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Dependence of positron lifetimes on the dielectric constants of some organic media
Volume 81A, number 7
PHYSICS LETTERS
9 February 1981
DEPENDENCE OF POSITRON LIFETIMES ON THE DIELECTRIC CONSTANTS OF SOME ORGANIC MEDIA M.G. GAIKWAD and S.K. DAVID Department of Physics, University of Poona, Pune 411007, Maharashtra State, India Received 21 November 1980
An attempt has been made to correlate the orthopositronium (o-Ps) pickoff annThllation lifetime with the dielectric constant for some organic liquids. A semi-empirical free volume model, developed by Thosar et al., is used.
1. Theoretical and empirical attempts have been made to understand the pickoff behaviour of positronium in molecular substances. In an empirical approach, many workers have tried to correlate the pickoffannihilation lifetime r2 with some physical parameters. Khan [1] suggested a correlation between the lifetime of o-Ps in a number of polymers with their molecular cohesive density. The most successful attempt was made by Gray et al. [3,4], who correlated the quenching cross sections of o-Ps in 193 liquids with their electron polarizabiities. However, no theoretical explanation was given by them. Tao [2] has correlated, on a good theoretical basis, pickoff lifetimes in liquids with the surface tension. Thosar et al. [5—8], have tried to correlate pickoff lifetimes with a transport property like viscosity, using their own semi-empirical free volume model. This correlation had a limited success because it seems that viscosity depends highly upon the hydrogen bonds in associated liquids. It has been suggested that the formation and decay of a Ps atom is mainly confinedto the free volume of the medium [9j. The free volume of the medium, comprisingregions of low electron density, plays an important role in determining the o-Ps pickoff annthilation lifetime r2. Another physical quantity which depends primarily on the free volume is the dielectric constant [10—12j. In the present work, a correlation between the pickoff lifetime r2 and the dielectric constant is suggested using the free volume model of Thosar et al.
2. The relevance of the free volume to positron annihilation has been recognized and discussed by Brant [13], by Wilson et al. [14] and by Thosar et al. [6,7]. They have developed a free volume model for simple molecular materials. According to this model, the formation and quenching of Ps atoms are confinedto the free volume, Vf, of the material. Vf = V5,~-~ ~ The specific free volume Vf is considered to be made up of a number of sites or void cells of average volume V. Ps atoms which preferentially occupy these sites are subsequently quenched by the pickoff process. l’he electron density overlap with the positron of o-Ps atoms at these sites is a function of V. The pickoff rate A2 hr2 which depends upon this overlap is a fucntion of the average site volume (V). According to Thosar et al. the expression —
r2
r1 exp[(V— V1)/(V0
—
V)]
(1)
,
where r1 = 0.7 ns is a constant, satisfies these conditions. V1 and V0 are the lower and upper limits of V, respectively. When V shrinks to V1, A2 decreases to A1 = 10~s~,the decay rate observed for Ps—molecular complexes. When V increases to V0, the quenching decreases and A2 becomes vanishingly small. The above expression can be written as
V1/V
[b(1 +lnr2/r1)/(b+lnr2/r1)]
,
(2)
where b = V1 /V0 = 0.02 (constant). Attempts have been made by the present authors [10—12] to obtain an understanding ofa constitutive
0 031—9163/81/0000--0000/$ 02.50 © North-Holland Publishing Company
415
Volume 81A, number 7
PHYSICS LETTERS
property of liquids, such as the dielectric constant, in terms of the free volume in the liquid. Gaikwad et al. [10,11] suggested that the changes in the dielectric constant with temperature in liquid n-alkanes could be correlated with changes in the free volume due to thermal expansion. The dielectric constants of liquid normal alkanes were found as a function of temperature and it was shown that the data could be best described by an empirical relation [10,11] of the type e_~P(V~/Vf)~
9 February 1981
Vm and the coordination number, i.e. V1 = k2 V11~. Since the pickoff lifetime r2 depends upon the free volume Vf, it is interesting to compare positron lifetime data with the dielectric constant which also depends upon the free volume Vf. Hence eq. (3) can be written as: hi =lnP+qln(V~/Vf), hi e = lnP+ q ln(V1 /k1k2V),
(3)
,
where is the dielectric constant, Vm the molecular volume, Vf the free volume per molecule, andF, q are constants for a single liquid, Now, Vf is the free volume per molecule with V the average site volume and they may be assumed to be related by Vf = V. Also eq. (2) shows that r2 depends upon two quantities, the average site volume V and the extent of the overlap due to the surrounding molecules which depends upon the molecular volume
Inc =ln(P/(k1k2)~)+q1n(V1/V).
(4)
In this equation V1/V can be evaluated from the lifetime r2 using eq. (2), while F, q, k1 and k2 are constants. ln(V1/V) is, therefore, expected to depend linearly on ln e. 3. The values of J’~/ V at various temperatures were calculated from the lifetime r2 using expression (2). Table 2
Table 1 A table of pickoff annihilation lifetimes and dielectric constants at various temperatures for liquids with e <3. Compound
Temp. r2(±0.1) e (V1/V) (°C) (ns) X 102 _____________________________________________________
n-octane
—20 25 35 45 55
3.1 3.6 3.8 3.9 4.1
2.000 1.942 1.929 1.916 1.903
3.299 3.182 3.145 3.128 3.096
n-pentane
—20 25
3.4 4.0
1.908 1.836
3.224 3.112
toluene
—75 —55 —35 —17 —8 25 35
2.6 2.7 2.8 2.9 3.1 3.2 3.3
2.637 2.583 2.529 2.482 2.459 2.379 2.355
3.471 3.430 3.393 3.360 3.299 3.273 3.248
—5 2 11 18 25
2.5 2.9 2.9 3.1 3.1
2.334 2.320 2.302 2.288 2.274
3.516 3.360 3.360 3.299 3.299
0 10 25
2.7 3.1 3.2
2.055 2.039 2.015
3.430 3.299 3.273
benzene
cyclohexane
416
A table of pickoff annihilation lifetimes and dielectric constants at various temperatures for liquids with e> 3. Compound Temp. r2(±0.l) e _________________________________ (°C) (ns)
(V1/V)
x
102
chlorobenzene
—50 —40 —20 25 .50
2.5 2.6 2.7 2.9 3.1
7.075 6.862 6.454 5.623 5.209
3.516 3.471 3.430 3.360 3.299
bromobenzene
—2 25 45 52 60
3.0 3.2 3.4 3.5 3.5
5.800 5.400 5.121 5.027 4.922
3.328 3~273 3.224 3.203 3.203
75
3.6
4.730
3.182
36 42 47 67
1.75 1.85 1.90 1.90
11.67 11.17 10.76 9.29
4.093 3.976 3.924 3.924
—15 0 .25 60 25
1.8 1.8 1.9 2.0 2.35
51.47 47.91 42.50 35.94 6.774
4.032 4.032 3.924 3.832 3.592
45 60 80
2.40 2.45 2.50
6.327 6.012 5.616
3.565 3.540 3.516
phenol
glycerol
aniline
Volume 81A,number 7
PHYSICS LETTERS
9 February 1981
1.000
TOLUENE~~~
BENZEN
(a 500 ~
~ -
-.
~ -
I
I -11.9
I
-11.7 LOG(VI/V)— Fig. 1. Graph of loge and log(V 1/V) for liquids with e < 3.
All the lifetime data at various temperatures are after Thosar et al. [5—8].The dielectric constants at these temperatures were obtained by extrapolating the dielectric constant data from the literature [15,16] .Tables 1 and 2 show the values of the pickoff lifetimes, the
.~00 -i4~
dielectric constants and V1 IV. Because of space limitations, we reproduce only part of the lifetime and dielectric constant data. Graphs of log and log(V1/V) are plotted in figs. 1 and 2. It can be seen very well that for all these nine liquids the plot is linear as expected.
—
20
-
1~5~
GLYCEROL
0
-J
PHENOL ANIL IRE CH
~
—~°~
—
..o,
10
BROHO BENZENE
~ -150
~
I I -11.6 —11.2 -~——L0G(V1/V)—
I
Fig. 2. Graph of loge and log(V1/ V) for liquids with e> 3.
417
Volume 81A, number 7
PHYSICS LETTERS
9 February 1981
4. The results have shown that the correlation between the pickoff annthilation lifetime r2 and the di-
[3] P.R. Gray, C.F. Cook and G.P. Sturm, J. Chem. Phys.
electric constant holds for all the nine liquids studied. Though the liquids studied belong to various systems like normal alkanes, (e.g. octane, pentane), unsubstituted cyclic alkanes (e.g. cyclohexane), benzene, and derivatives (benzene, toluene, chlorobenzene, bromobenzene, aniline, phenol etc.), more lifetime data at various ternperatures for a large number of liquids will be required to see perfectly the, validity of the correlation. We are engaged in a further development of the theoretical basis for this correlation.
[41 P.R. Gray, C.F. Cook and G.P.
The authors are thankful to the Head of the Department of Physics for providing facilities. One of the authors (M.G.G.) wishes to thank the University Grants Commission for a Teacher Fellowship and K.T.H.M. College, Nasik for study leave. References [1] M.N.G.A. Khan, Ph.D. Thesis, Univ. of New South Wales (1968). [2] S.J. Tao, J. Chem. Phys. 56 (1972) 5499.
418
46 (1967) 3487. Sturm, J. Chem. Ploys. 48 (1968) 1145. [5] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Lett. 33A (1970) 129.
[6] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Lett. 28A (1969) 760. [7] B.V. V.G. R.G. Lagu and G. Chandra, Phys.Thosar, Stat. Sol. 55 Kulkarni, (1973) 415. [8] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Stat. Sol. 58 (1973) K83. [9] R.A. Ferell, Rev. Mod. Phys. 28 (1956) 308. [10] M.G. Gaikwad, R. Chandrasekhar, S.K. David and V.G. Alwani, Phys. Lett. 80A (1980) 201. [11] M.G. Gaikwad and S.K. David, Phys. Lett. 81A (1981) 413. [12] M.G. Gaikwad, S. Deuskar and S.K. David, Study of dependence of dielectric constant of liquids on free volume, submitted to Solid State Commun. [13] W. Brant, S. Berko and W. Walker, Phys. Rev. 120 (1960) 1289. [14] R.K. Wilson, P.O. Johnson and R. Stump, Phys. Rev. 129 (1963) 2091. [15] R.C. Weast, Handbook of chemistry and physics, 59th Ed. (Chemical Rubber Co., Cleveland, OH, 1978—79). [16] N.A. Lange, Handbook of chemistry, 9th Ed. (Handbook Publishers Inc., Sandusky, OH, 1957).
PHYSICS LETTERS
9 February 1981
DEPENDENCE OF POSITRON LIFETIMES ON THE DIELECTRIC CONSTANTS OF SOME ORGANIC MEDIA M.G. GAIKWAD and S.K. DAVID Department of Physics, University of Poona, Pune 411007, Maharashtra State, India Received 21 November 1980
An attempt has been made to correlate the orthopositronium (o-Ps) pickoff annThllation lifetime with the dielectric constant for some organic liquids. A semi-empirical free volume model, developed by Thosar et al., is used.
1. Theoretical and empirical attempts have been made to understand the pickoff behaviour of positronium in molecular substances. In an empirical approach, many workers have tried to correlate the pickoffannihilation lifetime r2 with some physical parameters. Khan [1] suggested a correlation between the lifetime of o-Ps in a number of polymers with their molecular cohesive density. The most successful attempt was made by Gray et al. [3,4], who correlated the quenching cross sections of o-Ps in 193 liquids with their electron polarizabiities. However, no theoretical explanation was given by them. Tao [2] has correlated, on a good theoretical basis, pickoff lifetimes in liquids with the surface tension. Thosar et al. [5—8], have tried to correlate pickoff lifetimes with a transport property like viscosity, using their own semi-empirical free volume model. This correlation had a limited success because it seems that viscosity depends highly upon the hydrogen bonds in associated liquids. It has been suggested that the formation and decay of a Ps atom is mainly confinedto the free volume of the medium [9j. The free volume of the medium, comprisingregions of low electron density, plays an important role in determining the o-Ps pickoff annthilation lifetime r2. Another physical quantity which depends primarily on the free volume is the dielectric constant [10—12j. In the present work, a correlation between the pickoff lifetime r2 and the dielectric constant is suggested using the free volume model of Thosar et al.
2. The relevance of the free volume to positron annihilation has been recognized and discussed by Brant [13], by Wilson et al. [14] and by Thosar et al. [6,7]. They have developed a free volume model for simple molecular materials. According to this model, the formation and quenching of Ps atoms are confinedto the free volume, Vf, of the material. Vf = V5,~-~ ~ The specific free volume Vf is considered to be made up of a number of sites or void cells of average volume V. Ps atoms which preferentially occupy these sites are subsequently quenched by the pickoff process. l’he electron density overlap with the positron of o-Ps atoms at these sites is a function of V. The pickoff rate A2 hr2 which depends upon this overlap is a fucntion of the average site volume (V). According to Thosar et al. the expression —
r2
r1 exp[(V— V1)/(V0
—
V)]
(1)
,
where r1 = 0.7 ns is a constant, satisfies these conditions. V1 and V0 are the lower and upper limits of V, respectively. When V shrinks to V1, A2 decreases to A1 = 10~s~,the decay rate observed for Ps—molecular complexes. When V increases to V0, the quenching decreases and A2 becomes vanishingly small. The above expression can be written as
V1/V
[b(1 +lnr2/r1)/(b+lnr2/r1)]
,
(2)
where b = V1 /V0 = 0.02 (constant). Attempts have been made by the present authors [10—12] to obtain an understanding ofa constitutive
0 031—9163/81/0000--0000/$ 02.50 © North-Holland Publishing Company
415
Volume 81A, number 7
PHYSICS LETTERS
property of liquids, such as the dielectric constant, in terms of the free volume in the liquid. Gaikwad et al. [10,11] suggested that the changes in the dielectric constant with temperature in liquid n-alkanes could be correlated with changes in the free volume due to thermal expansion. The dielectric constants of liquid normal alkanes were found as a function of temperature and it was shown that the data could be best described by an empirical relation [10,11] of the type e_~P(V~/Vf)~
9 February 1981
Vm and the coordination number, i.e. V1 = k2 V11~. Since the pickoff lifetime r2 depends upon the free volume Vf, it is interesting to compare positron lifetime data with the dielectric constant which also depends upon the free volume Vf. Hence eq. (3) can be written as: hi =lnP+qln(V~/Vf), hi e = lnP+ q ln(V1 /k1k2V),
(3)
,
where is the dielectric constant, Vm the molecular volume, Vf the free volume per molecule, andF, q are constants for a single liquid, Now, Vf is the free volume per molecule with V the average site volume and they may be assumed to be related by Vf = V. Also eq. (2) shows that r2 depends upon two quantities, the average site volume V and the extent of the overlap due to the surrounding molecules which depends upon the molecular volume
Inc =ln(P/(k1k2)~)+q1n(V1/V).
(4)
In this equation V1/V can be evaluated from the lifetime r2 using eq. (2), while F, q, k1 and k2 are constants. ln(V1/V) is, therefore, expected to depend linearly on ln e. 3. The values of J’~/ V at various temperatures were calculated from the lifetime r2 using expression (2). Table 2
Table 1 A table of pickoff annihilation lifetimes and dielectric constants at various temperatures for liquids with e <3. Compound
Temp. r2(±0.1) e (V1/V) (°C) (ns) X 102 _____________________________________________________
n-octane
—20 25 35 45 55
3.1 3.6 3.8 3.9 4.1
2.000 1.942 1.929 1.916 1.903
3.299 3.182 3.145 3.128 3.096
n-pentane
—20 25
3.4 4.0
1.908 1.836
3.224 3.112
toluene
—75 —55 —35 —17 —8 25 35
2.6 2.7 2.8 2.9 3.1 3.2 3.3
2.637 2.583 2.529 2.482 2.459 2.379 2.355
3.471 3.430 3.393 3.360 3.299 3.273 3.248
—5 2 11 18 25
2.5 2.9 2.9 3.1 3.1
2.334 2.320 2.302 2.288 2.274
3.516 3.360 3.360 3.299 3.299
0 10 25
2.7 3.1 3.2
2.055 2.039 2.015
3.430 3.299 3.273
benzene
cyclohexane
416
A table of pickoff annihilation lifetimes and dielectric constants at various temperatures for liquids with e> 3. Compound Temp. r2(±0.l) e _________________________________ (°C) (ns)
(V1/V)
x
102
chlorobenzene
—50 —40 —20 25 .50
2.5 2.6 2.7 2.9 3.1
7.075 6.862 6.454 5.623 5.209
3.516 3.471 3.430 3.360 3.299
bromobenzene
—2 25 45 52 60
3.0 3.2 3.4 3.5 3.5
5.800 5.400 5.121 5.027 4.922
3.328 3~273 3.224 3.203 3.203
75
3.6
4.730
3.182
36 42 47 67
1.75 1.85 1.90 1.90
11.67 11.17 10.76 9.29
4.093 3.976 3.924 3.924
—15 0 .25 60 25
1.8 1.8 1.9 2.0 2.35
51.47 47.91 42.50 35.94 6.774
4.032 4.032 3.924 3.832 3.592
45 60 80
2.40 2.45 2.50
6.327 6.012 5.616
3.565 3.540 3.516
phenol
glycerol
aniline
Volume 81A,number 7
PHYSICS LETTERS
9 February 1981
1.000
TOLUENE~~~
BENZEN
(a 500 ~
~ -
-.
~ -
I
I -11.9
I
-11.7 LOG(VI/V)— Fig. 1. Graph of loge and log(V 1/V) for liquids with e < 3.
All the lifetime data at various temperatures are after Thosar et al. [5—8].The dielectric constants at these temperatures were obtained by extrapolating the dielectric constant data from the literature [15,16] .Tables 1 and 2 show the values of the pickoff lifetimes, the
.~00 -i4~
dielectric constants and V1 IV. Because of space limitations, we reproduce only part of the lifetime and dielectric constant data. Graphs of log and log(V1/V) are plotted in figs. 1 and 2. It can be seen very well that for all these nine liquids the plot is linear as expected.
—
20
-
1~5~
GLYCEROL
0
-J
PHENOL ANIL IRE CH
~
—~°~
—
..o,
10
BROHO BENZENE
~ -150
~
I I -11.6 —11.2 -~——L0G(V1/V)—
I
Fig. 2. Graph of loge and log(V1/ V) for liquids with e> 3.
417
Volume 81A, number 7
PHYSICS LETTERS
9 February 1981
4. The results have shown that the correlation between the pickoff annthilation lifetime r2 and the di-
[3] P.R. Gray, C.F. Cook and G.P. Sturm, J. Chem. Phys.
electric constant holds for all the nine liquids studied. Though the liquids studied belong to various systems like normal alkanes, (e.g. octane, pentane), unsubstituted cyclic alkanes (e.g. cyclohexane), benzene, and derivatives (benzene, toluene, chlorobenzene, bromobenzene, aniline, phenol etc.), more lifetime data at various ternperatures for a large number of liquids will be required to see perfectly the, validity of the correlation. We are engaged in a further development of the theoretical basis for this correlation.
[41 P.R. Gray, C.F. Cook and G.P.
The authors are thankful to the Head of the Department of Physics for providing facilities. One of the authors (M.G.G.) wishes to thank the University Grants Commission for a Teacher Fellowship and K.T.H.M. College, Nasik for study leave. References [1] M.N.G.A. Khan, Ph.D. Thesis, Univ. of New South Wales (1968). [2] S.J. Tao, J. Chem. Phys. 56 (1972) 5499.
418
46 (1967) 3487. Sturm, J. Chem. Ploys. 48 (1968) 1145. [5] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Lett. 33A (1970) 129.
[6] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Lett. 28A (1969) 760. [7] B.V. V.G. R.G. Lagu and G. Chandra, Phys.Thosar, Stat. Sol. 55 Kulkarni, (1973) 415. [8] B.V. Thosar, V.G. Kulkarni, R.G. Lagu and G. Chandra, Phys. Stat. Sol. 58 (1973) K83. [9] R.A. Ferell, Rev. Mod. Phys. 28 (1956) 308. [10] M.G. Gaikwad, R. Chandrasekhar, S.K. David and V.G. Alwani, Phys. Lett. 80A (1980) 201. [11] M.G. Gaikwad and S.K. David, Phys. Lett. 81A (1981) 413. [12] M.G. Gaikwad, S. Deuskar and S.K. David, Study of dependence of dielectric constant of liquids on free volume, submitted to Solid State Commun. [13] W. Brant, S. Berko and W. Walker, Phys. Rev. 120 (1960) 1289. [14] R.K. Wilson, P.O. Johnson and R. Stump, Phys. Rev. 129 (1963) 2091. [15] R.C. Weast, Handbook of chemistry and physics, 59th Ed. (Chemical Rubber Co., Cleveland, OH, 1978—79). [16] N.A. Lange, Handbook of chemistry, 9th Ed. (Handbook Publishers Inc., Sandusky, OH, 1957).