Positronium in radiation chemistry of liquids

Radiat. Phys. Chem. Vol. 28, No. 1, pp. 1-18, 1986 Int. J. Radiat. Appl. Instrum., Part C Printed in Great Britain. All rights reserved

0146-5724/86 $3.00 + 0.00 Copyright © 1986 Pergamon Journals Ltd

POSITRONIUM IN RADIATION CHEMISTRY OF LIQUIDS V. M. BYAKOVand V. I. GRAFUTIN Institute of Theoretical and Experimental Physics, Moscow, U.S.S.R.

Abstract--Unlike investigations of the behaviour of electrons, whose role in radiation chemistry of liquids has long been recognized, analysis of the possible contribution of the behaviour of its antiparticle namely, the positron, in condensed media to comprehension of radiation chemistry processes is still in its early stages. Excellent reviews of experimental data on positron annihilation in relation to radiation chemistry of liquids were published in the Proceedings of the VI International Conference on Positron Annihilation. °) The present work deals with the subjects mainly pertaining to the initial stage of radiolysis of liquids. The information accumulated in literature is considerable. To keep within reasonable bounds, we limited ourselves to the examination of primarily Russian publications that have had the poorest showing in English-language publications.

1. INTRATRACKSTAGE OF RADIOLYSISAND THE MECHANISM OF POSITRONIUMATOM FORMATION

the investigation of the essence of intratrack processes a task of great importance. Because of the extremely short duration ( < lO -~2 + lO-Ss in liquids at room temperature) of the intratrack stage, the information on the processes occurring in this time period is mainly obtained from theoretical examination of the interaction of charged particles with the medium. The possibility to study the processes occurring in the initial stage of radiolysis experimentally has emerged only with the appearence of picosecond pulse radiolysis techniques (1969). In 1973 the similarity of mechanisms of molecular hydrogen formation in water radiolysis, and of positronium (Ps) atoms when positrons (e +) pass through liquids was established. (2-4)It was shown that both H: and Ps are formed via intratrack recombination with participation of a presolvated (quasifree) electron (e-) in ~<10-~2s. The fact that the nonsolvated electron is one of positronium precursors made it possible to use the relatively easily accessible data on measurements of Ps formation probabilities in condensed media for investigations of various intratrack radiation chemistry processes occurring therein in the (sub) picosecond time period, ts) These data promote successful comprehension of intratrack processes in liquid media.

In the succession of physico-chemical transformations that occur in a condensed medium from the moment of the beginning of its interaction with the ionizing particle, it is convenient to reveal the stage which such processes as moderation and thermalization of delta electrons, their recombination with ions, the ion-molecular reaction and solvation occur and practically terminate. The main consideration for the isolation of this particular stage is the suggestion that all the above processes are of local, intratrack character; they occur in mutually isolated microvolumes of the irradiated medium, whose main part does not yet "realize" that the effect of irradiation has already occurred. For this time period, concentrations of radiolysis products are characterized by considerable space inhomogeneity. As a result, the main chemical reactions in this stage are reactions between the products themselves, and with the matrix; their interaction with trace substances becomes possible only if the latter are in high concentrations. This makes the processes occurring during this period almost independent of the kind of substance dissolved in the medium, and universal for most solutions. On the other hand, chemical transformations that take place in the following stage, occurring when the concentration of intermediate particles becomes comparable to the concentration of solutes, are specific for each system. Exerting a strong influence on the development of radiation chemistry transformations, intratrack processes occupy the key position in interpretation of irradiation effects on condensed medium, which is why their analysis is vital in establishing the mechanism of radiolysis of liquids and solids. This makes

2. SIMILARITY OF MECHANISMS OF POSITRONIUM AND RADIOLYTIC MOLECULAR HYDROGEN FORMATION IN WATER

The mechanism of hydrogen formation in the process of water radiolysis has for a long time remained a subject of discussion in literature on radiation chemistry. It had been assumed until the 1

2

V.M. BYAKOVand V. I. GP.AFUTIN

early 1960s that the formation of molecular hydrogen occurs according to the reaction H+H-,H2

SCHEME OF POSITRON(ELECTRON)TRACK END{DIFFUSION) PART ((~t < - Q)

(1)

INITIALPART (@.t>~) e e

but the discovery of the hydrated electron led to the following description of H2 formation: eaq + eaq -"+H 2 + 2 O H - ,

(2)

e~q + H ~ H 2 + O H - .

(3)

Time has proved, however, that both these descriptions are incorrect. Arguments in favour of the nonhydrated electron ( e - ) as the main precursor of molecular hydrogen were given:(3'6)

_

,ooi

.=a

HEO+ + e - --~H 2 + prod. In fact, if H2 formation followed reactions (1)-(3), then addition of substances that react with the Hatom or e~ to water should lead to suppression of H 2 yield, similarly as the addition of OH radicals suppresses H202 yield.(7) It turned out, however, that the addition of a large number of effective acceptors of e~q and H ( H +, Tl +, Zn 2+, La 2+, etc <8))has no effect on H 2 yield; on the contrary, some weak e~ acceptors (benzene) noticeably suppress H2.¢3'6)This means that neither e~q nor the H atom can be main H2 precursors. Some progress in the investigation of the mechanism of H2 formation became possible after comparing the effects of water solutes on H 2 and Ps formation probabilities.¢2-5) From the standpoint of the existing approaches to the mechanisms of positronium formation, this comparison had little sense. Indeed, according to the traditional concept [cf. for instance(9)] it was believed that the positronium atom forms in the process of charge exchange between molecule M of the medium and the "hot" positron (e + )*, in the energy interval from IM - I~ to IM (Iu, I~ are ionization potentials of molecule M and positronium, respectively): (e+) * + M --,Ps + M +.

(4)

This process bears no resemblance to intertrack reactions (1)-(3). However, the following considerations make the comparative analysis of data on the probability of Ps and H 2 formation meaningful. Initial positron energy emitted by the radioactive nucleus constitutes several hundred keV. The character o f positron movement in the time period, over which it suffers an energy loss from W~ to W2, is determined by the ratio of its (curved) path X to the transport length l~, which is the mean distance passed by the positron before it scatters by 90 °. At (X/lt) >>1 it is diffusive. Modeling the molecule of water t Principal possibility of positronium formation via e + electron recombination was originally mentioned by Tao and Green°3) (1968). However, they deemed the process hardly possible. Different versions of the recombination mechanism were given by Mogensen °4) (1974 and on), and in our works¢2-5)(1973 and on) in connection with the nature of radiolytic hydrogen precursors.

1

10 100 I 0 KINETIC ENERGY OF POSITRON {ELECTRON), eV

Fig. 1. The scheme of positron (electron) track. The transport length 1,, mean distance I between ionization acts, the total path X of positron in function of its kinetic energy W.

with the Ne atom, which is isoeleetronic to HzO, °°J and calculating It in the Born approximation 01) (cf. Fig. 1), we find 5 < X/I, < 10 at 7 < W < 500 eV, which means that the movement of such positrons is diffusive. Curve 1 in Fig. 1 yields the mean distance 1 between adjacent ionization acts. At W < 300 eV It < 1. This means that space distribution o f ions created by such a positron is of roughly spheric symmetry. Thus, in tracks of positrons emitted by radioactive nuclei, it is natural to distinguish between two parts: the initial, where kinetic positron energy exceeds 1 keV, and the end, where the energy changes from 1 keV to thermal. In the initial part of the track, positron movement can be described as "straight" one; at W < 1 keV it acquires a diffusion character, with the spherically symmetric space distribution of ions and electrons in the first approximation centered near the moderated positron, cm This creates in the end part of positron track favourable conditions for both ion-electron recombination, which is standard for tracks of all ionizing particles, and for electron and positron recombinations, which results in the formation of a positronium atom followed by the reaction of Ps with electrons. According to the suggestion made in, ¢2-s) in the diffusion part of positron track in 'liquid water, the following reactions with participation of presolvated electron e- are possible?: H20 + + e - --}H20*--}H2 + O e + + e - - - , P s . Evidently, the spherically symmetric cloud of ionelectron pairs appears not only in the end part of the positron track; each electron having the energy

Ps in radiation chemistry of liquids 1.0

I

I

1

3

I 00~

I

H 24-0

/

H20 + + e - ---¢H20*--~H + OH ~

(14)

\

0.8

H20 energy • +¢a~. • • ¢÷ kl

0.6

The following obvious conclusions can be drawn from the above reactions. @

I. The effect of electron accepting solutes in water on yields of radiolytic H2 and positronium should be 0.4 similar in character, i.e. the more a given solute M I) 0 suppresses H 2 yield, the stronger it should inhibit the formation of Ps. Data in Fig. 2 prove that this does 0.2 in fact occur. Figure 2 compares relative yields of H2 and Ps. Each point in the figure corresponds to a definite concentration of a solute; its coordinates are 0 relative H2 and Ps yields obtained at this concen0 0.2 04 06 08 l.O tration. An increase in the concentration of the p/pO electron acceptor reduces H2 and Ps yields. Stronger Fig. 2. A correlation of relative yield for radiolytic mole- inhibition of H 2 with some additives (acetone, glycular hydrogen with the probability for Ps formation cine) is explained by the fact that they also accept, in aqueous solutions: + CuSO4('s"9); • NOf('s-2°); • along with presolvated electrons, H~O + ions. Hg(ClO4)2t~2°; • NaBrO3~ng'22);• acetoneOg); 0 NaNO3°9); ZnSO4tS'lg);[] glycine(19);• CDSO4(8'22);• Na2CrO4(19,22); 2. The acceptor of presolvated electrons should reduce not only e~q yield but the yields of H 2 and Ps I ~ H202(19"23); CxlClO4tU);0 AgC104tt9~4); ~7 HC104(23'25); CDC12(19,26); ~ KIO3(22); ~ K2Cr2OT(2°'2227); ~ TI2SO4 (8'26). as well. Figures 3 and 4 show that this consequence also has an experimental confirmation. A more detailed analysis of data presented in Figs 2-4 will be given below. < 1 keV knocked out by the positron forms just such 3. Acceleration of the ion-molecular reaction (6) a cloud, or, in other words, "blob. ''t~5) It is in these should reduce H 2 yield and increase Ps yield. In fact, microvolumes that the highest probability for the this reaction transforms chemically active H20 + ions formation of radiolytical hydrogen exists. Naturally, into hydroxonium H30 + ions which are inert to the this proceeds not only in the positron track, but in presolvated electrons [cf. reaction (13)]. This concluany high velocity electron track. Along with the above reactions in tracks of high velocity electron and positron, the reactions of elec~.0 i 1 i i i i i )to 1 ~ tron soivation and capture with solutes occur as well. The most important for our purposes are the following intratrack reactions: tt

oO

®

O.B

H20 + + e H20++H20

kt2 , H 2 0 , --~H2 + O,

(5)

,h )H30+ + O H ,

(6)

H2O+ + S kts ) S+ + H20, e-

S + ee+ + e-

22

, S-,

(9)

k23) Ps,

(I0)

0.2

e+

kJS,(Se+),

(11)

a~ ) eaq+ )

(12)

H30+ + e - ~ ) H 3 0 +

+e -.

(13)

The formation of H2 is only one of possible channels of the e - + H2O÷ reaction; other possibilities are the formation of a hydrogen atom t'° and regeneration of water molecule: 07)

1

Q @-~m'

• O

I°

]

°!

O000

1 J

e+4-S

~..t~

N -gl ,~ + +

0.6

0.4 (8)

® -

n

(7)

, e&,

k~

~o

o,® ~°

0

i

0.2

i

i

0.4

i

i

0.6

i

i

0.8

i

1.0

Fig. 3. A correlation between relatives yields of Ps and e~ in aqueous solutions: (D AgCOOCF3CU'2s); O AgCIO4(U'2s);• acetone09,29); • CdCIO4, CdC12¢19'24"26~9); 0 + ~7 []

NaNO3°9'23'28-3D; CuC12(19,27,2&33); HC104(19"23"28"29); CC]3COOH(19'34);

(~ SeO42-(2s'32), • Na2CrO4(19'2s); [] NalO4(27,28); • KBrO309,34); (for C37 ~- 20 M); • N0~(19,28); • Hg2 + (21,28); ~ K2Cr2OT(27"28); t ~ KIO3(21).

4

V.M. BYAKOVand V. I. GRAFUTIN

sion is confirmed by a 50% decrease in H 2 yield, and a twofold increase of Ps yield upon transition from water to ice. (36'37) The ion-molecular reaction proceeds much faster in ice than in liquid water and a cornpetition between e ÷ and H2O÷ for the electron vanishes. 4. Addition of H2O÷ acceptors should have the same qualitative effect as the acceleration of the ionmolecular reaction. "6) This conclusion is confirmed by the data in Fig. 5. Addition of O H - ions that are hole acceptors (4~) increases Ps and e~q yields and reduce H2 yield.f

1.0 , ~

0.8 0

Im

0

•

•

•

•

0.6 o

.~

0.4

I

i

"

oo

0.2 3. ARGUMENTS IN FAVOUR OF QUASIFREE ELECTRON PARTICIPATION IN THE FORMATION OF POSITRONIUM IN VARIOUS LIQUIDS

*

&~e• •

0 '0'2

I

o'4

I

0'6

i

018

I

The conclusion that positronium has, as its main precursor, a quasifree electron, is confirmed by experiments pertaining to the effects of different factors on the probability of positronium formation in various media.

Fig. 4. A correlation between relative yields for radiolytic molecular hydrogen and e~ in aqueous solutions: • CdS04, CdC12(s'22'29);0 AgC104(19'28); • NO~-0s'2°'2a); O Na2CrO4(22'28); O CuNO3°9'28'29); • acetone°9'29.35);

Polar liquids

•

The time ~ of electron solvation (localization) in alcohols at low temperatures exceeds positron lifetime more than 1000 times; hence, under such conditions a reaction between e~- and e ÷ is impossible. If solvated electrons participated in positronium formation, it would have been understandable to expect the probability of Ps formation to decrease. However, this probability not only does not decrease, but actually slightly increases (Fig. 6). This result is against e~- being the precursor of Ps.

Hydrocarbons

C,e~l~e~

NaSO3(19,2s,29);

I:><1 a202(tg,29);

•

Hg2+ (21,2a);

KBrO3(22'34); V HC104(25'29) (for C37 = 20 M); + Cu(C104)2¢22.2s);[] glycine°9~s);~ ZnSO4(z2'34).

vation. Therefore, if a Ps atom really is the product of recombination of the positron with the nonsolvated electron, Ps yield should decrease with the addition of alcohol, while e~- yield should, on the contrary, grow. This actually takes place; considerable changes in the yields of e,- and Ps occur within the same range of alcohol concentrations. Yet another argument in favour of quasifree electron participation in positronium formation can be

The dominating role played by quasifree electrons in Ps formation in hydrocarbons becomes manifest in experiments on the effect of alcohol additions on Ps yield, t4s~ Polar molecules of alcohols are not appreciable effective electron acceptors, but their presence in hydrocarbons does result in efficient electron sol-

~-1.1 ? The data in Fig. 5 is also of interest from the standpoint of an estimate of the overall yield of water decomposition in the process of its radiolysis. At [OH] = 1 M, hydrogen yield G.: is reduced by 0.1, while the yield of the hydrated electron grows by approximately 0.3,(4o) if its yield in a neutral medium can be assumed equal to 4.8. (45'e~ With reference to an almost complete equality of the yields of molecular and atomic hydrogen that are formed via H2O+ + e- recombination,°6) we find that the yield of H:O regeneration in reaction (14) is also reduced by 0.1 mole per 100 eV. Thus, all the three channels of reaction (14) are equally probable. Since Gnz = 0.46 molecules per 100 eV, the overall yield of recombining ion-electron pairs constitutes - 1.5 per 100eV ion in agreement with the data of.~4~) This gives the overall yield of water decomposition at 4.8 + 1.5 -~ 6.3 molecules per eV. Unfortunately, it is difficult to name the error of the determination of this yield, since the accuracy of determining the yields of H2 and e~ in an alkali medium leaves much to be desired.

,.0

,.I

K)

2~0

3'() [OH']. M

•

10

~0

.9

•

0.8

•

o

Fig. S, The dependence of relative yields of Ps (1), radiolytic hydrogen (2) and e~q (3) formation on the NaOH o (39) concentration: O P/Po(38) ; • GH2/GH2 ; ~ G ~ / G o~ (4o) ; 0 (40. A GM~/Ge~ ,

0 (42) • • Ge.~/G~ (m 30ps),. []

• Ge~/Ge°"<42)(in 6 ns).

-0~(43). G.JG.~ ,

Ps in radiation chemistry of liquids

K

Temperature,

298 t

A

200 =

i

The effect of the ion-molecular reaction rate of proton transfer on the probability of positronium formation, described in the preceding section, leads to the conclusion that in liquid water the time taken by this reaction is commensurable with the time of intratrack ion--electron recombination and constitutes, therefore, roughly 10 -]2 s, rather than 10 -]4 s, as is sometimes claimed.

143 f

T

22

lc;'

/

5. QUANTITATIVE F O R M U L A T I O N O F T H E M O D E L F O R RADIOLYTIC H Y D R O G E N AND P O S I T R O N I U M F O R M A T I O N IN WATER AND A Q U E O U S S O L U T I O N S

J ...hl

10111 7 r/, 4

5

,

,

,

5

6

7

103/T (K "t) Fig. 6. Temperature dependences for Ps formation probability, electron solvation time and positron lifetime in n-propanol.(4)

drawn from the character of the dependence of Ps yield in gaseous xenon on its density (Fig. 7). In accordance with the diffusion-recombinational model of positronium formation, (5°) its yield climbs sharply up to a pressure of roughly 80 at, after which comes an abrupt fall. It is starting from this pressure that electron solvation in clusters resultant from density fluctuations becomes highly probable in gaseous xenon. °z'52) The solvation results in the reduction of Ps yield.

The reactions (5)-(14) were used as the basis for the diffusion-recombinational model of formation of positronium and radiolytic molecular hydrogen in water. The model takes into account the competition between the process of diffusion of ions and electrons that are formed in the track with reactions of their recombination and capture by solutes. As the conclusions drawn from this model are used in the interpretation of experimental results, we shall briefly describe the model. °9'6) Let Cl(r, t), C2(r, t), and C3(r, t) denote the concentrations of H20 ÷, e - , and e + particles, respectively, at time t at point r (r is the distance from the center of the diffusive part of the positron track). In cases when the solution contains only electron and positron acceptors, the equations describing diffusion and chemical intratrack reactions (5)-(12) of particles are as follows: dC! = DIAC 1 _ k I 2 C I C 2 - 21Cl ' tgt

(15)

C~C____22= D2AC2 - kl2C] C2 - k23C2C3 t3t

(16)

- k 2 s C s C 2 - 22C2, ~C3 = D z A C 3 - k 2 3 C 2 C 3 _ 2~C3 _ k3sCsC3. dt

4. TIMES OF POSITRONIUM FORMATION, PROCEEDING OF INTRATRACK

32

ION-ELECTRON RECOMBINATION, AND ION MOLECULAR REACTION

28

0 0 0 0

The fact that the solvated electron does not participate in Ps formation, makes it possible to evaluate an upper time limit of Ps formation. The probability of Ps formation should evidently decrease when the time of solvation becomes shorter than the time of e + and e - recombination. However, the constant Ps yield values on decreasing ~ in alcohols (47) leads to the conclusion that in these media the time of Ps formation, and consequently, that of ion-electron recombination in the track of the ionizing particles does not exceed I0 Ps, which is the lowest ~ , value for alcohols. (53J The lower limit of characteristic times of these reactions is evidently presented by the time of electron thermalization. As already mentioned, it constitutes almost I0 -t2 s.

(17)

24

,,

°

0

20 16 12 8 4

200K

/ 47 59 I

o

i

68

I

,;

~

T I

8

76 83 106 iI

I I

1(3 12 t4

Iolm}

Jv

16

(M)

Fig. 7. The dependence of Ps formation probability on xenon density (pressure) at room temperature. (49)

6

V.M. BYAKOVand V. I. GRAFUTIN

Here A is the radial part of Laplace operator, D~ is the diffusion coefficient of the/-type particle, k o.is the biomolecular rate constant for recombination of iand j-particles, k2s and k3s are rate constants of electron and positron interaction with acceptor S, Cs is acceptor concentration, ).~ is the rate constant of monomolecular reaction of/-particle. Owing to the radial symmetry of electron and ion space distribution in the diffusion part of the positron track, the following form of initial conditions may be used:

Cl(r, O)

nto

exp(_r2/a2),

(18)

C2(r,O ) n2° .exp(_r2/a2), -- (~a2)3/2

(19)

C3(r, O)

(20)

= (7ta2)3/------~ •

n3°

= (na])3/2

"exp(--r2/a2).

n,(t) "e-r2/(4D, t + a 2) (4Dit + a2) 3/2 ,

where n,(t) is the number of /-particles at time t. Integration of the equations over the whole volume gives:

dnl

-- = dt

dn____22= dt

dn--2 = dt

kl2ntn2 V12

kt2nln2 Vl2

,~1nl,

k23n2n3 k2sCsn2 - 22n2, (21) V23

Our aim is to find the amounts of recombined particles, i.e. to calculate the integrals Jr'z2 d¢'23

= f ~ klffl'(t)n2(t) dr,

Jo

Vl2(t)

(23)

= f ~ k2ffl2(t)n3(l) dl.

v~3(t)

J0

Here we shall attempt to find the expressions (23) for the case of "high" acceptor concentrations C s when ?130 k23 ~323 '~ k2sCs"

For this case the following asymptotic expression for probability Pr~ of positronium formation may be obtained: 1

Ph

n30

F_

fo~(k12ntn2/V23)dt = ( l + n30 -I" k23 V03 \ no k23 V~I2

3/2 k23C s "~ (k23no/V°3) ~23 -[" k23n--~V~23J'

Here, (22)

is the characteristic volume occupied at time t by particles of i- and j-type.

t Since the mean energy needed to form an ion-electron pair in liquid water is roughly 17 eV, no ~ 103eV/17 eV ~ 60 (103 eV is the energy lost by the positron in the diffusion part of the track). :t Ambipolar diffusion is known to occur when a~>/-~ ~ , where ro = e2/~KjT is Onsaser radius, # is iomc strength of the solution, Lv is the Debye screening length; a~ is determined in (18)-(20).

(24)

Similarly, for the probability PH2 of radiolytic hydrogen formation in one of the spherical clusters of ionizations (spur), constituting the track of high energy electron, we have: 1

no

Pn2

['®(k:an2n3/V23 ) dt Jo = 2+

Jkl2nol

2

k23 V~12.3o] k2sCs "~ k,2~23noJ "1 kl2no/V°2"

k23n2n3 k3sCsn3- 23n3. V23 VU---- VO(l + I 1 ~ ) 3/2

D~= (D, + Dj)/2.

4D U

no k12 ~ '~ k2sCs'

Here, n~o = n2o= no is the initial number of ionelectron pairs, no >>n30 = 1.t" Value of a~ characterizes the radius of a spherical volume initially occupied by i-particles. Equations (15)-(17) do not take into account electrostatic interaction between the charged partitles; we believe that the conditions of electroneutrality needed for ambipolar diffusion are fulfilled J6)~: Let us express C~(e,t) in eqns (15)-(17) as (cf. Ref. 6):

Ci(r, t) =

0 _ (2nau) 2 3/2., au -- [(a~ + a})/2] 112 Vii-

(25)

To obtain the dependence characterizing the probability of positronium formation under experimental conditions, it is necessary to average eqn (24) over the parameters of multiple positron tracks. Since this procedure is actually impossible, we shall assume in the future that the values in eqn (24) already provide a description of the average diffusion part of the position track. A similar averaging procedure with reference to track structure and energy distribution of Compton electrons should also be carried out to obtain an expression for hydrogen formation probability in the process of y-radiolysis of water. The parameters that characterize a spur averaged in this fashion of Compton electron track should differ from similar parameters for the diffusion part of positron track.

Ps in radiation chemistry of liquids We shall use the symbols n~0, a~i, V#, and ~ to describe mean parameters of the diffusion part of positron track, and N~o, bo, Vo, and T o shall be used to characterize the averaged spur of the Compton electron. We can then represent the equation for probability of hydrogen formation as

Similarly, by multiplying both parts of eqn (27) by G°2=0.45mol/100¢V (H2 yield in water), we get (assuming G°a2o = 6 reel/100 eV)

G°----z = 0.075.4 G}h

k2s' Qa2 = 0.075 Bs = 0.075 kl 2.helVen.

k12No/

Here, G°H2o ",, 6 molecules/100 eV (see above) is the overall radiation chemistry yield of water decomposition, Gu2 is the yield of formation of radiolytic molecular hydrogen. It follows from eqns (24)-(25) that in sufficiently concentrated solutions, the reciprocal yield of molecular hydrogen formation and the reciprocal value of probability of positronium formation increase linearly with growing solute concentration:

1

- - = ~ t + flsCs, 0 G_H~

+

(26) (27)

G_n2o

= 1 + n30 k12 V~23 3/2 no + ÷ (k12 o/ 3h3 +

It is seen from eqns (28) and (29) that reciprocal Ps and H2 yields should increase linearly with a growth of inhibitor concentration. However, this conclusion is true, strictly speaking, only for neutral inhibitors. Actually, the rate constants of reactions, k12, k23, k25, in eqns (26) and (27) depend on the ionic strength of the solution, i.e. on concentration of solute if the latter is dissociated into ions. For the bimolecular reaction (A + B -* product), whose rate constant kd is limited by diffusion, the kd dependence on ionic strength/z is determined by the equation: (4s) k d = 4~ DR f,

Bs . :---5----- Cs,

where L . ZA'ZB(ro/R ) exp(ZAZ,[ro/(L + r0)] f-~-L + ro exp(ZAZBro/R ) -- 1

where 3

re L + r0

k2s flS -~ k 23no/ V~23,

. 3/2\ 1 k23V°2N3o A = 2 + 21 + - -T12J k12no/V12 - o -I k12V~23N ° k2s

B s = k12no/V°2" The values of P ~ and G°2 should not be mistaken for -1 and A-1 values, as the former represent positronium and H2 yields in the absence of an acceptor, and the latter--the yields, extrapolated to zero acceptor concentration. It can be shown that the following inequalities are true for these values: ~-1 < p o ,

(29)

where

kz3 V~nNao k~Cs 4 - - ÷ - -

).!_= Gas

+ 0.075Bs' Cs =0.075A + Q a f C s ,

1 G ° 8 ~ ° = 2 + ( : h + 3/2'~ GH2 T12] kl2No/ V~12

k12 3No

7

0 A - I < GHJG_H:O"

Since in most experiments the relative positronium yield P ~ / P ~ (where p o is the probability of Ps formation in pure solvent) is measured, let us rewrite eqn (26) in relative units. Particularly for aqueous solutions ( p o =0.27) the positronium yield dependence on solute concentration takes the following form: P_.~_~_ 0.27 - 0.27. ~ + 0.27 Qr," Cs, P~ P~ where k2s

Q~ = 0.27fls = 0.27 k23no/V~-------~3.

(28)

(30)

D and R are the sums of the diffusion coefficients and reagent radii, respectively; ZA and ZB are the electric charges (in units of elemen._m~.~ charge) of reacting ions A and B, L =x/l/Snrolz is Debye screening length; ro = e2/keT.e is Onsager radius; k s is the Boltzmann constant; e is dielectric constant of the solution; T is absolute temperature. Therefore, strictly speaking, the changes in the ionic strength related to solute concentration Cs should distort the linear dependence of Ps and H2 reciprocal yields on Cs. This effect however, is negligibly small for Cs > 0.1 M, where eqns (28) and (29) are applicable. For instance, for NO~- and Cu 2+ ions (Fig. 8) in the (0.1-1)M concentration interval, the coefficient f, which takes into account the effect of the ionic strength, is changed by 20% for reactions with hydrated electron and by as little as 3% for reactions with the nonhydrated electron. (19) Thus, since Ps and radiolytic hydrogen formation takes place with participation of a nonhydrated electron, then the linear dependence of their reciprocal yields on concentration of inhibitor should be realized with fair accuracy, regardless of whether it is present in the solution in the form of neutral molecules or charged ions. At "low" solute concentrations, when k12 ~01°2, k23 ~n3° >>k2s Cs

8

V.M. BYAKOVand V. I. GRAV'VrlN

1.6

1.4

osI- r

. I

0.1

0.2

0.3

[S], M

Fig. 8. The dependence of coefficient f on solute concentration for reactions: (I) NOj- +eLt in aqueous solution of NaNO3; (2) NO~ + e- in aqueous solution of NaNO 3; (3) Cu 2+ + e- in a queous solution 2 _.. of Cu(C104)2; (4) Cu + + e~ in aqueous solution of Cu(CIO4)=.

calculation integrals (23) lead to the following equations: c6)

greater Cs concentrations of the acceptor, the GH~ hydrogen yield is adequately approximated by

GH2 = G% -- qrt2x/~s,

(31a)

a.~ = a% - r7.2" ',Y~,

pp, = p o _ q ~ x ~ s

(31b)

which is valid for a rather wide concentration region (?:/H2 has the same sense as qn2).

(qn2 and qrs are coefficients characterizing inhibitor ability). As shown in Ref. (6), eqn (31) is true for only extremely low acceptor concentrations, while molecular hydrogen yield is lowered by no more than several percent due to electron interception by acceptor.t At t Such a decrease lies within the accuracy margin for measuring radiolytic product yields, and it is quite natural that eqn (31a) finds no experimental confirmation in the process of water radiolysis. On the other hand, (31a) in hydrocarbons is true in a considerable range of Cs values. This is due to the fact that (2n)J/Zk~2no/SnDj2a12 is much more (and for water much less) than unity. (6)

6. COMPARISON OF THE RECOMBINATION DIFFUSION MODEL WITH EXPERIMENTAL DATA

Figure 9 presents experimental data on hydrogen and positronium yields. It is clear that at the acceptor concentrations over 0.1-0.2 M, linear dependence of reciprocal positronium and hydrogen yields on solute concentration is observed, in accordance with (28) and (29). Linear sections of curves were used for the calculation of Q~ and QH2 coefficients of eqns (28) and (29) (Fig. 10).

Reciprocol volues of relotive H2-ond Ps-yields i

i

f

i

i

i

~

i

p

I

H20 ')'-F e- ~ H z + ~ r o ~ _ e- + s -/---- H2 ~~ , . ~

50

'e+

P, .....

|

e - + s -¢-.- Ps

-15.0

No'~"/°

NO"

4.0

|

30

3.0

H202 20 1.0l Zn,

-

i

0

I

I

t

)

0.5

~

I

I

T~

I

(

(32)

I

I

1.0

I

I

I

0.5

I

I

I

I t

]

|

10

Solute concentrotion, M

Fig. 9. Linear dependences of reciprocal yields of Ps and that of hydrogen on concentration of solutes in water.

Ps in radiation chemistry of liquids

9

N°/V°2 "~k23/k12~- 0.3. no~v°23- 3.6 "t'tgz*

I0

2÷H2OZ* " / ~ N O ~ ~p b z ~

*Fe~"

Cu o /

1

COG-,~dx~

~

,0~

., lCHs )z CO

........

;

.......

i'o

K.,

Fig. 10. Comparison of Ps and radiolytic hydrogen inhibition by various solutes in water. The constant of inhibition for TI +, Zn2+, N +, and lantanoids ions lie inside the square.

(34)

Equation (34) indicates that the linear dependence (27) between Ps yield and inhibitor concentration should be realized at concentrations that are severalfold higher than for similar hydrogen dependence (28). This conclusion is illustrated in Fig. 9. The figure shows that, in accordance with the estimate (34), the region of transition to linear dependences (28) and (29) for each inhibitor is for Ps in the concentration range exceeding approximately three times that for hydrogen. 7. APPLICATIONOF THE RECOMBINATION DIFFUSION MODEL OF POSITRONIUM FORMATION TO INTRATRACK PROCESSES

7.1. Reactivity of quasifree electrons

Information on reactivity of nonsolvated electrons is mainly obtained from picosecond pulse radiolysis experiments. Data on inhibition of Ps formation can provide another source of such knowledge, which is Figure 10 shows that in most cases solutes exert a more accessible and therefore also attractive. Reducsimilar effect on Ps and H 2 yields. Moreover, Qr~ and tion of Ps yield can be used as a measure of solute QH2 as a rule coincide: Qr~ - Q.2" reactivity with respect to e-. According to the above This should be true for all solutes that react model of ion-electron recombination for the diffusion exclusively with electrons, and not with holes and part of the positron track, the reciprocal probability positrons. Deviations in one way or the other indicate of Ps formation linearly increases with growing Cs. that this particular substance is an aceeptor of posi- The slope of the straight line is directly proportional trons or holes. to the rate constant of the reaction of the quasifree Equality Qr~ ~ Qn2 makes it possible to establish electron with its acceptor. Relative rate constants the relation between initial ion concentrations in calculated in such a way (Fig. 10) were compared to tracks of high velocity e- and e + (with initial data on picosecond pulse radiolysis. Most of them are energies > 1 MeV). In fact, in fair correlation. Estimates of absolute constant values (~>10~2M-ls -l) are in fair correlation with ~ ~kp, = 3 . 6 ~fls = 3.6 ~ • No/V°2. (33) the known constants of reaction rates for quasifree 1n0/v°3 electrons in hydrocarbons. Notably, rate constants Assuming that reactions (1) and (6) are diffusion grow with growing gas phase cross sections of correcontrolled, the ratio of their rate constants sponding reactions. However, as opposed to the gas phase, the reaction k~2(e- + H20 +) rate constant in liquids depends not only on acceptor nature, but also, to a substantial degree, on interk23(e- + e + ) action of the reagents with a medium. An example is 41rNA (D,_ + DH, o÷) (Re_ + RH2O+ ) the dependence of benzene reactivity in alcohol solu4r~NA(Dc_ + D,+)(/~- + Re.) tions. Benzene is an electron acceptor and a positronium inhibitor, although only up to a certain limit De_ + DH2O+ in its concentration (Fig. 11). Should the concenDo_+D. tration of benzene exceed this limit, its reaction with where the electron no longer remains exothermic, ~ss) and it does not inhibit the positronium formation. e2 Additional data on reactivity of electrons and holes R,_ + RH2o+ & Re- + R,+ ~ kB T can be obtained from comparative analysis of the (NA is Avogadro's number). Under conditions of inhibiting effects of solutes on Ps, H2 and e~q_ yields assumed ambipolar diffusion, D,+ should roughly (Figs 2-4). In accordance with eqns (26) and (27), equal DH2o÷ and D,_. Thus, k23 "kz2. With this relative yield GnJG°2 is a function of Pr~/P°, estimate (33) can be used to find the ratio of mean ion A, ~t; Bs/k (S + e-), fls/k (S + e-). A and ~t are indeconcentration in spurs of the electron track to ion pendent of solute nature; Bs/k(S+e-) and concentration in the end part of positron (electron) fls/k (S + e - ) are also identical for those substances track: that accept only electrons. This is apparent in Fig. 2; '

I0

V.M. BYAKOVand V. I. GRAFUaaN

L

i

1

i

2.0

i

7.2. Reactions of hot electrons with solutes

Ai/Ai°

e

1.0

In a number of cases the dependence of probability P of positronium formation on solute concentration Cs can be described by the exponential eqn (Fig. 2): e = P0"exp(- Cs/C37).

1.6

1

20

.

2

~

10

1.0

5

I

0

0.2

I

I

I

0.4 0.6 0.8 Mole frQction ethanol

1.0

Fig. I 1. Relative probability of positronium formation (13/1~ EtOH), dielectric constant (() and relative yield of solvated electron (Ai/A°), Ref. (54), in ethanol-benzene mixtures. I ° EtOH, A ° are the probability of Ps formation and solvated electron yield in pure ethanol, respectively.

all experimental points (GnJG°2, Pr~/P ° ) that represent such substances, are accomodated by a single curve.t When solutes react not only with electrons (k s + a2o+ # 0) but with holes, the coefficients Bs/k (S + e - ) and fls/k (S + e - ) are governed by k (S + H20 +) and are therefore different for different substances. The greater k (S + H20 + ), the stronger a given substance should reduce (all other things being equal) H 2 yield as compared to Ps yield.:~ Acetone and glycine are the examples of such substances (Fig. 2). Figures 3 and 4 compare relative H E and Ps yields with relative e~q_ yields. Most points representing different substances are accomodated by a single curve, as in Fig. 2. However, some substances (Ag +, Cd 2+) inhibit the yields of H 2 and Ps much weaker than the yield of a hydrated electron.§ A possible explanation is that these substances are much more efficient acceptors of an infrared rather than dry electron which is a precursor of H2 and Ps.

t Stronger inhibition by small concentrations of solutes of the yield of H2 rather of Pr~ is due to the above-mentioned higher concentration of ion-electron pairs in the end part of positron track as compared to the concentration in most spurs of high velocity electron track. :[:Addition of substances that accept only holes [k (S + e - ) ~ 0] even leads to increased Ps yield.°'~) § An anomalous effect of acetone and glycine on Ps yield is due (Fig. 4) to their reactions with holes, that have already been discussed.

(35)

This empirical equation can also be obtained analytically, assuming that cross sections a s and (r+ of electron interaction with acceptor S and the positron decrease with growing energy W of electron in a different way: at W > E~' electrons mostly react with the acceptor, while at W < E* mostly with e + (or at W E* the rates of electron interaction with the acceptor and the positron become comparable). The fact that exponential dependence of P on Cs are in fact experimentally observed can be regarded as confirmation of reactions of hot electron with acceptor S. An exponential dependence similar to (35) is also realized for the yield G (e~-) of the solvated electron on solute concentration, assuming that at W > E s O"S" C S >~ otCt, and at W < E~, osCs <<.atCt (fit is cross section of quasifree electron capture by the trap, Ct is trap concentration), Electron yields, measured under conditions of picosecond pulse radiolysis, in polar liquids as a rule change according to the equation G (e~ ) = G°.exp(-Cs/C37).

7.3. Characteristic features of track structure of electrons having the energy less than l ke V in hydrocarbons csj A comparison of probability P for Ps formation with yield Ga of free ions, that have escaped intratrack recombination, in various liquids shows that P and G~ grow with increasing degree of branching of medium molecules (Fig. 13). The increase of G~ is due to growing initial distance between the ion and the electron in the pair, in other words, due to increasing probability for the electron, generated by molecule ionization, to escape from intratrack recombination. The reason for increased positronium yield is less obvious, since Ps is a product of e + and e- recombination. However, our approach, according to which Ps is formed in the diffusion part of e + track, makes it possible to give a reasonable explanation to these facts. In liquids (n-hexane, c-hexane) with small (<1) length of electron thermalization, only the electrons that had escaped from R H + + e - - - , R H geminal recombination contribute to the e++e-~Ps reaction. On the contrary, in neopentane and tetramethylsilane (TMS), the diffusion part of e ÷ track represents evenly mixed ions and electrons, rather than isolated ion-electron pairs. For this reason, Ps formation in such media proceeds simultaneously with ion-electron recombination. It is this factor that leads to growing P. The above features of the end part of positron track (and, therefore, of the track of an electron having the energy less than 1 keV) also become manifest in the

Ps in radiation chemistry of liquids

lI

1.0'

0.9

•

0.8 0.7

o*

o

o~0.6 0.5

+

o

0.4 0.3

0.2

0.1

0.2

C,, M

Fig. 12. Exponential dependence ofpositronium formation yield on solute concentration: O Cd(C104)2(~); • Ag'C104(z4); O SnC12(57); [] NaNO3(31); V K2CrO4(19);A FeCI3°~); + K2Cr2OT(2?);• KMnO4 (27).

different influence o f minor alcohol additions on Pe~ in hydrocarbons with different structures (Fig. 14). Alcohol prevents ion-electron recombination in e + track, thereby increasing P, which is rather considerable in hexan¢ and totally unobserved in TMS. It follows from the above that the introduction of such terms as blob and short track to characterize track structure is justified only in liquids with great length (-~ 1) of electron thermalization.

L,A 3001

-I

.

13% 50

~

7.4. Localization of quasifree electrons in hydrocarbon alcohol mixtures As was already mentioned, the presence of alcohols in hydrocarbons, even in minute amounts, can lead to effective electron solvation. The Ps yield is reduced at the same alcohol concentrations. The analysis of these data points to the following conclusions. °s) • It is not separate alcohol molecules that act as solvating quasifree electron traps, but rather alcohol polymers, associates that are composed of several molecules: reduction of Ps yield and increase of e~- yield occur in the alcohol concentration range where polymers are quickly accumulated.

t

P/Po

2.2~~...... n I 4o

!

100 -2.5

~"~.~.__o

_o;_

-0.4-93-q2-0~

°°

30

~ ,ql qz

:if 1.0~. 163

o .

I

10"z

I

10"~

--

t x~

Fig. 13. Positronkm-i formation probability (I3 = 3P/4) (1), yield Gn of free ions (2) and mean radius b of ion-electron pairs O) in hydrocarbons as function of V0, the energy of quasifrce electrons. R,P.C. 28/I--B

Fig. 14. Relative positronium formation probability P/Po in tetramethylsilane (1), isooctane (2), n-hexane (3), cyclohexane (4) versus mole fraction XROHof n-propanol; P0 is the positronium yield in n-propanol.

12

V.M. BYAKOVand V. I. GRAFLrnN

• Effective solvation commences only at alcohol concentrations above the critical point. The latter is characterized by a minimum correlation Kirkwood parameter, i.e. by maximally possible compensation of dipole momenta of molecules that form the polymer. • The condensation model of electron solvation, according to which thc clectron is initiallylocalizcd at a polar molecule with subscqucnt formation of a solvatc shcll, finds no experimental confirmation. • The bimolccular constant of the ratc of electron capture with alcohol polymers cannot be smaller than 1013M -l s -l. • Time of electron localization on polymers is equal to, or less than, 10 -t2 s.

to M I I m

7.5. Ortho-para-conversion of positronium and the rate of quasifree electron localization Excess electrons can exist in liquids in one of two states--localized or quasifree. It is assumed that entropy fluctuations that take place in liquids without the energy expenditure can lead to the cumulation in any site of the free volume that belongs to many molecules, forming a space where the quasifree electron can be captured. It is shown that Vo < Vo is a prerogative of quasifree electron localization with with free energy V0. Here T o = -Ol/~)2.Ry,

~/=1.2(1-~:~1)+(1-~o

~) (36)

(e0 is static dielectric permeability). In the liquids

{0.02)

0.8 ~LOCALIZATION

REGION OF ELECTRONS

EXCESS

I n eon 12" 10-31 0.4

Q n-dodecone {0.04) a n-octan (0.04) n - hexone { O.q~Lo n- heptene { 0.05) U " o cyclohexene [0.13) m " J] n-pentane (0.16)

0.0

>

\ \

o

\..n

II . ~.,-

o

n-butonol (7.10 -4) Q

i o.,- o.s)

rlncyciopentane(t.l) ~,~-.m~-II.II o toluene {0 54)

. organ ~ thylbutme u '-14751U ~i,(101 Isooctone171 _ n cz 2 , 2 , 4 , 4 - t e t r a he°pent one 1701"~ ]l _,,,, mathylbutone

-0.4

krlpton {tSO0)

n-proponol o (5" 10 -4)

tetra D ~ methylsilane (tO0) O xenon (2200) ~

ethanol t3.lO'~

-0.8

methanol (6.10-41 EXISTENCE REGION OF Q U A S I F R E E ELECTRONS

\

l water ( 2" 10"3)

-'1.2 I 0

I 0.2

I

i 0.4

I

I 0.6

,r/2 _..[ 1.2 (1 _ ~ 1 )+ ( ~ ,-1-

I

I 0.8

I

I t.0

z

=

~01) - "i 2

Fig. 15. Existence regions of quasifrce and localized electrons in function of 72- [1,20- eg~)+ (E~ - Eo~)]2;E~, ~0 arc high frequency and staticdielectricconstants,respectively.Values of P0 (dashed line) are calculated by Eel.(36). • are measured values of V0.

Ps in radiation chemistry of liquids for which this condition is fulfilled, electron localization is possible in a bubble with the radius in the R~n < R < Rm~ interval (Fig. 15). The region of parameters V0 and ~/2, located below line 1, 12o = -(Ry/n2)rl 2, provides for the existence of excess electrons only in the quasifree state; their localization is possible in the region above the line. Each point in the figure represents V0 and ~/2 values for a specific liquid. Mobility of excess electrons therein is given in brackets. The mobility values corresponding to the points above line 1 are much less than ones corresponding to the points located below the line. Together with condition (36) of quasifree electron locatization, analysis of the rate of their localization, i.e. the duration ~ of their existence in quasifree state, is also of interest. Apparently the measurement of the I2/I~ ratio of the amounts of reacting orthoand para-positronium atoms in liquids give the answer to this question in a number of cases. The processes that occur in the diffusion part of e + track after positronium formation in ion-electron pair environment (A +, e-), can be presented in the following way: k~

ortho -para i" }÷~,Ps+Tei- j c o n v e r s i o n

÷ J,PS ~"' , 2~ :1:J,Ps + A z2,2y + A

] f

e-

2~-annihilation

"] ion-electron ~- recombination J localization

A + + e- k~, A ) e/-

Calculation of integrals

Ii=

;o

21nl(t)dt,

12=

;:

22n2(t)dt

(37)

that characterize the intensities of para- and orthopositronium components, respectively, on the basis of eqns of the (15)-(17) type for diffusion kinetics of reactions (37), leads to: t59)

H2O+ + H20--,H30 + + OH

/ H20 + + (CH3)2CO

Here, Po is the radius of the reaction between i and j particles (i = 1, 2, 3, 4, 5 label para-Ps and ortho-Ps atoms, A+-ions, quasifree (e-) and localized (e() electrons, respectively); P25 < P24, 0.2 ~<. ~

87rD24a34

k34n30 < 1. 8nD34a~

It is apparent from (38) and (39) that rapid localization of electrons in "bubbles," such as occurs in liquid He, to a substantial degree prevents their interaction with the Ps atom: at ~ < ~ 4 J -~ ..~34(P25/P24) ~ 1 and 11/12 = 1. On the other hand, lengthy duration ~ of knocked out electron existence in quasifree state creates favourable conditions for o-Ps conversion into p-Ps: at ~1 > ~24 J = .£a24(1 - -£/34) ~ 1 and I1/I2 > 1. This is true for liquified Ar and Xe, where electron-medium interaction is not repulsive, with electrons existing in quasifree state. In fact, measurements give I~/I2 = 1 and 1.7 for Ar and Xe, respectively. (57-6°)

A comparison of the effect of additives on luminescence of aqueous acetone solutions that results from ionizing irradiation with the effect of identical additives on Ps yield (Fig. 16) forms the basis for the conclusion that the presolvated electron participates in the formation of excited acetone molecules. Measurements of luminescence were conducted by a pulse radiolysis technique, with a 22 MeV proton beam as the source of ionizing radiation. Such proton kinetic energy precludes the appearance of Cherenkov's radiation, thereby making observation of luminescence easier. The form of luminescence spectra and lifetime of excited states make it possible to assign the observed luminescence to triplet-excited acetone molecules. The dependence of luminescence intensity on acetone concentration signifies that in dilute aqueous solutions excitation occurs not through the direct action of radiation on acetone molecules, but rather due to the energy absorbed by the water component:

(CH3):CO + + e----} (CH3)2CO*

H20 + (CH3)2CO + (CH3)2CO* + H20 ----}(CH3)2CO + q

\

1,/I~ = (1 + ~ ) d - 1,

(38)

where =

(V

< ~34 -

OH + (CH3)2COH +

j

-

7.6. Nature of precursors of excited states appearing in radiolysis of aqueous solutions of organic compounds

-~TPs + ,[e - , , ~ ÷,L Ps + T e- ] ~TPs+~e-,

13

P25/P24+ 5tt/524 [1 + (~224/~334) ,~c°34] 1 + ~/,~'u ""Zu" (39)

(CH3)2CO* -} (CH3)2CO + ho~ Indirect ionization of acetone molecules is confirmed by suppression of luminescence by I-, Br-, CI- ions, acceptors of H20 + ions, which is the stronger, the weaker is the bond energy of the valent electron in the halogenide ion.

14

V.M. BYAKOV and V. I. GRAFUTIN

5 o a_

4

4 ._J

%

% 3

2

i

I

0

I

l

0.5

1.0 Cs,M

0

i

I

I

0.5

1.0

1.5

Cs,M

Fig. 16. Influence of electron accepters on (a) radioluminescence intensity of 0.86 M acetone aqueous solution(U): • Cu2+; [] IO~-; A Cd2+; • H+; O Zn2+; • NO2; • CH2CICOOH; (b) positronium formation probability (data from Refs (26,65) and this work): 0 Cu 2+; [] IOn-; A Cd2+; • H + ; © Zn 2+; • NO~-; • CH2CIOOH.

8. CHEMICAL REACTIONS OF POSITRONIUM AND THE SOLVATED ELECTRON

Useful information pertaining to radiation chemistry can be extracted not only from measurements of Ps yields, but also from Ps lifetime studies, i.e. Ps chemical reactions. The fact is that the positronium atom, similarly to the solvated electron, is an electron donor for a large number of substances dissolved in liquids.

8.1. The bubble model of positronium atom in liquids

(1 + /2mU. sin KR ~ sin4 KR q h2 ~ / KR(-E~--~KR) 27th2o -

~ mU

(40)

where K = 2 ~ - E - / h 2 , m is mass, E is energy of Ps atom measured from the bottom of the potential well having the depth U, formed by the bubble. The left part of eqn (40) has the maximum value at KR = 1.9. This means Ps bubble cannot exist in liquids whose surface tension surpasses a certain limit value. The latter at cr = 0 and U "~ I eV equals 55 erg/cm 2, although it reaches a value twice as great .at 6 = 2 A. Thus, it turns out that the formation of the Ps bubble is possible in liquids with surface tension values of up to 100erg/cm 2, including water and aqueous solutions.

A similar behaviour of Ps and e~- towards solutes and the similarity of the rate constants of their diffusion-controlled reactions serve to explain the similarity of models of P and e~- atoms in liquids. According to these models, it is assumed that Ps and e;- atoms are localized in microscopic cavities (bubbles). (52'~'~) Bubble radius is determined from the 8.2. The role of tunneling in fast liquid phase reactions of the positronium atom and excited molecules equality of repulsive pressure exerted by Ps atom on Although the role of tunneling in low temperature the surrounding molecules, and Laplace pressure compressing the bubble. It is usually considered that reactions of the trapped electron has been revealed, the bubble model for Ps is applicable to only those the role of sub-barrier transfer in the reactions of liquids that have small surface tension values. (6s) solvated electron and Ps atom is not quite clear. In However, such a conclusion results from the fact order to reveal the role of tunneling in liquid phase that the effect of bubble radius R on liquid surface reactions of e~- excited molecules and Ps atoms, tension a is ignored. Taking into account that eqn (41) was obtained in Ref. (71) for rate constant 0 = oo~R/(R + 6 ) (6 is the thickness of interface of diffusion-controlled reaction of positronium atom layer),(69)the condition of Ps bubble formation can be and solvated electron with accepter in a condensed ascertained; it can be demonstrated that the Ps phase taking into account the tunneling of Ps atom bubble formation is possible also in liquids with or electron [eqn (41) is suitable for describing e~considerable surface tension. Minimizing the free reactions with electroneutral molecules]. Owing to energy change in the process of Ps bubble formation the tunneling transfer existence, the rate constant k is the sum of two terms--tbe first of them is properresults in Pet'. (70):

Ps in radiation chemistry of liquids

Noncontact interaction in liquid phase reactions with electron-excited molecules is revealed by data on oxygen quenching of excited anthracene molecules (A*). Extrapolation of k~,p(A* + 02) at D --)0, as in Fig. 18, results in k~tr ~ 0.

Isn ! o

:E

6

8.3. Chemical reactions of Ps and e7

5 0 4-

o~

15

4

3

2 I

0

I

I

I

I

I

1

2

3

4

5

K0 (Ps + 0 2 ) ,

10-1°M-Is -I

Fig. 17. The correlation of experimental rate constants and calculated by eqn (41) values of k d for reaction Ps + 02 in various liquids. • data of Re£ ('/2), • data of Re('. (73). The angle between straight line and axis of k d is equal to 45 ° and intercept on axis of ordinates is equal to 1.6 x 101°M-I s-l.

tional to the coefficient D of mutual diffusion of reagents and the second one (characterizing the subbarrier reaction) does not include the diffusion coefficient:

k =kd+kt=4~DR +,r/4"v'~'(Rs/R) 2. (41) Here, Rs is radius of solute acceptor molecule, characterizes attenuatior~ of the e~- wave function (of the Ps atom) beyond the bubble, v is the frequency factor of tunneling. Equation (41) was used to calculate the rate constants of the Ps + 02 reaction in various liquids with known values of D (02). The diffusion coefficient for Ps was calculated according to the Rybezynski-Hadamard formula: DR = knT/ 41r R~. ~/(r/is the dynamic viscosity of the liquid). The radius Rps of the positronium bubble and 2 were determined according to formulae of the Ps bubble model t6s) using the data on the rate of pick-off o-Ps annihilation, v was equaled to x/E/2m~R~ (E is the energy level of Ps atom in the bubble). The values k = k d + k t calculated in this fashion are compared in Fig. 17 with measured values ko~ (Ps + 02). With due regard for the performed approximations, the correlation of the above simple model with experimental data is quite adequate.t

Figure 19 presents absolute constants of Ps and e/" reaction rates as a function of the oxidationreduction potential of the acceptors. At potentials below 2 eV, when e,- reactions are exothermic, their rate constants are practically independent of potential value, and limited by diffusion of reagents. In the potential range above 2 eV, the reactions are endothermic and their rate constants decrease sharply. The character of rate constant dependence on redox potentials is similar for Ps reactions as well although the threshold is shifted into the smaller potential range by 2.5 eV. This means that the energy level of the ground electron state in Ps atom is deeper than the ground level of the energy of the electron in hydrated state. It is interesting to point out that for the negative oxidation potentials the rate constants of Ps and efq have the same order of values. Similar dependences of the rate constants of the reaction of Ps atoms and efq with solutes implies that reactions of Ps have an oxidation character, i.e. are reactions of electron transfer from Ps to the acceptor. At present one often believes that positronium chemical reactions with an acceptor proceeds through the positronium complex formation. (77'78) It is however, possible that the concept of oxidation character of Ps reactions with solutes will prove its worth in interpreting experimental data on chemical Ps reactions in both water and weakly polar liquids. Figure 20 shows the dependences of rate constants of Ps reaction with nitrobenzene on temperature in different solvents. The most remarkable effect induced by the temperature changes is the appearance of a maximum in the observed reaction rate constant

"T

4

?i 3 2

( DA÷ 00, ) 10s cmZ/s

T Nonzero rate constant at D --, 0 was also observed for the reaction Ps + Co 2+ in water. <75) This result can be interpreted as another manifestation of tunneling in positronium reactions.

Fig. 18. The dependence of rate constant for quenching reaction of excited anthracene by dissolved oxygen on total diffusion coefficient D (02) + D (A).(7''74)

16

V.M. BYAKOVand V. I. GRAFUT[N

i;

T

I0m

T

"- - I

-o

°--+-+""-'~i.[~'

--

!

2t

0e

I

P,

The upper limit

.l

.+T l

i=

G)

1 I

~

¢'r (,~ I--

10 4 I

-2

I

I

l

I

I

-I 0 I 2 Redox polential for the reactions: _ n+ eeq + Aeq

3 E ° (eV)

.(n-I)+ - Aeq

Fig. 19. Rate constants of e~q (1) and Ps (2) oxidation reactions as functions of its redox potentials E°.(75)

(kobs) at a certain temperature. The location of the maximum depends essentially on the medium polarity. Rate constants of the reactions of Ps with other compounds are characterized by similar temperature behaviour. To explain medium polarity influence, and the extremal dependence of the rate constant of the Ps reaction on temperature, we used the concept of nonradiative electron transfer mechanism in a condensed medium. (32'3mThis concept specifies kob s =

10II

4nD .p. W(p). V 4:,tDp + W (p ). V"

Here, W is the probability of an electron tunneling from Ps to acceptor at reaction radius p; V is the volume of reaction layer; D is the coefficient of the mutual diffusion of Ps and its acceptor. At sufficiently high temperatures kV ~ exp

perature dependence of experimental kob~ values although the calculated temperature corresponding to maximum kob, is by about 100° higher than experimental. We believe that this discrepancy is caused by

[ (Er +AG+Gf)2] 4ErR T ,

where AG is the difference of the electron levels between which the electron transfer occurs. Er and Gf are the reorganization energy of medium and the energy required to separate the reaction products to infinity, respectively. At sufficiently high temperatures E r decreases faster than I/T. As a result, W decreases, and the reaction rate is no longer diffusion-controlled. On the contrary, at low temperatures 4nDp ,~ WV, and kot,,~-4rcDp, i.e. increases with temperature. In Fig. 20 the calculated values of kob, are compared with experimental ones in pentanol. Calculated curves satisfactorily reflect the character of the tern-

I

I

2

3

I

I

4

5

101° t~n

t 10 9

10 e 10SIT

Fig. 20. Comparison between the experimental and calculated values of the observed positronium-nitrobenzene reaction constant in pentanol vs I/T: (1) experimental c u r v e ; (79) (2 and 3) calculated by means of eqn for kob, and W using the following values of the paramaters. E~rs = 1.15 kcal/mol; E~ = 4.6 kcal/mol; p = 6 A; ~/2n=5x 10t3 and 3.3x 10t2 for curves 2 and 3, respectively.

Ps in radiation chemistry of liquids approximation of the ion surrounding medium as a continuous dielectric used in the expression for the reorganization energy. This assumption underestimates the temperature coefficient of Er.~s~) CONCLUSION Positrons, once introduced into a medium, inevitably annihilate either in a free state on collision with electron molecular shells, or in an electron-bonded state, namely the positronium atom. Positron in a condensed medium behaves as an acceptor of a quasifree (dry) electron; the product is the Ps atom. Positronium, similarly to the solvated electron, is an electron donor in relation to many solutes. The above characteristic features of positron and positronium behaviour present novel opportunities for the investigation of dry and solvated electrons, w h i c h are two important reagents yielded by radiolysis of liquids. These features determine the value and outlook of positron studies in radiation chemistry.

REFERENCES

1. VI International Conference on Positron Annihilation, Denmark, 1982. 2. L. T. Bugaenko, V. M. Byakov and E. P. Kaliazin, Symposium on Radiation Chemistry of Aqueous System, Moscow, 10-12 December, 1973. Abstracts, Nauka, Moscow, 1973, p. 4. 3. V. M. Byakov, Int. J. Radiat. Phys. Chem. 1976,$, 283. 4. V. M. Byakov, V. I Grafutin and O. V. Koldaeva, Int. J. Radiat. Phys. Chem. 1977, 10, 239. 5. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva et al., Preprint ITEP 1976, Moscow, N 62. 6. V. M. Byakov, Mekhanizm Radioliza Vody (in Russian) (The Mechanism of Water Radiolysis ), Proc. Symp. on Radiation Chemistry, Moscow, 1970, p. 5-40. Moscow State University, 22-24 May 1968. 7. N. I. Tretjakova and V. M. Byakov, Preprint ITEP, Moscow, 1982, No. 115. 8. M. Faraggi and J. Desalos, Int. J. Radiat. Phys. Chem. 1969, 1, 335. 9. V. I. Goldanskii, At. Energy Rev. 6, 1968, 3. I0. A. Mozumder, in Proc. 3rd Tyhany Syrup. on Radiation Chemistry, Vol. 2, p. 1123. Budapest. Academiai Kiado, 1972. 11. N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions. 3rd edn. Clarendon Press, Oxford, 1965. 12. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva, E. V. Minaichev and O. P. Stepanova, Kh/m, vys. energij, 1983, 17, 506. 13. S. J. Tao and J. H. Green, J. Chem. Soc. A, 1968, 408. 14. O. E. Mogensen, J. Chem. Phys. 1974, 64, 998. 15. A. Mozumder and J. L. Magee, Radiat. Res. 1966, 28, 203. 16. V. M. Byakov and F. G. Nichiporov, Radiachem. Radioanal. Left. 1981, 48, N2, 101. 17. L. T. Bugaenko, V. M. Byakov and C. S. A. Kabakchi, Khim. vys. energij 1985, 19, N4, 312. 18. H. A. Schwarz, J. Am. Chem. Soc. 1955, 77, 4960. 19. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva, E. V. Nirlaichev, F. G. Nichiporov and O. P. Stepanova, Preprint ITEP, Moscow, 1984, No. 37. 20. E. Hayon and M. Moreau, J. Phys. Chem. 1965, 69, 4058.

17

21. G. Duplatre and C. D. Jonah, Radiat. Phys. Chem. 1985, 24, 557. 22. E. Peled and G. Chapski, d. Phys. Chem. 1970, 74, 2903. 23. S. J. Tao and J. H. Green, J. Phys. Chem. 1969, 73, 882. 24. A. G. Maddock, J. Ch. Abbe, G. Duplatre et al.,Chem. Phys. 1977, 26, 163. 25. D. Katakis and A. O, Alien, J. Phys. Chem. 1964, 68, 3108. 26. L. J. Bartal, J. B. Hicholas and H. J. Ache, d. Phys. Chem. 1972, 76, 1124, 27. M. Eldrup, V. P. Shantarovich and O. E. Mogensen, Chem. Phys. 1975, II, 129. 28. C. Jonah, J. Miller and M. Matheson, d. Phys. Chem. 1977, 81, 1618. 29. R. K. Wolff, M. I. Bronskill and J. W. Hunt, J. Chem. Phys. 1970, 53, 4211. 30. R. E. Green and R. E. Bell, Can. d. Phys. 1957, 35, 398. 31. A. G. Maddock, J. Ch. Abbe and A. Haessler, Chem. Phys. 1976, I7, 343. 32. J. Ch. Abbe, G. Duplatre, A. G. Maddock and A. Haessler, Strasbourg, CRN/CPR, 794)9. 33. O. A. Anisimov and Yu. N. Molin, Khim. vys. energij 1975, 9, 541. 34. K. Y. Lam and J. W. Hunt, Int. J. Radiat. Phys. Chem. 1975, 7, 317. 35. A. Appleby, in The Chemistry of lonisation and Exitation (Edited by I. R. A. Johnson and I. Scholes), p. 269. Tailor and Francis, London, 1967. 36. O. A. Anisimov and Yu. N. Molin, Khim. vys. energij 1975, 9, 376. 37. B. G. Ershov and A. K. Pikaev, Radiat. Res. Rev. 1969, 2, 101. 38. C. D. Beling and F. A. Smith, Radiat. Phys. Chem. 1984, 23, 571. 39. E. Hayon, Trans. Faraday Soc. 1965, 61, 734. 40. A. Singh, Faraday Discussion, 1977, 63. 41. M. Haissinsky, Rendements radiolyiques primaires en solution aqueuse, neutre ou alcaline. In Actions Chimiques et Biologiques des Radiations. Collection dirig~e par M. Haissinsky, Masson, Paris, 1967. 42. R. K. Wolff, J. E. Aldrich, T. L. Pennew and J. W. Hunt, J. Phys. Chem. 1975, 79, 210. 43. F. S. Dainton and W. S. Watt, Proc. Roy. Soc. A, 1963, 275, 447. 44. V. N. Belevskii, S. I. Belopushkin and L. T. Bugaenko, Khim. vys. energij 1975, 9, 252. 45. V. M. Byakov and V. R. Petuchov, Radiochem. Radioanal. Lett. 1982, 51, 143. 46. C. D. Jonach, M. S. Matheson and J. R. Miller, J. Phys. Chem. 1976, 80, 1267. 47. T. Sumiyoshi and M. Katayama, Chem. Lett. 1982, 12, 1887. 48. V. M. Byakov, V. L. Bugaenko, V. I. Grafutin, O. V. Koldaeva, E. V. Minaichev and Yu. V. Obukhov, Khim vys. energ O" 1978, 12, 412. 49. V. I. Goldanskii, B. M. Levin and V. P. Shantarovich, Khim. vys. energO" 1976, 10, 192. 50. V. M. Byakov, Positronium formation in noble gases (in Russian). Preprint ITEP, 1983, Moscow, No. 57. 51. S. S.-S. Huang and G. R. Freeman, J. Chem. Phys. 1973, 44, 4120. 52. A. G. Khrapak and I. T. Yakubov, Elektrony vplotnych gazach, Nauka, Moscow, 1981. 53. G. A. Kenney-Wallas and C. D. Jonach, Chem. Phys. Left. 1976, 39, 596. 54. K. Okazaki and G. Freeman, Can. J. Chem. 1978, 56, 2313. 55. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva and E. V. Minaichev, Radiochem. Radianal. Lett, 1982, 54, 283. 56. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva et al., Chem. Phys. 1977, 24, 91. 57. L. J. Bartal and H. J. Ache, Radiochim. Acta, 1972, 17, 205.

18

V.M. BYAKOVand V. I. GRAFUTIN

58. V. M. Byakov and V. I. Grafutin, Khim. vys. energy] 1980, 14, 263. 59. V. M. Byakov and V. R. Petuchov, Radiochem. Radioanal. Lett. 1983, 58, 75. 60. V. I. Goldanskii, B. M. Levin and V. P. Shantarovich, Khim. vys. energy] 1976, 10, 192. 61. C. V. Briscoe and A. T. Stewart, in Positron Annihilation, p. 377 Academic Press, New York, 1967. 62. P. Varlashkin, Phys. Rev. 1971, A3, 1230. 63. E. A. Gorbaty and R. W. La Bahn, Phys. Rev. 1971, A4, 1425. 64. O.P. Stepanova, F. G. Nichiporov, I. G. Aksenov et al., Khim. vys. energij 1981, 15, 51; O. P. Stepanova and F. (3. Nichiporov, Proc. 4th Tyhany Syrup. on Radiation Chemistry, Budapest, 1977. p. 731. 65. V. M. Byakov, V. I. Goldanskii and V. P. Shantarovich, Dokl. Akad. Nauk SSSR, 1974, 219, 633. 66. R. A. Ferrel, Phys. Rev. 1957, 108, 167. 67. L. O. Roellig, in Positron Annihilation (Edited by L. O. Roellig and A. T, Stewart), p. 127. Academic Press, New York, 1965. 68. A. P. Buchikhin, V. I. Goldanskii, A. O. Tatur and V. P. Shantarovich, Zh. Eksperim. Teor. Fiz. 1971, 60, 1136.

69. R. C. Tolman, J. Chem. Phys. 1949, 17, 333.

70. V. M. Byakov and V. R. Petuchov, Radiochem. Radioanal. Lett. 1983, 58, 91. 7 l. V. M. Byakov and V. R. Petuchov, J. Radioanal. Nucl. Chem. Lett. 1984, 85, 67. 72. J. Lee and G. J. Celitance, J. Chem. Phys. 1966, 44, 2506. 73. P. Levay and P. Hautojarvi, J. Phys. Chem. 1972, 76, 1951. 74. W. R. Ware, J. Phys. Chem. 1962, .66, 455. 75. R. Zana, S. Millan, J.-C. Abbe and G. Duplatre, J. Phys. Chem. 1982, 86, 1457. 76. V. M. Byakov, V. I. Goldanskii and V. P. Shantarovich, Elektrokhimija 1977, 13, 804. 77. V. I. Goldanskii and V. P. Shantarovich, Appl. Phys. 1974, 3, 335; V. P. Shantarovich and P. Jansen, Chem. Phys. 1978, 34, 39. 78. E. Hall, W. J. Madia and H. J. Ache, Radiochem. Radioanal. Lett. 1975, 23, 283. 79. V. M. Byakov, V. I. Grafutin, O. V. Koldaeva and E. V. Minaichev, J. Phys. Chem. 1980, 84, 1864. 80. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Eksp. Teor. Fiz. 1969, 56, 1287. 81. R. R. Dogonadze and A. M. Kuznetsov, Rezuitaty nauki, Vol. 2, VINITI, Moscow, 1974. 82. M. Ya. Ovchinnikova and A. A. Ovchinnikov, Opt. Spektrosk. 1970, 28, 964.