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Muonium formation in nonpolar liquids
Radiat. Phys. Chem. Vol. 28. No. 1, pp. 85-89, 1986 Int. J. Radiat. AppL Instrum., Part C
0146-5724/86 $3.00+ 0.00 Copyright © 1986 PergamonJournals Ltd
Printed in Great Britain. All rights reserved
MUONIUM FORMATION IN NONPOLAR LIQUIDS O. E. MOGENSEN Chemistry Department, Rise National Laboratory, 4000 Roskilde, Denmark and P. W. PERCIVAL Department of Chemistry and TRIUMF, Simon Fraser University, Burnaby, British Columbia V5A IS6, Canada Aimtraet--Despite the very wide acceptance of the spur model of positronium formation there has been continuing criticism of the analogous model for muonium formation. After critical comparison of the muonium and positronium formation processes it is shown how differences in muonium and positronium behaviour can be explained by differences in their properties and the way in which measurements are made. It is concluded that the sparse data presently available for muonium are consistent with a spur formation process.
INTRODUCTION The mechanism of muonium formation is currently a matter of considerable debate. 0-12) Opposing schools of thought favour the "hot model ''03,14'4,7~ and the "spur model. ''~1,5'9)In the hot model muonium (Mu) is formed during the slowing clown of the muon (/t + ) as it sheds its last 104eV or so in matter. In this regime there is a series of charge-exchange cycles in which electrons are successively abstracted from the medium and returned to it. As it approaches thermal energy the muon is expected to spend more and more of its time as the neutral muonium atom, and it is commonly assumed that all muons are converted to muonium before thermalization (except where the~ ionization energy of the medium exceeds that of muonium). However, experimental results indicate that many of the muons are incorporated into diamagnetic species within picoseconds of thermalization, Accordingly, an integral part of the hot model is the existence of "hot atom, ''05:0 or epithermal reactions, in which energetic (10°-10-1 eV) muonium atoms can indulge in reactions forbidden to them at thermal energies. The main elements of the hot model are summarized in the following scheme: (/t + )* + M - * M u * + M +, charge exchange, Mu* +
(1)
R--,diamagnetic hot atom reaction, (2) products,
Mu*~Mu,
thermalization.
(3)
In contrast, muonium formation in the spur model results from the reaction of a stopped muon (i.e. a muon at or close to thermal energy) with one of the excess electrons created during the slowing down process. A crucial assumption inherent in this model is that the muon stops within the terminal spur of the
muon track. Competitive intraspur reactions and outdiffusion of/, ÷ and e - both reduce the probability of Mu formation: (/~+ + e-)spur~Mu, (e- + R)sp,r--*products,
(5)
(/~+)spur--'(/~+)b~i-*diamagnetic species. (6) There is a close analogy here with the spur model of positronium formation, which also met initial scepticism but has since found general acceptance. "7-22) Recently, proponents of the hot model have used the correlation between Ps and Mu formation to argue against the spur model for the l a t t e r : 's) In this paper we critically examine their arguments and consider the similarities and differences expected within the Mu and Ps spur models. We shall restrict our discussion to nonpolar liquids since aqueous systems have received attention elsewhere. C3,9~
EXPERIMENTAL CONSIDERATIONS In comparing the experimental data on positronium and muonium formation, account must be taken of the differences in experimental technique. Although both muon spin rotation (~SR) and positron annihilation lifetime spectroscopy (LS) involve single-particle lifetime counting, there is the additional factor in # S R of the muon spin polarization. The intensities (I) of the various lifetime components in LS spectra are usually taken as direct measures of the formation probabilities (h3 of the corresponding positron species:
h,= I~/~l,. 85
(4)
(7)
86
O.E. MOGENSENand P. W. PERCIVAL
However, different muon species are distinguished, not by their lifetimes, but by their precession frequencies; there are three classes of "signal," denoted by subscripts D, M and R, which arise from muons in diamagnetic states, muonium and radicals, respectively. The amplitudes of these/~ SR signals are determined by their share of the original muon spin polarization. The polarization fractions (P~) are easily measured by comparing signal amplitudes (Si) with those of a standard (So) for which P = 1, but they are not necessarily equal to the formation probabilities: S,/So = P~ = f A ,
0 <<.f~<. 1,
(8)
where f is a factor which accounts for the loss of ensemble polarization. For example, if chemical reaction is accompanied by a change in muon precession frequency, such as would occur if muonium reacts to give diamagnetic products, then there is a loss of phase coherence unless the reaction rate is much faster than any frequency differences. Loss of phase coherence constitutes transverse relaxation (depolarization), Longitudinal depolarization is also possible, as for example, when muonium encounters other paramagnetic species in solution. The fraction of polarization not accounted for by observed signals is usually labelled PL, SO that PM + PD h- PL = 1 for a system in which no radical is formed. In view of the potential for depolarization it is only permissible to equate measured polarization fractions with formation probabilities when all the initial muon polarization can be accounted for. Accordingly, attempts to correlate Pr, alone with bond energy (in tetrahedral tetrachlorides), with degree of unsaturation (e.g. in hexane, hexene and hexyne), and degree of substitution (e.g. in chlorinated methanes) are at best speculative, and their inclusion in a recent discussion of muonium formation ~7)is inappropriate.
points out that the situation in condensed matter may be quite different. In our opinion it is unrealistic to expect the muon to slowly lose its last 100 eV or so of energy while retaining its neutral form (Mu). In the 103 eV region energy loss is primarily by charge exchange, but by 100 eV the muon velocity is much smaller than that of the bound electrons, so transient binding of the muon to the molecules is possible. Muon affinities are expected to be similar to proton affinities, i.e. typically 5-8 eV ~25)for nonpolar molecules. In condensed matter the energetic muon will move so quickly through the electron cloud of one molecule to the next that it is hardly meaningful to consider the formation of discrete molecular ions. Nonetheless, to conform to conventional chemical symbology we can write the "reactions" as Mu + M ° ~ M u M + + e - , # + + M°--*MuM + MuM+--*/~ + + M,
(9) (10) (11)
where M ° represents a molecule of the medium in its ground state, and all the other entities may be excited. The high muon affinity of typical liquids is also the reason why the thermalized g+ cannot form Mu simply by picking up a valence electron from the medium. The binding energy of Mu (13.5 eV) is not large enough to overcome the sum of the ionization energy of the medium (typically 9 eV) and the muon affinity. (If the solvent has fully relaxed around a localized #+ there would be a further contribution of 0.5-1.0 eV from the polarization energy.) Thus, just as for e +, there is a lower energy threshold below which charge exchange will not occur. Claims to the contrary (7'23) are incorrect.
INHIBITION AND ANTI-INHIBITION MUON AND POSITRON TERMINAL SPURS
The nature of the end of a radiation track is largely unknown. In the hot model of muonium formation it is proposed that charge exchange will be complete at sufficiently high energy (10a-102eV) that the neutral atom can escape the charged species of the muon track/7) In contrast, the positron continues to much lower energies to ionize and excite molecules of the medium. Thus, it is claimed, ~7,23)there is a high probability that the positron will expend its last 102 eV to form a spur, and that this is unlikely for the muon. Support for this point of view is drawn from plots presented by Mozumder, (24)which suggests that in the crucial energy regime the LET for muons is a factor 102 less than for electrons. However, the curves in this limit have little foundation, being mere extrapolations from higher energies where the theory is well understood. Furthermore, the calculations explicitly apply to the gas phase. Mozumder, himself,
The spur model of positronium formation has been highly successful in explaining and predicting Ps yields for solutions of electron scavengers in nonpolar liquids319,2°'26-29) It would seem reasonable, therefore, to test models for muonium formation in the same systems. Miyake et al/s) took this approach, and measured Pt~ for mixtures of CC14 and C6F6 in cylohexane. On comparing their results with the known Ps yields they came to the conclusion that muonium formation is different from positronium formation and does not agree with a simple spur model. We are of the opposite opinion, as will become evident by our interpretation of Miyake's results below. First, it is necessary to describe and explain the situation for positronium. The yield of positronium is determined by competition between the large number of nonhomogenous reactions which can occur in a positron spur. The most important are: C19)
87
Muonium formation in nonpolar liquids M.,.,,.,~M + + e - ,
ionization,
(12)
recombination.
(13)
Ps formation,
(14)
e - scavenging,
(15)
(CCI~-)*--*C1- + CCI 3,
dechlorination,
(16)
e+ +CI---.[CI-,e+].
e + scavenging,
(17)
e + + CCI4~[CCI4,e+].
e + scavenging.
(18)
e - + C6F6-"*C6F 6,
e - scavenging,
(19)
e - pick-off,
(20)
recombination,
(21)
M + +e-~M, e - + e ÷--*Ps, e - "4- CCl4"*(CC14)*,
e + + C6F 6 ---~Ps+ C6F6,
M + + C6F6 ---.M + C6F 6,
CCI 4 + C6F 6 ~(CCI~')* + C6F6,
e - transfer.
(22)
inhibition, although it may be necessary to take account of the reaction of Mu with CCl4, Mu + CCI4-,MuCI + CCI 3
(23)
which can potentially make an additional contribution to PD. Miyake et al. cs) made a quantitative analysis of the effect of CCI 4 (and other scavengers) on PD for cyclohexane solutions, and contrasted the results with those for Ps inhibitiont22) and electron scavenging.
(~c) ° F ( c ) = -1- -+- -(ow) ~'
(24)
where c represents the concentration of the electron scavenger (CCID and ~ and a are fit parameters. They found that values of ct and a for Mu inhibition do not agree with those found for electron scavenging and positronium inhibition. Superficially, these results seem to support the view that Ps formation is similar to geminate recombination in conventional radiolysis, and that Mu formation is not. However, before drawing conclusions it is instructive to consider the basis for eqn (24). In its original form, with a = 0.5, the WAS scavenging function was presented merely as an empirical description of electron scavenging results, ol) However, the total scavenging probability is related to the lifetime distribution of the geminate pair, f (t) (in the absence of scavenger) by a Laplace transform
In addition, each of the transients is subject to diffusion out of the spur. Once an electron is scavenged by CC14 it is no longer available for Ps formation, because e + cannot pick off the electron from CI- (it is trapped instead) or from (CCI~-)*, which is too short-lived. Accordingly CC14 is a strong Ps inhibitor in hydrocarbon solutions. In contrast, C6F6 enhances the Ps yield in hydrocarbons. This is because the electron adduct C6F~- is a stable anion from which e + can pick off an electron to form Ps. Furthermore, positron mobility must be higher than that of the positive ion, M +, so that Ps formation can occur before recombination. The ability of C6F# to act as a temporary "bank" of spur electrons is so efficient that it can even counteract the Ps inhibiting effect of CC14. This is known as anti-inhibition in the 1 - F(c) = f ( t ) e - k " dt, (25) literature. To explain this effect it is essential to postulate that e + has a higher mobility than heavy where k is the rate constant for the scavenging ions, so that reaction (20) is favoured over (21) and reaction. In principle, then, f (t) can be calculated for (22). Preliminary experiments on positron mobility°°) any F(c), and the decay of the geminate pair is found support this assumption. by integrating f ( t ) : To discuss muonium formation in the same systems as for Ps, above, we consider the analogous V(t) = t) dt. (26) flU( reactions to (12)-(22) but with e + and Ps replaced by tz+ and Mu. In addition, there is muon attachment to For the WAS scavenging function [eqn (24) with the hydrocarbon (10). Inhibition by CC14 is still ~ = 0.51 predicted to occur, but we have seen above that enhancement and anti-inhibition depend crucially on F ( t ) = ea'erfc(2t) la, (27) particle mobility. The heavier g+ can reasonably be expected to have lower mobility than e +, but even where 2 = k / e is the reciprocal of the geminate more importantly, once #+ is attached to a hydro- pair lifetime (in the absence of scavenger). °2) F o r carbon molecule its mobility is reduced to that of the a = 1 eqn (24) becomes the common Stern-Volmer other positive ions. Accordingly, no enhancement or expression, and anti-inhibition effects can be expected. Miyake's reF ( t ) = e -at. (28) sults are limited to measurements of the diamagnetic Since the lifetime of an individual geminate pair is muon fraction, Pp. However, since it is known that Mu is long-lived in sufficiently pure cyclohexane(23) determined by the initial separation of the ions, F ( t ) will depend on the distribution of cation-anion sepawe can take h = Po. Furthermore, if it is assumed that the missing fraction PL represents depolarized rations. There is no intrinsic reason why the e + . . . e muonium, we arrive at hM = l -- Pp. The lack of effect separation in the terminal spur should resemble of C6F6 on PD in pure cyclohexane and solutions of that of a geminate pair in conventional (low LET) CC14 is then as predicted. The increase in PD with radiolysis. Although the results of Ito et al. ~22) were analyzed with the WAS equation (o =0.5), many CC14 concentration may be interpreted as muonium
f0 o
88
O.E. Mo~tqs~rq and P. W. PmtOVAL
other authors (~s'2~'zT-zg~report tr values from 0.5-1.0 for Ps inhibition in other systems. Despite the tenuous connection between positronium formation and geminate pair recombination, it is still valid to speculate that muonium inhibition and positronium inhibition should be similar, since they both relate to competitive processes in terminal spurs. However, once more we draw attention to the importance of the muon attachment reaction (10). For a hydrocarbon solvent (RH) attachment may be followed by fast processes which leave the muon trapped in a neutral diamagnetic molecule: # + + R H ~ R H M u +, R H M u ÷--*R + + HMu, R H M u + + RH--#RMu + R H f .
its highest precession frequency (3 x 10~°rads-t), with the result that all the muon polarization will be in the diamagnetic fraction. This is in complete accord with Miyake's finding
(29)
e ÷ + CCI4~PsC1 + CCI~
(30)
as a path for positron scavenging in addition to electron scavenging by CC14. The equivalent reaction for #+
(31)
(33)
Thus the lifetime d i s t r i b u t i o n f (t) for # ÷ . .. e - pairs # + + CCI4~MuCI + CCI~ (34) is governed, not only by the physical characteristics of the terminal spur (number of ion pairs and their could also account for the higher • determined from separations), but also by the chemistry of the me- Po. Thus, the ~ value that is most likely to properly describe Mu inhibition by electron scavenging is the dium. Given that there are additional processes that limit value of 0.1 M-1 found for C2HsBr. Since the rate the lifetime of the # + . . . e - pair compared to the constant for electron scavenging is k = 2 . 7 x l0 n e ÷ . . . e- pair, it is not surprising that the shape of the M -t s-~,t~) we estimate the mean lifetime of the lifetime distribution (characterized by a) is different. #+ . . . e - pair to be ;t -~ = ~/k = 0.04ps. This seems Furthermore, it is to be expected that the reactivity remarkably short, but not unreasonable when muon parameter ct will be smaller for Mu inhibition, attachment (29) and subsequent reactions (30) or (31) are in competition with muonium formation. reflecting the narrower distribution of lifetimes. However, ct should be independent of the nature of the electron scavenger, provided no other reaction is ADDITIONAL REMARKS affected by it. Miyake's results (s~ are not in accord with this principle--~ = 9.0, 0.1 and 0.65M -t for In the previous section we dealt with what seem to C2H~I, C2HsBr and CC14, respectively. This would be us to be the strongest arguments against the spur a serious challenge to the spur model were it not model of muonium formation in nonpolar liquids. for the fact that Mu reacts with these scavengers, as Other arguments have arisen, and have been repeated in reaction (23). The rate constant for Mu reaction many times. ~z4,6'7.n'35,3° Detailed discussion of each with CC14 is about 5 x 10SM-~s -~ (determined in point should not be necessary, since the principles methanol, (33)) so for concentrations of CC14 greater underlying our replies have already been expounded. than 2 M muonium will be converted to diamagnetic Thus, once the limited lifetime of # + . . . e - pairs is appreciated it is easy to see why solvent properties, product at a rate in excess of the lower muonium such as the static dielectric constant, are irrelevant, precession frequencies ( ~ 7 x 10Srads -~ at 8 0 G applied field), resulting in an increase in Po. Thus the and why Pv and PM do not depend on the probability variation of Po with CC14 concentrations in excess of of electrons surviving intraspur reactions. Further2 M will not just be due to muonium inhibition. For more, it is not surprising that applied electric fields up this reason Roduner ~331restricted the fit of eqn (24) to to 20 kV/cm failed to change PD" If, in the simplest concentrations ~<1 M in his study of muonium inhi- possible model, the effect of this field on a single ion bition by CC14 in benzene. His results were a = 0.42 pair is calculated, the absence of effect implies that and 1.2 M - ~according to whether hM was assumed to the initial charge separation is less than 35 A. ° ° This be equal to 1 - PD (as in Miyake's work) or propor- does not mean that the mean # + . . . e - separation is tional to PR, the muon polarization resident in r a d i - as small as this, merely that the fraction that successcals created by reaction of muonium with benzene. fully combine to form muonium in a fraction of a These results bracket the value found by Miyake for picosecond are close together. This paper focuses on muonium formation in CC14 in cyclohexane. The situation for C2H5I is far nonpolar liquids, but a few final remarks will be made more serious. The reaction with Mu: concerning the situation in other phases. In gases the Mu + C2HsI~MuI + C2H5 (32) fast reactions with neighbouring molecules [e.g. (29)is probably diffusion-controlled, i.e. we estimate (31)] which serve to "freeze" #+ in a diamagnetic the rate constant to be 5 x 10~°M -~s -1, Thus at molecule will largely be absent, due to the low prob1 M C2HsI the muonium reaction rate is faster than ability of termolecular collisions. Instead, attachment
Muonium formation in nonpolar liquids reactions will more often lead to dissociation (9)-(11). Thus, higher muonium fractions and lower diamagnetic fractions are predicted for gases, in full accord with the experimental findings of Arsenau et aL, (m who concluded that "radiation-induced spur effects play a dominant role in determining thermal muon fractions in condensed media." The comparison between liquids and solids is less clear-cut. We merely draw attention to a preliminary report by Ito et al., 07) who find that muon fractions are similar in liquids and plastic crystal, but that formation of the nonplastic crystalline state results in a drop in Po, which further decreases as the temperature is lowered. SUMMARY
The foregoing discussion of muonium formation in nonpolar liquids has exposed many errors in the principal arguments against the spur model. Experimental data are sparse, but are in full accord with the model. It must be emphasized that changes in the measured muon fraction, PD, do not always represent changes in the muonium formation probability, hM. Furthermore, the yield of muonium (or positronium) is determined by a complex set of competing spur processes, so that general statements such as "good electron scavengers should be strong inhibitors of Mu (or Ps) formation" are at best naive. The original model of muonium formation, the hot model, can also "explain" the experimental data, but only because it is too ill-defined to make any predictions. That is the reason why it is the spur model that is debated in the literature. REFERENCES
1. P. W. Percival, E. Roduner and H. Fischer, Chem. Phys. 1978, 32, 353. 2. D. C. Walker, Y. C. Jean and D. G. Fleming, J. Chem. Phys. 1979, 70, 4534; 1980, 72, 2902. 3. P. W. Percival, J. Chem. Phys. 1980, 72, 2901. 4. Y. C. Jean, B. W, Ng, J. H. Brewer, D. G. Fleming and D. C. Walker, J. Phys. Chem. 1981, 85, 451. 5. P. W. Percival, Hyperfine Interactions 1981, 8, 315. 6. D. C. Walker, Hyperfine Interactions 1981, 8, 329. 7. D. C. Walker, Muon and Muonium Chemistry. Cambridge Univ. Press, Cambridge, 1983. 8. Y. Miyake, Y. Tabata, Y. Ito, B. W. Ng, J. M. Stadlbauer and D. C. Walker, Chem. Phys. Lett. 1983, 101, 372. 9. P. W. Percival, J. C. Brodovitch and K. E. Newman, Hyperfine Interactions 1984, 17-19) 721. 10. E. Roduner, Hyperfine Interactions 1984, 17-19, 785.
89
11. B. W. Ng, J. M. Stadlbauer and D. C. Walker, J. Phys. Chem. 1984, 88, 857. 12. D. J. Arsenau, D. M. Garner, M. Senba and D. G. Fleming, 3. Phys. Chem. 1984, 88, 3688. 13. J. H. Brewer, F. N. Gygax and D. G. Fleming, Phys. Rev. A. 1973, 8, 77. 14. D. G. Fleming, D. M. Garner, L. C. Vaz, D. C. Walker, J. C. Brewer and K. M. Crowe, in Positronium and Muonium Chemistry (Edited by H. J. Ache), Chap. 13, pp. 279-334. Advances in Chemistry Series No. 175. Am. Chem. Soc., Washington, DC, 1979. 15. D. S. Urch, in Radiochemistry (Edited by A. G. Maddock), p. 49. MTP International Review of Science, Inorganic Chemistry Series Two, Vol. 8. Butterworths, London, 1975. 16. T. Matsuura (Ed.), Hot Atom Chemistry. Studies in Physical and Theoretical Chemistry Vol. 31. Elsevier, Amsterdam, 1984. 17. O. E. Mogensen, J. Chem. Phys. 1974, 60, 998. 18. O. E. Mogensen, in Proc. 6th Int. Conf. Positron Annihilation, Fort Worth, Texas, 1982, p. 763. NorthHolland, Amsterdam, 1982. 19. G. Wikander, O. E. Mogenscn and N. J. Pederscn, Chem. Phys. 1984, 87, 149. 20. B. Levay and O. E. Mogensen, J. Phys. Chem. 1977,81, 373. 21. O. A. Anisimov, S. V. Vocel and Yu. N. Molin, Chem. Phys. 1981, 56, 261. 22. Y. Ito, Y. Katsumara, T. Azuma and Y. Tabata, J. Phys. Chem. 1984, 88, 1921. 23. Y. Ito, B. W. Ng, Y. C. Jean and D. C. Walker, Can. J. Chem. 1980, 58, 2395. 24. A. Mozumder, Adv. Rad. Chem. 1969, 1, 1. 25. A. J. Swallow, Radiation Chemistry, p. 75. Longman, London, 1973. 26. Yu. N. Molin and O. A. Anisimov, in Positronium and Muonium Chemistry (Edited by H. J. Ache), Chap. 9, p. 179. Advances in Chemistry Series No. 175. Am. Chem. Soc., Washington DC, 1979. 27. O. E. Mogensen, Chem. Phys. Lett. 1979, 65, 511. 28. G. Wikander, Chem. Phys. 1979, 38, 181. 29. G. Wikander and O. E. Mogensen, Chem. Phys. 1982, 72, 407. 30. F. Heinrich and A. Schiltz, in Proc. 6th lnt. Conf. Positron Annihilation, Fort Worth, Texas, 1983, p. 705. North-Holland, Amsterdam, 1983. 31. J. M. Warman, K. D. Asmus and R. H. Schuler, J. Phys. Chem. 1969, 73, 931. 32. S. J. Rzad, P. P. Infelta, J. M. Warman and R. H. Schuler, J. Chem. Phys. 1970, 52, 3971. 33. E. Roduner, Hyperfine Interactions 1984, 17-19, 785. 34. A. O, Allen, T. E. Gangwer and R. A. Holroyd, J. Phys. Chem. 1975, 79, 25. 35. B. W. Ng, J. M. Stadlbauer, Y. C. Jean and D. C. Walker, Can. J. Chem. 1983, 61, 671. 36. Y. Ito, B. W. Ng, Y. C. Jean and D. C. Walker, Hyperfine Interactions 1981, 8, 355. 37. Y. Ito, Y. Miyake, Y. Tabata, B. W. Ng, J. M. Stadlbauer and D. C. Walker, Hyperfine Interactions 1984, 17-19, 733.
0146-5724/86 $3.00+ 0.00 Copyright © 1986 PergamonJournals Ltd
Printed in Great Britain. All rights reserved
MUONIUM FORMATION IN NONPOLAR LIQUIDS O. E. MOGENSEN Chemistry Department, Rise National Laboratory, 4000 Roskilde, Denmark and P. W. PERCIVAL Department of Chemistry and TRIUMF, Simon Fraser University, Burnaby, British Columbia V5A IS6, Canada Aimtraet--Despite the very wide acceptance of the spur model of positronium formation there has been continuing criticism of the analogous model for muonium formation. After critical comparison of the muonium and positronium formation processes it is shown how differences in muonium and positronium behaviour can be explained by differences in their properties and the way in which measurements are made. It is concluded that the sparse data presently available for muonium are consistent with a spur formation process.
INTRODUCTION The mechanism of muonium formation is currently a matter of considerable debate. 0-12) Opposing schools of thought favour the "hot model ''03,14'4,7~ and the "spur model. ''~1,5'9)In the hot model muonium (Mu) is formed during the slowing clown of the muon (/t + ) as it sheds its last 104eV or so in matter. In this regime there is a series of charge-exchange cycles in which electrons are successively abstracted from the medium and returned to it. As it approaches thermal energy the muon is expected to spend more and more of its time as the neutral muonium atom, and it is commonly assumed that all muons are converted to muonium before thermalization (except where the~ ionization energy of the medium exceeds that of muonium). However, experimental results indicate that many of the muons are incorporated into diamagnetic species within picoseconds of thermalization, Accordingly, an integral part of the hot model is the existence of "hot atom, ''05:0 or epithermal reactions, in which energetic (10°-10-1 eV) muonium atoms can indulge in reactions forbidden to them at thermal energies. The main elements of the hot model are summarized in the following scheme: (/t + )* + M - * M u * + M +, charge exchange, Mu* +
(1)
R--,diamagnetic hot atom reaction, (2) products,
Mu*~Mu,
thermalization.
(3)
In contrast, muonium formation in the spur model results from the reaction of a stopped muon (i.e. a muon at or close to thermal energy) with one of the excess electrons created during the slowing down process. A crucial assumption inherent in this model is that the muon stops within the terminal spur of the
muon track. Competitive intraspur reactions and outdiffusion of/, ÷ and e - both reduce the probability of Mu formation: (/~+ + e-)spur~Mu, (e- + R)sp,r--*products,
(5)
(/~+)spur--'(/~+)b~i-*diamagnetic species. (6) There is a close analogy here with the spur model of positronium formation, which also met initial scepticism but has since found general acceptance. "7-22) Recently, proponents of the hot model have used the correlation between Ps and Mu formation to argue against the spur model for the l a t t e r : 's) In this paper we critically examine their arguments and consider the similarities and differences expected within the Mu and Ps spur models. We shall restrict our discussion to nonpolar liquids since aqueous systems have received attention elsewhere. C3,9~
EXPERIMENTAL CONSIDERATIONS In comparing the experimental data on positronium and muonium formation, account must be taken of the differences in experimental technique. Although both muon spin rotation (~SR) and positron annihilation lifetime spectroscopy (LS) involve single-particle lifetime counting, there is the additional factor in # S R of the muon spin polarization. The intensities (I) of the various lifetime components in LS spectra are usually taken as direct measures of the formation probabilities (h3 of the corresponding positron species:
h,= I~/~l,. 85
(4)
(7)
86
O.E. MOGENSENand P. W. PERCIVAL
However, different muon species are distinguished, not by their lifetimes, but by their precession frequencies; there are three classes of "signal," denoted by subscripts D, M and R, which arise from muons in diamagnetic states, muonium and radicals, respectively. The amplitudes of these/~ SR signals are determined by their share of the original muon spin polarization. The polarization fractions (P~) are easily measured by comparing signal amplitudes (Si) with those of a standard (So) for which P = 1, but they are not necessarily equal to the formation probabilities: S,/So = P~ = f A ,
0 <<.f~<. 1,
(8)
where f is a factor which accounts for the loss of ensemble polarization. For example, if chemical reaction is accompanied by a change in muon precession frequency, such as would occur if muonium reacts to give diamagnetic products, then there is a loss of phase coherence unless the reaction rate is much faster than any frequency differences. Loss of phase coherence constitutes transverse relaxation (depolarization), Longitudinal depolarization is also possible, as for example, when muonium encounters other paramagnetic species in solution. The fraction of polarization not accounted for by observed signals is usually labelled PL, SO that PM + PD h- PL = 1 for a system in which no radical is formed. In view of the potential for depolarization it is only permissible to equate measured polarization fractions with formation probabilities when all the initial muon polarization can be accounted for. Accordingly, attempts to correlate Pr, alone with bond energy (in tetrahedral tetrachlorides), with degree of unsaturation (e.g. in hexane, hexene and hexyne), and degree of substitution (e.g. in chlorinated methanes) are at best speculative, and their inclusion in a recent discussion of muonium formation ~7)is inappropriate.
points out that the situation in condensed matter may be quite different. In our opinion it is unrealistic to expect the muon to slowly lose its last 100 eV or so of energy while retaining its neutral form (Mu). In the 103 eV region energy loss is primarily by charge exchange, but by 100 eV the muon velocity is much smaller than that of the bound electrons, so transient binding of the muon to the molecules is possible. Muon affinities are expected to be similar to proton affinities, i.e. typically 5-8 eV ~25)for nonpolar molecules. In condensed matter the energetic muon will move so quickly through the electron cloud of one molecule to the next that it is hardly meaningful to consider the formation of discrete molecular ions. Nonetheless, to conform to conventional chemical symbology we can write the "reactions" as Mu + M ° ~ M u M + + e - , # + + M°--*MuM + MuM+--*/~ + + M,
(9) (10) (11)
where M ° represents a molecule of the medium in its ground state, and all the other entities may be excited. The high muon affinity of typical liquids is also the reason why the thermalized g+ cannot form Mu simply by picking up a valence electron from the medium. The binding energy of Mu (13.5 eV) is not large enough to overcome the sum of the ionization energy of the medium (typically 9 eV) and the muon affinity. (If the solvent has fully relaxed around a localized #+ there would be a further contribution of 0.5-1.0 eV from the polarization energy.) Thus, just as for e +, there is a lower energy threshold below which charge exchange will not occur. Claims to the contrary (7'23) are incorrect.
INHIBITION AND ANTI-INHIBITION MUON AND POSITRON TERMINAL SPURS
The nature of the end of a radiation track is largely unknown. In the hot model of muonium formation it is proposed that charge exchange will be complete at sufficiently high energy (10a-102eV) that the neutral atom can escape the charged species of the muon track/7) In contrast, the positron continues to much lower energies to ionize and excite molecules of the medium. Thus, it is claimed, ~7,23)there is a high probability that the positron will expend its last 102 eV to form a spur, and that this is unlikely for the muon. Support for this point of view is drawn from plots presented by Mozumder, (24)which suggests that in the crucial energy regime the LET for muons is a factor 102 less than for electrons. However, the curves in this limit have little foundation, being mere extrapolations from higher energies where the theory is well understood. Furthermore, the calculations explicitly apply to the gas phase. Mozumder, himself,
The spur model of positronium formation has been highly successful in explaining and predicting Ps yields for solutions of electron scavengers in nonpolar liquids319,2°'26-29) It would seem reasonable, therefore, to test models for muonium formation in the same systems. Miyake et al/s) took this approach, and measured Pt~ for mixtures of CC14 and C6F6 in cylohexane. On comparing their results with the known Ps yields they came to the conclusion that muonium formation is different from positronium formation and does not agree with a simple spur model. We are of the opposite opinion, as will become evident by our interpretation of Miyake's results below. First, it is necessary to describe and explain the situation for positronium. The yield of positronium is determined by competition between the large number of nonhomogenous reactions which can occur in a positron spur. The most important are: C19)
87
Muonium formation in nonpolar liquids M.,.,,.,~M + + e - ,
ionization,
(12)
recombination.
(13)
Ps formation,
(14)
e - scavenging,
(15)
(CCI~-)*--*C1- + CCI 3,
dechlorination,
(16)
e+ +CI---.[CI-,e+].
e + scavenging,
(17)
e + + CCI4~[CCI4,e+].
e + scavenging.
(18)
e - + C6F6-"*C6F 6,
e - scavenging,
(19)
e - pick-off,
(20)
recombination,
(21)
M + +e-~M, e - + e ÷--*Ps, e - "4- CCl4"*(CC14)*,
e + + C6F 6 ---~Ps+ C6F6,
M + + C6F6 ---.M + C6F 6,
CCI 4 + C6F 6 ~(CCI~')* + C6F6,
e - transfer.
(22)
inhibition, although it may be necessary to take account of the reaction of Mu with CCl4, Mu + CCI4-,MuCI + CCI 3
(23)
which can potentially make an additional contribution to PD. Miyake et al. cs) made a quantitative analysis of the effect of CCI 4 (and other scavengers) on PD for cyclohexane solutions, and contrasted the results with those for Ps inhibitiont22) and electron scavenging.
(~c) ° F ( c ) = -1- -+- -(ow) ~'
(24)
where c represents the concentration of the electron scavenger (CCID and ~ and a are fit parameters. They found that values of ct and a for Mu inhibition do not agree with those found for electron scavenging and positronium inhibition. Superficially, these results seem to support the view that Ps formation is similar to geminate recombination in conventional radiolysis, and that Mu formation is not. However, before drawing conclusions it is instructive to consider the basis for eqn (24). In its original form, with a = 0.5, the WAS scavenging function was presented merely as an empirical description of electron scavenging results, ol) However, the total scavenging probability is related to the lifetime distribution of the geminate pair, f (t) (in the absence of scavenger) by a Laplace transform
In addition, each of the transients is subject to diffusion out of the spur. Once an electron is scavenged by CC14 it is no longer available for Ps formation, because e + cannot pick off the electron from CI- (it is trapped instead) or from (CCI~-)*, which is too short-lived. Accordingly CC14 is a strong Ps inhibitor in hydrocarbon solutions. In contrast, C6F6 enhances the Ps yield in hydrocarbons. This is because the electron adduct C6F~- is a stable anion from which e + can pick off an electron to form Ps. Furthermore, positron mobility must be higher than that of the positive ion, M +, so that Ps formation can occur before recombination. The ability of C6F# to act as a temporary "bank" of spur electrons is so efficient that it can even counteract the Ps inhibiting effect of CC14. This is known as anti-inhibition in the 1 - F(c) = f ( t ) e - k " dt, (25) literature. To explain this effect it is essential to postulate that e + has a higher mobility than heavy where k is the rate constant for the scavenging ions, so that reaction (20) is favoured over (21) and reaction. In principle, then, f (t) can be calculated for (22). Preliminary experiments on positron mobility°°) any F(c), and the decay of the geminate pair is found support this assumption. by integrating f ( t ) : To discuss muonium formation in the same systems as for Ps, above, we consider the analogous V(t) = t) dt. (26) flU( reactions to (12)-(22) but with e + and Ps replaced by tz+ and Mu. In addition, there is muon attachment to For the WAS scavenging function [eqn (24) with the hydrocarbon (10). Inhibition by CC14 is still ~ = 0.51 predicted to occur, but we have seen above that enhancement and anti-inhibition depend crucially on F ( t ) = ea'erfc(2t) la, (27) particle mobility. The heavier g+ can reasonably be expected to have lower mobility than e +, but even where 2 = k / e is the reciprocal of the geminate more importantly, once #+ is attached to a hydro- pair lifetime (in the absence of scavenger). °2) F o r carbon molecule its mobility is reduced to that of the a = 1 eqn (24) becomes the common Stern-Volmer other positive ions. Accordingly, no enhancement or expression, and anti-inhibition effects can be expected. Miyake's reF ( t ) = e -at. (28) sults are limited to measurements of the diamagnetic Since the lifetime of an individual geminate pair is muon fraction, Pp. However, since it is known that Mu is long-lived in sufficiently pure cyclohexane(23) determined by the initial separation of the ions, F ( t ) will depend on the distribution of cation-anion sepawe can take h = Po. Furthermore, if it is assumed that the missing fraction PL represents depolarized rations. There is no intrinsic reason why the e + . . . e muonium, we arrive at hM = l -- Pp. The lack of effect separation in the terminal spur should resemble of C6F6 on PD in pure cyclohexane and solutions of that of a geminate pair in conventional (low LET) CC14 is then as predicted. The increase in PD with radiolysis. Although the results of Ito et al. ~22) were analyzed with the WAS equation (o =0.5), many CC14 concentration may be interpreted as muonium
f0 o
88
O.E. Mo~tqs~rq and P. W. PmtOVAL
other authors (~s'2~'zT-zg~report tr values from 0.5-1.0 for Ps inhibition in other systems. Despite the tenuous connection between positronium formation and geminate pair recombination, it is still valid to speculate that muonium inhibition and positronium inhibition should be similar, since they both relate to competitive processes in terminal spurs. However, once more we draw attention to the importance of the muon attachment reaction (10). For a hydrocarbon solvent (RH) attachment may be followed by fast processes which leave the muon trapped in a neutral diamagnetic molecule: # + + R H ~ R H M u +, R H M u ÷--*R + + HMu, R H M u + + RH--#RMu + R H f .
its highest precession frequency (3 x 10~°rads-t), with the result that all the muon polarization will be in the diamagnetic fraction. This is in complete accord with Miyake's finding
(29)
e ÷ + CCI4~PsC1 + CCI~
(30)
as a path for positron scavenging in addition to electron scavenging by CC14. The equivalent reaction for #+
(31)
(33)
Thus the lifetime d i s t r i b u t i o n f (t) for # ÷ . .. e - pairs # + + CCI4~MuCI + CCI~ (34) is governed, not only by the physical characteristics of the terminal spur (number of ion pairs and their could also account for the higher • determined from separations), but also by the chemistry of the me- Po. Thus, the ~ value that is most likely to properly describe Mu inhibition by electron scavenging is the dium. Given that there are additional processes that limit value of 0.1 M-1 found for C2HsBr. Since the rate the lifetime of the # + . . . e - pair compared to the constant for electron scavenging is k = 2 . 7 x l0 n e ÷ . . . e- pair, it is not surprising that the shape of the M -t s-~,t~) we estimate the mean lifetime of the lifetime distribution (characterized by a) is different. #+ . . . e - pair to be ;t -~ = ~/k = 0.04ps. This seems Furthermore, it is to be expected that the reactivity remarkably short, but not unreasonable when muon parameter ct will be smaller for Mu inhibition, attachment (29) and subsequent reactions (30) or (31) are in competition with muonium formation. reflecting the narrower distribution of lifetimes. However, ct should be independent of the nature of the electron scavenger, provided no other reaction is ADDITIONAL REMARKS affected by it. Miyake's results (s~ are not in accord with this principle--~ = 9.0, 0.1 and 0.65M -t for In the previous section we dealt with what seem to C2H~I, C2HsBr and CC14, respectively. This would be us to be the strongest arguments against the spur a serious challenge to the spur model were it not model of muonium formation in nonpolar liquids. for the fact that Mu reacts with these scavengers, as Other arguments have arisen, and have been repeated in reaction (23). The rate constant for Mu reaction many times. ~z4,6'7.n'35,3° Detailed discussion of each with CC14 is about 5 x 10SM-~s -~ (determined in point should not be necessary, since the principles methanol, (33)) so for concentrations of CC14 greater underlying our replies have already been expounded. than 2 M muonium will be converted to diamagnetic Thus, once the limited lifetime of # + . . . e - pairs is appreciated it is easy to see why solvent properties, product at a rate in excess of the lower muonium such as the static dielectric constant, are irrelevant, precession frequencies ( ~ 7 x 10Srads -~ at 8 0 G applied field), resulting in an increase in Po. Thus the and why Pv and PM do not depend on the probability variation of Po with CC14 concentrations in excess of of electrons surviving intraspur reactions. Further2 M will not just be due to muonium inhibition. For more, it is not surprising that applied electric fields up this reason Roduner ~331restricted the fit of eqn (24) to to 20 kV/cm failed to change PD" If, in the simplest concentrations ~<1 M in his study of muonium inhi- possible model, the effect of this field on a single ion bition by CC14 in benzene. His results were a = 0.42 pair is calculated, the absence of effect implies that and 1.2 M - ~according to whether hM was assumed to the initial charge separation is less than 35 A. ° ° This be equal to 1 - PD (as in Miyake's work) or propor- does not mean that the mean # + . . . e - separation is tional to PR, the muon polarization resident in r a d i - as small as this, merely that the fraction that successcals created by reaction of muonium with benzene. fully combine to form muonium in a fraction of a These results bracket the value found by Miyake for picosecond are close together. This paper focuses on muonium formation in CC14 in cyclohexane. The situation for C2H5I is far nonpolar liquids, but a few final remarks will be made more serious. The reaction with Mu: concerning the situation in other phases. In gases the Mu + C2HsI~MuI + C2H5 (32) fast reactions with neighbouring molecules [e.g. (29)is probably diffusion-controlled, i.e. we estimate (31)] which serve to "freeze" #+ in a diamagnetic the rate constant to be 5 x 10~°M -~s -1, Thus at molecule will largely be absent, due to the low prob1 M C2HsI the muonium reaction rate is faster than ability of termolecular collisions. Instead, attachment
Muonium formation in nonpolar liquids reactions will more often lead to dissociation (9)-(11). Thus, higher muonium fractions and lower diamagnetic fractions are predicted for gases, in full accord with the experimental findings of Arsenau et aL, (m who concluded that "radiation-induced spur effects play a dominant role in determining thermal muon fractions in condensed media." The comparison between liquids and solids is less clear-cut. We merely draw attention to a preliminary report by Ito et al., 07) who find that muon fractions are similar in liquids and plastic crystal, but that formation of the nonplastic crystalline state results in a drop in Po, which further decreases as the temperature is lowered. SUMMARY
The foregoing discussion of muonium formation in nonpolar liquids has exposed many errors in the principal arguments against the spur model. Experimental data are sparse, but are in full accord with the model. It must be emphasized that changes in the measured muon fraction, PD, do not always represent changes in the muonium formation probability, hM. Furthermore, the yield of muonium (or positronium) is determined by a complex set of competing spur processes, so that general statements such as "good electron scavengers should be strong inhibitors of Mu (or Ps) formation" are at best naive. The original model of muonium formation, the hot model, can also "explain" the experimental data, but only because it is too ill-defined to make any predictions. That is the reason why it is the spur model that is debated in the literature. REFERENCES
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