1, which, therefore, must be discussed in some detail. Positronium reacts with molecules having an elec-
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tron affinity (EA) above 0.5-l eV. Experiments show that the reaction is influenced by other properties of the molecule and solvent than the EA value of the molecule, in particular by the surface tension of the solvent. The reactions with diamagnetic and paramagnetic molecules seem to be determined by the same properties. although only paramagnetic molecules can cause spin conversion of Ps. The rate constant of the Ps-molecule reaction, k, has been measured as a function of temperature for about 30 combinations of reactant and solvent [ 68 1. In most cases k increases with temperature at low temperatures, reaches a maximum, &, at t,, and then decreases. A plot of log k versus T- ’is roughly linear below I,, corresponding to an activation energy which always seems to be lower than that expected from the temperature dependence of the viscosity, in some cases a factor of two lower. At temperatures above t, logk versus T-i varies roughly according to a negative activation energy [ 71. These results have been interpreted as due to formation of a complex of Ps and the molcculcs, (Ps, M ), according to the equation Ps + M = (Ps, M) At low temperatures the complex is assumed to be stable, and the variation of k with temperature is assumed to be caused mainly by the temperature dependence of the diffusion and kinetic processes of the forward reaction, as is normally found for many chemical reactions. At temperatures above t, the complex is assumed to be increasingly unstable for increasing temperature, because it splits up by thermal activation. As shown below, this interpretation is somewhat too simple. Nitrobenzene solutions show this typical behaviour for seven solvents [ 6-8 1, including hexane [ 8 1. Nitrobenzene/hexane gives a maximum in k, k,=l.5x10’0 M-l SK’, at t,=--50°C. The benzene solutions have only been studied above room temperature, t, [ 61. However, nitrobenzene/benzene shows a typical decrease of k with temperature above t, and a clear saturation of k at a value close to the diffusion-controlled rate constant at t,, indicating that t, is close to t,. Nitrobenzene/cyclohexane solutions have only been studied at t,, where k= 1.2~ IO” M-’ s-’ has been measured [ 11. Two facts indicate strongly that t, is close to t, for cyclohexane. First, the measured k value at t, is only a little smaller than the diffusion-controlled rate con146
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stant, which is expected to be about (2-3) X 10” M-’ SK’. Such high rate constants have been found only at temperatures close to t, for hydrocarbons. Secondly, a strong correlation oft, with the surface tension for hydrocarbons has been found [ 9,101 (see below). This correlation Indicates that t, is expected to be close to t, for nitrobenzene/cyclohexane solutions. Carbon tetrachloride and biphenyl do not react with Ps.
3. The proposed explanation We can now formulate the proposed new explanation of the anomalous magnetic quenching of Ps. The basic assumption is the following: The anomalous magnetic quenching effect is caused by magnetic quenching of particular Ps-molecule states which exist in a temperature interval above t,, where the energy of free Ps is comparable to, or somewhat lower than, the energies of the intermediate, weakly bound Ps molecule, and the fully bound positron-electron-molecule, states. Roughly speaking, these states are populated in the temperature interval where Ps is beginning to be squeezed out of the free Ps bubble and onto the acceptor, if the temperature is lowered and the surface tension thereby increased. The main idea is not that the effect is found if Ps is squeezed, as it is in pure hexane too. It must be not far from being squeezed out of a Ps-like state into a state which is not Ps like.
4. Discussion Let us now discuss the physics of the Ps interaction with a molecule which is a strong electron acceptor, i.e. which has a positive electron affinity, EA>0.5-1 eV. The structures of the states of the positron-electron-molecule during or after Ps attachment and during Ps detachment have been studied very little experimentally. Hence, it is a difficult discussion. At first, it must be realized that the Ps-molecule complex, on which all previous discussions of the Psmolecule reaction have been based, should be called a Ps state only in certain particular cases. Further-
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more, several positron-electron-molecule compounds are probably playing a role (seebelow), while only one Ps complex was assumed in previous discussions in the literature. This is partly a nomenclature problem. Except for the mentioned Ps-molecule complexes, the compounds of the positron are only designated Ps states in the literature if certain attributes are experimentally measured. Normally, annihilation from positron compounds called Ps states results in a long lifetime due to o-Ps pick off annihilation and a narrow peak in the angular correlation spectrum due to p-Ps intrinsic annihilation. In certain casts (c.g.. in some ionic crystals and selected solid hydrocarbons) only one of these attributes is detected. In most cases, annihilation of the compounds formed as a result of a Ps-molecule reaction gives neither a long lifetime nor a narrow peak. This situation is also found if the annihilation occurs from a positron bound state with Cl-, Br-, and I-, for which the designation Ps state is not used [ 1 I]. Hence, if the two attributes are not measured, the name “Ps complex” is misleading, and should be avoided. On the other hand, the particular Ps-molecule compounds which give the anomalous magnetic quenching effect according to our proposed explanation, can be designated “Ps complexes”. Their annihilation must result in a long lifetime due to o-Ps decay, if the explanation is valid. Can we reasonably assume that such states exist? To answer this question it is necessary to discuss the structureofthe Ps states in some detail. The state of free Ps in most liquids is the so called Ps bubble, which seems to be fairly well described by the assumption that Ps is bound in a square-well potential caused by the Ps-molecule repulsion [ 12,131. An equilibrium between the inward pressure on the bubble wall due to the surface tension and the outward pressure caused by the zero-point motion of Ps and the Ps-molecule repulsion is assumed. We cannot discuss the model here, but we can illustrate the principles by stating the results of a calculation in which an infinitely deep square well is used, and the pressure-volume energy disregarded [ 121. The energy is given by the sum of the zero-point energy and the surface energy, E= n2fz2/4mr2f4nr20,
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CHEMICALPHYSICSLETTERS
(1)
where m is the electron mass, r the radius of the well (~0.5 nm in hexane), and CJthe surface tension. Minimizing the energy determines the radius r=(nfi2/16mo)“4, which with ( 1) gives the energy versus surface tension: E=0.24288d’*
eV, for uindynfcm,
(2)
Other models, e.g., a harmonic potential with extra assumptions, give another constant in (2). The surface tension decreases linearly with rising temperature in most liquids [ 141. For example, in hexane a=20.44-0.1022t dyn/cm (t in “C), which gives E= 1.35, 1.23, 1.10, 1.04, and 0.85 eV at t= - 100, -50, 0, 20, and 80°C respectively. Hence, by decreasing the temperature from 80 to - 100”C we can increase the energy of the Ps bubble state by 0.5 eV in hexane. If Ps encounters an acceptor which provides a lower energy state for Ps, it will normally react with the acceptor. For nitrobenzene the energy of “bound Ps” (see below) is comparable to that of the bubble at roughly -5O”C, where the rate constant begins to decrease for increasing temperature. For benzene this happens at roughly 2O”C, where according to (2) E= 1.31 eV. In cyclohexane E= 1.23 eV at 20°C. Hence, the Ps bubble energy at t, is roughly the same in the three solvents, as expected. Of course, other effects than that of the surface tension play minor roles, too, such as details in the structure of the acceptors and their participation in secondary bonding, e.g., hydrogen bonding and CT complexes. Actually, only that part of the surface tension which is not due to hydrogen bonds seems 10 determine the size of the l’s bubble [ 151. Spin conversion of Ps is expected to occur if Ps and the molecule encounter, stay together, and split up again, i.e. above t,, in agreement with experiments. This is a new interpretation of the temperature effect in Ps reactions with molecules and ions in liquids. A preliminary test indicates that it is in good agreement with the experimental data. Other models are found in refs. [ 6-91. The new explanation of the anomalous magnetic quenching follows naturally from the above interpretation. The effect is probably found at temperatures where Ps is just energetically stable, i.e. where it is not far from being transferred into a non-Ps state. Only one other similar case has been reported, namely that of Ps in solid naphthalene [ 16 1.In most 147
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molecular crystals Ps cannot make a large bubble, and it is not delocalized either. It makes a smaller cage (hole ) by self-trapping, from where it is normally transferred into vacancies if their concentration becomes high at higher temperatures. In naphthalene, polyphenyls, etc., Ps cannot be formed at low temperatures, but the cage state is formed over about 60°C ifthe temperature is increased. In naphthalene this happens between -80 and -20°C [ 171. Hence, at t,.Ps is just stable, as in the nitrobenzene/hexane solutions at high enough concentration. The magnetic quenching in naphthalene at t, is the strongest ever measured. It can be fitted by a model where roughly speaking two “swollen” Ps, having a ratio of the electron density at the positron to that in vacuum, LY =0.25 and 0.13, are assumed. The nitrobenzene/hexane results can be fitted roughly if we assume that about 15% of Ps is “swollen” Ps with cry 0.1, and 85% is “normal” Ps with acx 0.9. These results seem to show that only about 15% of Ps is in a strongly “swollen” state in nitrobenzene/ hexane at 0.1 M. This appears to be a reasonable result, as we expect that many states of the Ps-molecule-solvent system, connecting the free Ps bubble and the fully non-Ps bound state, can exist and be populated at t,, and only some of these states may give long o-Ps lifetimes and be “swollen”, too. Of course, many of these states will have an associated “bubble” which is smaller than that of free Ps. In the nitrobenzene case the positron is expected to bc attracted to the negative NO* group and the electron to the benzene ring, resulting in part of the swelling of Ps. The structure of the stable pasitron-electron-molecule state at low temperatures, which give no Ps attributes. probably depends on the molecule, e.g., its polarity and available lowest unoccupied electron orbitals. It is important to realize that the positron (or electron) might even be totally outside the molecule, and the electron (positron) bound on the molecule, in such a state. The available experimental facts require that the positron and electron leave the state together to form Ps on thermal excitation at higher temperatures, but this is assured by the strong Coulomb attraction in nonpolar liquids with low dieiectric constants. The structure of the “swollen”-Psmolecule-solvent states is expected to depend strongly on the properties of the molecule, and hence 148
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the anomalous magnetic quenching effect is ex-
pected to depend on the molecule. too. Angular correlation (AC) spectra are given by the Fourier transform of the electron-positron overlap at annihilation. They are therefore “fingerprints” of the state at annihilation, and, except for fortuitous resemblance in shapes, different AC spectra correspond to different states. Concerning the new cxplanation, the following AC results [ IO,18 ] measured at t, are of interest. High-concentration (z 1 M) benzene solutions of nitrobenzene, S02, dibenzoyl, quinone, and DPPH give identical AC spectra. if the spectra are properly corrected for small components due to Ps annihilation. These spectra are very different from high-concentration nitrobenzene/hexane, nitrobenzene/heptane, nitrobenzene/methanol, and the pure nitrobenzene AC spectra. Thts seems to indicate that the I M nitrobenzene/benzene AC spectrum is a property of the solvent only, and it has been ascribed to the annihilation of the “free” (i.e. solvated) positron state [ 181. However, nitrobenzene/ heptane gives an AC spectrum very different from that of SOz/heptane at high concentrations [lo], which indicates that for heptane, opposite to benzene, the AC spectra of these two solutes depend on the solute molecules. Nitrobcnzene/ heptane and nitrobenzeneihexane solutions give very similar AC spectra, apart from a somewhat higher rate constant in heptane, which has t, z - 17 “C [ 81. This indicates that benzene and heptane, and hence hexane, as solvents for nitrobenzene behave differently with respect to annihilation of Ps and the positron at t,, in agreement with the proposed explanation. Furthermore, a 2 M nitrobenzene/methanol solution gives an AC spectrum which is close to that of pure nitrobenzene [ IO], a result which seems to indicate that the positron annihilation is only a property of the nitrobenzene molecule at high concentrations in methanol In summary, these, and other [ lo], AC spectra seem to indicate that the result of the Ps reaction with electron acceptors depends on the solute and solvent in a complicated way, as expected from the proposed model. Another complicating effect is that the product of the Ps-electron acceptor reaction is expected to form also by some of the spur processes leading to Ps inhibition. Ps is formed by a reaction of the positron
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with an excess electron in the positron spur, which is the terminal spur of the positron track formed when the positron loses the last 100-500 eV of its kinetic energy [ 151. Ps formation competes with the other spur reactions, in particular with the electron-positive-ion recombination. Added electron acceptors react with the excess electrons, and provided the anion formed has a long enough lifetime and the electron is bound in a deep enough trap, the positron will have a good chance to react with the anion before it annihilates (~400 ps), resulting in a state equal to that formed by the Ps-electron-acceptor reaction. It is also likely that the positron is trapped on the acceptor and that the electron reacts with the positronmolecule state. These reactions lead to inhibition of Ps formation. Except for very small molecules, such as SO, and 02, all electron acceptors reacting with Ps seem to inhibit Ps formation. In particular, these processes probably occur in nitrobenzene solutions. At temperatures where the product formed is unstable and breaks up due to thermal activation, the states formed as a result of Ps inhibition break up, too. The particular states assumed to give the anomalous magnetic quenching will, of course, also be present in the cases where they are results of Ps inhibition processes. Furthermore, Ps forms “collision cages” (i.e. something like a Ps bubble with the electron acceptor on the ill-defined surface) with the acceptors at the diffusion-controlled rate, which may be much bigger than the actually measured rate of reaction of o-Ps at temperatures where the positron-electronmolecule slate is unstable. If these “collision cages” have reasonably long Lifetimes they may perhaps also contribute “ swollen” Ps states giving anomalous magnetic quenching. For nitrobenzcne/hexane k= 1.6 x I O8M-’ s- ’at t,, which is roughly ten times lower than lc,. Consequently, the details of the anomalous magnetic quenching might depend on the properties of the solutions in a complicated way. The proposed explanation predicts new measurable results, as mentioned above. Variation of the temperature will probably strongly influence the anomalous magnetic quenching of Ps. It will probably disappear on lowering the temperature below I,. However, it is difficult to predict in detail the variation with temperature, in particular at higher
10November 1989
temperatures. It is predicted that the effect depends mainly on t,. For example, in heptane the effect is shifted roughly 30°C upward in temperature compared to hexane. Pentane solutions have a lower t, than hexane, and for tetradecane t,,, is probably close to t,, as for cyclohexane. The benzene case might give surprises, as benzene interacts more strongly with the acceptors. As discussed above, the effect is expected to depend on the structure of the electron acceptor. It could be conjectured that polar acceptors, such as nitrobenzene, will show the strongest effects. The effect of changing the solute should be correlated to the structure of the solutes, as is done for electron affinities [ 191. In general, the effect can probably be expected, whenever Ps is close to being unstable, as shown by the naphthalene case, and its variation, if parameters, such as temperature, pressure, electric field, etc., are changed, and should be explainable in terms of the dependence of the stability of Ps on these parameters.
5. Conclusion The anomalous magnetic quenching of Ps in nitrobenzene/hexane solutions can be reasonably well explained as caused by magnetic quenching of particular “swollen”-Ps-nitrobenzene states occurring at temperatures where the positron-electron-nitrobenzene states are unstable and break up frequently due to thermal activation. This effect is expected for particular electron-acceptor solutions. The measured differences between solutions of nitrobenzene in hexane and in benzene and cyclohexane are well explained. Furthermore, it is easy to understand in terms of the model why CC& and biphenyl solutions do not show the effect, as these solutes arc unrcactive towards Ps. The effect has been related to a similar effect in solid naphthalene. New AC spectra support the model. It must be emphasized that the new explanation is only a proposal. It explains the available data but might be in disagreement with future results. On the other hand, it seems to be a useful hypothesis for the selection of new experiments.
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References
[ I] S. Rochanakij and D.M. Schrader, Radiat. Phys. Chem. 32 (1988) 557. [2] I. Billard. J.Ch. Abbe and Cl. Duplltre, in: Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 19g8, eds. L. Dorikens-Vanpraet, M. Dorikens and D. Segers (World Scientific, Singapore, 1988) p. 540. [ 31 L. Dorikcns-Vanpraet, M. Dorikens and D. Segers, eds., Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 1988 (World Scientific, Smgapore, 1988). [4] 1. Billard, J.Ch. Abbe and G. DuplPtre, J. Chem. Phys. 91(19X9) 1579.
] D.A. Diehl and D.M. Schrader, in: Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 1988 (World Scientific, Singapore, 1988 ) p. 537. ‘1V.I. Goldanskii and V.P. Shantarovich, Appl. Phys. 3 (1974) 335.
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[7] W.J. Madia, A.L. Nichols and H.J. Ache, J. Am. Chem. Sot. 97 (1975) 5041. [8] V.M. Byakov, V.I. Grafutin, O.V. Koldaeva and E.V. Minaichev, J. Phys. Chem. 84 (1980) 1864. [9] Y. Kobayashi and Y. Ujihira, I. Phys. Chem. 85 (1981) 2455. [IO] O.E. Mogensen, unpublished. [ II] O.E. Mogensen and N.J. Pedersen, Radiat. Phys. Chem. 28 ( 1986) 33. [ 121 R.A. Ferrell, Phys. Rev. 108 (1957) 167. [ 131 A.P. Buchikhin, V.I. Goldanskii and V.P. Shantarovieh. JETP Letters 13 (1989) 444. [ 141 J.J. Jasper, J. Phys. Chem. Ref. Data 1 (1972) 841. [ 151 O.E. Mogensen, Electrochim. Acta 33 (1988) 1203. [ 161 A. Bisi, G. Consolati and L. Zappa, Hyperfine Interaction 36 ( 1987) 29. [ 171 T. Goworek, C. Rybka, R. Wasiewicz and J. Wawryszczuk, Phys. Stat. Sol. B 113 ( 1982) K9. [ 181 O.E. Mogensen, in: Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 1988, eds. L. Dorikens-Vanpraet, M. Dorikens and D. Segers (World Scientific, Singapore, 1989) p. 552. [ 191 P. Kebarie and S. Chowdhury, Chcm. Rev. 87 (1987) 5 13.
CHEMICAL PHYSICS LETTERS
10 November 1989
ON THE ANOMALOUS MAGNETIC QUENCHING OF POSITRONIUM IN SOLUTIONS OF NITROBENZENE IN HEXANE O.E. MOGENSEN Ugiebakken28, DK-4040Jyllinge, Denmark Received 4 July 1989: in final form 22 August 1989
A new explanation is proposed of the anomalous magnetic quenching ofpositronium found by others for solutions of nitrobenzene in hexane, but not benzene and cyclohexane. It is assumed that particular “swollen”-Ps-nitrobenzene states occurring during attachmen&detachment of Ps on nitrobenzene at higher temperatures, give the effect. Depending on temperature and acceptor structure, the effect is expected for certainother solutionsof electronacceptors. The effect is related to a similar effect in solid naphthalene.
1. Introduction Positronium (Ps) is the bound state of a positron and an electron. An external magnetic field mixes the para positronium (p-Ps) and the ortho positronium (o-Ps) (m = 0) states of Ps in vacuum, resulting in a reduction ofthe o-Ps (m=O) lifetime, which can be measured by the positron lifetime technique. In condensed matter the magnetic quenching of Ps may be somewhat stronger (swelling of Ps), or, in rare cases, weaker (compression of Ps), than in vacuum. Recently Rochanakij and Schrader [ I] discovered an anomalous magnetic quenching of o-Ps (m=O) in hexane solutions of nitrobenzene, but not in nitrobenzene/benzene, nitrobenzene/cyclohexane, carbon tetrachloride/hexane, and biphenyl/hexane solutions, and not in the pure solvents. The weighted average, TV, of the o-Ps (m = 0 ) and o-Ps (m = I 1) lifetimes could be measured. For example, for a 0.1 M nitrobenzene/hexane solution 753decreased from 2.6 to 2.33 ns for a magnetic field increase from 0 to 2 kG, while the theoretically expected decrease in 73 would be only about 0.02-0.04 ns. At higher magnetic fields the difference between the expected and measured 7, values decreased, and at 15 kG it was only just detectable. The anomalous magnetic quenching was found also for a 0.05 M, but it was not significant for a 0.01 M, nitrobenzene/ hexane solution. A similar strong effect had not been found
for any liquid before; however, only a few organic solutions have been studied. The anomalous effect has been reproduced by Billard et al. [ 2,4] in both lifetime and Doppler broadening of the annihilation radiation measurements. Three attempts to explain the anomalous magnetic quenching of Ps in nitrobenzene/hexane have been published [2-51. However, these attempts do not explain why, in particular, the hexane,but not the benzene and cyclohexane, solutions of nitrobenzene give the anomalous effect. The purpose of the present paper is to propose another explanation of the anomalous magnetic field effect. It will be shown that the proposed model explains why only the nitrobenzenejhexane solution gives the effect. Furthermore, the model predicts that the effect is present in some other solute/solvent combinations as well, and that it depends on the temperature.
2. Positronium-molecule
reactions
The proposed explanation of the anomalous magnetic quenching of Ps is based on the available knowledge of the reaction of Ps with diamagnetic and paramagnetic molecules in organic liquids [ 6-9 1, which, therefore, must be discussed in some detail. Positronium reacts with molecules having an elec-
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tron affinity (EA) above 0.5-l eV. Experiments show that the reaction is influenced by other properties of the molecule and solvent than the EA value of the molecule, in particular by the surface tension of the solvent. The reactions with diamagnetic and paramagnetic molecules seem to be determined by the same properties. although only paramagnetic molecules can cause spin conversion of Ps. The rate constant of the Ps-molecule reaction, k, has been measured as a function of temperature for about 30 combinations of reactant and solvent [ 68 1. In most cases k increases with temperature at low temperatures, reaches a maximum, &, at t,, and then decreases. A plot of log k versus T- ’is roughly linear below I,, corresponding to an activation energy which always seems to be lower than that expected from the temperature dependence of the viscosity, in some cases a factor of two lower. At temperatures above t, logk versus T-i varies roughly according to a negative activation energy [ 71. These results have been interpreted as due to formation of a complex of Ps and the molcculcs, (Ps, M ), according to the equation Ps + M = (Ps, M) At low temperatures the complex is assumed to be stable, and the variation of k with temperature is assumed to be caused mainly by the temperature dependence of the diffusion and kinetic processes of the forward reaction, as is normally found for many chemical reactions. At temperatures above t, the complex is assumed to be increasingly unstable for increasing temperature, because it splits up by thermal activation. As shown below, this interpretation is somewhat too simple. Nitrobenzene solutions show this typical behaviour for seven solvents [ 6-8 1, including hexane [ 8 1. Nitrobenzene/hexane gives a maximum in k, k,=l.5x10’0 M-l SK’, at t,=--50°C. The benzene solutions have only been studied above room temperature, t, [ 61. However, nitrobenzene/benzene shows a typical decrease of k with temperature above t, and a clear saturation of k at a value close to the diffusion-controlled rate constant at t,, indicating that t, is close to t,. Nitrobenzene/cyclohexane solutions have only been studied at t,, where k= 1.2~ IO” M-’ s-’ has been measured [ 11. Two facts indicate strongly that t, is close to t, for cyclohexane. First, the measured k value at t, is only a little smaller than the diffusion-controlled rate con146
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stant, which is expected to be about (2-3) X 10” M-’ SK’. Such high rate constants have been found only at temperatures close to t, for hydrocarbons. Secondly, a strong correlation oft, with the surface tension for hydrocarbons has been found [ 9,101 (see below). This correlation Indicates that t, is expected to be close to t, for nitrobenzene/cyclohexane solutions. Carbon tetrachloride and biphenyl do not react with Ps.
3. The proposed explanation We can now formulate the proposed new explanation of the anomalous magnetic quenching of Ps. The basic assumption is the following: The anomalous magnetic quenching effect is caused by magnetic quenching of particular Ps-molecule states which exist in a temperature interval above t,, where the energy of free Ps is comparable to, or somewhat lower than, the energies of the intermediate, weakly bound Ps molecule, and the fully bound positron-electron-molecule, states. Roughly speaking, these states are populated in the temperature interval where Ps is beginning to be squeezed out of the free Ps bubble and onto the acceptor, if the temperature is lowered and the surface tension thereby increased. The main idea is not that the effect is found if Ps is squeezed, as it is in pure hexane too. It must be not far from being squeezed out of a Ps-like state into a state which is not Ps like.
4. Discussion Let us now discuss the physics of the Ps interaction with a molecule which is a strong electron acceptor, i.e. which has a positive electron affinity, EA>0.5-1 eV. The structures of the states of the positron-electron-molecule during or after Ps attachment and during Ps detachment have been studied very little experimentally. Hence, it is a difficult discussion. At first, it must be realized that the Ps-molecule complex, on which all previous discussions of the Psmolecule reaction have been based, should be called a Ps state only in certain particular cases. Further-
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more, several positron-electron-molecule compounds are probably playing a role (seebelow), while only one Ps complex was assumed in previous discussions in the literature. This is partly a nomenclature problem. Except for the mentioned Ps-molecule complexes, the compounds of the positron are only designated Ps states in the literature if certain attributes are experimentally measured. Normally, annihilation from positron compounds called Ps states results in a long lifetime due to o-Ps pick off annihilation and a narrow peak in the angular correlation spectrum due to p-Ps intrinsic annihilation. In certain casts (c.g.. in some ionic crystals and selected solid hydrocarbons) only one of these attributes is detected. In most cases, annihilation of the compounds formed as a result of a Ps-molecule reaction gives neither a long lifetime nor a narrow peak. This situation is also found if the annihilation occurs from a positron bound state with Cl-, Br-, and I-, for which the designation Ps state is not used [ 1 I]. Hence, if the two attributes are not measured, the name “Ps complex” is misleading, and should be avoided. On the other hand, the particular Ps-molecule compounds which give the anomalous magnetic quenching effect according to our proposed explanation, can be designated “Ps complexes”. Their annihilation must result in a long lifetime due to o-Ps decay, if the explanation is valid. Can we reasonably assume that such states exist? To answer this question it is necessary to discuss the structureofthe Ps states in some detail. The state of free Ps in most liquids is the so called Ps bubble, which seems to be fairly well described by the assumption that Ps is bound in a square-well potential caused by the Ps-molecule repulsion [ 12,131. An equilibrium between the inward pressure on the bubble wall due to the surface tension and the outward pressure caused by the zero-point motion of Ps and the Ps-molecule repulsion is assumed. We cannot discuss the model here, but we can illustrate the principles by stating the results of a calculation in which an infinitely deep square well is used, and the pressure-volume energy disregarded [ 121. The energy is given by the sum of the zero-point energy and the surface energy, E= n2fz2/4mr2f4nr20,
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(1)
where m is the electron mass, r the radius of the well (~0.5 nm in hexane), and CJthe surface tension. Minimizing the energy determines the radius r=(nfi2/16mo)“4, which with ( 1) gives the energy versus surface tension: E=0.24288d’*
eV, for uindynfcm,
(2)
Other models, e.g., a harmonic potential with extra assumptions, give another constant in (2). The surface tension decreases linearly with rising temperature in most liquids [ 141. For example, in hexane a=20.44-0.1022t dyn/cm (t in “C), which gives E= 1.35, 1.23, 1.10, 1.04, and 0.85 eV at t= - 100, -50, 0, 20, and 80°C respectively. Hence, by decreasing the temperature from 80 to - 100”C we can increase the energy of the Ps bubble state by 0.5 eV in hexane. If Ps encounters an acceptor which provides a lower energy state for Ps, it will normally react with the acceptor. For nitrobenzene the energy of “bound Ps” (see below) is comparable to that of the bubble at roughly -5O”C, where the rate constant begins to decrease for increasing temperature. For benzene this happens at roughly 2O”C, where according to (2) E= 1.31 eV. In cyclohexane E= 1.23 eV at 20°C. Hence, the Ps bubble energy at t, is roughly the same in the three solvents, as expected. Of course, other effects than that of the surface tension play minor roles, too, such as details in the structure of the acceptors and their participation in secondary bonding, e.g., hydrogen bonding and CT complexes. Actually, only that part of the surface tension which is not due to hydrogen bonds seems 10 determine the size of the l’s bubble [ 151. Spin conversion of Ps is expected to occur if Ps and the molecule encounter, stay together, and split up again, i.e. above t,, in agreement with experiments. This is a new interpretation of the temperature effect in Ps reactions with molecules and ions in liquids. A preliminary test indicates that it is in good agreement with the experimental data. Other models are found in refs. [ 6-91. The new explanation of the anomalous magnetic quenching follows naturally from the above interpretation. The effect is probably found at temperatures where Ps is just energetically stable, i.e. where it is not far from being transferred into a non-Ps state. Only one other similar case has been reported, namely that of Ps in solid naphthalene [ 16 1.In most 147
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molecular crystals Ps cannot make a large bubble, and it is not delocalized either. It makes a smaller cage (hole ) by self-trapping, from where it is normally transferred into vacancies if their concentration becomes high at higher temperatures. In naphthalene, polyphenyls, etc., Ps cannot be formed at low temperatures, but the cage state is formed over about 60°C ifthe temperature is increased. In naphthalene this happens between -80 and -20°C [ 171. Hence, at t,.Ps is just stable, as in the nitrobenzene/hexane solutions at high enough concentration. The magnetic quenching in naphthalene at t, is the strongest ever measured. It can be fitted by a model where roughly speaking two “swollen” Ps, having a ratio of the electron density at the positron to that in vacuum, LY =0.25 and 0.13, are assumed. The nitrobenzene/hexane results can be fitted roughly if we assume that about 15% of Ps is “swollen” Ps with cry 0.1, and 85% is “normal” Ps with acx 0.9. These results seem to show that only about 15% of Ps is in a strongly “swollen” state in nitrobenzene/ hexane at 0.1 M. This appears to be a reasonable result, as we expect that many states of the Ps-molecule-solvent system, connecting the free Ps bubble and the fully non-Ps bound state, can exist and be populated at t,, and only some of these states may give long o-Ps lifetimes and be “swollen”, too. Of course, many of these states will have an associated “bubble” which is smaller than that of free Ps. In the nitrobenzene case the positron is expected to bc attracted to the negative NO* group and the electron to the benzene ring, resulting in part of the swelling of Ps. The structure of the stable pasitron-electron-molecule state at low temperatures, which give no Ps attributes. probably depends on the molecule, e.g., its polarity and available lowest unoccupied electron orbitals. It is important to realize that the positron (or electron) might even be totally outside the molecule, and the electron (positron) bound on the molecule, in such a state. The available experimental facts require that the positron and electron leave the state together to form Ps on thermal excitation at higher temperatures, but this is assured by the strong Coulomb attraction in nonpolar liquids with low dieiectric constants. The structure of the “swollen”-Psmolecule-solvent states is expected to depend strongly on the properties of the molecule, and hence 148
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the anomalous magnetic quenching effect is ex-
pected to depend on the molecule. too. Angular correlation (AC) spectra are given by the Fourier transform of the electron-positron overlap at annihilation. They are therefore “fingerprints” of the state at annihilation, and, except for fortuitous resemblance in shapes, different AC spectra correspond to different states. Concerning the new cxplanation, the following AC results [ IO,18 ] measured at t, are of interest. High-concentration (z 1 M) benzene solutions of nitrobenzene, S02, dibenzoyl, quinone, and DPPH give identical AC spectra. if the spectra are properly corrected for small components due to Ps annihilation. These spectra are very different from high-concentration nitrobenzene/hexane, nitrobenzene/heptane, nitrobenzene/methanol, and the pure nitrobenzene AC spectra. Thts seems to indicate that the I M nitrobenzene/benzene AC spectrum is a property of the solvent only, and it has been ascribed to the annihilation of the “free” (i.e. solvated) positron state [ 181. However, nitrobenzene/ heptane gives an AC spectrum very different from that of SOz/heptane at high concentrations [lo], which indicates that for heptane, opposite to benzene, the AC spectra of these two solutes depend on the solute molecules. Nitrobcnzene/ heptane and nitrobenzeneihexane solutions give very similar AC spectra, apart from a somewhat higher rate constant in heptane, which has t, z - 17 “C [ 81. This indicates that benzene and heptane, and hence hexane, as solvents for nitrobenzene behave differently with respect to annihilation of Ps and the positron at t,, in agreement with the proposed explanation. Furthermore, a 2 M nitrobenzene/methanol solution gives an AC spectrum which is close to that of pure nitrobenzene [ IO], a result which seems to indicate that the positron annihilation is only a property of the nitrobenzene molecule at high concentrations in methanol In summary, these, and other [ lo], AC spectra seem to indicate that the result of the Ps reaction with electron acceptors depends on the solute and solvent in a complicated way, as expected from the proposed model. Another complicating effect is that the product of the Ps-electron acceptor reaction is expected to form also by some of the spur processes leading to Ps inhibition. Ps is formed by a reaction of the positron
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with an excess electron in the positron spur, which is the terminal spur of the positron track formed when the positron loses the last 100-500 eV of its kinetic energy [ 151. Ps formation competes with the other spur reactions, in particular with the electron-positive-ion recombination. Added electron acceptors react with the excess electrons, and provided the anion formed has a long enough lifetime and the electron is bound in a deep enough trap, the positron will have a good chance to react with the anion before it annihilates (~400 ps), resulting in a state equal to that formed by the Ps-electron-acceptor reaction. It is also likely that the positron is trapped on the acceptor and that the electron reacts with the positronmolecule state. These reactions lead to inhibition of Ps formation. Except for very small molecules, such as SO, and 02, all electron acceptors reacting with Ps seem to inhibit Ps formation. In particular, these processes probably occur in nitrobenzene solutions. At temperatures where the product formed is unstable and breaks up due to thermal activation, the states formed as a result of Ps inhibition break up, too. The particular states assumed to give the anomalous magnetic quenching will, of course, also be present in the cases where they are results of Ps inhibition processes. Furthermore, Ps forms “collision cages” (i.e. something like a Ps bubble with the electron acceptor on the ill-defined surface) with the acceptors at the diffusion-controlled rate, which may be much bigger than the actually measured rate of reaction of o-Ps at temperatures where the positron-electronmolecule slate is unstable. If these “collision cages” have reasonably long Lifetimes they may perhaps also contribute “ swollen” Ps states giving anomalous magnetic quenching. For nitrobenzcne/hexane k= 1.6 x I O8M-’ s- ’at t,, which is roughly ten times lower than lc,. Consequently, the details of the anomalous magnetic quenching might depend on the properties of the solutions in a complicated way. The proposed explanation predicts new measurable results, as mentioned above. Variation of the temperature will probably strongly influence the anomalous magnetic quenching of Ps. It will probably disappear on lowering the temperature below I,. However, it is difficult to predict in detail the variation with temperature, in particular at higher
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temperatures. It is predicted that the effect depends mainly on t,. For example, in heptane the effect is shifted roughly 30°C upward in temperature compared to hexane. Pentane solutions have a lower t, than hexane, and for tetradecane t,,, is probably close to t,, as for cyclohexane. The benzene case might give surprises, as benzene interacts more strongly with the acceptors. As discussed above, the effect is expected to depend on the structure of the electron acceptor. It could be conjectured that polar acceptors, such as nitrobenzene, will show the strongest effects. The effect of changing the solute should be correlated to the structure of the solutes, as is done for electron affinities [ 191. In general, the effect can probably be expected, whenever Ps is close to being unstable, as shown by the naphthalene case, and its variation, if parameters, such as temperature, pressure, electric field, etc., are changed, and should be explainable in terms of the dependence of the stability of Ps on these parameters.
5. Conclusion The anomalous magnetic quenching of Ps in nitrobenzene/hexane solutions can be reasonably well explained as caused by magnetic quenching of particular “swollen”-Ps-nitrobenzene states occurring at temperatures where the positron-electron-nitrobenzene states are unstable and break up frequently due to thermal activation. This effect is expected for particular electron-acceptor solutions. The measured differences between solutions of nitrobenzene in hexane and in benzene and cyclohexane are well explained. Furthermore, it is easy to understand in terms of the model why CC& and biphenyl solutions do not show the effect, as these solutes arc unrcactive towards Ps. The effect has been related to a similar effect in solid naphthalene. New AC spectra support the model. It must be emphasized that the new explanation is only a proposal. It explains the available data but might be in disagreement with future results. On the other hand, it seems to be a useful hypothesis for the selection of new experiments.
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References
[ I] S. Rochanakij and D.M. Schrader, Radiat. Phys. Chem. 32 (1988) 557. [2] I. Billard. J.Ch. Abbe and Cl. Duplltre, in: Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 19g8, eds. L. Dorikens-Vanpraet, M. Dorikens and D. Segers (World Scientific, Singapore, 1988) p. 540. [ 31 L. Dorikcns-Vanpraet, M. Dorikens and D. Segers, eds., Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 1988 (World Scientific, Smgapore, 1988). [4] 1. Billard, J.Ch. Abbe and G. DuplPtre, J. Chem. Phys. 91(19X9) 1579.
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[7] W.J. Madia, A.L. Nichols and H.J. Ache, J. Am. Chem. Sot. 97 (1975) 5041. [8] V.M. Byakov, V.I. Grafutin, O.V. Koldaeva and E.V. Minaichev, J. Phys. Chem. 84 (1980) 1864. [9] Y. Kobayashi and Y. Ujihira, I. Phys. Chem. 85 (1981) 2455. [IO] O.E. Mogensen, unpublished. [ II] O.E. Mogensen and N.J. Pedersen, Radiat. Phys. Chem. 28 ( 1986) 33. [ 121 R.A. Ferrell, Phys. Rev. 108 (1957) 167. [ 131 A.P. Buchikhin, V.I. Goldanskii and V.P. Shantarovieh. JETP Letters 13 (1989) 444. [ 141 J.J. Jasper, J. Phys. Chem. Ref. Data 1 (1972) 841. [ 151 O.E. Mogensen, Electrochim. Acta 33 (1988) 1203. [ 161 A. Bisi, G. Consolati and L. Zappa, Hyperfine Interaction 36 ( 1987) 29. [ 171 T. Goworek, C. Rybka, R. Wasiewicz and J. Wawryszczuk, Phys. Stat. Sol. B 113 ( 1982) K9. [ 181 O.E. Mogensen, in: Positron Annihilation, Proceedings of the Eighth International Conference on Positron Annihilation, Gent, Belgium, 29 Aug. - 3 Sept. 1988, eds. L. Dorikens-Vanpraet, M. Dorikens and D. Segers (World Scientific, Singapore, 1989) p. 552. [ 191 P. Kebarie and S. Chowdhury, Chcm. Rev. 87 (1987) 5 13.