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Electron-ion geminate escape probability in anisotropic media
Volume 142, number 5
ELECTRON-ION
CHEMICAL PHYSICS LETTERS
GEMINATE ESCAPE PROBABILITY IN ANISOTROPIC
18 December 1987
MEDIA
A. MOZUMDER =, SM. PIMBLOTT b, P. CLIFFORD ’ AND N.J.B. GREEN a ’ Department of Chemistry and the Radiation Laboratory ‘, Universityof Notre Dame. Notre Dame, IN 46556, WSA b Physical Chemistry Laboratory, Universityof Oxford, Oxford, UK
’ Mathematical Institute, St. Giles, Universityof Oxford, Oxford, UK Received 17 September 1987
The diffusion-controlled electron-hole geminate escape probability in anisotropic media has been calculated by numerical solution of the adjoint equation for that probability after a series of simplifying transformations. Relative to the crystalline axes the anisotropies of both dielectric constant and mobility, which generally have different principal directions, are important. The escape probability depends on the magnitude and direction of the initial electron-hole separation. Applied to anthracene the implied experimental escape probability ~0.01 is attained at initial separations 4.0, 4.0 and 3.2 nm along the a, b, c’ crystal directions respectively
1. Introduction
In a variety of condensed media absorption of lowLET radiation generates mainly geminate ion pairs, i.e. isolated pairs of electrons and positive ions or holes [ 1,2 1. In hydrocarbon liquids Onsager’s theory [ 31 provides the basis for calculating the probability of escaping geminate recombination under the joint influence of diffusion and mutual Coulombic attraction. The theory compares favorably with the results obtained from experimentally determined freeion yields [ 41. Onsager’s solution, derived from the isotropic Smoluchowski equation or diffusion in a potential field, gives in the limit that the reaction radius a CCr,, @iso =exrd
-
~J~o)
,
(1)
where 4i,, is the geminate escape probability in isotropic media, r. is the initial separation of ions and rC=e21tkT with e as electronic charge, t, dielectric constant of the medium, k, Boltzmann’s constant and T, the absolute temperature. In crystals Onsager’s ’ The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-3030 from the Notre Dame Radiation Laboratory.
0 009-2614/87/$
treatment needs extension since the diffusion is anisotropic; furthermore the Coulombic field is also anisotropic. A prime example is provided by the photoionization free-carrier yield in anthracene crystals where, for reasons of simplicity, experimental results have been analyzed using the Onsager theory [ 5,6], which in principle is not applicable to this case. Our aim in this paper is to give a systematic derivation of the escape probability in anisotropic media and to indicate approximately the directiondependent values of the initial (or, interpretively the thermalization) distance of an electron-hole pair in anthracene that would be comparable with the experimentally deduced escape probability. Onsager [ 31 notes the self-adjoint character of the diffusion equation written in terms of w exp(u/kT) where w is the density function and v is the potential, By virtue of this property he then asserts that the probability @ will satisfy the backward equation: V - [ exp( - v/kT) V $1 = 0. This equation is readily generalized to anisotropic media as follows [ 71 V.[exp(-u/kT)D*V#]=O,
(2)
where D, the diffusion tensor, is related to the experimentally determinable mobility tensor p by the Nemst-Einstein relation, p= (e/kT) D. Boundary conditions for solving eq. (2), consistent with those
03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
385
Volume 142, number
5
CHEMICAL
PHYSICS LETTERS
for the Smoluchowski equation, and considering that + is the escape probability, may be given by @=O at the origin and #= 1 at infinite separation. The mutual electrostatic potential v with the hole at the origin and X’denoting the electron coordinates is given by 181
~=--*(I,I,,~'x~x~)-"~=-~~(~~E,)-"*. (3) In eq. (3) 1t 1 is the determinant of the dielectric tensor with elements t,k, r is the radius vector, rT the transpose of r and E, called here the symmetric dielectric matrix, is related to the dielectric tensor by E= ]E]E-‘. The solution of eq. (2) with the potential of eq. (3) will be developed in section 2.
2. Method and procedure Let Q be the orthogonal transformation that diagonalizes the diffusion tensor, i.e. D = QT diag( A,)Q where A, (i= 1,2,3) are the eigenvalues. Making the transformation
s=QT diag(d;“2)Qr~D-1’2r with D “2 =QT diag(lf’*)Q and substituting in eq. (2) one obtains
V~VqL(e*/kT)(Vqb*Ms)(s~Ms)-3'2=0, (4) where M-D”*ED’/*
(5)
and the differential operators of eq. (4) operate in the s space. Note that the matrix M has all the required combination of the elements of the diffusion and dielectric tensors. Let P diagonalize M with eigenvalues mi (i= 1,2,3), i.e. M=PTdiag(mi)P. Then transforming s to f by t= Ps, eq. (4) may be written as V -V$-(e21kT)V#-diag(m,)t[tTdiag(mi)l]-3’2 =o,
(6)
where t, (i = 1,2,3) are the principal directions of 1. In eq. (6) the differential operators now operate in the t space. Finally the linear transformation u=cf, with c chosen such that c* = XT=I m,(e*/kT)-*, simplifies eq. (6) into 386
3a2$
18 December
1987
2=o (7) c --,~~Ui~(j~,u:P,)-3’ ,=,a~5 3
where p, = milC :=, m, and Zpi= 1. The significance of the first transformation (cf. eq. (4)), effectively a rotation, is to represent 9 in a coordinate system where diffusion is diagonal; the second transformation (cf. eq. ( 6)) then extends or contracts the axes to afford further simplification. The solution of eq. (7) can now be represented in dimensionless form by a family of two parameters. To construct the reduced equation (7) in a given case one proceeds by (i) diagonalizing the diffusion matrix, (ii) constructing the M matrix (cf. eq. (5)) and then (iii) diagonalizing the M matrix. However, since we only need the ratios pi and not the absolute mi values we are permitted to replace E by c-’ and D by P, which really contains the experimental results on transport. Schematically this is represented by diag( m:) such that diag( m:) =PQT diag(,&)“*Qe-‘QT
diag(&)“2QPT
,
with the understanding that p, = m,/I & I mi = m:lC~,l m:. After obtaining the pi values the numerical solution of eq. (7) is facilitated by further transforming the infinite u space into a finite one and then applying the relaxation method to the so-obtained discretized form of eq. (7). Several transformations of u have been tried, however, w=exp( -3”*u-‘)
(8)
has been found to be most effective for the numerical work reported here. The polar and azimuthal angles, with the usual meanings referred to the respective Cartesian coordinates, remain unaltered. This transformation maps the infinite u space onto a unit sphere in the w space and the boundary conditions are #( w+O) =0 and $( w+ 1) = 1. While, subject to these boundary conditions, any reasonably continuous form of @is acceptable for starting the relaxation procedure, in point is simplest starting practice the esran= exp( v&T) with the potential v of eq. (3). This @stanis the exact analogue of the Onsager probability (cf. eq. (1)) except only that the potential refers to the anisotropic medium. Some details of the relax-
Volume 142, number 5
ation method employed together with the criterion of convergence, procedure for acceleration of convergence, etc., will be reported elsewhere [ 91.
3. Results and discussion In this paper we carry out numerical computations for the anthracene crystal. The experimental determinations and theoretical rationalization of the elements of the dielectric [lo] and mutual (electron-hole) mobility [ 111 tensors for this crystal (12&c directions) are given below. 2.9 r= 0 ( -0.57
p=
2.70 0 -0.0458 i
0 2.94 0
18 December 1987
CHEMICAL PHYSICS LETTERS
-0.57 , 0 3.841
0 3.00 0
-0.0458 . 0 1.20 i
While the elements of B admit errors of <4W, considerable uncertainties still exist about the elements of p and their interpretation. In the calculations reported here we use the values as given above. Using high-energy X-rays it has been established that z 3000 eV is required at a vanishing collecting field to create a free carrier in anthracene over z-25” to +25” [12]. With a Wvalue for ionization of 30 eV [ 131 one then gets a geminate escape probability P,,, x 0.01. With UV irradiation Chance and Braun [ 13 ] indicate P,,, x 0.001-0.07 and representative electron-hole separation z lo-70 8, depending on the UV wavelength and the form of assumed initial distribution. However these studies are indirect and based on activation energy measurement; also an uncorrected Onsager equation is used for P,,,. The interpretation is therefore dubious. On the other hand the quantum yield for photocarrier generation in the 240-280 nm region is indicated as = 1.5~ 10m4[ 131. If this is rationalized with a Gaussian distribution [ 131 one gets a quantum yield of photoionization x 0.0 1 and also P,,, NN 0.0 1. We may thus take P,,, x 0.0 1 in anthracene at room temperature both for UV and X-rays. Our calculation using the E and p matrices given earlier and ini-
tial a-distribution gives the corresponding initial electron-hole separation along the a, b, and c’ directions as 4.0, 4.0 and 3.2 nm respectively. The effect of an initial distribution will be reported elsewhere [ 91. Pope and Swenberg [ 141 argue an initial electron-hole separation z 50 A (spherically symmetrical) under similar circumstances, However this, being based on the isotropic Onsager equation, is also dubious. Although the backward equation for the escape probability is readily generalized to the anisotropic media (eq. (2)) its solution is not straightforward. The present treatment is the first complete generalization of the Onsager problem to such cases. The most important finding is that P,,, depends on D as well. Actually it depends on the ratios of the elements of D; therefore its absence in the isotropic case is easily understood. For recombination of neutral radicals (v=O) an exact solution of eq. (2), also depending Z
Fig. I. Contour plot of escape probability of a geminate electron-hole pair in anthracene at 290 K calculated according to the present theory and using the experimentally determined P and t. The contours represent closed curves of equal probability (numbered 1 through 10 denoting equally spaced probabilities 0.02 through 0.20) in the X-Zplane ofthe transformed w space. Each division on the X and 2 axes represent a dimensionless unit of 0.02. The closed curve A denotes the escape probability for 50 A electron-hole separation distance in real space. The directional dependence of P,, is evident at the same magnitude of the separation.
387
Volume 142, number 5
CHEMICAL PHYSICS LETTERS
on the ratios of the elements of D, is possible and will be reported elsewhere [ 91. In the past Pest for geminate pairs in crystals has been computed using the Onsager equation (1) and a diagonally averaged E [ 5,6,13,14], Apart from the lack of theoretical justification this procedure ignores the contribution of D and therefore cannot be valid. The anisotropy of the Coulombic potential has also been often ignored introducing another dimension of uncertainty [ 151. Our procedure pays due attention to both the B and D anisotropies. Furthermore we have checked our numerical computation against the Onsager equation in the isotropic limit and a high degree of agreement has been found with a difference < 1 part in 104. The remaining difference is due to the finite mesh size used for numerical calculation; the agreement can be made much better at the cost of computer time. Fig. 1 shows a contour plot of PC,, for anthracene in the reduced (w) coordinates. The closed curve A denotes the escape probability calculation for an initial 50 A separation of electron-hole pair in the geometrical (crystal) space. The directional dependence PC,, at the same magnitude of the separation is apparent and is characteristic of anisotropic diffusion.
Acknowledgement SMP is grateful to Harwell Laboratory, United Kingdom Atomic Energy Authority, for the award of a research studentship. AM thanks Harwell Laboratory for a visiting appointment and the Wolfson College, Oxford University for a visiting fellowship
388
I8 December 1987
which enabled him to perform part of the work at the physical chemistry laboratory of Oxford University. References [ 11 A. Hummel, in: Advances in radiation chemistry, Vol. 4, eds. M. Burton and J.L. Magee (Wiley, New York, 1974) [2] i. lMozumder and J.L. Magee, J. Chem. Phys. 47 (1967) 939. [3] L. Onsager, Phys. Rev. 54 (1938) 554. [4] W.F. Schmidt and A.O. Allen, J. Phys. Chem. 72 (1968) 3730; J. Chem. Phys. 52 (1970) 2345. [ 51K. Kato and C.L. Braun, J. Chem. Phys. 72 (1980) 172. [6] R.R. Chance and C.L. Braun, J. Chem. Phys. 59 (1973) 2269; R.H. Batt, CL. Braun and J.L. Hornig, J. Chem. Phys. 49 (1968) 1967; Appl. Opt. Suppl. 3 (1969) 20; N.E. Geacintov and M. Pope, in: Proceedings of the 3rd International Conference on Photoconductivity, ed. EM. Pell (Pergamon Press, Oxford, 1971) p. 289. [ 71 M. Tachiya, J. Chem. Phys. 69 (1978) 2375. [ 81 L. Landau and E.M. Lifshitz, Electrodynamics of continuous media ( Pergamon Press, Oxford, 1960) pp. 6 1,62. [9] SM. Pimblott, N.J.B. Green, P. Clifford and A. Mozumder, to be published. [lo] R. Munn, J. Nicholson, H. Schwab and D. Williams, J. Chem. Phys. 58 (1973) 3828. [ 1l] G.D. Thaxton, R.C. Jamagin and M. Silver, J. Chem. Phys. 66 (1962) 2461; J.J. Katz, S.A. Rice, S. Choi and J. Jortner, J. Chem. Phys. 31 (1963) 1683. [ 121 R.G. Kepler and F.N. Coopage, Phys. Rev. 151 ( 1966) 6 IO: R.C.Hughes, J.Chem.Phys. 55 (1971) 5442. [ 131 R.R. Chance and C.L. Braun, J. Chem. Phys. 64 (1976) 3573. [ 141 M. Pope and C.E. Swenberg, Electronic processes in organic crystals (Clarendon Press, Oxford, 1982) p. 49 1. [ 151 P.J. Bounds and W. Siebrand, Chem. Phys. Letters 75 (1980) 414.
ELECTRON-ION
CHEMICAL PHYSICS LETTERS
GEMINATE ESCAPE PROBABILITY IN ANISOTROPIC
18 December 1987
MEDIA
A. MOZUMDER =, SM. PIMBLOTT b, P. CLIFFORD ’ AND N.J.B. GREEN a ’ Department of Chemistry and the Radiation Laboratory ‘, Universityof Notre Dame. Notre Dame, IN 46556, WSA b Physical Chemistry Laboratory, Universityof Oxford, Oxford, UK
’ Mathematical Institute, St. Giles, Universityof Oxford, Oxford, UK Received 17 September 1987
The diffusion-controlled electron-hole geminate escape probability in anisotropic media has been calculated by numerical solution of the adjoint equation for that probability after a series of simplifying transformations. Relative to the crystalline axes the anisotropies of both dielectric constant and mobility, which generally have different principal directions, are important. The escape probability depends on the magnitude and direction of the initial electron-hole separation. Applied to anthracene the implied experimental escape probability ~0.01 is attained at initial separations 4.0, 4.0 and 3.2 nm along the a, b, c’ crystal directions respectively
1. Introduction
In a variety of condensed media absorption of lowLET radiation generates mainly geminate ion pairs, i.e. isolated pairs of electrons and positive ions or holes [ 1,2 1. In hydrocarbon liquids Onsager’s theory [ 31 provides the basis for calculating the probability of escaping geminate recombination under the joint influence of diffusion and mutual Coulombic attraction. The theory compares favorably with the results obtained from experimentally determined freeion yields [ 41. Onsager’s solution, derived from the isotropic Smoluchowski equation or diffusion in a potential field, gives in the limit that the reaction radius a CCr,, @iso =exrd
-
~J~o)
,
(1)
where 4i,, is the geminate escape probability in isotropic media, r. is the initial separation of ions and rC=e21tkT with e as electronic charge, t, dielectric constant of the medium, k, Boltzmann’s constant and T, the absolute temperature. In crystals Onsager’s ’ The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-3030 from the Notre Dame Radiation Laboratory.
0 009-2614/87/$
treatment needs extension since the diffusion is anisotropic; furthermore the Coulombic field is also anisotropic. A prime example is provided by the photoionization free-carrier yield in anthracene crystals where, for reasons of simplicity, experimental results have been analyzed using the Onsager theory [ 5,6], which in principle is not applicable to this case. Our aim in this paper is to give a systematic derivation of the escape probability in anisotropic media and to indicate approximately the directiondependent values of the initial (or, interpretively the thermalization) distance of an electron-hole pair in anthracene that would be comparable with the experimentally deduced escape probability. Onsager [ 31 notes the self-adjoint character of the diffusion equation written in terms of w exp(u/kT) where w is the density function and v is the potential, By virtue of this property he then asserts that the probability @ will satisfy the backward equation: V - [ exp( - v/kT) V $1 = 0. This equation is readily generalized to anisotropic media as follows [ 71 V.[exp(-u/kT)D*V#]=O,
(2)
where D, the diffusion tensor, is related to the experimentally determinable mobility tensor p by the Nemst-Einstein relation, p= (e/kT) D. Boundary conditions for solving eq. (2), consistent with those
03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
385
Volume 142, number
5
CHEMICAL
PHYSICS LETTERS
for the Smoluchowski equation, and considering that + is the escape probability, may be given by @=O at the origin and #= 1 at infinite separation. The mutual electrostatic potential v with the hole at the origin and X’denoting the electron coordinates is given by 181
~=--*(I,I,,~'x~x~)-"~=-~~(~~E,)-"*. (3) In eq. (3) 1t 1 is the determinant of the dielectric tensor with elements t,k, r is the radius vector, rT the transpose of r and E, called here the symmetric dielectric matrix, is related to the dielectric tensor by E= ]E]E-‘. The solution of eq. (2) with the potential of eq. (3) will be developed in section 2.
2. Method and procedure Let Q be the orthogonal transformation that diagonalizes the diffusion tensor, i.e. D = QT diag( A,)Q where A, (i= 1,2,3) are the eigenvalues. Making the transformation
s=QT diag(d;“2)Qr~D-1’2r with D “2 =QT diag(lf’*)Q and substituting in eq. (2) one obtains
V~VqL(e*/kT)(Vqb*Ms)(s~Ms)-3'2=0, (4) where M-D”*ED’/*
(5)
and the differential operators of eq. (4) operate in the s space. Note that the matrix M has all the required combination of the elements of the diffusion and dielectric tensors. Let P diagonalize M with eigenvalues mi (i= 1,2,3), i.e. M=PTdiag(mi)P. Then transforming s to f by t= Ps, eq. (4) may be written as V -V$-(e21kT)V#-diag(m,)t[tTdiag(mi)l]-3’2 =o,
(6)
where t, (i = 1,2,3) are the principal directions of 1. In eq. (6) the differential operators now operate in the t space. Finally the linear transformation u=cf, with c chosen such that c* = XT=I m,(e*/kT)-*, simplifies eq. (6) into 386
3a2$
18 December
1987
2=o (7) c --,~~Ui~(j~,u:P,)-3’ ,=,a~5 3
where p, = milC :=, m, and Zpi= 1. The significance of the first transformation (cf. eq. (4)), effectively a rotation, is to represent 9 in a coordinate system where diffusion is diagonal; the second transformation (cf. eq. ( 6)) then extends or contracts the axes to afford further simplification. The solution of eq. (7) can now be represented in dimensionless form by a family of two parameters. To construct the reduced equation (7) in a given case one proceeds by (i) diagonalizing the diffusion matrix, (ii) constructing the M matrix (cf. eq. (5)) and then (iii) diagonalizing the M matrix. However, since we only need the ratios pi and not the absolute mi values we are permitted to replace E by c-’ and D by P, which really contains the experimental results on transport. Schematically this is represented by diag( m:) such that diag( m:) =PQT diag(,&)“*Qe-‘QT
diag(&)“2QPT
,
with the understanding that p, = m,/I & I mi = m:lC~,l m:. After obtaining the pi values the numerical solution of eq. (7) is facilitated by further transforming the infinite u space into a finite one and then applying the relaxation method to the so-obtained discretized form of eq. (7). Several transformations of u have been tried, however, w=exp( -3”*u-‘)
(8)
has been found to be most effective for the numerical work reported here. The polar and azimuthal angles, with the usual meanings referred to the respective Cartesian coordinates, remain unaltered. This transformation maps the infinite u space onto a unit sphere in the w space and the boundary conditions are #( w+O) =0 and $( w+ 1) = 1. While, subject to these boundary conditions, any reasonably continuous form of @is acceptable for starting the relaxation procedure, in point is simplest starting practice the esran= exp( v&T) with the potential v of eq. (3). This @stanis the exact analogue of the Onsager probability (cf. eq. (1)) except only that the potential refers to the anisotropic medium. Some details of the relax-
Volume 142, number 5
ation method employed together with the criterion of convergence, procedure for acceleration of convergence, etc., will be reported elsewhere [ 91.
3. Results and discussion In this paper we carry out numerical computations for the anthracene crystal. The experimental determinations and theoretical rationalization of the elements of the dielectric [lo] and mutual (electron-hole) mobility [ 111 tensors for this crystal (12&c directions) are given below. 2.9 r= 0 ( -0.57
p=
2.70 0 -0.0458 i
0 2.94 0
18 December 1987
CHEMICAL PHYSICS LETTERS
-0.57 , 0 3.841
0 3.00 0
-0.0458 . 0 1.20 i
While the elements of B admit errors of <4W, considerable uncertainties still exist about the elements of p and their interpretation. In the calculations reported here we use the values as given above. Using high-energy X-rays it has been established that z 3000 eV is required at a vanishing collecting field to create a free carrier in anthracene over z-25” to +25” [12]. With a Wvalue for ionization of 30 eV [ 131 one then gets a geminate escape probability P,,, x 0.01. With UV irradiation Chance and Braun [ 13 ] indicate P,,, x 0.001-0.07 and representative electron-hole separation z lo-70 8, depending on the UV wavelength and the form of assumed initial distribution. However these studies are indirect and based on activation energy measurement; also an uncorrected Onsager equation is used for P,,,. The interpretation is therefore dubious. On the other hand the quantum yield for photocarrier generation in the 240-280 nm region is indicated as = 1.5~ 10m4[ 131. If this is rationalized with a Gaussian distribution [ 131 one gets a quantum yield of photoionization x 0.0 1 and also P,,, NN 0.0 1. We may thus take P,,, x 0.0 1 in anthracene at room temperature both for UV and X-rays. Our calculation using the E and p matrices given earlier and ini-
tial a-distribution gives the corresponding initial electron-hole separation along the a, b, and c’ directions as 4.0, 4.0 and 3.2 nm respectively. The effect of an initial distribution will be reported elsewhere [ 91. Pope and Swenberg [ 141 argue an initial electron-hole separation z 50 A (spherically symmetrical) under similar circumstances, However this, being based on the isotropic Onsager equation, is also dubious. Although the backward equation for the escape probability is readily generalized to the anisotropic media (eq. (2)) its solution is not straightforward. The present treatment is the first complete generalization of the Onsager problem to such cases. The most important finding is that P,,, depends on D as well. Actually it depends on the ratios of the elements of D; therefore its absence in the isotropic case is easily understood. For recombination of neutral radicals (v=O) an exact solution of eq. (2), also depending Z
Fig. I. Contour plot of escape probability of a geminate electron-hole pair in anthracene at 290 K calculated according to the present theory and using the experimentally determined P and t. The contours represent closed curves of equal probability (numbered 1 through 10 denoting equally spaced probabilities 0.02 through 0.20) in the X-Zplane ofthe transformed w space. Each division on the X and 2 axes represent a dimensionless unit of 0.02. The closed curve A denotes the escape probability for 50 A electron-hole separation distance in real space. The directional dependence of P,, is evident at the same magnitude of the separation.
387
Volume 142, number 5
CHEMICAL PHYSICS LETTERS
on the ratios of the elements of D, is possible and will be reported elsewhere [ 91. In the past Pest for geminate pairs in crystals has been computed using the Onsager equation (1) and a diagonally averaged E [ 5,6,13,14], Apart from the lack of theoretical justification this procedure ignores the contribution of D and therefore cannot be valid. The anisotropy of the Coulombic potential has also been often ignored introducing another dimension of uncertainty [ 151. Our procedure pays due attention to both the B and D anisotropies. Furthermore we have checked our numerical computation against the Onsager equation in the isotropic limit and a high degree of agreement has been found with a difference < 1 part in 104. The remaining difference is due to the finite mesh size used for numerical calculation; the agreement can be made much better at the cost of computer time. Fig. 1 shows a contour plot of PC,, for anthracene in the reduced (w) coordinates. The closed curve A denotes the escape probability calculation for an initial 50 A separation of electron-hole pair in the geometrical (crystal) space. The directional dependence PC,, at the same magnitude of the separation is apparent and is characteristic of anisotropic diffusion.
Acknowledgement SMP is grateful to Harwell Laboratory, United Kingdom Atomic Energy Authority, for the award of a research studentship. AM thanks Harwell Laboratory for a visiting appointment and the Wolfson College, Oxford University for a visiting fellowship
388
I8 December 1987
which enabled him to perform part of the work at the physical chemistry laboratory of Oxford University. References [ 11 A. Hummel, in: Advances in radiation chemistry, Vol. 4, eds. M. Burton and J.L. Magee (Wiley, New York, 1974) [2] i. lMozumder and J.L. Magee, J. Chem. Phys. 47 (1967) 939. [3] L. Onsager, Phys. Rev. 54 (1938) 554. [4] W.F. Schmidt and A.O. Allen, J. Phys. Chem. 72 (1968) 3730; J. Chem. Phys. 52 (1970) 2345. [ 51K. Kato and C.L. Braun, J. Chem. Phys. 72 (1980) 172. [6] R.R. Chance and C.L. Braun, J. Chem. Phys. 59 (1973) 2269; R.H. Batt, CL. Braun and J.L. Hornig, J. Chem. Phys. 49 (1968) 1967; Appl. Opt. Suppl. 3 (1969) 20; N.E. Geacintov and M. Pope, in: Proceedings of the 3rd International Conference on Photoconductivity, ed. EM. Pell (Pergamon Press, Oxford, 1971) p. 289. [ 71 M. Tachiya, J. Chem. Phys. 69 (1978) 2375. [ 81 L. Landau and E.M. Lifshitz, Electrodynamics of continuous media ( Pergamon Press, Oxford, 1960) pp. 6 1,62. [9] SM. Pimblott, N.J.B. Green, P. Clifford and A. Mozumder, to be published. [lo] R. Munn, J. Nicholson, H. Schwab and D. Williams, J. Chem. Phys. 58 (1973) 3828. [ 1l] G.D. Thaxton, R.C. Jamagin and M. Silver, J. Chem. Phys. 66 (1962) 2461; J.J. Katz, S.A. Rice, S. Choi and J. Jortner, J. Chem. Phys. 31 (1963) 1683. [ 121 R.G. Kepler and F.N. Coopage, Phys. Rev. 151 ( 1966) 6 IO: R.C.Hughes, J.Chem.Phys. 55 (1971) 5442. [ 131 R.R. Chance and C.L. Braun, J. Chem. Phys. 64 (1976) 3573. [ 141 M. Pope and C.E. Swenberg, Electronic processes in organic crystals (Clarendon Press, Oxford, 1982) p. 49 1. [ 151 P.J. Bounds and W. Siebrand, Chem. Phys. Letters 75 (1980) 414.