1264
J . Phys. Chem. 1986, 90, 1264-1270
Relation between Hole-Burning Parameters and Molecular Parameters: Free-Base Phthalocyanine In Polymer Hosts L. Kador, G . Schulte, and D. Haarer* Physikalisches Institut, Universitat Bayreuth, 0-8580 Bayreuth, F.R.G. (Received: September 17, 1985)
A general analytical description of the lineshape theory is presented. The theory links experimental parameters like Debye-Waller factors, Debye temperatures, and photochemical quantum yields to hole-burning parameters. For some experimental optical
parameters, like Debye-Waller factors and Debye temperatures, hole burning and fluorescence line narrowing seem to be the only accessible optical methods due to the large bandwidth of straightforward optical experiments. Our model system is free-base phthalocyanine (H2Pc) in polyethylene (PE), polystyrene (PS), and poly(methy1 methacrylate) (PMMA).
Introduction In the following paper we present analytical calculations describing the line shape of photochemical holes, as they are measured in a real experiment. It will be shown by comparing experiments and calculations that the assumptions of Lorentzian line shapes and isotropic optical media are rather crude and do not in general hold. Therefore a quantitative analysis of the experiments has to take into account two phenomena: One is the frequency-specific bleaching of the photochemical species, which can take place long before the hole depth approaches the height of the absorption line under consideration; we call this phenomenon “saturation broadening” (in contrast to “power broadening”, which is due to high electric field intensities of the laser and which will not be dealt with in this article). The other phenomenon is the optical anisotropy, which creates a dichroic sample after irradiation with a polarized laser beam. The latter phenomenon has been dealt with in a separate paper;’ in this paper we will treat the polarization aspects only to the extent, to which they are needed, to treat the analytical description of the measured line shapes. Before we present the formalism of our line-shape analysis, we would like to point out reasons why it is useful to develop the mathematical framework which will be presented below. It will be shown in the experimental part of the paper that, with the pertinent equations at hand, one can obtain molecular parameters, such as Debye-Waller factors, effective Debye temperatures, and photochemical quantum yields with rather high accuracy. It will also be shown that the present theory can be used to describe photochemical holes which are more than 50% deep within the experimental uncertainties. Hence, we argue that it is easier to investigate deep holes under favorable signal to noise conditions and use our mathematical line-shape analysis, rather than burning very shallow holes and assuming that saturation and the dichroic character of the medium can be neglected. Analytical Description of the Zero-Phonon Hole For the following calculations we use the normalized line-shape function, which had been used previously to characterize a photochemical hole under conditions, which we called “short burning time” limit.2 In this limit all power series were truncated after the first term. The line-shape function g(w-w’) is characterized by a normalized zero-phonon origin z(w-w’) and a normalized, displaced phonon tail p(o-w’-A). A is the mean displacement of the phonon band from the zero-phonon origin. We have g ( w - d ) = ~ z ( w - w ’ ) + (1 - a ) p ( ~ - w ’ - A ) (1) a is the Debye-Waller factor, which characterizes the fraction of oscillator strength belonging to the zero-phonon part of the transition. If a is one, the whole line intensity is in the zero-phonon origin and we have negligible electron-phonon coupling. Figure 1 gives a schematic view of a typical line-shape function. For most of our experimental examples, the Debye-Waller factor is close to one ( a > 0.5) and the phonon tail is much less pronounced than
the one depicted in Figure 1 . It will become evident in the experimental section that hole burning is a very accurate method for determining a-values, which are close to one and which cannot be measured easily in a straightforward absorption experiment, where the low intensity phonon tail is usually buried under the base line (because the phonon tail has a much larger line width than the zero-phonon origin). Photochemical (or photophysical) hole burning is an experiment requiring two subsequent “photon processes”. First the photochemical change has to be induced by a (laser) photon leading to an altered site distribution (in case of a two-photon process, two photons are involved in this step; our paper will deal with one-photon processes). Second, the new site distribution has to be monitored via a fluorescence excitation or absorption experiment (in our case we will present absorption data). Consequently, the spectrum of a photochemical (or photophysical) hole is described by a convolution of the site distribution function W I N , representing the first “photochemical” photon event and a line-shape function g(o-w’), representing the second photon event of monitoring the line shape. Since both the photochemistry and the monitoring are performed with polarized light, one has to introduce the corresponding angular functions. For that purpose we use polar coordinates and choose the polar axis along the electric field of the (laser) light source initiating the photochemistry. The molecular absorbers in the sample are therefore characterized by a polar angle 29 and an azimutual angle p, where 29 is the angle between the laser field and the molecular transition dipole of an individual molecule. Therefore, one has to write the absorption cross section udiffin the following differential form
where u is the total absorption cross section, integrated over all frequencies. If we perform the photochemistry with linearly polarized light and collinear light beams, we have to introduce a polarization factor offB(8) of cos’ 29, taking into account that the electric laser field is only inducing photochemistry proportional to its projected intensity along the axis of the molecular transition dipole. A second polarization factor fsc(d,p) (SC, scanning) takes into account that the “monitoring experiment” is usually performed with the electric vector of the fluorescence excitation or absorption experiment being either parallel or perpendicular to the electric vector of the (laser) light source inducing the photochemistry. Therefore we have fscll = cos2
8, f s c i = sin’ 8 sin2 p
With the above definitions, one gets for the line shape L’(w) of a photochemical hole after a burning time T : L‘(0) =
(1) Kohler, W.; Breinl, W.; Friedrich, J. J . Phys. Chem. 1985, 89, 2473 (2) Friedrich, J.; Haarer, D. J . Chem. Phys. 1982, 76, 61
0022-3654/86/2090-1264$01.50/0
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 7, 1986
Hole-Burning Parameters and Molecular Parameters
1265
We assume that the burning intensity I is low enough that power broadening can be neglected; then is the natural homogeneous line width. Furthermore, we introduce a dimensionless variable B which is proportional to the integrated time intensity of the hole-burning experiment:
Next we expand the exponential function in ( 4 ) into a Taylor series:
A’“ dpS,“ sin
1:-
29 dt9
1 [(WL
cud
Figure 1. Homogeneous absorption line profile for a specific site at frequency w’ under an inhomogeneously broadened absorption band. The homogeneous profile has been enlarged with respect to the inhomogeneous one.
A,, is the absorption intensity before the (laser) photochemistry and N is the total number of molecules in the sample volume. In a steady-state experiment, NrCfB(t9),w’) represents the number of photoreactive molecules which have the polar angle 8 and which remain in the sample after a burning time T has elapsed. It is = No(W’) eXp
1
[
h f B ( 0 ) g(WL-W’) (3) hiL I is the laser intensity irradiated into the sample (measured outside the sample), 3 the quantum yield of the photoreaction, and No(w’) is the inhomogeneous distribution before laser irradiation. Equation 3 is valid for one-photon photochemistry. It had been shown earlier (ref 2 and references therein) that the zero-phonon line z of a spectrum is a sharply peaked function, compared to the phonon sideband p and the inhomogeneous site distribution. Therefore, the total hole spectrum L‘(w) can be regarded as a sum of four contributions. Only one of these contributions, namely the zero-phonon hole z z ( w ) represents a narrow line and thus dominates the spectrum. In the following, we will focus our attention on this part of the spectrum. It can be written as N’CfB(8)W’)
z z ( w ) = ua-
NdWL)
N
-1’“ 1 47r o
dp
x“
sin 29 dt9 J+-
dw’ (1 -
-m
The above expression contains the site distribution function nzCfs(t9),w’), which describes the frequency distribution of photochemically converted centers. The site distribution function can be calculated easier than the line shape of the measured hole spectrum, which is a convolution of the site distribution function with the zero-phonon line profile (scaled with the two factors a and f s c ) . It is given by
In order to obtain the real line shape of the photochemical hole, one has to integrate eq 4 after introducing a Lorentzian line-shape function for ~(w-w’) with a full half-width of y (fwhm; full width at half-maximum). Y/2
1
z(w-w’) = -
=
(w
- w’)2
+ y’/4
(6)
+
- w’)2
- 1 -(-yZB)PCfB(t9))P x p! 1 (8) y ’ / 4 ] p f s C ( t 9 q W - w’)2 y’/4 dw’
p=l
+
The integrations over the angular coordinates can now easily be performed yielding
with
Fp =
4p2
1 8p
(case I)
+ +3
(lob)
The w’-integration can also be carried out analytically, but the procedure is a little lengthy; it will be shown in the Appendix. As a result of our calculation we get zz(w#q) = 1
(
1
i y k+
z z ( w = q ) = -ua-
l)”k
-
NO(WL) 1 -(”)F,(-B)‘ N p=lp!
(1 1b )
The above equations are valid for all burning times. Although the hole shape function is real, there seems to be no way of writing it down without using complex expressions. The convergence of the above series is comparatively poor, as can immediately be seen counteracts the coeffrom (1 Ib): the binomial coefficient ficient 1/ p ! (Le., slower convergence than an exponential function). Equations 1l a and 11b yield the known result of a Lorentzian line with twice the homogeneous line width if one takes only the lowest term with p = 1 (which had been published earlier2). For investigating the general line-width behavior we define the half-width I’(=fwhm) of the hole as
(y)
and we can show that our model calculation predicts the following power series for the experimental half-width (up to the third order):
No. 7, 1986
B Figure 2. Convergence of the calculated four lowest order terms describing the saturation broadening of a photochemical hole in case 11 (according to eq 14a). The curves show the hole width with the terms up to the 0 (a), 1 (b), 2 (c), and 3 (d) order in the burning dose included. r
Kador et al.
The Site Distribution Function Whereas the calculation of the real hole shape is rather involved, it can be shown that the corresponding site distribution function can be calculated quite readily. Its photochemically induced line width can, for instance, be derived analytically. Also the site distribution function is, in one way, easier to visualize: The site distribution function reaches the bottom of the band surprisingly early; in most instances this happens at burning times, at which the "real hole" is still far from reaching the bottom of the band. The real hole usually does not reach the bottom of the band because it is shifted toward a finite absorption through all the phonon wings of neighboring sites which sum up to a broad background signal allowing a rather accurate determination of the Debye-Waller factor a (see below). If we assume, for simplicity, thatfB(tY) = 1 (eq 5) then we get the following expression for the site distribution function. nz(1,w) =
I
1 (w - WL)2
u -
N
+ y2/4 (15)
.et
I
/
It can be shown easily that this curve has a half-width of
For B 0 this function approaches the value of ]/,-independent offB($). Therefore, it is obvious that in the short burning time limit the distribution function contributes half the hole width. If one compares the depth of nz as given by the following equation -+
0.0
.5
1.5
1.0
2.0
B Figure 3. Dependency of various features of a photochemical hole on irradiated light dose E . The upper curve shows the maximally bleached distribution function (Le., withf, = l), and the medium and lower curves correspond to the depth of the observed hole in the cases I/ and I, respectively. Note that all three curves are normalized and, hence, converge to the limiting values of 1.0 for B a.
-
In our two specified polarization directions this general formula yields:
rl, -- 1 + -B5 + -B2 5 _ 2Y 14 168
425 11319
- -B3
(case
11)
(14a)
nz(l,w=q) =
u
No(uL)(l - e-4B) N
with the depth of the observed photochemical hole, one obtains the result which is graphically shown in Figure 3. As has been pointed out the distribution function reaches the bottom of the absorption band very quickly, quite in contrast to the observed hole. One important fact should be emphasized at this point: the site distribution function really approaches the bottom of the absorption band for large burning times, Le.: NO(WL) N The depth of the observed hole, however, approaches only the fraction a of the height of the absorption band (see eq l l b ) : lim nz(lrw=wL) =
u
B-m
Figure 2 shows the magnitude of the various contributions to the line width for rll.As one can see, eq 14a is valid to about B = 1.0, a t which point the third-order term is less than 3%. This procedure allows the evaluation of holes whose depth is up to about 54% of the total absorption, as can be seen in Figure 3. (For Debye-Waller factors a < 1, this number has to be multiplied by a.) Note, however, that at B = 1.0 the linear corrections contribute 26% to the line width and the quadratic corrections contribute 2%. In the case of rl our formula for the hole width (eq 14b) is valid to about B = 1.8, the corresponding depth being 34%.
The fact that the hole width is not constant, but grows linearly with the number of irradiated photons, is due to the photochemical saturation: while the sample is exposed to the laser light, more and more absorbers in the vicinity of the laser frequency are bleached and photochemistry increasingly goes over to such molecules which are not quite in resonance but are affected by the laser radiation in the wings of their Lorentzian zero-phonon lines. The linear terms in eq 14 demonstrate that this effect shows up already in a very early stage of burning. As we have seen, our power expansion method allows the evaluation of holes for small to intermediate burning times. At m) it was shown earlier' that a square long times (Le., for B root dependence of the hole width as a function of time is to be expected.
-
NO(WL) 3N Therefore, if a is not equal to one, it is not possible to burn holes which reach the bottom of the band. The remaining absorption is due to the phonon wings of molecules whose zero-phonon lines are located at lower frequencies outside the hole and which have therefore not been burnt away. In the next section we shall use this phenomenon t o determine accurate values for t h e DebyeWaller factor and its temperature dependence. lim zz(w=wL) = ua
R--
-
Comparison with Experiment and Discussion We performed hole-burning experiments in the SI So 0-0 absorption band of free-base phthalocyanine (H,Pc) in the hosts polyethylene (PE), polystyrene (PS), and poly(methy1 methacrylate) (PMMA). For details of the hole-burning mechanism of this molecule we refer to ref 3. The concentrations of the dye molecules were about 10-3-104 mol/mol. For cooling the samples we used either a flow cryostat Oxford CF204 or a custom-built 4He bath cryostat. The holes were burnt with a single-mode CW (3) GutiCrrez, A. R.; Friedrich, J.; Haarer, D.; Wolfrum, H. ZBM J . Res. Deu. 1982, 26, 198.
The Journal of Physical Chemistry, Vol. 90, No. 7, 1986 1267
Hole-Burning Parameters and Molecular Parameters
r
t 0 ~
H~PC/PMMA
0.2
0.4
0.6
0.8
1.0
1.2
E [ IO-~Jcm-2I
,,/*---
Figure 4. Comparison of the measured saturation broadening of photochemical holes with the polarization vectors of the burning and the probing beam either parallel (upper curve) or perpendicular (lower curve)
’
------
e-
T=BK
to each other. The sample was H,Pc in PMMA, the temperature 4.2 K. Hole widths r are plotted vs. irradiated energy per unit area E . The slopes of the curves for E 0 are displayed.
-
dye laser Coherent 599-21 (bandwidth < 1 X lo4 cm-’) and detected with a 3-m monochromator Jobin-Yvon THR (resolution =3 X cm-l). As light source we used a high-pressure Xe arc lamp. Burning and scanning beam were linearly polarized; they penetrated the sample in a collinear geometry. By using a quarter-wave plate and a Glan-Thompson prism we were able to rotate the polarization plane of the laser beam relative to that of the monochromator beam so that the two cases considered theoretically could be realized in experiment. The laser intensity used for burning was always kept below 100 pW/cmZ and was therefore at least one order of magnitude below those values, at which power broadening of the holes begins to play a role! The corresponding irradiation times were in the range between 20 s and 4 min. In order to test the different bleaching behavior of photochemical holes predicted by eq 14, when the burning beam is polarized either parallel or perpendicular to the probing beam, we measured the hole width as a function of the irradiated light dose in PMMA for both polarization directions. The results shown in Figure 4 are in agreement with the predictions. Clearly the hole broadens slower in the perpendicular case than in the parallel case. We denote with E the energy per unit area irradiated into the sample ( E = 17) and define the slopes of the hole widths as a function of E in the short burning time limit as
T=12K
0
1
I
I
I
0.5
1
15
E [IC3J cm-’]
H2Pc/PS
4,
I
drll,l. E+ dE We determine with our experiment the ratio of the two slope functions and get
so,l,l= lim
which is in excellent agreement with the theoretical value of
(2)
lhcor
=k
8-
-* I
T=4K
______
L-------------d-
I
I
(see eq 14).
Debye-Waller Factors, Debye Temperatures Figure 5 shows the saturation broadening of photochemical holes in all three polymer hosts for several temperatures, where the burning beam was polarized perpendicular with respect to the probe beam. The slopes sol for E 0 have been marked at every curve. In Figure 6 we present the same data in a different manner, showing the hole widths as a function of the measured relative depths of the holes. Additionally, we have plotted in these figures the expected theoretical curve for the case in which the Debye-
-
(4) Kador, L.; Oswald, G.; Haarer, D., to be published.
Waller factor is one. This curve has been obtained from Figure 3 of this paper and from Figure 2 of ref 1, where the hole width was calculated numerically for arbitrary burning times. Obviously, the hole depths do not reach the bottom of the absorption bands but approach smaller limiting values, which, according to eq 19, give the Debye-Waller factors. We obtain CY = 0.90f 0.1 for H,Pc in PE
1268 The Journal of Physical Chemistry, Vol. 90, No. 7, 1986 3
a = 0.72 f 0.1
I
H ~ PIcPE O
T=8K
A
T =12K T =16K T :23K
I
Kador et al.
I
(a)
a = 0.77 f 0.1
for H2Pc in PMMA ( T I 12K)
a = 0.57 f 0.1
for H,Pc in PMMA ( T = 20 K)
It is interesting to note that, within the measured temperature interval of 4 K IT I 20 K, the Debye-Waller factors for PE and PS seem to be temperature-independent within our experimental uncertainty. For PMMA, however, we observe a small variation of a. Within the Debye approximation one can describe the measured temperature dependence by the following equation?
-T 2 w
. L
LE
a(T) = exp{-S[l 1
0
for H2Pc in PS
0.2
0.4
0.6
0.8
1.0
02
3.4
06
0.8
1C
+ Y3r2( ~ p , ) ~ ] (21) )
If we fit our data for PMMA with the above equation, we obtain an “effective” Debye temperature of about 50 K. It is commonly known that polymers show a rather striking deviation from the straightforward Debye model of the heat capacity. The Tarasov theory, for instance, defines four different Debye temperatures, two of which are due to the one-dimensional €IlT). The other two temcharacter of the polymer chains peraturw reflect the three-dimensional nature of the matrix (e3L, The indices L and T stand for the longitudinal and transverse modes, respectively.6 The published values of €13for PMMA are f33T= 51 K and 03L= 88.4 K;7 they are in rather good agreement with our optical values. It should be noted that the published e3-values are parameters which follow from an analysis of thermal data with the aid of the Tarasov model. Our experimental e3-value is, in contrast to the above thermal values, a parameter resulting from an optical line-shape analysis. Comparable data on Debye temperatures from optical experiments in polymers are still rather scarce-to our knowledge, the above data are the first optical data on that subject. We think that it will be interesting to collect more data in the future and obtain @values from optical experiments. It is likely that dye molecules in polymer hosts will preferentially couple to “soft” optical modes and, hence, our data may be pertinent to the interpretation of thermal anomalies. The latter are frequently interpreted by assuming low-frequency modes of the polymer In this context it was suggested that the low-frequency relaxation modes of PMMA may be due to the vibration of the methyl group of the p01ymer.~ More optical data on Debye temperatures may shed more light on the hitherto rather sketchy picture. The above described method of determining the Debye-Waller factors of host molecules in polymers is rather sensitive. It is especially suited for cases in which a is close to one. Here a “normal” optical experiment is not very sensitive because the area under the phonon sideband is very small. In the holeburning experiment, however, the area of many broad phonon sidebands accumulates under the zero-phonon line making the experiment rather sensitive. One condition, however, which has to be fulfilled is that there is no measurable back-reaction in the investigated photochemical reaction scheme. This condition is, in our case, well fulfilled (see below).
Photochemical Quantum Yield Usually the photochemical quantum yield of a reaction is difficult t o m e a ~ u r e . One ~ could, in principle, burn a hole and integrate its area to obtain the quantum yield. To get a correct answer, one would have to correct for saturation phenomena which occur, as we have seen, already in the “early burning” state of a photochemical experiment. One straightforward way to obtain the photochemical quantum yield is the determination of the slope relative depth Figure 6. Relative hole width vs. relative hole depth for H2Pc in PE (a), PS (b), and PMMA (c). The data are the same as plotted in Figure 5. The hole width is given in units of the residual width for E 0 (Le., 27). the depth in units of the total absorbance of the inhomogeneous absorption band at the wavelength considered. The solid curves represent the data, whereas the dashed curves show the expected behavior for a Debye-Waller factor of unity (see text).
-
( 5 ) Pryce, M. H. L. In ’Phonons in Perfect Lattices and in Lattices with Point Imperfections”; Stevenson, R. W., Ed.; SUSSP Publications: Edinburgh, 1966. (6) Tarasov, V. V . Zh. Fir. Khim. 1950, 24, 111. (7) Reese, W. J . Macromol. Sei. Chem., 1969, A3, 1257. (8) Choy, C. L.; Hunt, R. G.; Salinger, G. L. J. Chem. Phys. 1970, 52, 3629. (9) Moerner, W. E.; Gehrtz, M.; Huston, A. L. J . Phys. Chem. 1984,88, 6459.
The Journal of Physical Chemistry, Vol. 90, No. 7, 1986 1269
Hole-Burning Parameters and Molecular Parameters of the hole-burning curves. Figure 5 shows that these slopes are temperature-independent. To understand this behavior, we have to rewrite eq 14b. We neglect, at this point, second- and thirdorder terms and pick up only the “linear” slope factor. This yields
4
n
73 w
-
In the above equation we can replace the value of
which is not commonly used in optical spectroscopy, with the more familiar quantity of the oscillator strengthfI2. This value describes the integrated optical line intensity between the states 1 and 2 (in our case the lowest electronic transition of the phthalocyanine molecule, not including vibrational transitions, yet including phonon transitions). Inserting the oscillator strength, we can derive the following equation in a rather straightforward fashion:I0 cr =
3a
e2 1
(
n2+ 2
cr,
L \
L
2
1
B
2
2 cOmcn 4 f I 2
(23)
Here e and m are the electron charge and mass, respectively, and to is the dielectric constant of vacuum (in SI units), c the velocity of light, and n the refractive index of the sample. With the above relation we can calculate the slope of the broadening curve and get
Our final expression for the quantum yield is I
0.1
100
10
1
1000
B Note that this equation is independent of the line width and, therefore, independent of the temperature (see Figure 5). In the above equations we take as laser frequency wL = 2.7 X 1015 s-l and we insert the Debye-Waller factors obtained above. For the refractive indices we take the following data:” PE, n = 1.52; PS, n = 1.57; PMMA, n = 1.50. The largest uncertainty is due to the oscillator strength for which we find literature values between 0.1I2 and 0.52.13 The difficulty with getting accurate data on the oscillator strength is the poor solubility of phthalocyanine in conventional solvents and its tendency to form colloidal particles rather than dissolving on a molecular level. This tendency is likely to give too small values for the experimentally determined oscillator strength. Taking as a mean valuef12 = 0.3 we get for the proton tautomeric reaction of phthalocyanine in polymer matrices the following values for the quantum yield: @ = 1.1 X low3
in PE (f70%)
@ = 1.3 X
in PS (f70%)
@ = 1.1 X
in P M M A (f70%)
The large error margins mostly arise from the uncertainty of the f-value. The main advantage of our experimental approach of measuring the photochemical quantum yield by measuring the onset of the saturation broadening of a photochemical hole is the fact that the numerical value is based on an extrapolation to zero burning time and, hence, is not falsified by the bleaching behavior (see above). Finally we want to show that there is no measurable photochemical back-reaction in our hole-burning systems. In Figure 7 we have plotted the data of Figure 5 semilogarithmically and compared them with theoretical curves which have been obtained by the mathematical procedures as outlined in ref 1. The authors (10) Dexter, D. L. Phys. Rev. 1956, 101, 48. (1 1) Vieweg, R., Schley, A., Schwarz, A., Daumiller, G., Esser, F., Eds. ‘Kunststoff-Handbuch”; Hanser: Munich, 1969, 1975; Vol. IV, V, and IX. (12) Lyons, L. E. J . Chem. SOC.1958, 1347. (13) Lyons, L. E.; Walsh, J. R.; White, J. W. J . Chem. SOC.1960, 167.
H,Pc4 IT=12K TPMMA =4K n
= 100
A:,=,:,?(
4 -
73 -
T=20K
7‘
W
I
L \
L
2
-
8 .
. I
_ _ _ - - -A:10_---K)
B Figure 7. Relative hole width vs. burning dose B (time-intensity product) for HzPc in PE (a), PS (b), and PMMA (c) in a semilogarithmical representation. The data are the same as plotted in Figure 5. The curves are theoretical and have been obtained following the procedures of ref 1; they show the expected saturation broadening in the absence (solid curves) and presence (dashed curves) of photochemical back-reactions. The parameter A is the quotient of forward and backward reaction rate.
of ref 1 calculated the hole width as a function of normalized burning time numerically for arbitrary times, based on a similar phenomenological model as that presented above, yet allowing for a backward reaction. They show that the broadening behavior is quite different for different back-reaction rates. A comparison of our measured data with the solid curves shown in Figure 7 shows that the back-reaction has to be at least 200 times less efficient than the forward reaction. This justifies our simplified model which neglects the backward reaction completely. Summary and Conclusions We have presented a complete analytical model of the zerophonon line shape of photochemical holes. This theory allows the quantitative evaluation of photochemical holes, which are up to 60% deep. Our model predicts a linear increase of the hole width with burning time. The slope of the line width as a function of burning time has been shown to be temperature-independent. It
1270
J. Phys. Chem. 1986, 90, 1270-1275
allows a rather accurate determination of the photochemical quantum yield of the investigated hole-burning process. We have presented first preliminary data of the Debye-Waller factors and Debye temperatures, which may lead into an interesting new direction, namely, relating Debye temperatures from thermal data to comparable values from optical data. In case of the optical data one has to assume that the “lattice” modes, which contribute to the specific heat, also couple to the optical transition. Our first experimental data seem to justify this assumption. More data, however, are needed to prove this point. Acknowledgment. We thank our colleague J. Friedrich for many helpful discussions and we acknowledge the support of the Stiftung Volkswagenwerk without whose support this work would not have been possible.
Appendix: Evaluation of the Convolution Integral in the Hole Shape Function In eq 8 we had derived an expression which describes the spectral shape of a zero-phonon hole. It can be written in the following form
The sum has to be taken over all the poles in the upper complex half-plane. The residue of a functionflz), which at zo has a pole of order p , can be calculated from the following formula:I4
In order to get general expressions for the derivatives of our functions fp, we use the partial fraction analysis 1
1
The further calculation is straightforward. It yields
-
xlyp(w’) dw’ =
1
y
(w - wL
+ iy)P(w - wL)P + (‘46)
with f p ( ~ ’ ) = ([(w’
+ w - o L ) ~+ y2/4]”(~’*+ y2/4)]-l
(A21
Here the integrations over the angular coordinates have been carried out. The functions fp(w’) can be. extended over the whole complex w’ plane; they have infinities of order one at w’1,2 = 2=iy/2 and infinities of order p at = wL - w f iy/2, respectively. Therefore the integral in ( A l ) can be solved by using the residue iaw:14
Inserting this formula in (Al) directly leads to the final expression for the hole shape given in (1 la). At the burning frequency (Le., for w = wL) eq l l a is not applicable because of zero denominators. In this case we start out with eq A2 and w = wL; with the residue law we can calculate
+I ,dw’,=,, f p ( 4
= 2+)-
1 y2P+1
(‘47)
which leads to (1 lb). R e m NO. H~Pc,574-93-6;PE, 9002-88-4;PS, 9003-53-6;PMMA, 901 1-14-7.
f m
Jm
fp(w’) dw‘ = 2rriCRes Cfp,dj) J
(A31
(14) Endl, K.; Luh, W. ‘Analysis 111”; Endl, K., Ed.; Akademische Verlagsgesellschaft: Frankfurt/Main, 1974; pp 207, 203.
Electron Spfn Resonance in Phosphorescent Triplet States of 2,P’-Blpyridine and 2,2’-Biquindine Complexes with Diamagnetic Metal Ions and of 2,2‘-Binaphthaiene Jiro Higuchi,* Kazuhiro Suzuki, Hiroyuki Arai, Akira Saitoh, and Mikio Yagi Department of Chemistry, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya- ku, Yokohama, 240 Japan (Received: September 17, 1985)
The phosphorescent triplet states of 2,2’-bipyridine and 2,2’-biquinoline complexes with several kinds of diamagnetic group I1 (2 and 12) metal ions M2+ (M = Mg, Ca, Sr, and Zn) have been studied by ESR and phosphorescence spectroscopies. For the 2,2’-bipyridine (bpy) complexes, the ESR spectra of [M(bpy)I2+ were clearly observed. In general, the zero-field splitting parameters of these complexes are fairly close to those of (Z)-2,2’-bipyridine, which has been detected by ESR spectroscopy. For the ESR and phcsphoresknce spectra of 2,2‘-biquinoline (bq) complexes, only [Zn(bq)I2+could be detected. In connection with its conformation, the ESR and phosphorescence spectra of 2,2’-binaphthalene were studied as the parent hydrocarbon of 2,2’-biquinoline. Its ESR signals originating from either its E or Z conformer could be assigned separately by using a stretched poly(viny1 alcohol) film as a host. As a result, the zero-field splitting parameters of the undetected (Z)-2,2’-biquinoline could be discussed by comparison with those of [Zn(bq)12+and also those of (Z)-2,2’-binaphthalene. From the analysis of the phosphorescence spectra, all the lowest triplet states of the molecular species studied in the present work are estimated to have mr* character, and the electronic structure of the ligand in these metal complexes is not largely different from the corresponding metal-free ligand with Z conformation. On the basis of all the present experimentalobservations, the zero-field splitting parameters appear to be more sensitive and useful for detecting the formsltion of a metal complex with a ligand whose conformation is changeable upon coordination, in comparison with the phosphorescence spectrum.
Introduction Various kinds of metal chelates in their ground (G)state have long been studied by ESR spectroscopy with fruitful results. Because of the short lives of excited paramagnetic wamition-metal complexes, however, the phosphorescent ESR works of metal 0022-3654/86/2090-1270.$01.50/0
chelates have been restricted to the lowest excited triplet (TI) states of complexes with a diamagnetic ion. The results obtained for these molecular species may usually be explained by a relatively small perturbation of ligand by the metal ion. On the other hand, there are Some different types of molecules whose geometries are 0 1986 American Chemical Society
