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The source of excess linewidth in EPR of triplet excitons in molecular crystals
1 June 1978
Volume 56, number 2
THE SOURCE OF EXCESS LINEWIIBTH IN EPR OF TRIPLET EXCITONS IN MOLECULAR CRYSTALS J. ROSENTHAL,
L. YARMUS, N.F. BERK and W. BIZZARO
Depattment of Pfiysics, New York University, New York City, New York IOOO3, USA
Received9 Jannary 1978 Revisedmanuscriptreceived7 March 1978
It is shown that the EPR linewidthspectrum of triplet excitons in molecular crystabscan be severelydistorted by weak orientationaldisorder.The demonstrationemploysa one-parametercorrection procedure based ou the assumptionthat the angulardependenceof excess width is directly’proportionalto the angulargradientof the resonant field Applicationto the discrepantdata of I-barer and Wolf brings them into agreementwith theory and with more recent experiments. A phenome nologiealmodel of the disorderis used to interpret the distortion parameter. These resultssuggestthe potential vabreof exciton EPR as a probe of structural imperfections in molecular crystals.
1. Introduction
2. Analysis
In a recent publication we have reported au experimental study of the EPR liuewidth of triplet excitons iu crystalhue authracene measured at room temperature [ 11. The motivation for our experiment was the appearance of Reineker’s theory of the linewidth [2] and its failure to fit the data of Haarer and Wolf [3] _ Inasmuch as our results did conform to the theoretical predictions, the earlier experiments remained a puzzle in our paper. We also noted in passing that in a particular angular region of the ac plane the lineshapes in our experiment-deviated distinctly from the anticipated lorentzian shape. The fact that the distortions were most severe where the spectrum was most anisotropic caused us to suggest that theg resulted from orientational disorder. Now we should like to present evidence that such disorder can in fact create the discrepancies in the data of ref. [3 3 and thus it seems that our two previously uurelated observations have a common origin*. We shall also point out certain implications of the hypothesis of orientational disorder as a source of excess liuewidth.
Since the spectrum of Zeeman levels for triplet excitons in molecular crystals is a function of the orientation of excited molecules in the unit cell, the dispersion in energy that arises from orientational disorder will cause an inhomogeneous broadening of a magnetic resonance line. To see what determines the excess width, and therefore how it may be dealt with in analyzing a spectrum, the disorder can be thought of as a distribution of slightly misoriented “domains”, each with a volume that must be at least as large as would be traversed by an exciton during the lifetime of its spin state (not the lifetime of the exciton). Let H(6) be the resonant magnetic field in the direction of the unit vector fi and consider the ith domain slightly misaligned from the perfect crystal by the relative orientation Siii. The resulting small shift in resonance position is given by
*-2= emeker remarks in ref. (41, attributing the suggestion to D. Haarer, that “an additionallinewidthwould arise if the orientation of the crystal is not homogeneous”_
214
Ski
= H(ri + SPi) - H(~) = 6r7i =(S;ii-~)IViiH(~)l)
l
vit H(ri) (1)
where & is a unit vector in the direction of the gradient. We are thus led to postulate a simple prescription for correcting a disorder-broadened linewidth:
Volume 56. mrnber 2 AH(corr.)
1 June 1978
CHEMICAL PHYSICS LETTERS
= AH(obs.)
- k 1VAH(ii) 1,
I
(2)
I
I
I
ANTHRACENE
where AH represents linewidth and k is a parameter
I
I
I
I
bc’ PLANE
to be determined. H(k) is given by the resonance condition for the monoclinic lattice, * ;(Fa,cos2a
hu=gSH(ii)
+ Fbbcos2$
+F c.,.cos2 y + 2F,=. cos CYcos 7)) where Q, /3, and y are the angles between
ii and the
corresponding crystal axes. The Fii aLecomponents of the excitonic fine structure tensor referred to the same axes. Although the applicability of (2) must be tested experimentally, an expression fork that is consistent with (1) can be given by noting that the total width of the resonance arises from the integrated contributions of the 6 Hi averaged over the crystal. It follows
I
I
1
I
I
I
ANTHRACENE
ac’
I
I
Fig. 2. The same as fig. 1 but for the k’ plane with rl = 3200 G, residual width = 0.7 G, and k = 0.13” in (2).
PLANE
that
.
k = [((G-i - 1;)2 ,] l/2 ,
.
I
I
I
30
FIELD
I
I
I 60
I
ORIENTATION (degrees)
I x2
Fig. 1. Augular dependenceof the EPR linewidthof triplet excitons for the uc’ plane of singlecrystal anthracene at room
temperature. The width refers to the half-width at half-maximum of a Xorentzian shape- The solid curve is the theoretical prediction for 35 GHz deduced from the au&& in ref. [I]. The tiiangIesarethe averages of the data given in ref. [ 3 ] ; thesearereported with a 10% error. The circles are the data corrected according to expression (2) with k = 0.17”.
(3)
where the angular brackets denote the appropriate average. Obviously an evaluation of k requires knowledge of the disorder. We shall suggest below, as an example, a simple model of the disorder for which k may be given a geometric interpretation. In order to test (2) it is necessary to know the correct linewidth spectrum. For antbracene we appeal to our earlier analysis [l] in which the effective hopping rate r1 of the linewidth theory was deduced from measurements at 24 GHz. In fact theoretical predictions were made in ref. [I] for the width that shollld be observed at 35 GHz, the frequency of Haarer and Wolfs experiments. These predictions were given for the QC’ and ab planes and are extended here to the bc’ plane. All are shown in figs. 1-3 as are the linewidth data reported in ref. [3]. Expression (2) was applied to the broadened linewidths by means of a conventional one-parameter (k), least squares, fit to the predicted
widths; the corrected
215
VoIume 56, number 2
I
CHEMICAL PHYSICS LETTERS I
I
I
ANTHRACENE A.*
14 -
ab a
*
I
I
I
order is very slight since k,an overall measure of disorder according to (3), ranges from 0.13O to 0.3Oe. In ‘this regard it may be helpful to examine an explicit
I
PLANE
model of the disorder to see that although (3) appears
.
. 12 -
.
.
2-
I 0 a-(IXIS
I
I
I
I
30 FIELD
I
I
60 ORIENTATlOh
1 June 1978
(degrees)
I b-z
Fig. 3. The same as fig_ 1 but for the ob plane and wrth k = 0.30” in (2).
linewidths are shown in figs_ l-3 *. The values of reduced &i-square h2), the figure of merit characterizing a tit in the procedure are 0.53,0_1, and 1.5 for the UC’,bc’, and ab planes respectively. These values imply that the fit is significant, as is clearly evident in figs. 1-3.
3. Discussion Our goal has been to demonstrate that the source of excess line broadening in EPR of triplet excitons in molecular crystals is to be sought in some form of orientationai disorder. The evidence is compelling since the Iinewidth correction depends only on a oneparameter subtraction procedure we&ted directly by a function that is a measure of the anisotropy of resonance. Furthermore it should be apparent that the dis-
to depend on field direction, it actually characterizes disorder only. Note tha_t for any direction of the magnetic field the Gii and h lie approximately in the same plane since h is normal to ii and the ;ii = ri + 6iii are nearly parallel to I?. Thus assumingthat the 6rii are isotropicalLy distributed, the disorder defmes for every ii an effective cone of angle (([6G12})1/2 c Qrms. Then ((66 - &)2> = q5~ms(cos2(Sji, i)) = $$zms; i.e. k= qSrms/ 2’/*. The experimental values of k imply a very slender “cone of disorder” in ‘&is model. In this discussion it should be understood that disorder will not inevitably produce inhomogeneous broadening. Since relaxation arises from the hopping among differently oriented molecules, orientational disorder will create a homogeneous mechanism when al1 the disorder is sampled during a spin state lifetime. We should like fmally to enumerate several impiications of this work: (i) Perhaps the most significant deduction from these preliminary results is the prospect of using EPR lineshape studies of triplet excitons as a probe of structural imperfections arising from orientational Ssorder?. Because the triplet ranges over relatively large spatial dimensions and the spectrum itself is insensitive to localized defects, the technique is well suited to this purpose. It is apparent from our results that the deviation from perfect crystaliinity may be very small and yet observable: not only are widths measurable to fractions of a width but minute disorder is effectively magnified by the gradient, which can be as large aa tens of gauss per degree. In using the linewidth spectrum as a probe the highest microwave frequency is most suitable since intrinsic widths decrease with increasing field while the gradient remams constant_ (ii) Linewidth experiments in general are most easily interpreted when there is a single broadening mechanism or at least a single strongly dominant one. In the i The value of using the triplet exciton as a probe of structural
* The shght variation among thevalues of k, which are given rn the figure captions for the three planes, is consistent with the fact that the data were obtained from three different crystals [3], doubtfess some statistical variation is also present.
216
imperfections has been examined by Goode et ah [S]. lhey discuss the way in which phosphorescence and delayed fluorescence from crystallineanthracene may be used to demonstrate the nature, and determine the depth, of many traps.
Volume 56, number 2
CHEMICAL PHYSICS LETTERS
latter case, the minor contributions, if not understood, are generally characterized as creating a “residual” width. In ref. [1] it was necessary to invoke such a width to fit the measurements, the justification being that it was a relatively small fraction of the measured widths and was independent of angle while the full angular dependence was accounted for by theory. Based on the discussion given in this communication, it appears likely that the residual width also arises from disorder. In fact, if this is so it will add a second, much weaker angular dependence to the linewidth spectrum making it possible to test the theoretical predictions more closely. In this context we may view the disorder mech-
anism as producing a range of effects, from the source of residual width and lineshape distortion in ref. [l] to the more severe effect of distorting the linewidth spectrum in ref. [3]. (iii) Since resonance saturation provides a method for measuring relaxation rates, it may be asked whether or not the technique can be applied with confidence to distorted lines. This question arises in practice since it is our experience with many samples of different systems (anthracene, pyrene, tetracene) that deviations from l&entzian shapes are frequently encountered. For lines that are broadened in the manner discussed here the saturation method should be reliable since relaxation will be virtually constant over the angular ranges that contribute to broadening_ (iv) Our results have some bearing on the measurements of relaxation parameters made in ref. 133. Of course the values of T2 deduced from the linewidth measurements on anthracene must be revised as implied in ref. El]. The saturation data however are not seriously affected by this revision as we explain in (iii)_ Despite the excess broadening of the anthracene lines, the shapes are reported as iorentzian. A judgment
1 June 1978
on this apparent &lemma might best await model simulations since they may reveal that disorder broadening will preserve a predominant lorentzian component under certain conditions. (Total lorentzian character is not required since we have found that with the signal levels often met in exciton EPR as much as 20% non-lorentzian cannot be resolved.) The problem is significant however since it suggests that experimental lorentzian shape in itself need not be a guarantee against excess broadening_ (v) Since the broadening we have identified arises only from anisotropy in resonant field, it would not be present in a zero-field experiment. (vi) In the light of these results it is now appropriate to regard the linewidth measurements of Haarer and Wolf as a second validation of the linewidth theory for anthracene.
Acknowledgement This research has been supported by a grant from the National Science Foundation (Grant #DMR 7709206).
References [I] We Biua-o, J. Rosenthal, N-F. Berk and L. Yam&,
Phys.
Stat Sol. 84b (1977)
[2] [3] [4] [S]
27. P. Reineker, Phys. Stat. Sol. 70b (1975) 189,471. D. Haarer and H.C. Wolf, Mol. Cryst. LiquidCryst. 10 (1970) 359. P. Reineker, Phys. Stat. Sol. 74b (1976) 121. D. Goose, Y. Lupien, W. Siebrand. D-F. Williams, J-M. Thomas and J-0. Williams, Chem. Fhys. Letters 25 i1974) 308.
217
Volume 56, number 2
THE SOURCE OF EXCESS LINEWIIBTH IN EPR OF TRIPLET EXCITONS IN MOLECULAR CRYSTALS J. ROSENTHAL,
L. YARMUS, N.F. BERK and W. BIZZARO
Depattment of Pfiysics, New York University, New York City, New York IOOO3, USA
Received9 Jannary 1978 Revisedmanuscriptreceived7 March 1978
It is shown that the EPR linewidthspectrum of triplet excitons in molecular crystabscan be severelydistorted by weak orientationaldisorder.The demonstrationemploysa one-parametercorrection procedure based ou the assumptionthat the angulardependenceof excess width is directly’proportionalto the angulargradientof the resonant field Applicationto the discrepantdata of I-barer and Wolf brings them into agreementwith theory and with more recent experiments. A phenome nologiealmodel of the disorderis used to interpret the distortion parameter. These resultssuggestthe potential vabreof exciton EPR as a probe of structural imperfections in molecular crystals.
1. Introduction
2. Analysis
In a recent publication we have reported au experimental study of the EPR liuewidth of triplet excitons iu crystalhue authracene measured at room temperature [ 11. The motivation for our experiment was the appearance of Reineker’s theory of the linewidth [2] and its failure to fit the data of Haarer and Wolf [3] _ Inasmuch as our results did conform to the theoretical predictions, the earlier experiments remained a puzzle in our paper. We also noted in passing that in a particular angular region of the ac plane the lineshapes in our experiment-deviated distinctly from the anticipated lorentzian shape. The fact that the distortions were most severe where the spectrum was most anisotropic caused us to suggest that theg resulted from orientational disorder. Now we should like to present evidence that such disorder can in fact create the discrepancies in the data of ref. [3 3 and thus it seems that our two previously uurelated observations have a common origin*. We shall also point out certain implications of the hypothesis of orientational disorder as a source of excess liuewidth.
Since the spectrum of Zeeman levels for triplet excitons in molecular crystals is a function of the orientation of excited molecules in the unit cell, the dispersion in energy that arises from orientational disorder will cause an inhomogeneous broadening of a magnetic resonance line. To see what determines the excess width, and therefore how it may be dealt with in analyzing a spectrum, the disorder can be thought of as a distribution of slightly misoriented “domains”, each with a volume that must be at least as large as would be traversed by an exciton during the lifetime of its spin state (not the lifetime of the exciton). Let H(6) be the resonant magnetic field in the direction of the unit vector fi and consider the ith domain slightly misaligned from the perfect crystal by the relative orientation Siii. The resulting small shift in resonance position is given by
*-2= emeker remarks in ref. (41, attributing the suggestion to D. Haarer, that “an additionallinewidthwould arise if the orientation of the crystal is not homogeneous”_
214
Ski
= H(ri + SPi) - H(~) = 6r7i =(S;ii-~)IViiH(~)l)
l
vit H(ri) (1)
where & is a unit vector in the direction of the gradient. We are thus led to postulate a simple prescription for correcting a disorder-broadened linewidth:
Volume 56. mrnber 2 AH(corr.)
1 June 1978
CHEMICAL PHYSICS LETTERS
= AH(obs.)
- k 1VAH(ii) 1,
I
(2)
I
I
I
ANTHRACENE
where AH represents linewidth and k is a parameter
I
I
I
I
bc’ PLANE
to be determined. H(k) is given by the resonance condition for the monoclinic lattice, * ;(Fa,cos2a
hu=gSH(ii)
+ Fbbcos2$
+F c.,.cos2 y + 2F,=. cos CYcos 7)) where Q, /3, and y are the angles between
ii and the
corresponding crystal axes. The Fii aLecomponents of the excitonic fine structure tensor referred to the same axes. Although the applicability of (2) must be tested experimentally, an expression fork that is consistent with (1) can be given by noting that the total width of the resonance arises from the integrated contributions of the 6 Hi averaged over the crystal. It follows
I
I
1
I
I
I
ANTHRACENE
ac’
I
I
Fig. 2. The same as fig. 1 but for the k’ plane with rl = 3200 G, residual width = 0.7 G, and k = 0.13” in (2).
PLANE
that
.
k = [((G-i - 1;)2 ,] l/2 ,
.
I
I
I
30
FIELD
I
I
I 60
I
ORIENTATION (degrees)
I x2
Fig. 1. Augular dependenceof the EPR linewidthof triplet excitons for the uc’ plane of singlecrystal anthracene at room
temperature. The width refers to the half-width at half-maximum of a Xorentzian shape- The solid curve is the theoretical prediction for 35 GHz deduced from the au&& in ref. [I]. The tiiangIesarethe averages of the data given in ref. [ 3 ] ; thesearereported with a 10% error. The circles are the data corrected according to expression (2) with k = 0.17”.
(3)
where the angular brackets denote the appropriate average. Obviously an evaluation of k requires knowledge of the disorder. We shall suggest below, as an example, a simple model of the disorder for which k may be given a geometric interpretation. In order to test (2) it is necessary to know the correct linewidth spectrum. For antbracene we appeal to our earlier analysis [l] in which the effective hopping rate r1 of the linewidth theory was deduced from measurements at 24 GHz. In fact theoretical predictions were made in ref. [I] for the width that shollld be observed at 35 GHz, the frequency of Haarer and Wolfs experiments. These predictions were given for the QC’ and ab planes and are extended here to the bc’ plane. All are shown in figs. 1-3 as are the linewidth data reported in ref. [3]. Expression (2) was applied to the broadened linewidths by means of a conventional one-parameter (k), least squares, fit to the predicted
widths; the corrected
215
VoIume 56, number 2
I
CHEMICAL PHYSICS LETTERS I
I
I
ANTHRACENE A.*
14 -
ab a
*
I
I
I
order is very slight since k,an overall measure of disorder according to (3), ranges from 0.13O to 0.3Oe. In ‘this regard it may be helpful to examine an explicit
I
PLANE
model of the disorder to see that although (3) appears
.
. 12 -
.
.
2-
I 0 a-(IXIS
I
I
I
I
30 FIELD
I
I
60 ORIENTATlOh
1 June 1978
(degrees)
I b-z
Fig. 3. The same as fig_ 1 but for the ob plane and wrth k = 0.30” in (2).
linewidths are shown in figs_ l-3 *. The values of reduced &i-square h2), the figure of merit characterizing a tit in the procedure are 0.53,0_1, and 1.5 for the UC’,bc’, and ab planes respectively. These values imply that the fit is significant, as is clearly evident in figs. 1-3.
3. Discussion Our goal has been to demonstrate that the source of excess line broadening in EPR of triplet excitons in molecular crystals is to be sought in some form of orientationai disorder. The evidence is compelling since the Iinewidth correction depends only on a oneparameter subtraction procedure we&ted directly by a function that is a measure of the anisotropy of resonance. Furthermore it should be apparent that the dis-
to depend on field direction, it actually characterizes disorder only. Note tha_t for any direction of the magnetic field the Gii and h lie approximately in the same plane since h is normal to ii and the ;ii = ri + 6iii are nearly parallel to I?. Thus assumingthat the 6rii are isotropicalLy distributed, the disorder defmes for every ii an effective cone of angle (([6G12})1/2 c Qrms. Then ((66 - &)2> = q5~ms(cos2(Sji, i)) = $$zms; i.e. k= qSrms/ 2’/*. The experimental values of k imply a very slender “cone of disorder” in ‘&is model. In this discussion it should be understood that disorder will not inevitably produce inhomogeneous broadening. Since relaxation arises from the hopping among differently oriented molecules, orientational disorder will create a homogeneous mechanism when al1 the disorder is sampled during a spin state lifetime. We should like fmally to enumerate several impiications of this work: (i) Perhaps the most significant deduction from these preliminary results is the prospect of using EPR lineshape studies of triplet excitons as a probe of structural imperfections arising from orientational Ssorder?. Because the triplet ranges over relatively large spatial dimensions and the spectrum itself is insensitive to localized defects, the technique is well suited to this purpose. It is apparent from our results that the deviation from perfect crystaliinity may be very small and yet observable: not only are widths measurable to fractions of a width but minute disorder is effectively magnified by the gradient, which can be as large aa tens of gauss per degree. In using the linewidth spectrum as a probe the highest microwave frequency is most suitable since intrinsic widths decrease with increasing field while the gradient remams constant_ (ii) Linewidth experiments in general are most easily interpreted when there is a single broadening mechanism or at least a single strongly dominant one. In the i The value of using the triplet exciton as a probe of structural
* The shght variation among thevalues of k, which are given rn the figure captions for the three planes, is consistent with the fact that the data were obtained from three different crystals [3], doubtfess some statistical variation is also present.
216
imperfections has been examined by Goode et ah [S]. lhey discuss the way in which phosphorescence and delayed fluorescence from crystallineanthracene may be used to demonstrate the nature, and determine the depth, of many traps.
Volume 56, number 2
CHEMICAL PHYSICS LETTERS
latter case, the minor contributions, if not understood, are generally characterized as creating a “residual” width. In ref. [1] it was necessary to invoke such a width to fit the measurements, the justification being that it was a relatively small fraction of the measured widths and was independent of angle while the full angular dependence was accounted for by theory. Based on the discussion given in this communication, it appears likely that the residual width also arises from disorder. In fact, if this is so it will add a second, much weaker angular dependence to the linewidth spectrum making it possible to test the theoretical predictions more closely. In this context we may view the disorder mech-
anism as producing a range of effects, from the source of residual width and lineshape distortion in ref. [l] to the more severe effect of distorting the linewidth spectrum in ref. [3]. (iii) Since resonance saturation provides a method for measuring relaxation rates, it may be asked whether or not the technique can be applied with confidence to distorted lines. This question arises in practice since it is our experience with many samples of different systems (anthracene, pyrene, tetracene) that deviations from l&entzian shapes are frequently encountered. For lines that are broadened in the manner discussed here the saturation method should be reliable since relaxation will be virtually constant over the angular ranges that contribute to broadening_ (iv) Our results have some bearing on the measurements of relaxation parameters made in ref. 133. Of course the values of T2 deduced from the linewidth measurements on anthracene must be revised as implied in ref. El]. The saturation data however are not seriously affected by this revision as we explain in (iii)_ Despite the excess broadening of the anthracene lines, the shapes are reported as iorentzian. A judgment
1 June 1978
on this apparent &lemma might best await model simulations since they may reveal that disorder broadening will preserve a predominant lorentzian component under certain conditions. (Total lorentzian character is not required since we have found that with the signal levels often met in exciton EPR as much as 20% non-lorentzian cannot be resolved.) The problem is significant however since it suggests that experimental lorentzian shape in itself need not be a guarantee against excess broadening_ (v) Since the broadening we have identified arises only from anisotropy in resonant field, it would not be present in a zero-field experiment. (vi) In the light of these results it is now appropriate to regard the linewidth measurements of Haarer and Wolf as a second validation of the linewidth theory for anthracene.
Acknowledgement This research has been supported by a grant from the National Science Foundation (Grant #DMR 7709206).
References [I] We Biua-o, J. Rosenthal, N-F. Berk and L. Yam&,
Phys.
Stat Sol. 84b (1977)
[2] [3] [4] [S]
27. P. Reineker, Phys. Stat. Sol. 70b (1975) 189,471. D. Haarer and H.C. Wolf, Mol. Cryst. LiquidCryst. 10 (1970) 359. P. Reineker, Phys. Stat. Sol. 74b (1976) 121. D. Goose, Y. Lupien, W. Siebrand. D-F. Williams, J-M. Thomas and J-0. Williams, Chem. Fhys. Letters 25 i1974) 308.
217