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J. Phys. Chem. 1994,98, 1342-1349
Lattice Model of Inhomogeneous Broadening in Crystals: Correlation of Frequency Distributions for Different Transitions D.L. Orth and J. L. Skinner' Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706 Received: February 1 1 , 1994; In Final Form: May 3, 19940
Inhomogeneous broadening of a dilute chromophore's optical transition can provide detailed information about disorder in crystalline systems. Certain nonlinear experiments such as fluorescence and phosphorescence line narrowing and spectral hole burning, which probe the correlation between the inhomogeneous frequency distributions of two different transitions, are capable of yielding even more information. In this paper we present a microscopic lattice theory of inhomogeneous broadening by point defects, focusing specifically on the correlation between a pair of transitions. Wedevelop a series of approximations for the joint frequency distribution for the two transitions that are valid for all possible defect densities, and which are shown to be accurate in comparison with exact numerical calculation. The model is capable of describing frequency distributions with any amount of correlation, We discuss several experiments within the context of this theory.
1. Introduction The optical absorption line shape of a dilute chromophore in a crystal at low temperatures has two prominent and distinct features, a relatively sharp zero-phonon line and a broader phonon side band. The zero-phonon line arises from a purely electronic transition, while the phonon side band is due to the excitation of one or more phonons in addition to the electronic transition. Typically, zero-phonon lines are inhomogeneously broadened. This means that the observed smooth line shape is actually the superposition of many much sharper lines due to individual chromophores, each of which has a slightly different transition frequency as a result of its slightly different microscopic environment. The different environmentsare produced by defects, such as vacancies, interstitials, substitutional impurities, dislocations, domain walls, or grain boundaries, in an otherwise perfect crystal lattice. They may also result from long-range strains induced by mounting the sample. Theoretical calculationsshow that the line shape is very sensitiveto the nature and concentration ofdefectsand the formof their interaction with thechromophore.' Therefore,inhomogeneous line shapesprovide a very useful probe of thevariouskinds of disorder that arealways inherent in crystals.2 Quite recently, a variety of nonlinear optical experimentshave been developed that hold even more promise for probing disorder in crystals. All of these nonlinear experiments involve two distinct transitions of the same chromophore, each of which is characterized by an inhomogeneous distribution of transition frequencies. (If the lower level of a particular transition is the ground state of the chromophore, then the corresponding distribution of transition frequencies is simply the inhomogeneous absorption line shapefor that transition.) In fact, these nonlinear experiments measure the correlation between the inhomogeneous frequency distributions for the two transitions. Thus, for example, in nonresonant fluorescence line narrowing (FLN), a particular transition from the ground state to some excited state of the chromophoreis selectively excited with a narrow-band laser, and then the red-shifted fluorescence from the excited state to a third level is frequency analyzed. If the frequency distribution of this fluorescence is independent of the excitation frequency and is identical to the inhomogeneous distribution of frequencies for this second transition (which can be obtained from the emission spectrum after broad-band excitation), then one would say that the frequencydistributions for the two transitions are uncorrelated. Abstract published in Aduance ACS Abstracts, July 1, 1994.
0022-3654 f 94f 2098-1342$04.50 f 0
If this is not the case, then the two frequency distributions are correlated. Typically the fluorescence is indeed narrower than the distribution of frequencies for this second transition, and thus thedistributionsare correlated. As with theabsorption lineshape itself, thedetails of this fluorescence frequencydistribution depend upon the concentrationof defects and the form of their interaction with the chromophore. An experimental example of the above involves excitation of the 3H4 'Po transition followed by nonresonant fluorescence to the 3H6 state of Pr3+ in LaF3.3 Other related experimental techniques include phosphorescence line narrowing (PLN), excitation line narrowing (ELN), and spectral hole burning (SHB). In PLN a particular ground to excited state transition of the chromophoreis selectively excited, and following radiationless relaxation to a third level, the ensuing emission back to the ground state is frequency-analyzed. For the level schemes appropriate for FLN or PLN, the same information can be obtained from an ELN experiment, where the emission is selectively monitored and the excitation frequency is varied. An experimentalexampleof PLN involves the 7F0 sD2and 5D0 7F0transitions of Sm2+in BaFC1,Brl,.4 Another example of the PLN level scheme within the ELN detection mode involves the 7F6 SG4 and 5G6 7F6 transitions of Cf4+in CeF4.5.6 In a photochemical SHB experiment a particular ground to excited state transition is excited selectively, and then a photoinduced process occursthat removes population from theoriginal transition by producing a new chromophore with different ground and excited states. From an absorption experiment one can then frequency resolve the hole shape of a second transition of the original chromophore or measure the frequency distribution of the new chromophore's transition (the "antihole"). An example of the former involvesa phthalocyanine chromophorein crystalline n-nonane,' and experiments on the antiholefrequency distribution were performed on pentacene in benzoic acid.8 As in the FLN experiments,all of these PLN or SHB experimentsshow that the two frequency distributions in question are correlated. These experiments can all be discussed within the context of phenomenologicalmodels of correlated frequencydistributions."7 With this approach, however, the experiments cannot provide any informationregarding the microscopic disorder in the system. The goal of this paper is to develop a microscopic theory of the correlation between a pair of inhomogeneous frequency distributions for a chromophore in a disordered crystal. Twenty-five years ago Stoneham reviewed continuum theories of inhomogeneous broadening in crystals.' He assumed that the
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0 1994 American Chemical Society
Lattice Model of Inhomogeneous Broadening defects responsible for the inhomogeneous broadening were sufficiently dilute that the details of the crystal lattice were unimportant. In thecaseof point defects (vacancies, interstitials, substitutionalimpurities) whose interaction with the chromophore is of the elastic strain or electrostatic dipolar form, Stoneham found that the chromophore’sabsorption line shape is Lorentzian. This model has been discussed further by DavieP and Kador.lg Kikas and Ratsep20generalized Stoneham’s low-density theory to discuss the frequency distributions for several different transitions. They found that, in general, these distributions are indeed correlated. Quite recently, the present authors generalized Stoneham’s continuum model for the inhomogeneous line shape to the case where the defects are not necessarily dilute. In that work2’ the defects occupy the sites of a regular lattice with probability p, where, completely generally, 0 I p I 1/2. We found21 that for p = 1/2 the line shape was approximately Gaussian, forp E 0.1 the line shape was structured due to different configurations of nearest-neighbor defects, and that for p
The double Fourier transform of eq 7 can be performed analytically, and then eq 10, together with eq 21, gives the approximate conditional probability
(13)
and (from eq 10)
fg,(@’l@) = P&’) (14) It then follows that R5. = 1 and S,qa= 0. We will see below under what circumstancesthis occurs. If eq 13 does not hold, then the two transitions arecorrelated, at least to some extent. To examine the completely correlated limit, suppose that for all sites i, u g l = Xuai, where X is a constant. In this case it is easy to show from = XGa, w5 - wgo = X(wa - w d ) , and eqs 6-10 that
again valid forp = 112. We see that the conditionalprobability is Gaussian in w’, centered at w b = w5 + (w - wa)D&/Daa, with FWHM & = (8 In 2)’/*(&@ $4/Daa)1/2. From eqs 11, 12, and 25 we see that
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fBa(W’1W)= 6(w’ - wBo- X(w - Wd))
(15) Thus we see that, indeed, in this case the two frequency distributions are completely correlated in that a particular value of w implies a unique value of 0’. We also see that Rga = 0 and Sga’ 1.
3. Approximate Results Let us begin by rewriting eqs 8 and 9 as
+
Ga(t) = e x p ( x l n ( 1 p(e4‘ud - 1)))
(16)
and
First considering the case where p H 112, we found in our previous treatment of the inhomogeneous line shape21 that, by expanding eq 16 in powers of uai and truncating at second order we obtained a
(18)
where
and
DaB
(P - P’)
UaPBi
Note that in this case neither of these quantities depends on w and that they are related by Sia
i
G (2) = e-&4‘,42/2
and
(20)
I
+ Ria = 1
Two important limiting cases of the above are the following: (1) if for some reason (to be discussed in more detail below) Da5 = 0, then we see that eq 25 reduces to eq 14, indicating that the two frequency distributions are completely uncorrelated and that R5. = 1 and Spa = 0; (2) if u51= &I, then 0 5 5 = AD4 = X’Daar and we recover the exact (completely correlated) result of eq 15, along with the results that Rga = 0 and Sga = 1. In our previous work21 on inhomogeneousline shapes, we found that for smaller values of p (p N 0.1) the line shapes are not Gaussian; nonetheless, we were able to develop reasonably accurate approximations. Since the defect-chromophore perturbation falls off with distance (as r3in the case of electrostatic or strain dipoles), the nearby defects will make the largest perturbations. If we treat some small number of nearby defect sites exactly, then for the remaining sites the central limit theorem is more likely to apply (since even for relatively smallp there will be a sizable number of roughly equal strength perturbations), and we can use the above approximationsonthe rest of the lattice. For the “generating function” of the frequency distribution this approach gives
Upon Fourier transformation (using eq 6) this gives a Gaussian for the frequency distribution, where where
w, = w,,+
0,
(22)
and the FWHM, ia, is
Ga = m&
and (23)
For dipolar defect-chromophoreinteractions, this approximation was shown21 to be quite accurate for p = 112. This is a result of the central limit theorem, which states that a large number of uncorrelated perturbations of roughly equal magnitude produce a Gaussian distribution.
The primed summations exclude the S nearby sites. The inhomogeneous distribution is then obtained from numerical Fourier transformation of Ga(t). For Gd(t,t’) we take a similar approach,treating S nearby sites exactly and making the Gaussian
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7345
Lattice Model of Inhomogeneous Broadening approximation for the rest of the lattice. This gives G (t
Lorentzian frequency distribution
9 = e-ir;.-itl;;p(~nnr'+D"z+~~t")/z
a8
n S
[1
(44)
+ p(e-i(t"d""@)- I)]
(32)
i- 1
Performing the double Fourier transform in eq 7 numerically gives, using eqs 6,10, and 29,fga(w'lw). For a given value ofp, Scan be increased until convergence is obtained. For the models we have considered, for 0.05 Ip I0.5,this requires S I26. To obtain approximate results for p
