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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
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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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