Tb(III) in β″-alumina crystals

Volume 168, number 2

CHEMICAL PHYSICS LETTERS

27 April 1990

ENERGY MIGRATION AND ENERGY TRANSFER IN THE SYSTEM Ce(III)/Tb(III) IN fl”-ALUMINA CRYSTALS R. TWARDOWSKI I, M. EYAL, R. REISFELD 2 Department ofAnalytical and Inorganic Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

L.A. MOMODA and B. DUNN Materials Science and Engineering Department, University of California. Los Angeles, CA 90024, USA Received 6 February 1990

The parameters of energy migration and energy transfer between Ce(III) and Tb( III) ions in p”-alumina were found by Monte Carlo simulation ofexpetimental decay curves. The polarization measurements reveal significant energy migration in the Ce(II1) systems. The value of the critical radius for Ce( lIl)-Ce(II1) energy migration is 20.9 t 1 A and for energy transfer from Ce(II1) to traps is about 24 A. The critical radius for Ce (111)-l%(III) energy transfer .is 3.1 f 0. I A.The results are consistent with twodimensional space.

1. Introduction

Energy transfer between Ce( III ) and Tb( III) and the behaviour of Ce(II1) have been studied in various crystals [ l-6 ] and glasses [ 7 1. Blasse and Bril [ 1 ] found that concentration quenching of Ce(II1) depends remarkably on the host matrix. In recent works by Momoda et al. [ 5,6] concentration quenching of Ce(II1) and energy transfer from Ce(II1) to Tb(III), Nd(II1) and Pr(II1) in B”alumina were studied. The authors found weak, albeit significant, dependence of the lifetime of Ce( III) on its concentration. A number of theoretical papers have dealt with the problem of the influence of the excitation energy migration among donor ions on the donor fluorescence in donor-acceptor systems [ 8-2 11. Burshtein [ lo] and Huber [ 141 distinguish two models for the energy transfer in such systems. When there are many donors in the vicinity of an acceptor, the transfer of energy has a diffusive character and the model of ’ Permanent address: Department of Technical Physics and Applied Mathematics, Technical University of Gdansk, 80-952 Gdansk, Poland. * Emique Berman Professor of Solar Energy.

Yokota and Tanimoto [ 8 ] is relevant. In the opposite case, when the excitation is hopping from one surrounding to another, the hopping models are adequate [ 9,10,14,18,19,2 13. Practically, the hopping model is appropriate when the donor-acceptor transfer is slow in comparison with donor-donor transfer. In the opposite limit the diffusion model is more suitable. If dipole-dipole interaction is responsible for the energy transfer, the necessary condition for applying the hopping model is the fulfilment of the inequality R. < R,, where R. and R, are critical radii for donor-acceptor and for donor-donor interactions respectively [ 221. However, if Re * R, the diffusion model holds [ 10,141. While the qualitative aspect of energy transfer from Ce ( III ) to Tb (III) was already described in ref. [ 6 1, it is the purpose of the present paper to address the following issues: ( 1) The existence of energy migration between Ce(II1) ions. (2) The critical radius for this energy migration. ( 3 ) The critical radius for quenching of the Ce (III) ions, whereby the excitation energy is lost. (4) The critical radius for energy transfer from Ce(II1) to Tb(II1).

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Conduction

( 5 ) The dimensionality of the system with respect to these processes. In the present paper we compare the measured decay curves of Ce(II1) luminescence with the theoretically obtained survival probabilities of Ce( III) in p”-alumina in the presence and in the absence of Tb( III) ions.

Plane

2. Structure and properties of exchanged Na’(RE)-fY’-alumina l -Beevers-Ross

Sodium P”-alumina, nominally Na1.67Mg0.67A110.33 -O,,, is well known for its ability to transport cations [ 23,241. The crystal consists of distorted spine1 blocks 11.3 A thick separated by relatively open planes, where the Na+ ions are organized in hexagonal arrays (honeycomb structure) around the bridging oxygen. The Na+ occupy Beevers-Ross (BR) sites having CsVsymmetry (one oxygen in the spine1 block above and three in the spine1 block below [ 5,251). The Na+ occupy an average of 5/6 of the positions available, so a large value for the ionic conductivity, 2.5 x 10e2 Q-’ cm-’ at room temperature [ 231, results. The mobile Na+ ions can be readily exchanged by other ions [ 261, in particular by rare earths such as Nd(III), Ce(III), Th(III), etc. In this case, conservation of charge requires that for each rare-earth ion introduced, three sodium ions are replaced. The preferred site for the rare-earth ions is at the mid-oxygen (MO) position which is midway between two adjacent BR sites [ $27,281. The honeycomb structure is shown in fig. 1. The distance between the ions at adjacent MO sites is 2.8 A and that between ions at adjacent BR sites is 3.2 A. The probability of occupation of a BR site by a rare-earth ion is less than 1% [ 27,29 1. Thus the MO sites form small hexagons with “empty” spaces at the BR sites, whereby each MO site belongs to two such hexagons. The probability of occupation of a MO site by a rare earth is therefore pi=& in the case of complete exchange, and pi = & c./cf , in the case of incomplete exchange (where cf= 1.8~ 102’ cm-’ is the maximum concentration of the rare earth in the case of complete exchange [ 27 ] and ci is the actual concentration of the rare earth). Because of this essentially planar distribution of the rare earths in the crystal, the migration of energy and the energy transfer 212

0 -Mid-Oxygen

sites sites

Fig. 1. Schematicstructure of planar organization of sodium and rare-earth ions in 0”-alumina. Eachthree sodium ions (placed in the 3R positions) are replacedby one rare-earth ion (occupying the position MO). There is low probability that a rare-earth ion will enter the BR position.

among rare-earth ions should have a twodimensional character, similar to the ionic conductivity [ 231 provided that the “vertical” energy transfer, i.e. energy transfer between adjacent planes is much less probable than the energy transfer in the plane.

3. Experimental The p”-alumina crystals were prepared by the ionexchange process. In this approach, precursor single crystals of Na+p”-alumina (grown by the flux evaporation technique [23]) were immersed in the appropriate molten salt to obtain ion exchange of the desired dopant ( s ) . Ce (III ) : p” -alumina crystals were prepared using CeCl,-NaCl molten salts while the Ce( III) /Tb (III ) : j3”-alumina crystals were prepared by a double-exchange method. Details of crystal preparation and the ion exchange technique are outlined in ref. [ 6 1. Compositions of the substituted crystals are as follows: (1) Sample7A:2.9~10~~cm-~Ce(III) (16%exchange ) _ (2) Sample 7B: 3.8x10Zocm-3 Ce(II1) (21%exchange). (3) Sample 4Tb: 1.6x IO*’ cm-3 Ce(III), 6.3x 102’ cm- 3 Tb (III ) ( 44% exchange).

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CHEMICAL PHYSICS LETTERS

(4) Sample 1OTb: 9.9X 10L9cm-3 Ce(III), 9.9X 10” cm- 3 Tb (III ) ( 60% exchange). Excitation and emission spectra were measured on a FP-770 Jasco spectrofluorimeter. Polarized emission spectra were measured on the same instrument. Luminescence decay curves of the samples were measured on a system consisting of a PRA nitrogen laser (1.4 mJ/pulse, 3-5 Hz repetition rate, 0.6 ns fwhm), a MacPhenon monochromator equipped by a Hamamatsu microchannel plate, a Tektronix signal averager (0.1ns resolution) and a PC computer for data storage. The signals were collected at 400 nm and averaged over 32 pulses. The stored signals were then transferred to another computer and compared to various synthetic curves convoluted with the shape of the laser pulse

4. Results and discussion 4.1. Anisotropymeasurementsfor Ce(IZI)): jl”alumina (samples 7A and 78) and for Ce(Ill)/ Tb(1II):jP-alumina(samples4Tb and 1 OTb) In order to verify the existence of migration among the Ce (III) ions (the donors) we studied these samples by the polarized luminescence technique [ 301. This technique allows to detect the migration of energy in the donor system by gradual loss of anisotropy as a function of concentration. The anisotropy of the system is given by r= (I,-Z,)/(Z,+21,) [30], wherelpdenotestheemission polarized parallel to the polarization of the exciting light and 1, denotes the emission polarized perpendicular to the exciting light. In practice, the parameter r varies between 0.4 for energy migration much slower than the radiative decay rate, and 0 for energy migration much faster than the radiative decay rate. We have detected a complete loss of anisotropy for the four samples studied with r equal to 0 in the entire spectrum range of the emission, which indicated that the energy migration within the Ce( III) system is predominant.

4.2. Simulationsfor Na+-Ce(III)-/3”-alumina (samples 7A and 7B) To describe the fluorescence decay in the Na +Ce(III)+“-alumina we use the continuous approximation for the energy transfer, and we assume random distribution of the Ce(II1) ions. This is a reasonable approximation since the results of anisotropy measurements indicate a fast energy migration even in the diluted sample 7A ( 16% exchange). The critical radius for the energy migration is much larger than the nearest-neighbour distance in the MO lattice. This allows to ignore the lattice details in the model [ 141. It has been shown experimentally [31] that the hopping model describes fluorescence decay fairly well in a three-dimensional donor-acceptor system consisting of organic dyes. In the present paper we apply this model to our systems consisting of trivalent ions in the p”-alumina crystal. In the hopping model the donor decay function d(t) satisfies the general relation [ 9,lS ]

where 4’(t) denotes the averaged decay function of donor ions originally excited by a short pulse of light at time t~0, and A(r) is the probability density of energy transfer from donor to donor averaged over the spatial distribution of donors and traps. These traps are responsible for the quenching of luminescence of Ce(II1) in the absence of Tb(II1). It has been shown that [ 171 @O(t)=&(t) #g(t)

(2)

and A(t) = -g:(t)

dd.o(t)ldt >

(3)

where @p(t) is the averaged decay functions of initially excited donors transferring energy to another donor (without any nonradiative relaxation) and #t(t) is the averaged decay function of the initially excited donors losing energy to a trap. The radiative decay function is omitted in eqs. ( 1 ), (2) and (3) since the full decay functions can be obtained by multiplying these functions by exp ( - t/q,), where ~~ is the mean decay time of the donor in the absence of energy transfer. 213

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In our case (critical radii much larger than the minimum distance between the ions) these functions are conveniently expressed as (subscript a stands for “donor” and b for “trap”)

~~tt)=exp[-Yb(tlT~O)d’nl,

(4)

9~(1)=exp[-r,(tlzo)d’~l ,

(5)

where d and n are the Euclidian dimensionality and the order of the multipolar interaction, respectively,

‘0

200

loo Channel

Yb=(CLJGN)Ul-d/n)

3

(6)

where cb is the concentration of traps and cob is the Critical Concentration Of traps given by cob= 1/fleb, where f= 1, IC, 41c/3 for d= 1, 2, 3, respectively, r( 1 -d/n) is the gamma function, and Y,-2d/“-I (G/CO,) r( 1-dh)

,

(7)

where c, is the concentration of donors and co. = 1/ fRO,, f= 1, x, 4x/3 for d= 1, 2, 3, respectively; here, Rob is the critical radius for energy transfer to traps and &, is the critical radius for energy migration. The functions (6) and (7) differ by the factor 2d’“-’ since the migration of energy among the donors can be reversible, contrary to the irreversible transfer from donors to traps [ 14,19,32,33 1. The theoretical decay curves were generated in parallel by use of eq. ( 1) and the radiative lifetime of Ce(II1) of 46 ns [6] with the Monte Carlo method, and by iteration of eq. ( 1) (the Monte Carlo method is outlined in the Appendices) for two and three dimensions choosing a variety of h and h pairs. Both methods give identical results for the same gamma values as shown in fig. 2. The Monte Carlo method is, however, much faster than the iteration method. These theoretical curves, convoluted with the laser pulse, were compared with the experimentally measured decay curve for one of the samples. The gamma values giving the best tit to the decay curve were than used to calculate the R, and Rb vahes. These critical radii were then inserted into eqs. (6) and ( 7) for the second concentration of Ce(II1) and the theoretical curve so obtained compared with the experimental decay curve of Ce(II1) at the second concentration. When these values did not yield a good fit, a new pair of gamma values was generated for one concentration and applied for the second concentration, until 214

300

Iterations - Mlgrollon

I----++++

Monte-Carlo-

2- . . . . .

Iterations - Forster Monte-Carlo-Forster

Migration

Fig. 2. Comparison of iterative methodwith Monte Carlomethod. Curve 1: Migration t trapping. ya= 1S, ybzO.5; broken line: Iteration,crosses:MonteCarlo.Curve 2: Trapping without migration (Fijrster model). y.=O, %=0.5 solid line: iteration, dots: Monte Carlo.

critical radii simultaneously satisfying both decay curves were found. The goodness of fit was determined by using an autocorrelation method [ 3 11. For dimension = 2 we found only one pair of critical radii distinctly satisfying a good fit for both decay curves. For dimension= 3 we found a few pairs satisfying a moderate fit. However, as seen in table 1, the correlation factors for two dimensions are 0.960.99, while for three dimensions the correlation factors were not more than 0.81. Fig. 3 (two upper pictures) presents the comparison of the experimental curves (samples 7A and 7B) with the theoretical curves for dimension=2. The critical radius Rb of 24 A was obtained from the best values of &,assuming that the energy trapping occurs at BR sites occupied by Ce( III), the concentration of which is about lob of the total Ce( III) concentration, the majority of Ce(II1) ions being situated at MO sites [29]. This critical radius is similar to that determined by Stevels, 22-25 A, for energy transfer from Ce(II1) to Ce-0 associates in magnetoplumbites [ 41. Having determined the migration parameters for the donors, we proceed now to determination of the parameters for energy transfer Ce(III)-Tb( III) in the presence of energy migration.

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CHEMICAL PHYSICS LETTERS

27 April 1990

Table 1 Interaction parameters calculated for energy transfer and energy migration in Ce (III)/Tb(III) Sample

1A 7B 4Tb lOT%

Concentration

Migration

doped j3”-alumina

Trapping

Ce(III)

Tb(II1)

1:

A.~)

2.9 x 1O-2o 3.8 x10-20 1.6 x~O-‘~ 0.99x10-20

6.3~10-~” 9.9x lo-zD

3.75 5.20 -

20.7 21.2 21 21

(A)

H

RI,') (A)

0.075 0.109

27 23

Energy transfer Ro”) (A)

Correlation R

_ 3.1 3.1

0.999 0.997 0.970 0.960

4.3. Simulations for Na+-Ce(III)-Tb(II+jl”alumina

Time

Inset]

Fig. 3. Decay curves of Ce(III), experimental (solid lines) and fitted (broken lines). From top to bottom: sample 7A, Tb(III) absent, sample 7B, Tb(II1) absent. Critical radius for migration: 20.9 f 1 A, estimated critical radius for trapping: about 24 A. Sample 4Tb, sample lOTb, Critical radius for energy transfer to Tb( III ): 3-l+ 0.1 A.Allcurvesgenerated for two dimensions and dipole-dipole interactions.

In the case of the more complicated system Ce(III)/Tb(III) (samples4Tband 10Tb) weapply a different procedure than that for energy migration among the Ce(II1) ions. The reason for this is that the critical radius, Ro, for Ce(III)-Tb(II1) energy transfer is, by a first estimate, comparable with the nearest-neighbour distance between Ce( III) and Tb(II1). In such a case, the continuous approximations (eqs. (6) and (7 ) ) do not hold. The Monte Carlo method used for simulation of such system originates from refs. [ 34-361 and proceeds as follows: A two-dimensional honeycomb lattice is generated. This lattice has the same lattice parameters as the hexagonal two-dimensional lattice of MO sites in b”-alumina. Next, the lattice is seeded randomly by donors (Ce (III) ) and acceptors (Tb ( III) ) according to the respective probabilities of occupation, p=$ c/cr where cf= 1.8x 10” cmm3 (cris the maximum concentration of rare-earth ions achieved by lOOohexchange) and c is the actual concentration of the donors or acceptors. Further details of the method are given in the Appendix. In a manner similar to the procedure given in section 4.2, we simulate a decay curve for a chosen critical radius at given concentrations of donors and acceptors (sample 4Tb or 1OTb) and for a dimension of 2 or 3. The decay curve is then convoluted with the laser pulse and compared with the decay curves for samples 4Tb and 1OTb until the critical radius gives a good fit for both samples simultaneously. In these simulations we already make use of the value of the critical raius for energy migration between the Ce (III) ions, R, = 2 1 215

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A, which was found in section 4.2. In the above treatment we have not taken into account the energy transfer from Ce (III) to traps, since, although its estimated critical radius is large, the very low concentration of traps allows one to ignore their influence on energy transfer from Ce (III) to Tb (III). Fig. 3 (second from bottom, solid line) presents the measured decay curve for the system 4Tb, and the bottom picture, solid line, presents the decay curve for the system 1OTb. The fitted curves (broken lines) were generated from the Monte Carlo model for R,,= 3.1 t 0.1 A, for two dimensions and for dipole-dipole interaction and convoluted with laser pulse. With this critical radius, a significant energy transfer could occur only in the presence of a strong energy migration in the donor system, similar to the systems of Mn(II)-Tm(Il1) [ 111 and Mn(II)Nd(II1) in fluoride glasses [37].

5. Conclusions We studied the energy transfer among Ce (III) and Tb(II1) ions in fi”-alumina. We found, by polarization measurements, that significant energy migration takes place in all the systems studied in the present work. Monte Carlo simulations in a continuous approximation were performed for the energy migration, and by lit of the generated curves we found that the critical radius for donor-donor energy migration is 20,9 + 1 A, and for energy trapping it is about 24 A. The traps are probably BR sites occupied by Ce (III) ions. Energy transfer in the presence of acceptors (Tb (III) ions ) was studied by use of the Monte Carlo model in a discrete mode and on an organized lattice seeded randomly by donors and acceptors. We found that the critical radius for the energy transfer from Ce(II1) to Tb(II1) is 3.1-10.1 A. The model was performed for dipole-dipole interactions. We found also, that most of the energy-transfer events occur in a two-dimensional space, this result being consistent with the ionic conductivity in $“alumina.

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Acknowledgement These studies were in part funded by the Offlce of Naval Research, Washington DC. One of us (RT ) thanks the Enrique Berman Fund and the Polish Fund CPBP-01.06 for support.

Appendix 1. Monte Carlo method for energy migrationbetweenCe(II1) ions in the presence of quenching traps The Monte Carlo method was developed by Gillespie for chemical reactions [ 351 and adapted to energy transfer by Kusba et al. [ 36,371. Here we briefly outline the method: The probability density for fluorescence, PF, is given by PF={-d[exp(-tlT,)lldt}~~$~.

(10)

The averaged probability density for transfer of energy from an excited donor to a trap, Pm, can be represented in the form &=(-d&d0

exp(-t/r01

CZ

(11)

and the averaged probability density for migration of the energy from an excited donor to another donor, PEM,is &

= ( - d&/d0

@i?ev( - tboo) ,

(12)

where the functions #f and @Ehave the same meaning as in eqs. ( 1)-( 8). The Monte Carlo method allows us to answer two questions: When will one of the processes describedin eqs. ( lo)-( 12) take place and what kind of process will it be? The time t,, at which one of these processes occur can be determined from the equation -

1

dtd[&$exp(-t/~o)]/dt=6,,

(13)

where 6, is a random number from the interval 0 to 1. In the next step we choose the kind of process by using another random number &. If the inequality S, QPF/ ( PF t PET+ PEM) is satisfied, the process of emission of photon, PF, is chosen. Otherwise, a migration of energy or energy transfer to an acceptor occurs. If P~/(P~+P~~+P~~)
CHEMICAL PHYSICSLETTERS

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(PF+PET+PEM) exists, then the energy transfer from donor to acceptor is chosen. Otherwise if S,> (PF +PET)/(PF+PET+PEM), the migration occurs. The computer simulation consists of N trials. The process starts by excitation of a donor. Then a random time t, is chosen according to eq. ( 13). With the use of inequalities as described above and a random number S,, one of the possible processes, i.e. emission of a photon, energy transfer to another donor or energy transfer to an acceptor, is chosen. The result of the choice is placed in a suitable counter. Then, the next donor is excited and the process is repeated. After the number of trials reaches N, the sampling is stopped and the random times tl are &ted in ascending order and placed in an indexed array, The decay curve is constructed by assigning to each index j in this array a value (N-j) /N, where N is the number of trials.

Appendix 2. Monte Carlo method for energy transfer Ce(III)-Tb(II1) accompaniedby energy migration between Ce(II1) ions Since in this case the transfer of excitation occurs on a well-defined lattice, the interactions are summed over the surroundings of the excited Ce(II1) ion. The fate of the excitation is then determined by probabilities of various outcomes (PF, fluorescence; PET, energy transfer to Tb(II1); PEM,energy migration) as given by &=(1/G

exp(-at),

PET=(~/~o) exp(-at) PEM=(lhO)

exP(-at)

(14) C

CROIRj)",

(15)

c (&JR,)“,

(16)

where a is defined as a=

(

1 +C (&IRj)“+C

(&IRj)m

>

/TO,

(17)

m and n are the multipolar orders for interactions among the donors and between donors and acceptors, respectively, and the sums are over the MO sites occupied by unexcited donor or acceptor ions, respectively. R. is the critical radius for energy transfer from Ce(II1) to Tb(II1) and R, is the critical radius for energy migration between Ce( III) ions. This value is equal to 2 1 8, as determined previously. We

27 April 1990

have approximated the sums taking into account all ions within the range of three critical radii from the excited donor. The time tl at which one of the processes described by eqs. ( 14)-( 16) is chosen is determined from eq. (13). In this case eq. ( 13) gives tl = - (ln 6)/a.

(18)

Further treatment is almost the same as described before, the only difference being that in the case of choice of energy transfer to a donor, we must also specify which donor was excited. This is done by choosing the index k of the donor from the following inequality,

(W&)m

c

k


b

c

k+l

(RdRtJ” b

’

119)

where S, is again a random number from the uniform unit interval and b denotes the sum of energytransfer rates from the excited donor to all other donors within the distance 3R,, b=C

(RJR,)“‘.

(20)

[ 1 ] G. Blasse andA. Bril, J. Chem. Phys. 5 I ( 1969) 3252. [2] J.L. Sommerdijk, J.A.W. van der DoesdeBye and P.H.J.M. Verberne, J. Luminescence 14 (1976) 91. [ 3 ] B. Blanzat, J.P. Denis and R. Reisfeld, Chem. Phys. Letters 51 (1977) 403. [4] A.L.N. Stevels, J. Electrochem. Sot. 125 (1978) 588. [ 5 ] L.A. Momoda, J .D. Barrie, B. Dunn and O.M. Stafsudd, in: Proceedings of the International School on Excited States of Transition Elements, Wroclaw, 1988, eds. B. JeiowskaTrzebiatowska, J. Legendziewicz and W. Stqk (World Scientific, Singapore, 1989) p. 368. [ 6 ] L.A. Momoda, J.D. Barr&B. Dunn and O.M. Stafsudd, in: Proceedings of the Electrochemical Society Meeting, Chicago, 1988, eds. S.W. Struck, B. DiBartolo, M.W. Yen, Vol. 88-24, p. 235. [ 71 L. Boehm, R. Reisfeld and B. Blanzat, Chem. Phys. Letters 45(1977)441. [ 8 ] M. Yokota and 0. Tanimoto, J. Phys. Sot. Japan 22 ( 1967) 779. [9] C. Bojarski and J. Domsta, Acta Phys. Hung. 30 (1971) 145. [ 101A.I. Burshtein, Zh. Eksp. Tear. Fiz. 62 (1972) 1695 [Soviet Phys. JETP 35 (1972) 8821. [ 111 M. Eyal, R. Reisfeld, A. Schiller, C. Jacoboni and C.K. Jorgensen, Chem. Phys. Letters 140 ( 1987) 595.

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[ 12) R. Reisfeld and M. Eyal, Acta Phys. Polon. A 71 ( 1987) 799. [ ¶3] B.E. Vugmeister, Fiz. Tverd. Tela 18 ( 1976) 819 [Soviet Phys. Solid State 18 (1978) 4691. [ 141 D.L. Huber, Phys. Rev. B 20 ( 1979) 2307,5333. [ 151A. Biumen, J. KIafter and R. Silbey, J+ Chem. Phys. 72 (1980) 5320. [ 161H.C. Chow and R.C. Powell, Phys. Rev. B 21 (1980) 3785. [ 171R.F. Loring, H.C. Andersen and M.D. Fayer, J. Chem. Phys. 76 (1982) 2015. I 181 R. Twardowski, 1. Kusba and C. Bojaraki, Chem. Phys. 64 (1982) 239, [ 19 ] AI. Burshtein, Zh. Eksp. Tear. Fix. 83 ( 1983) 2001 [Soviet PhysJETP57 (1983) 11651. 1201 S.G. ~o~~oand A.I. B~te~,~em. Phys. 128 f 1988) 185. [Zl ] R. Twardowski and J. Kusba, 2. Naturlorsch. 43a (1988) 627. [ 22 ] V.M. Agranovich and M.D. Galanin, Electronic excitation energy transfer in condensed matter (North-Holland, Amsterdam, i982). [23] IL. Briant and G.C. Farrinpon, J. Solid State Chem. 33

(1980) 385. 1241R. Hilfer and R. Orbach, Chem. Phys. 128 ( 1988) 275.

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[25] M. Bettman and C!.R Peters, J. Phys. Chem. 73 (1969) 1774. [26]B. Dunn and G.C. Farrington, Solid State Ionics 9/10 (1983) 223. [27] B. Dunn, D.L. Yangand D. Vivien, J. Solid State Chem. 73 (1988) 235. [ 281A.J. Alfrey, O.M. Stafsudd, B. Dunn and D.L. Yang, J. Chem. Phys. 88 (1988) 707.

f29] D. Barrie,L.A. Momoda, B. Dunn, D. Gourier, G. Aka and D. Vivien, submitted for publication. 1301J.R. Lakowicz, Principles of fluorescence spectroscopy (Plenum Press, New York, 1983). [ 311R. Twardowski and R.K. Bauer, Chem. Phys. Letters 93 (1982) 56. 1321D.L. Huber, D.S. ~milton and D. Barnett, Phys. Rev. B 16 (1977) 4662.

[ 331 I. Rips and J. Jortner, Chem. Phys. 128 ( 1988) 237. [ 341 D.T. Gillespie, J. Phys. Chem. 8 1 ( 1977) 2340. 1351J. Kusba and C. Bojarski, in: Conference Digest, Szegcd, Hungary, 1979, Vol. 1, p. 41. [ 361 J. Kusba and R. Twardowski, in: intermolecular interaction in excited states, eds, C. Bojarski and A. Kawski (Ossolineum, Wrociaw, 1986) in Polish. [ 37 ] R. Reisfeld, M. Eyal, C.K. Jorgensen and C. Jaeoboni, Chem. Phys. Letters 129 ( 1986) 392.