Theoretical and experimental study of dynamic triplet-triplet annihilation in an organicone-dimensional motion system

A. Benfredj,1 F. Henia,1 L. Hachani,1 S. Romdhane,1,2 and H. Bouchriha11Unité Matériaux Avancés et Optronique, Faculté des Sciences de Tunis, 1008 Campus Universitaire Tunis, Tunisia

2Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, TunisiasReceived 4 May 2004; revised manuscript received 30 August 2004; published 25 February 2005d

Pair-density-matrix theory has been adapted to the triplet-pair exciton annihilation under microwave exci-tation by including the dimensionality of exciton motion and applied to the fluorescence detected magnetic-resonance spectrum observed for a one-dimensional molecular crystal. This theory gives a satisfactory fit of theobserved effects on a 1,4 dibromonaphtalene crystal and leads to the determination of the singlet annihilationrate constantl and the triplet exciton pair lifetimeb−1.

DOI: 10.1103/PhysRevB.71.075205 PACS numberssd: 71.35.2y, 32.30.Dx, 32.50.1d

I. INTRODUCTION

In molecular crystals made of small organic moleculessuch as anthracene, tetracene, naphtalene, or 1,4 dibro-monaphtalenes1,4 DBNd, the exciton is essentially confinedto a single moleculesFrenkel excitond, leading to a bindingenergy of the order of 1 eV. The formation and annihilationof the electronic Frenkel triplet exciton and triplet-triplet ex-citons pairsFrenkel biexcitonsd in these materials has beenrepeatedly discussed.1–5 The formed biexciton can annihilateto a singlet exciton which relaxes to the lowestsluminescentd11Bu state resulting in so-called delayed fluorescencesDFd.The following schema is used to interpret this phenomenon.

T stands for free excitons,sT,Td for bound pairs,S* for theexcited singlet,S0 for the ground singlet state, andhn corre-sponds to the delayed fluorescence.

Such studies are motivated by the important role whichFrenkel biexcitons could play in the area of optical nonlin-earity. In this situation, further experimental as well as theo-retical studies certainly present interest, particularly when theincreasing significance of organic materials in photonics isconsidered.6–9 Direct experimental evidence for triplet exci-ton formation in polymers has recently been available10–12

and triplet-triplet interaction was observed.13–20 Also manytheoretical21–24 and experimental25–27 investigations indicatethe possibility of extension of the molecular models top-conjugated polymers. The theoretical discussions in manycases can be considered as continuations of those found inthe early exciton literature of polyenes.28 However, dynamictriplet-triplet annihilation was studied experimentally byfluorescence detected magnetic resonancesFDMRd tech-niques and theoretically by a Suna-like approach, only foranthracene29 and tetracene crystals30 which have a bidimen-sional exciton motion. Discussions of the dynamic triplet-triplet annihilation in one-dimensional organic systems,

where confinement greatly enhances exciton binding, are notavailable. However, dynamical studies of intramolecular andintermolecular electronic energy transport in solids are ofgreat importance for understanding the details of energy-transfer mechanisms in many different systems.31–37A well-known example of molecular crystal where the triplet exci-tons motion is one-dimensionals1Dd, as demonstrated inRef. 38, is the 1,4-dibromonaphthalenesFig. 1d. The mol-ecules are piled up as stacks alongc8=a∧b. Intrastack andinterstack hopping of excitons can occur.

There are eight molecules per cell arranged into four unitsof two molecules each notedi and j8 si =1, . . . ,4 andj8=1, . . . ,4d related by no symmetry elements. The lattice con-stants area=27.23 Å,b=16.42 Å,c=4.05 Å, and the tripletexcitons propagate mainly along thec axis having a lifetimeof a few ms. The energy of the lowest triplet exciton level isET1

=20 192 cm−1 sET1=2.37 eV, Fig. 2d with a bandwidth of

about 30 cm−1.We present FDMR effects in 1, 4 DBN crystal for a static

magnetic field applied in thesac8d crystal plane. Experimen-tal results are analyzed with a Kinematic approach adapted to1D motion system.

II. EXPERIMENTAL SETUP AND RESULTS

We have obtained crystalline samples of 1- to 3-mm thickby cutting and polishing sections from crystal ingots grownfrom the melt of highly purified 1,4 -DBN. The faces areparallel to thesac8d crystal plane, as determined by conos-copy. The experimental setup is given in Fig. 3.

The experiments were performed using a cylindrical mi-crowave cavity, adapted to resonate in the TE011 mode. Themicrowave source was a gun diode of 30 mW and 9.4 GHzfrequency followed by a wave oscillator amplifiersBWOVariand delivering a maximum power of 10 W.

The electromagnet’s power supply was digitally con-trolled by microcomputer. The samples were excited througha flexible light guide in the singlet exciton band, with thefiltered 4880 Å line of an argon laser. The light excitation

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intensities are low and correspond to the monomolecular DFregime.

The delayed fluorescencefF~ sgn2dg, which is propor-tional to the square of the triplet concentration, was fed to alock-in amplifier tuned to the 2 KHz microwave-power-modulating frequency. The output of the lock-in amplifierwas on-line digitalized and data fed to computer for process-ing and plotting of the relative microwave effectDF /FsDF=F8−F, F8, andF being the DF signals in the presenceand absence of the microwave field, respectivelyd as a func-tion of the static magnetic field strengthH. The experimentswere carried out under the following conditions: the PM ispolarized at 1200 V and the time constant is 300 ms.

Figure 4 shows the relative variation of the DF that hasbeen observed at room temperature for the direction of thestatic magnetic field;su=150°,w=0°d in the 1,4 DBN crys-tal. The spectrum presents two equidistant FDMR resonancesfrom a central positionH0=3350 Oe which gives the Larmorfrequency of the microwavev=2p39.4 GHz. The two cen-tral FDMR resonances have a maximum height ofDF /F

=0.50/00 and a width at half maximum of about 25 Oewhen the two others present a maximum height ofDF /F=0.30/00 and a width at half maximum of about 50 Oe.

III. KINEMATIC THEORY

In this paper, we focus on system in which exciton motionwas described by hopping model and adopt the smooth ap-proximation for nearest neighbors. The two-particle densitymatrix, for a uniform system, depend only on the vectordifferenceR of the positions of the two excitons and is de-noted byrsRd satisfying the equation

FIG. 1. Projection of DBN crystal structure parallel to thecaxis.

FIG. 2. The lowest energy levels of DBN crystalsRef. 41d.

FIG. 3. Experimental setup:s1d Cavity and sample,s2d laser,s3dphotomultiplier, s4d PM power supply,s5d lock-in amplifier, s6dpicoampere meter,s7d computer,s8d signal generator,s9d electro-magnet power supply,s10d PIN diodes,s11d attenuator,s12d back-ward wave oscillator,s13d circulator, s14d charge resistance,s15dand s19d detectors,s16d oscilloscope,s17d microwave source,s18dfrequency meter,s20d modulator,s21d laser power supply.

FIG. 4. Microwave modulation of delayed fluorescence as afunction of a static magnetic field, lying in theac8 plane of a onedimension 1,4 DBN crystal, at room temperature. The orientationfield is from thec axis.

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s1d

wheref…,…g stands for commutator andf. . . , . . .g+ for anti-commutator,rsRd being the density matrix for a pair sepa-rated byR, H=H0+H1 in which H0 is the sum of the ZFSHamiltonian and the Zeeman Hamiltonian of the two exci-

tons:H0=2D* sSZ2− 1

3S2d+2E*sSX2 −SY

2d+2gmBHW ·SW, whereSW isthe spin of the triplet exciton;SX, SY, andSZ are its compo-nents andS its module;D* and E* are the ZFS exciton pa-rameters; andH1=gmBH1Sx cossvtd is the microwave pertur-bation Hamiltonian.

The quantityb is the effective single exciton decay rate,CsRd is the incoherent jump rate for an exciton hopping bya lattice vectorR along thec axis, the ratel is the nearest-neighbor pair annihilation rate leading to singlets, ands isthe matrixs= uS0lkS0u which have as diagonal elements thesquares of the singlet amplitudes of the pair statesss11= 1

3ands22= 2

3d. Finally Q=admn is the pair-source term.Equation s1d can be solved in steady state conditions

f]rsRd /]t=0g or in modulated conditions to take the effectof the microwave of frequencyv into account. In the steadystate conditions, withrsRd=r00sRd andH=H0, Eq. s1d canbe written as

0 = −i

"fH0,r

00sRdg − 2br00sRd + 2oR8

csR8dfr00sR + R8d

− r00sRdg − dsRdl

2fs,r00s0dg+ + Q s2d

which just describes the static-field effect. To solve Eq.s2done first considers the Green’s functionGsb ,Rd, satisfyingthe hopping diffusion equation

oR8

csR8dfGsb,R + R8d − Gsb,Rdg − bGsb,Rd = dsRd,

s3d

wheredsRd is functionds0d=1 anddsRd=0 for RÞ0.Calling Em the pair state eigenvalues in the static field

H0uml=Emuml, and introducing the notationvmn=sEm

−End /", Vmn=vmn−v, bmn=b+ isvmn/2d, and Gsbmn,0d=Gsbmnd:

rmn00 sRd =

admn

2b+ Gsbmn,Rd

l

4fs,r00s0dgmn

+ . s4d

Eigenvectors and eigenvalues of the unperturbed Hamil-tonianH0 at high field are given in Table I, where«0 is the

energy of thems=0 level for the triplet exciton:«0=D* s 13

−n2d+E*sm2− l2d. l, m, andn are the cosines directors of the

static field directionHW with respect to the principal ZFS ex-citon axes.

In the smooth approximation bothGsRd andrsRd do notvary much over those values ofR for which eitherl andcsRd are nonzero for one lattice spacing only, and replace

them by the constantsGsR̄d andrsR̄d for all suchR, for the

next R̄ will be notedR.Equations4d becomes

rmn00 sRd =

a

2bdmn−

1

2kGsbmn,Rd

Gsbdfs,r00s0dgmn, s5d

wherek=−12lGsbd andGsbd=Gsb ,Rd−1/c given from Eq.

s3d for b!cfc=oR8csR8dgSolving Eq.s5d first for R=0, one gets, for the two states,

rnn00sRd =

a

2bF1 − Gr

An − 1

AnG, n = 1,2, s6d

rnn00sRd =

a

2b, n . 2.

The notation isGr =Gsbmn,Rd /Gsbd andAn=1+nk/3.Considering now the microwave HamiltonianH1, as a

second order perturbation inH1, the solution of Eq.s1d canbe expressed as

rsRd = r00sRd + fZsRde−ivt + Z̄sRdeivtg + H12r02sRd, s7d

where one has three matrices, namely,r00sRd for the usualstatic field effects, an auxiliary matrixZsRd, which allowsone to get ther02sRd matrix which, in turn, determines, tosecond order in microwave field strengthH1, the relativeeffect DF /F.

Introducing Eq.s7d in Eq. s1d, one gets a system of threeequations. The first one is identical to Eq.s2d, the two othersare coupled to the first equation and are given by Eqs.s8dand s9d:

TABLE I. Triplet pair states energy levels.

Pair states Energies

u1l= 1/Î3uS0l+Î2/3uQ0l 2«0

u2l=Î2/3uS0l− 1/Î3uQ0l −«0

u3l= uQ+2l= uT+1T+1l −«0+2gmBH

u4l= uQ+1l= 1/Î2fuT+1T0l+ uT0T+1lg «0/2 +gmBH

u5l= uQ−1l= 1/Î2fuT−1T0l+ uT0T−1lg «0/2 −gmBH

u6l= uQ−2l= uT−1T−1l −«0−2gmBH

u7l= uT+1l= 1/Î2fuT+1T0l− uT0T+1lg «0/2 +gmBH

u8l= uT0l= 1/Î2uT+1T−1l− uT−1T+1l −«0

u9l= uT−1l= 1/Î2uT−1T0l− uT0T−1l «0/2 −gmBH

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− ivZsRd = −i

"fH0,ZsRdg − 2bZsRd

+ 2oR8

csR8dfZsR + R8d − ZsRdg

− dsRdl

2fs,Zs0dg+ − i

gmBH1

2"fSx,r

00sRdg,

s8d

0 = −i

"fH0,r

02sRdg − 2br02sRd + 2oR8

csR8dfr02sR + R8d

− r02sRdg − dsRdl

2fs,r02s0dg+ − i

gmB

2"H1fSx,ZsRd + Z̄sRdg.

s9d

With the notationb̃mn=b+ isVmn/2d, Eq. s8d for ZmnsRd canbe solved as

ZmnsRd = Gsb̃mn,Rdl

4fs,Zs0dgmn

+

+ igmBH1

4"oR8

Gsb̃,R − R8dfr00sR8d,Sxgmn

s10d

andrmn02 sRd matrix is obtained from Eq.s9d:

rmn02 sRd =

l

4Gsb̃mn,Rdfs,r02s0dgmn

+

+i

2"H1oR8

Gsb̃mn,R − R8dfZsR8d + Z̄sR8d,Sxgmn.

s11d

As will be seen below, the algebra can be simplified byeliminating the sums overR8 using the following property ofthe Green’s functions. IfGsb2,Rd and Gsb1,Rd are twoGreen’s functions whereGsb1,Rd satisfies Eq.s3d, then theequation forGsb2,Rd can be written as

oR8

csR8dfGsb2,R + R8d − Gsb2,Rdg − b1Gsb2,Rd

= dsRd + sb2 − b1dGsb2,Rd s12d

which may be solved as

Gsb2,Rd = Gsb1,Rd + sb2 − b1doRW8

Gsb1,R − R8dGsb2,R8d

s13d

so that

oR8

Gsb1,R − R8dGsb2,R8d =Gsb2,Rd − Gsb1,Rd

b2 − b1.

s14d

The task is then to substitute for each elementr00sR8dappearing in Eq.s10d the corresponding expressions4d, andthen use Eq.s14d to obtain the sum overR8. The resultingequations10d for R=0, is then the equation forZmns0d itssolution yieldsZmnsRd for all R, via Eq.s10d, a result whichmust be used in Eq.s11d for r02sRd where a similar proce-dure is to be applied.

According to Table I, six microwave induced transitionssDms= ±1d are possible; three of thems3↔4, 1↔4, and2↔5d correspond to the lower field resonancesHl ,H0d oc-curring when

"Vl = −3

2«0 + gmBHl − "v. s15d

The otherss2↔4, 1↔5, and 5↔6d correspond to higherfield resonancesHh.H0d occurring when

"Vh = +3

2«0 + gmBHh − "v. s16d

Two symmetric FDMR resonances with respect to thecentral fieldH0=v /gmB are expected. The separationDHbetween the two resonances is given byDH=3«0. For thetransitionsVl andVh one then gets, from Eq.s10d and withthe aid of Eq.s14d,

Z14sRd =1

6lgH1r11

00s0dFGsb,Rd − Gsb̃14,RdV14

G , s17d

Z52sRd =Î2

6lgH1r22

00s0dFGsb,Rd − Gsb̃52,RdV52

G , s18d

whereg=gmB/".This result is now used in Eq.s11d which, using Eq.s14d

again, allows one to obtain the diagonal elements of the ma-trix r02sRd needed to get the relative microwave effect on thedelayed fluorescence signal obtained from

DF

F=

Trfsr02sRdgTrfsr00sRdg

H12. s19d

One finally gets

DF

F=

2

9

kg2H12

s1 − GrdA1A2 + S1 +4

9kDGr

FA2

A1fA1s1 − GrdGrg

3Fsb̃14d +2A1

A2fA1s1 − GrdGrgFsb̃52dG s20d

in which

Fsb̃mnd =1 − RefGsb̃mn,Rd/Gsbdg

Vmn2 . s21d

IV. APPLICATION TO FDMR SPECTRUM OF AQUASI-ONE-DIMENSIONAL CRYSTAL (1,4 DBN)

According to Sternlicht and Mac Connel,39 the ZFSHamiltonian of an exciton, jumping from a molecule to an-

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other, placed in a nonequivalent position, is an average onthe non-equivalents sites ifts

−1@ uDu and uEu, ts being thetime of jump which characterizes the movement,D andE theZFS parameters. In bidimensional or three-dimensional mo-lecular crystals, such as anthracene, tetracene or pyrene, thets are about 10−11–10−12 s, that is to say ts

−1 from1 to 10 cm−1, D andE are about 1/10 cm−1. The inequalityis often respected, and it is licit to calculate the excitonicHamiltonian as an average of the molecular Hamiltonians.The average is made over all the sites of the crystal if themovement is three-dimensional and only in a plane if themovement is two dimensional. In one dimension, as in the 1,4 dibromonaphtalene,38 the time of jump from one stack toanother is larges10−8 sd, one can average only on one stack.We give in Table II the ZFSsRefs. 40 and 41d tensor princi-pal axis of molecules 1 and 18 sFig. 1d. The molecule 2 isdeduced from 1 by a rotation aroundb, the molecule 4 byinversion from 1 and 3 by inversion from 2, and it is thesame relation for the primed molecules. We call the two sitesI for the unprimed molecules and II for the primed once.

The difference between the triplet exciton energy levels ofI and II is 50 cm−1.42

Three types of triplet interactions can be consideredsTable IIId intrastacki − i or i8− i8 se.g., 1-1 or 18-18, 8 inter-actionsd, interstack intrasitei − j or i8− j8 se.g., 1-2 or 18-28,12 interactionsd and interstack intersitei − i8 and i − j8 se.g.,1-18 or 1-28, 16 interactionsd.43

Strictly speaking, we ought to find 36 values oflsRd if weconsider only the interactions between excitons having a dif-fusion motion on the nearest stackswhatever the sited. Due tothe symmetry relations in unit cell the number of interactionsis reduced; stacks related by inversion give identical resultsse.g., 1-2 is equivalent to 3-4, it is the same for 18-28 and

38-48d. Delayed fluorescence is then the sum of four contri-butions and its modulation by the microwave power can beexpressed by

DF

F= Ci−iSDF

FD

i−i+ Ci8−i8SDF

FD

i8−i8+ Ci−jSDF

FD

i−j

+ Ci−j8SDF

FD

i−j8. s22d

Cl−m represents the normalized probability for each family ofinteractionssTable IVd.

To calculate the relative microwave effect, we have usedthe Green’s function corresponding to one dimensional dif-fusion motion which has the form2

TABLE II. Triplet exciton ZFS parameters and the ZFS tensor principal axis.

INTERACTION Dscm−1d Escm−1d D*scm−1d E*scm−1dTriplet exciton ZFS tensor

principal axis Ref.

intrastack 1-1 0.0968 0.0000 s a b c8

x −0.5414 −0.7300 +0.4171

y +0.6954 −0.6676 −0.2659

z +0.4726 +0.1463 +0.8690d 40

intrastack 18-18 0.0968 0.0000 s a b c8

x −0.3594 +0.8633 +0.3543

y −0.8105 +0.4770 −0.3400

z −0.4625 −0.1649 +0.8711d 40

interstack intrasite 1-2 0.0968 0.0000 −0.0453 −0.0474 s a b c8

x +0.8785 +0.0000 −0.4777

y +0.4777 +0.0000 +0.8785

z 0.0000 +1.0000 +0.0000d 41

interstack intersite 1-18 0.0968 0.0000 −0.0484 0.0249 s a b c8

x +0.0058 −0.0107 +0.9999

y +0.9488 +0.3157 −0.0021

z −0.3157 +0.9487 +0.0120d 41

TABLE III. Molecules involved in the three annihilationtypes.

Type of annihilation Molecules involved

intrastacksi − i , i8− i8d 1-1, 2-2, 3-3, 4-4,18-18, 28-28, 38-38, 48-48

interstack intrasitesi − j , i8− j8d 1-2, 1-3, 1-4, 2-3,2-4, 3-4, 18-28, 18-38,

18-48, 28-38, 28-48, 38-48

interstack intersitesi- j8d 1-18, 1-28, 1-38, 1-48,2-18, 2-28, 2-38, 2-48,3-18, 3-28, 3-38, 3-48,4-18, 4-28, 4-38, 4-48

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Gsb̃mn,Rd = −

expF− S b̃mn

DD1/2

RG2sDb̃mnd1/2

, s23d

where D=Dcc=3,5310−4 cm2 s−1 sRef. 44d is the excitondiffusion constant along thec axis.

Since we have four types of interactions we expectFDMR spectra to be made of four symmetrical pairs ofFDMR resonances compared to the central fieldH0 or ex-perimental spectrum exhibit only two pairs of FDMR reso-nancessFig. 4d. Figure 5sad represents the calculated FDMRspectra for the different interaction types. They show thatsi − id se.g., 1-1d interaction is more efficient than the otherones. We notice that the FDMR resonances positions, corre-sponding to the intrastack intrasitesi − id and interstack intra-site si − jd se.g., 1-2d interactions are juxtaposed. The contri-bution of si8− i8d se.g., 18-18d interaction is negligiblecompared to others, since the exciton pair lifetime is theshortest for this interactionsTable IVd. That limits the num-ber of pair FDMR resonances to 2.

Using Eqs.s20d ands22d we obtain the FDMR spectrum,represented in Fig. 5sbd sopen circled, in the sac8d plane of1,4 DBN crystal at room temperature. The best-fit with theexperimental FDMR spectrumfFig. 5sbdg lets us determinethe parameters valuesl sthe annihilation rated and ß sthedecay rated for the different interactionssTable IVd; the hop-ping rate value was determined using the relationDij

= 12oRCsRdRiRj,

2 which giveCc=Dcc/c2=231011 s−1 sjump

rate along thec axisd; C=2Cc.The average value of the determined effective decay rates

b is about 23108 s−1, since the monomolecular triplet decayrate is b0<103 s−1 then the effective decay ratefb=Cout

+ 12b0+ sCoutb0+ 1

4b02d1/2; Cout being the jump rate out of the

stack of motiong2 for the exciton motion in one stack isb=Cout. This result confirms the value ofCout=23108 s−1

determined from the angular dependance of ESR line shapesat 300 K.40 That the anisotropyCc/Cout is only 103, despitethe anticipated very small static transfer integrals betweenmolecules in thesabd plane, can be understood by taking intoaccount nonlocal scattering which is known to make a non-negligible contribution to triplet diffusion in anthracene,45,46

and may be dominant in pyrene.47 In the latter crystal, it mayexplain the relatively small experimental anisotropy, withD.3310−5 cm2 s−1 in all directions. Therefore, diffusioncoefficients in thesabd plane of 1,4 DBN of the order of10−6 cm2 s−1 can be reasonably expected.

l is the rate of singlet generation by the annihilation ofnearest neighbor triplets, and it should be at least comparableto the hopping rate for exciton motionfanthracenel=2.531011 s−1, C=231011 s−1 sRef. 2dg, if not, one can not havea competition between the evolution towards the singlet stateand the decorrelation of the pair. Ifl is much largerscase oftetracenel=1.231012 s−1, C=1011 s−1d, any pair of tripletexcitons, reacts immediately to give a singletsmode strictlylimited by the diffusiond. If l is much smaller, the reaction isnot immediate and the mechanism is not controlled by thediffusion scase of 1,4 DBNl<3.531010 s−1d.

V. CONCLUSION

Fluorescence detected magnetic resonance measurementsare performed on DBN crystals. Triplet excitons in this sys-tem have a one-dimensional motion. FDMR spectrum showsfour FDMR resonances for magnetic field orientationsu=150° ,w=0°d, which are related directly to triplet-tripletinteractions occurring in intrastack, interstack intrasite, andinterstack intersite configurations. The positions of theseFDMR resonances and their line shapes are analyzed by in-

TABLE IV. Used parameters for the best fit to experimentalcurve.

INTERACTIONS C1−m b s108 s−1d l s1010 s−1d

Intrastacksi − id 0.075 0.8 5

Intrastacksi8− i8d 0.15 5 1.4

Interstack intrasitesi − j , i8− j8d 0.075 3 5

Interstack intersitesi − j8d 0.7 2.1 3.15

FIG. 5. sad Calculated FDMR spectra for each interaction.sbdBest fit of experimental FDMR spectra for one dimension 1,4 DBNcrystal. Solid curve is the FDMR experimental spectra at roomtemperature with the field lying in theac8 plane and making theangle with the crystalc axis. Open circles correspond to the fitobtained with expressionss20d and s22d.

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corporating a time dependent perturbation in the spin systemHamiltonian. The analysis of the FDMR resonances posi-tions shows that annihilations of triplet excitons can occur inthe same one-dimensional stack and between triplet excitonslocalized on different neighboring stacks. Values of deter-mined effective decay rates are in good agreement with theESR measurements and let us to estimate the life time oftriplet exciton pair of about 5 ns. The rate of singlet genera-

tion by the annihilation of nearest-neighbor triplet excitonslindicates that the mechanism is not controlled by the diffu-sion.

ACKNOWLEDGMENTS

We are most grateful to J. L. MongesGroupe de Physiquedes Solides, Universités Paris 6 et Paris 7d for useful discus-sions.

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