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Effect of temperature on the formation of a boundstate of positronium with

nitrobenzene and on the related hyperfine interaction parameters

I. Billard, J. Ch. Abbé, and G. Duplâtre 

Citation: The Journal of Chemical Physics 97, 1548 (1992); doi: 10.1063/1.463230 

View online: http://dx.doi.org/10.1063/1.463230 

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/97/2?ver=pdfcov 

Published by the AIP Publishing 

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Effect of temperature on the formation o a bound state o positronium

with nitrobenzene and on the related hyperfine interaction parameters

I Billard, J. Ch. Abbe, and G Duplatre

Centre de Recherches Nucieaires, Chimie Nucieaire, B.

P.

20, 67037 Strasbourg Cedex

2

France

Received 9 December 1991; accepted 8 April 1992)

The strong magnetic field effect on the chemical bound-state formation of positronium with

nitrobenzene in n-hexane previously studied at 294 K has been further examined as a function

of temperature, from 264 to 328

K. The

model proposed before

to

explain the data, implying

the definition

of

triplet

and

singlet positronium bound-state, appears

to

be valid at all

temperatures studied.

The

variation with temperature

of the

parameters descriptive

of

the

bound-state are derived; the hyperfine splitting

is

constant at 1.06 X

10-

5

eV

while the contact

density parameter,

7]c

shows an abrupt change from 0.43 to 1.56 between 286 and 292 K.

These experiments give new insight on the structure

of

the bound-state formed, which may be

considered as a charge-transfer complex.

I. INTRODUCTION

Although the possibility has been known for some

time, it is only during the past decade that the effects of an

external magnetic field,

B

on positronium Ps) annihilation

have been used

to

study the physical properties of Ps in

li

quids,2 and also, more recently, the chemistry of PS 3 4 5 a

domain which is still poorly understood.

In

this context, the

strong magnetic field effect on Ps in solutions of nitroben

zene <P-N0

2

) in n-hexane, first observed by Rochanakij

et

al. 6 has given new insights on the mechanism of this reac

tion.

3

The

present paper reports on the extension

of

this

work to the study

of the

influence

of

emperature, in

order

to

assess the validity

of

the formerly proposed kinetic scheme

and to gain more information on the parameters descriptive

of the positron-electron hyperfine interaction within the

bound-state.

II. EXPERIMENTAL

The chemicals, reagent grade from Aldrich, were used

as received. In a first paper, we studied the effect of the mag

netic quenching on solutions

of

nitrobenzene in n-hexane

at

294 K.

3

The concentration selected for the present work was

0.025 M. The temperature

n of

the sealed ampoule con

taining the freshly prepared solution was thermostatically

controlled within 0.1

K,

from 264 to 328K.

The

influence of

B on the pure solvent, already studied at 294 K,

3

was com

plementarily examined at 268 K. Due to the additional geo

metric constraints imposed by the thermostatic system, the

maximum value

of

the magnetic field intensity, B was 1 T.

The experimental setup for lifetime spectroscopy LS)

was essentially the same as that previously described.

2

 3

However, two modifications were made to improve the time

resolution

and

counting efficiency;

i) the

length

of

the

Plexiglas light guides was reduced from 15 to 11.5 cm; ii)

owing

to the

aging

of he

silicon elastomer Rhodovsil R

TV

2)

disks inserted between the light guides and the photomul

tipliers to enhance light collection, these have been replaced.

In this way, the time resolution, measured as the full-width

at half-maximum

of

the 60Co prompt curve, has been im

proved to 380 ps.

Whatever the value of the field, the LS spectra should

include the contributions from several components, the two

Ps states, singlet p Ps) and triplet

0-

Ps), the free positron

e+) and the bound-state of Ps with a solute molecule. As

was shown previously,

3

this bound-state also presents singlet

and triplet substates. In a lifetime spectrum or when dealing

with theoretical expressions, the presence

of

these various

components results in an equivalent number

of

decay rate

constants,

A

and associated intensities,

I.

Due to the com

plexity of the problem, the notations which will be used in

the following are summarized in Table I. SUbscript c will

denote parameters related to the bound-state, superscript 0

will denote values in the pure solvent and subscripts i = 1 3,

4,

will refer to the singlet, triplet

m = 0) and

triplet

m

=

±

1)

substates whether

of

Ps

or of

the bound-state,

respectively, while subscript 2 will be for free e+. Whether

the field

is

on or off will be indicated by the symbols B) or

0) ,

respectively.

When the field

is

off a three component analysis

of

the

LS spectra appears to be satisfactory, implying that the com

ponents relative to the bound-state are unresolved from the

short lived components; most probably, as shown in the fol

lowing, A3c

0 ) is

close to A

2

( 0) ,

and Alc

0) ,

to Al

0) .

When the field is on, due to

the

mixing

of

the spin sub

states as explained in the next section, additional compo

nents appear in the LS spectra. As previously,3 the LS spec

tra

were therefore analyzed in four components, fixing the

parameters related to

0-

Ps m =

±

1) and the intensity of0-

Ps m = 0), in order to derive reliable values of A4

B).

Even

so, the error on lifetime 1/

A4 B) at

specified T was rather

large, ranging from 0.04 ns up to 0.1 ns.

As proposed by Bisi

et al. 7

however, the LS data will be

mostly expressed in terms

of

R, defined as the ratio

of

the

normalized integrals of counts in a time window ta,t

b

) of

the

spectra, when the field is on and

off.

To focus on the

variation with B

of

the longest-lived components,

ta

was

chosen so that the contributions of free e+

and of other

short-lived components are negligible,

and

tb was taken at

the limit of statistically significant countings. The values

of

1548

J. Chern. Phys. 97 (2), 15 July 1992

0021-9606/92/141548-06 06.00 © 1992 American Institute

of

PhYSics

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Billard, Abbe, and Duplatre: Bound-state

of

positroniul')1 with nitrobenzene

1549

TABLE

I.

Summary

of

he symbols used to denote the decay rate constan ts

arising

in

the

LS

spectra because

of

the various posi tron states involved, in

the absence

(C

= 0)

or

in the presence

(C

0)

of

the solute and of the

magnetic field

(B).

The star points out those states whose decay rate con-

stant s are not affected by the field. The intrinsic lifetimes of he singlet states

ofPs

and

of

the Ps bound-state with

l>N

2

, in vacuo

are

A,

and

A

respec-

tively.

Positron stat e

C O C O

singlet Ps

(p Ps)

A'I(B) AI(B)

free positron*

A ; 0 ) A

2

(0)

triplet Ps (o-Ps, m = ± 1)*

A; 0) A,(O)

triplet Ps (o-Ps,

m =

0) A ~ B )

A.(B)

singlet Ps bound-state

p-C)

A,c (B)

triplet Ps bound-state o-C, m = ±

I)

A

3c

(O)

triplet Ps bound-state

o-C, m =

0)

A.c(B)

ta and tb for the solution at the various temperatures studied

are

gi

ven in Table II.

The

theoretical expression forR

is

(this

corrects a misprint in Ref. 3)

R

=/ B) I / O)

with

/ B)

=

L

r h I;A; B) exp[ - A; B)t

]dt

/

o

1)

~ f

IjAj B)exp[ -A j B) t ]d t 2)

The experimental

error

on

R

was within 0.012.

III THEORETICAL BASIS

A Effects o the magnetic field

When the field

is

on, in the absence

of

any reaction with

the solute, the two magnetic substates m = 0 of Ps are

mixed, leading to an increase in the Oops (m = 0) decay rate

constant as (a similar expression holds for the poPs decay

rate constant)

A (B) = [A; 0)

+

a

2

A·I

0)

] /(1

+

a

2

) , (3)

with a = [ 1

+ x

2

) 1/2 -

l]1x

and

x

= 4f-lBB

1

AE

m

, where

f-l

B is the Bohr magneton and AEm ' the y p e r f i n e splitting of

Ps in the medium. The latter parameter is related to the Ps

hyperfine splitting in vacuo through a phenomenological pa

rameter,

1/

AEm

=

1/AEvacuo

.

4)

The

measurable decay rate constants are A; (0), neglecting

the three-gamma annihilation, and

A 1 0) = A 0 ) +

1/As

,

(5)

where As (8 ns- I ) is the intrinsic singlet decay rate constant.

As noted by Mills,9 the mathematical form ofEq. (3) is

valid ifone considers a more complex e+1

e-

system than Ps,

provided that e

+

is bound to a set of electrons with total spin

1/2. This is the case when dealing with a bound-state

of

Ps

with a diamagnetic molecule, as presently. Then, specific

parameters descriptive

of the Ps bound-state are to be used,

A

4c

(D)

=

[A3e(0)

+ b

2

Ale 0)]/ l

+ b

2

  ,

(6)

TABLE

II. Experimental LS data for pure n-hexane,

I ;

and

1/A; (0),

and

for the 0.025 M nitrobenzene solution,

1/A)(0),

at zero field;

(ta ,t

b

)

are the

values for the t ime windows defining parameter

R,

used for the solution.

T

I

l /A;

(0)

l/A)(O)

ta,t

b

)

(K)

( )

(ns)

(ns)

(ns,ns)

264 40.1 3.57

2.29

4.21, 5.80

268 40.4

3.58

2.35 4.30, 5.93

273 40.7 3.64 2.57 4.71, 6.49

279 40.9 3.65 2.78 5.06, 6.97

286 41.1 3.74 2.99 5.49, 7.57

292 41.7 3.74 3.30 6.05, 8.34

294 41.8 3.75

3.30

6.15, 8.50

300

42.0

3.76

3.27

6.00, 8.27

303

42.0

3.76

3.32 6.08, 8.38

313 42.6 3.79 3.46 6.34, 8.74

328

43.0

3.87

3.59 6.57, 9.06

with

b

= [ I + f)

1/2

- l]1y and

y

= 4f-lBD

1

AE

me

,

where

AEme

is

the hyperfine splitting of

the

Ps bound-state in the

medium. Similarly to the case ofPs,

AEme

can be related to

the hyperfine splitting of the bound-state in vacuo, AE

c

 

through the following equation:

AEme = 1/cAEc . (7)

To complete the parallel in the formal treatments for

both Ps and the bound-state, A Ie

0)

in Eq.

(6)

is given by

A1c(0) =1/cAse +A

 

0)

,

(8)

where Asc is the intrinsic decay rate constant of the singlet

state of the Ps bound-state.

It may be noted

that

in the case of Ps alone, the only

unknown parameter expressing the field dependence is

1/

in

Eqs. (3 ) - 5 ) , while in the case of he Ps bound-state both 1/c

andAE

c

'

in Eqs. 6)- 8) , are unknown but couldapriori be

determined through a fit of experimental data because they

appear in distinct parts of the equations.

B

Kinetic model

The

reaction

ofPs

with nitrocompounds, leading to a Ps

bound-state, has been the subject of extensive studies in var

ious solvents. \0 11

On

the basis of

an

irreversible reaction and

in the absence of any magnetic field effect, the apparent reac

tion rate constant, k I can be obtained from

(9)

where C is the solute concentration. However, it has been

repeatedly shown

that

the reaction

of

Ps with nitrobenzene

is, in fact, reversible,12 involving the forward (k

l

)

and re

verse (k

2

)

reaction rate constants. In our previous study of

the magnetic field effects, it appeared that the data could be

explained on the basis of the following hypotheses

3

:

(i) Similarly as for Ps proper, one can define a singlet

(p-C) and a triplet (o-C, with substates m = 0 ±

1)

state

for the complex. As in the case ofPs, the decay rate constants

of the two m = 0 substates of the complex

[Ale (D)

and

A4c (D)] are liable to magnetic field effects, while all intensi

ties and the decay rate constant of the

m

= ± 1 substates

[A3e

0)]

are unaffected. (ii) A particular (sub ) state

of

Ps

J. Chern. Phys., Vol. 97,

No.2,

15 July 1992

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1550

Billard, Abbe, and Duplatre: Bound-state of positronium with nitrobenzene

can only lead, through the chemical reaction, to the corre

sponding (sub) state of the complex.

The kinetic equations are then written as follows (the

rate constants for the reversible reaction, not noted, are kl

and k

2

  :

<;

Bl -<le(Bl

2y < -

p-Ps

+ <I>-N02

p-C -+ 2y,

I)

< ~ B l -< e(Bl

2y

+ -

o-Ps m

=

0)

+

<I>-N02

o-C

m

=

0)

-

2y,

-<;(0)

2y

+-

o-Ps m = ±

1)

+ <I>-N02

-<3e(0)

o-C m

=

±

1) -

2y.

IV. RESULTS

A. Pure solvent

II)

III)

The LS parameters

of

pure n-hexane at zero field and for

various temperatures are reported in Table II. Both I; and

1/ ; 0) increase smoothly with temperature, in agreement

with previous observations. 13

The experimental variations

of

R with B at 268 K are

shown in Fig. 1, for a time window set

at

fa = 6.56 ns and

fb

= 9.04 ns. A best fit to the data according to

Eq. 2)

gives

l/ = 0.83. Comparing with l/ = 0.82 found at 294 K,14,15 it

thus appears

that l/ is

temperature independent; in the var

ious calculations to follow, a constant value of l/

=

0.82 has

been used.

B Solution

Lifetime 1/ 2 0 ) appears to be constant at all tempera

tures, at (0.43

±

0.03) ns, while

1/A

3

  0 )

increases with

T

(Table II).

The

values

of

the apparent reaction rate con

stants,

k ,

derived from Eq.

9),

are given in Table III; it may

be noted that the value at 294 K is in fairly good agreement

with other determinations.

3

,4 The continuous decrease of

k

with increased temperature

is

indicative of the reversibility

of

the reaction; the maximum in the apparent reaction rate

constant has been reported to occur

at

220

K.12

1.0

R

0.9

ci>err.

o

FIG. 1.

Variation

of parameter R

with magnetic field intensity.

B n.

in

pure

n-hexane

at

268

K. The

solid line is calculated with

TJ

=

0.83.

TABLE III.

Kinetic parameters derived from

the data

in Table II; apparent

k

). forward

[k,;

calculated from Eq.

10)], and

reverse

k2)

reaction rate

constants.

T

K )

264

268

273

279

286

292

294

300

303

313

328

6.26

5.86

4.58

3.53

2.68

1.43

1.45

1.59

1.41

1.01

0.81

29.3

30.6

32.4

34.4

37.0

39.2

40.0

42.2

43.4

47.4

55.7

7.68

8.75

12.8

19.3

28.6

58.1

51.2

53.9

65.5

103

158

The variation of R with B are shown in Figs. 2 and 3 for

all the temperatures studied; as the curve at 294 K is very

close to that reported in our first study, it is not presented.

The changes of 1/

4

B) with B are illustrated in Fig. 4. By

contrast with what

is

observed in pure solvents,2,3 it may be

seen

that

the curvature

of

the variation of

R

at low field

is

most generally positive, as was found before for the solution

1.0

R

0.9

0.8

o

o

0.5 B(T)

FIG.

2.

Variation of parameter R with magnetic field intensity. B n. in a

0.025 M solution of nitrobenzene in n-hexane. for temperatures ranging

from 264

to

286

K. The

solid lines are calculated

on the

basis

of the

model

proposed (see

text).

when fitting

the

variation

of

R

alone.

J. Chern. Phys

•

Vol. 97. No.2. 15 July 1992

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Billard, Abbe, and Duplatre: Bound-state

of

positronium with nitrobenzene

1551

1.0

R

0 9

0.8

1.0

0.9

0.8

p err.

1.0

__

0.9

328 K

0.8

o

0.5

B

T)

FIG.

3.

Variation of parameter

R

with magnetic field intensity,

B

n, in a

0.025 M solution

of

nitrobenzene in n-hexane, for temperatures ranging

from

292

to 328 K. For the solid lines, see Fig. 2.

at 294

K.3

At all temperatures, there is a very abrupt de

crease of

I/A

4

 B)

at low field, as was the case at 294

K.3

V. DISCUSSION

On the basis of the model previously proposed,

3

the

present

data

which combine the variation

of R

and

of

1/

A4 B) with Tand B, can be fitted using the various param

eters descriptive of the Ps bound-state with nitrobenzene

[7]c>AEc.A.sc,A3c 0)] and of its reversible formation k

l

and k

2

) .

Note

that the parameters characteristic of Ps [7]

and

A; 0) ] are

fully determined from the study of the pure

solvent.

For

a specified temperature, due to the complexity of

the problem resulting in strong correlation between the pa

rameters, several sets

of the latter can lead to equally satis

factory fits. The quality

of

the fits can be assessed by consid

ering the

standard

deviations,

UA and U

R

,

accordingly,

which should be close to the experimental errors in favorable

cases.

In

the following, the conclusions

drawn

from the var

ious attempts made to fit the data are presented concerning

the temperature dependence

of

i) those parameters which

can be studied at zero field, essentially the kinetic param

eters; (ii) the parameters descriptive

of

the Ps complex with

nitrobenzene,

]c

and

AEc>

which

are

derived by examining

the magnetic field effects.

2.5

1/.\.o ns)

2.0

err.

o

0.5

264

K

1/.\.0 ns)

3.0

2.5

2.0

1/.\.0 ns)

_ 2.0

1.5

o

0.5 B

T)

FIG.

4.

Variation of

1/

A4 B) with magnetic field intensity, B n, in a 0.025

M solution

of

nitrobenzene in n-hexane. The solid lines are calculated on the

basis of the model proposed (see text), when fitting the variation of 1/

A

4

 B)

alone.

A.

Temperature

effects on the

parameters

derivable at

zero field

In all cases,

u

R

appears to be largely insensitive to the

valueofA

3e

0) , within the range 1-5 ns-

I

•

As

no otherlong

lifetime than

1/

A3

0)

is apparent at any temperature in the

LS spectra,A

3e

0 ) was fixed

at

2.3 ns-

I

, the valueofA2(

0) .

For similar reasons, Ase was fixed at 8 ns - I the value of

As

As previously observed, the fits

are

sensitive to the ratio

k J k2 rather than to the absolute values of hese reaction rate

constants. Therefore, the variation of kl with

T

was sup

posed

to

obey the Smoluchowski equation, wich has been

repeatedly used in Ps chemi

stry

l6

kl

T) =

2I3000)N

A

kB 2

+

rp.lrs

+

rslrps T lv

10)

with

11)

where

vis

the viscosity of the solvent

l2

with activation ener

gy Ev

=

0.045 eV and preexponential factor Vo = 0.054 cp,

N

is the Avogadro number,

kB

is the Boltzmann constant,

and rps = 0.053 nm and rs = 0.3 nm

are

the radii

of

the

reactive species, Ps and nitrobenzene, respectively. The val

ue

of rs

is only a crude estimate; however, it is consistent

with rs = 0.23 nm implicit in the fitting parameters derived

to recover the variation

of k

I with

T

for nitrobenzene in to

luene.

17

Furthermore,

the

absolute value

of

kl

calculated

from Eqs.

10)

and

11)

are in agreement with the experi

mental values

of k I

at temperatures around

and

below 200

K 18

when the reverse reaction k

2

  is negligible, even

though the slope of the k I vs Tvariation appears somewhat

higher

than

for

k I

On this basis, the value of k2 can be determined at each

temperature from the value

of

I/A3 0),

through

the rel

evant kinetic equation

not

reported

here)Y

Table

III

col

lects the calculated values for

kl

together with those for

k

2

•

Excepting

T

= 292 K, for which 1/A3 0) appears some

what too high, (see Table

II),

the latter values give a very

good Arrhenius plot, showing that

k2

can be written as

J. Chern. Phys., Vol. 97,

No.2,

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1552

Billard, Abbe, and Duplatre: Bound-state

of

positronium with nitrobenzene

(12)

with

k; =

9x

10

7

ns-

I

and

E

=

0.37 eV.

Knowing

E

z

, a rough approximation to the value of the

Ps affinity of nitrobenzene PA) can be made by considering

the following: (i) the forward reaction requires little

or

no

activation energy as is implicit in Eq. (10); (ii) before reac

tion, the energy level ofPs Eo), which is quite possibly in a

bubble within the solvent, is ~ 0 2 2 eV, as derived from the

width

of

the

p-Ps

momentum

distribution.

19

,2o

Neglecting

the solvation terms for nitrobenzene and for the Ps complex,

this gives PA

=Eo

+ E

2

=O.59 eV which is consistent with

the theoretically predicted value, PA

=

(1.11

±

0.81) eV.

21

B Temperature effects on the hyperflne interaction

parameters

The remaining parameters to consider are f

and

b.E

which are only derivable

through

the variations R and

1/A

4

 B) with B, either independently or simultaneously.

Several fitting procedures were attempted.

Most generally the values for fc and b.E

c

derived from

the variations with

B

of

either

R

or

1/..1 4

B)

alone are con

cordant, although the uncertainty is usually higher in the

latter case.

Figure 5 shows the change with Tof f as derived when

fit.ting

Rand

1/..1 4 B) together. Two regions are apparent

With

constant values of

Tfc

above 289 K, with a mean value

at (1.56

±

0.05) and below, at (0.43

±

0.04); thevalueofTfc

at 264 K, (0.75 ± 0.2) is not included in the latter mean

value, and possibly indicates an increase in

fc

at lower tem

perature.

For

comparison, the figure also includes

fc

= (1.77 ± 0.2), the average of the values previously de

rived at C

=

0.1

and

0.05 M; this average inserts well in the

present plot.

Contrary

to

Tfc

b.E

c

was most generally very badly de

fined, particularly in the temperature range between 268 and

286 K, where the variation with B

of

both

Rand

1/A

4

 B)

2.0

I1c

1.5

1.0

0.5

300

T K)

FIG.

5

Variation

of

parameter

7<

with temperature,

T K),

as derived when

fitting

Rand

l / i l

4

 B) together (open circles), in a 0.025 M solution of

nitrobenzene in n-hexane. Filled circle

=

average value obtained previously

(Ref. 3) for nitrobenzene concentrations

of

0.05 and

0 1

M. Solid lines

= fitting values of 7< for all data, below and above 289 K (see text).

appear to be completely insensitive to this parameter. As in

the high temperature region and at 264 K a somewhat con

stant value was found for b.E

c

at 10-

5

eV, a fitting of all

the data at all temperatures was attempted with only three

parameters, fc above 289 K and below 289 K, and b.E

c

. The

resulting values for

fc

are the same as the average values

given above, obtained when fitting the

data

for each tem

perature separately (solid line in Fig.

5),

together with b.E

c

=

1.06 X 10-

5

eV. Considering the relevant changes in

0

and O-A. this last value is only meaningful within, roughly,

factor

of2.

The values

of

b.E

mc

=

fc

b.E

c

= 1.65 X 10-

5

eV

(Tfc

= 1.56)

or

4.56x 10-

6

eV

(Tfc

= 0.43) for the

Ps

bound-state in n-hexane are very low, compared to b.E

m

= f b.E

vacuo

=

0.82 X 8.45 X 10-

4

eV

=

6.93 X 10-

4

eV ob

tained for Ps in pure n-hexane. These give a ratio

b.Emc/ b.E

m

of 2.4 down to 0.6 , according to the tem

perature region considered for Tfc t may be noted that theo

retical calculations lead to a value of 2.5 ,22 which com

pares well with the present data. t is worth noting that the

same

order

of magnitUde is experimentally found for the

ratio of the hyperfine splitting energies for

muonium

bound

states

and

muonium.

23

The sharp variation in

fc

around 289 K is not under

stood at the moment. The implication of interactions be

tween solvent molecules and the

Ps

bound-state at the level

of the

N 0

2

moiety cannot be excluded; the change from

fc

= 0.43 at low T to fc = 1.56 at high temperature might

therefore reflect a temperature-assisted deeper penetration

ofPs into N0

2

accompanied by a stronger overlap of the e+

and

e

wave functions. On this basis, whether the concept

of

bubble and bubble shrinkage at the moment

of

the bound

state formation 17,24 or, more generally, changes in the dielec

tric constant and solvation shell are dominant factors has yet

to be established. Anyhow, it is clear that the proposed mod

el is somewhat simplified in its use

of

parameter

fc

because

Eq.

(4),

and its equivalent for the

Ps

bound-state, Eq.

(7),

are only valid as long as the wave function ofPs is spherical,

which implies values of fc not too far from unity,

9

In spite of

this simplification, the quantitative

treatment

has been suc

cessfully applied to all temperatures investigated.

C.

omparison with other models

The non-Arrhenius behavior

of

the reaction rate con

stant of Ps with nitrobenzene in n-hexane and in other sol

vents is now a well-established fact, which has been exam

ined

and

explained on kinetic grounds.II,IS As concerns the

magnetic field effects, two alternative approaches have been

proposed, besides the treatment described in this paper.

One of these models involves various spur reactions and

does not allow to fit the various data, as a function

of

either

B,

Cor T, with a limited number of parameters.

15

With the

present model, any of these variations can be recovered by

simply considering, besides the kinetic constants

kl

and k2

the parameters descriptive of the hyperfine interaction of the

Ps bound-state with nitrobenzene.

The other model is based on

the

idea that e+ in Ps, when

bound to nitrobenzene, interacts with more than one ele-

t

25 26

D b

h

.

ron . escn mg t ese mteractlOns through contac t den-

J. Chern. Phys., Vol. 97, No.2, 15 July 1992

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Billard Abbe and Duplatre: Bound-state

of

positronium with nitrobenzene 1553

sity parameters, calculations have been performed on a 0 1

M solution of nitrobenzene in n-hexane at 294 K, with the

assumption that all Ps atoms form a bound-state.

26

With this

crude hypothesis, it is found that the interaction

of

the posi

tron with only two electrons is sufficient to recover the

anomalous variation of R with B, and the values of the con

tact density parameters for each of the two electrons are

(Ref. 27) 1/a = 1.65 and 1/b = 0.14.

Whereas Brusa et

al

6

have considered essentially the

mathematical description

of

a distorted positronium .at

tached to a molecule to form a bound-state, our approach,

which is based on a wealth of experimental data, is

of

a more

phenomenological nature. However, the two explanations

are very similar. Actually, the high value of 1/0 = 1.65 as

compared to

1/b =

0.14 shows that one

of

the electrons in

volved interacts strongly with the positron, while the contri

bution from a second electron

is

only marginal; this fits in

very well with our description. Furthermore, within the er

rors, the value of 1/0 coincides with the average found in this

work, 1/c = (1.56 ± 0.05). Summarizing, the bound-state

can be described as presenting a singlet and a triplet state, in

such a way that the identities of the components,

l>N

2

and

Ps are, to a large extent, preserved. Chemically speaking,

this resembles a loose charge-transfer complex,28 even

though one of the partners of the association is somewhat

unusual, with l>N

2

as the electron acceptor and Ps as the

electron donor.

VI CONCLUSION

The strong sensitivity

of

the solution

of

nitrobenzene in

n-hexane to the magnetic field appears to be dependent on

temperature. These complementary experiments have been

rewarding not only to assess the assumptions of

our

model,

but also to gain information on the temperature dependence

ofthe hyperfine parameters of he Ps bound-state with nitro

benzene. The absence of sensitivity of he hyperfine splitting,

AE

c

  to temperature is not surprising, as this is an intrinsic

parameter. Conversely, some variation of1/c with Tmight be

expected as this parameter can be influenced by the solvent.

However, the abruptness with which 1/c changes around 289

K is not yet understood.

At

present, the effect of temperature and

of

magnetic

field have been studied extensively and i t seems necessary to

consider other possible factors. In this context, examining

the influence of pressure and of the nature of the solvent

appears promising.

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