Photoluminescence Decay Curves: An Analysis of the Effects of Flash Duration and Linear Instrumental Distortions J. N. Demas and G . A. Crosby Department of Chemistry, Washington State University, Pullman, Wash. 99163 Several mathematical techniques are described for calculating lifetimes of photoluminescent materials from data taken under diverse experimental conditions. The effects of finite flash duration coupled with various degrees of linear instrumental distortion are discussed and mathematically analyzed. Specific criteria are given for choosing appropriate data reduction procedures to handle practical experimental results. In particular it is shown that improvement in signal-to-noise ratios by the introduction of large RC time constants into the detection system does not necessarily prohibit the accurate measurement of decay phenomena if appropriate methods are employed to analyze the results. The mathematical techniques presented here can be readily implemented with a digital computer and, in some cases, are amenable to desk-calculator or hand computation. These methods can extend the useful range of decaytime measuring equipment by an order of magnitude.
EXPERIMENTAL STUDIES on luminescent molecules often involve the measurement of the decay time of an excited species after optical pumping. For many luminescent substances it is possible to use a single flash technique to obtain these data. A common procedure is to flash the sample, detect the resulting luminescence by a phototube, display the signal on an oscilloscope, and photograph the resulting trace. Data reduction consists of making a semilogarithmic (semilog) plot of the signal intensity (in arbitrary units) us. time and calculating the slope, which is proportional to the reciprocal of the desired mean lifetime. This simple procedure works well when the excitation flash is much shorter than the lifetime to be measured, but if the lifetime is comparable to the decay of the flash, serious errors are introduced. For lifetimes shorter than the flash, the method fails completely. Accurate lifetime measurements on a luminescent material are possible, however, even when the simple semilog method is unreliable. We describe three mathematical techniques for extracting information from such data and discuss the relative merits of each. Two are statistical methods developed by Cooper (I, 2), and the third is a numerical curve simulation method (3, 4). All these procedures have been used previously for the evaluation of fluorescence lifetimes where the problems are somewhat different from those discussed here. To verify the usefulness of these methods, we present the results of experimental measurements of the decay time of a luminescent molecule under conditions where the merits of the data reduction techniques and error analyses described here can be clearly recognized and assessed. Another recurrent problem in the measurement of lifetimes is the selection of load resistor and filter capacitor size for any given experiment in order to maximize signal-to-noise ratios while keeping distortion and consequent losses in (1) S . S. Brody, Reu. Sci. Instrum., 28, 1021 (1957); I. H. Munro and I. A. Ramsay, J. Sci. Instrum. ( J . Phys. E ) , Ser. 2,1,147 (1968). (2) D. H. Cooper, Rea. Sci. Instrum., 37, 1407 (1966). (3) L. Hundley, T. Coburn, E. Garwin, and L. Stryer, ibid., 38, 488 (1967). (4) 0. J. Steingraber and I. B. Berlrnan, ibid., 34, 524 (1963).
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accuracy within acceptable limits. We discuss here errors introduced by instrumental RC time constants and present ways of detecting, minimizing, or avoiding them. EXPERIMENTAL
The lifetime of a sample of tris(2,2'-bipyridine)ruthenium (11) chloride hexahydrate ([Ru(bipy)~]Cl,)(G.Frederick Smith Chemical Co.) was measured in an ethanol-methanol (4/1;
v/v) glass at 77 O K . The absolute ethanol (U.S. Industrial Chemical Co., U.S.P.-N.F. grade) and methanol (Baker spectrophotometric grade) were used without further purification. In all experiments the sample was excited by a flash lamp whose light was filtered through a Corning 7-60 glass filter and 1 cm of aqueous CuS04.5Ha0 (200 g/liter). The complementary filters over the phototube were 2 cm of aqueous tris(2,2'-bipyridine)iron(II) chloride (100 mg/liter) and a Corning 2-58 filter. Leakage of exciting light to the detector was negligible. The phototube viewing the luminescence was an EM1 9558QC (S-20 response) operated at 600-800 V from a well-regulated power supply. Phototube current was kept below 100 p A to prevent nonlinearities. The RC time constant of the measuring circuit was -0.4 psec, and the oscilloscope (Tektronix 535A with a 1Al plug-in unit) had a band width of -15 MHz which produced negligible instrumental distortion of the signals. A Tektronix C-14 oscilloscope camera was used to record the data. For purposes of comparison an accurate measurement of the mean lifetime, r , of the sample was obtained by use of a fast flash whose duration was much shorter than r. The instrumentation was the same as described elsewhere (5) except that the preionizer was omitted. In order to simulate an experiment where a measurement is carried out on a substance whose r is comparable to the duration of the exciting light, an E G & G FX-12 flash lamp was used in the high voltage hold-off mode with a series inductor and resistor inserted deliberately to prolong the flash. The lamp was triggered by a 20-kV pulser previously described (5). Under the conditions used, this spoiled flash had a duration of 20 psec between the 1/3 peak amplitude points and an exponential decay on the trailing edge of 6.5 psec. To be able to reduce the spoiled flash data by all the methods described, it was essential to obtain the representations of the flash and the luminescence decays to the same time-base zero. This was accomplished by triggering the oscilloscope externally with a pulse from an RCA 2020 photomultiplier which viewed the flash lamp directly. This arrangement permitted both decays to be observed to the same time base as long as the triggering controls were not disturbed between flashes. For each measurement the triggering level was set, an exposure was taken of the decay of the compound, the base line was recorded, the sample was replaced with a solution of Rhodamine B dye, the dye decay was photographed, and, finally, time marks were superimposed on the photograph. Rhodamine B was chosen as a color converter to shift the flash output to a region of the spectrum which could be monitored by our phototube without changing filters. The (5) J. N. Demas and G. A. Crosby, J , Mol. Spectrosc., 26, 72 (1968).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
0
+
1.0-
H
- 6.5 = -0.0923 psec-’
b. slope
P (3
s
y : - L :
2.303(b)
4.71 prec
0
.0.5 -
I
0 T 1 ME
(psec)
I
I
I
5
IO
15
TIME
Figure 1. Time dependences of the fast flash and the luminescence decay of tris(2,2 ‘-bipyridine)ruthenium(II) chloride
I
20
(psec)
Figure 2. Calculation of a mean lifetime from a semilogarithmic plot of intensity us. time
Curve A (x) is the fast flash. Curve B (0)is the decay curve. The x’s and 0’s are experimentally measured points; the smooth curves have been drawn in by referring to the original photographs. (Intensity in arbitrary units)
luminescence of this dye has a decay time of several nanoseconds, and, for our purposes, the fluorescence of the dye can be considered to follow the flash faithfully in real time. (The flash lasts microseconds.) EVALUATION OF DECAY TIMES rflaah < < rsample.Figure 1 shows data obtained from a lifetime measurement on [Ru(bipy),]C12 using the fast flash. Curve A displays the flash intensity as monitored by the detector system (using the Rhodamine B color shifter). Curve B is the response curve of the detector to the observed luminescence. The height of each curve is arbitrary, having been adjusted to put both traces on the same photograph. The important point to notice is that the flash has fallen to less than 5 of its maximum value at -7 psec, whereas the luminescence has a measurable intensity for considerably longer times. We recognize such a situation as a proper one in which to use the simple semilog plot method to obtain the mean decay time. Figure 2 is a semilog plot of Curve B of Figure 1. Part of the curve has been approximated by a straight line which was fit visually. The slope yielded 4.71 psec for T ; a leastsquares fit from 8 to 19 psec yielded 4.90 psec for r . Two more photographs taken under identical conditions gave similar plots whose least-squares fits yielded 4.97 and 4.88 psec. An average of these results gave r = 4.92 psec with a standard deviation of 0.05 psec. A much simpler method of obtaining T is to plot intensity us. time directly on semilog paper and visually draw in a straight line over the linear part. The half life, r1/2,is obtained by choosing an intensity value on this line and noting the time interval between this time and the point where the signal has fallen to half the initially chosen value. The mean life is related to 7 1 1 2 by
=
~1/2/0.693
(1)
A quicker (and less accurate) estimate of can be obtained directly from a photograph merely by choosing an intensity value on the exponential part of the trace and measuring the time interval necessary for the intensity to fall to half this value. the results of an experirflssh > ~ ~ . ~Figure ~ l 3~ shows . mental measurement on [Ru(bipy),]Cl, with the spoiled flash.
T I M E (ysec)
Figure 3. Observed and calculated luminescence decay curves of tris(2,2’-bipyridine)ruthenium(II) chloride after excitation with a spoiled flash Curve F (x) is the spoiled flash. Curve D ( 0 ) is the observed decay. Curve Dcsl0(-) is the calculated decay for T = 5.15 fisec, the best fit value. Data for this plot are given in Table I. All curves have been normalized
Curve F, defined by the x’s, is the flash intensity; Curve D, defined by the Q’s, is the sample decay. [Curve Dealc, the solid line, is defined in a later section.] All curves have been normalized to a peak value of 100, a procedure which does not affect the final results. One recognizes that both curves have nearly the same exponential falloffs, and there is no part of the luminescence decay which is not strongly influenced by the protracted pumping by the flash. In this typical case all the simple methods so far described fail to yield reliable results; in fact, they may yield the decay time of the flash rather than rsample. There is, however, an obvious delay of the sample luminescence with respect to the flash, and we turn to statistical techniques and curve simulation to sift out the desired quantity, i.e., Tssmple. Moment Methods. The ith moment of a curve is defined by the equation
where Y represents the flash, F , or the sample decay, D. For obtaining rssmple = T from moments, two methods are available :
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9 , AUGUST 1970
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Table I. Calculation of Decay Time of Photoluminescent Compounds by Statistical and Curve Simulation Methods Compound, [R~(bipy)~]Cl~; solvent, ethanol-methanol (4/1; v/v); temperature, 77 "K; spoiled flash, TF (trailing edge) = -6.5 Nsec. ti -
Time (rsec) 0.0
1.o 2.0 3.0 4.0
F(tJ
Flash intensity (normalized) 0.0
D(ti)
Dos'@(ri)
Observed sample intensity (normalized)
Calculated sample intensityo (normalized)
0.0
0.0
1.5 0.2 0.0 4.1 0.8 0.4 12.8 2.5 2.4 28.1 6.5 6.5 12.6 39.3 13.8 5.0 21.1 59.5 22.2 6.0 69.1 31.2 32.2 7.0 83.5 42.1 43.8 8.0 89.0 53.2 54.9 9.0 95.4 63.7 66.5 10.0 73.4 76.0 100.0 11.o 81.8 99.2 84.0 12.0 99.0 88.7 91.1 13.0 96.4 94.0 95.5 14.0 91.3 97.5 98.0 15.0 87.3 99.5 99.8 16.0 100.0 81.2 100.0 17.0 98.1 99.0 74.4 18.0 96.8 67.2 96.3 19.0 93.4 92.9 60.6 20.0 89.2 89.1 53.3 21 .o 84.2 47.2 22.0 84.9 78.8 41.2 79.3 23.0 73.2 36.2 24.0 74.5 67.6 31.9 25.0 68.2 62.8 62.0 26.0 27.0 23.6 56.4 27.0 57.4 51.2 52.1 20.6 28.0 46.4 17.2 46.2 29.0 41.5 41.4 14.3 30.0 37.0 36.7 12.2 31 . O 33.0 10.3 32.9 32.0 8.8 29.2 29.1 33.0 7.3 25.7 25.7 34.0 22.7 35.0 6.4 22.3 5.4 19.9 36.0 19.6 4.7 16.8 17.5 37.0 15.4 3.9 14.5 38.0 3.4 13.4 39.0 12.1 Based on an assumed value of r = 5.15 psec; for this fit G = 1.16. Moments for flash: OF = 0.9167 X lo4; lF = 0.1433 X lo6; 2F = 0.2796 X lo7. Moments for luminescence: OD = 0.1136 X lo6; lD = 0.2333 X lo6; 2 D = 0.5743 X lo7. Decay time of luminescence: Moment Method I: 4.905 psec; Moment Method 11: 4.794 psec; Curve simulation method: 5.15 psec.
proximation. Alternatively a functional form can be assumed between pairs of data points on the curve, and the integrations can be carried out analytically. We have chosen to approximate each curve by straight line segments between consecutive data points and to integrate exactly. This method was chosen because a limited set of data points is required for accurately defining the decay curves, the calculated moments are reasonably accurate, and the equations are well adapted for computer computation. For comparable accuracy, numerical integration would demand a closer spacing of data points, especially for evaluation of the higher moments. A problem arises in the moment calculations because F(r) and D(t) are usually known accurately only over one decade, whereas, owing to the slow falloff of the integrand in 1Y and 2Y, the signal intensity must be known accurately over several decades. Obtaining data from oscilloscope photographs for such long times is virtually impossible, and we resorted to an extrapolation technique to extend the traces. We made use of the fact that F(t) and D(t) were very nearly exponential on the trailing edges, as recognized from the semilog plots. This will generally be the actual experimental situation unless the lamp has a long afterglow. From the semilog plots we selected the linear part and determined the decay parameters by a linear least-squares fit on five points in this region. Assuming this exponential form for the trailing edge of Y(t),we carried out the integrations exactly. The equations used in our computations of the moments are given in Appendix C. An alternative, but somewhat inferior, method is to extrapolate the semilog plots visually and read off the additional data points required. Extension of the data could be done experimentally rather than analytically. One could use the pulse-sampling technique developed by Berlman (4,6), the ingenious gated phototube methods of Bennett (7), or, for slower lifetimes, a commercial signal averager. Table I summarizes the pertinent information for the calculation of from a single photograph by both moment methods. Curve Simulation. The more sophisticated method of curve simulation can be used on the data of Figure 3 to obtain the decay time of the sample. Equation 5 is the pertinent relation.
5
Moment Method 1(2) (See Appendix B) (3)
Moment Method I1 ( I , 2) (See Appendix B)
Computing r by either method is trivial, once the moments have been evaluated. To obtain 'Y, numerical integration can be carried out using the trapezoidal rule, Simpson's rule, or some other ap1012
F(t)
=
0, t
I 0.
Dcalc(t)is the response of an ideal detector (one whose output is directly proportional to the number of emitted photons) viewing a compound with mean life r which decays by only first-order or pseudo first-order processes under excitation by a flash of time dependence F(t). The constant K takes into account the absolute intensity of the flash, the fraction of photons absorbed, the experimental geometry, the transmittance of the filter systems, the spectral distribution of the emission, and the sensitivity characteristics of the detector. In practice a value (rgues9) is assumed for the compound, DcalC(t)is computed from F(t), and the calculated curve is compared with the experimental D ( t ) . Based on the fit obtained a new guess for 7 is made, and the cycle is repeated until the fit is acceptable. To judge the quality of the fit we chose the simple approach of normalizing the calculated and
(6) I. B. Berlman and 0. J. Steingraber, IEEE Trans. Nucf. Sci., NS-11, 27 (1964). (7) R. Bennett, Rev. Sei.Imtrum., 31, 1275 (1960).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
observed decay curves, Dcalc(ti)and D(ti)[i = 1, N], to the same maximum value (arbitrarily set at 100 in our computation) and then computed G, our goodness-of-fit parameter.
I
0
0
2ol 0
This simple procedure gives undue weight to a single point, the maximum value. If the maximum is poorly defined because of noise or extreme curve sharpness, large errors in the calculation of 7 may result. An alternate method of comparing the curves would be first to normalize D(t) to a maximum value of 100, then normalize the curve Dcalc(t)so that N
0
0
I
0 0
N
0 I
I
Dcalc(tn) =
+ +
r(M,(t, - r ) B, e - ht/T t M n ( 7 - fn-1) - Bn1)
+
If all time intervals are equal, only one exponential need be evaluated for each trial decay curve. This exponential can be readily computed even for quite small r values. The same data used in the moment computations (Table I) were used in the calculation of r by the curve simulation tabulated in column 4 were genertechnique. The DcalC(tf) ated for an assumed lifetime of 5.15 psec, our best fit value. The corresponding G value was 1.16. Figure 4 shows the values of G calculated for decay curves generated from the data of Table I for assumed lifetimes (8) J. G.Becsey, L. Berke, and J. R. Callan, J. Chem. Educ., 45, 728 (1968).
I
5 LIFETIME
4
This routine equalizes the area under both curves and eliminates excessive emphasis on any one point. We have carried out curve analyses using both methods of normalization. It is our experience that the equal area method normally gives slightly better fits as judged by the G value; however, rarely is the final r value changed significantly. Although normalization to equal area was not employed in the calculations here, we are presently using this technique for our data reduction. If no prior knowledge of the lifetime is available, it is difficult to make a first guess for T . A powerful technique which we employed was first to estimate 7 by Moment Method I1 and then to calculate a series of decay curves and associated G values for a grid of 7’s ranging from 0.5 rguess to 1.5 rguess in steps of 0.05 rguesa.Another option would be to carry out a one-dimensional grid search for Gmi, (8). The formula used for generating the decay curves is given by Equation 7.
0 I
I
6
psec
Figure 4. A typical plot of goodness-offit cs. assumed lifetime. Results are for the observed data from Table I
ranging from 4.00 to 6.30 pec. Clearly the best fit for this grid is T = 5.15 psec. A slightly better fit could be obtained by choosing a finer grid around 5.15 psec. DISCUSSION
Table I1 gives the lifetime of [Ru(bipy),]Cl2 measured by the fast and spoiled flashes. Each value is the average of three different data sets; the indicated error is the standard deviation. All three methods applied to the spoiled flash data yield results close to the fast flash value, with comparable error spread. These data show that reliable lifetimes can be obtained readily even when the flash duration is significantly longer than the lifetime to be measured. The “best fit” simulated curve, DCalC(t), is plotted in Figure 3 from the data of Table I. The excitation pulse and the observed response are also displayed. The high quality, judged both visually and analytically, of the fit is evidence that the observed decay can be described by a single emitting species decaying by first-order processes. The result indicates that emissive impurities, if present, are only making weak contributions to the signal or have lifetimes close to that of the main contributor. This ability to test for the presence of several species of differing lifetimes is one of the principal advantages of curve simulation, A disadvantage of the curve-fitting method is the heavy computational requirements which necessitate access to a digital computer. A few comments on the expected Gminvalue are in order. For data having the fine grid and close spacing of Table I, a G value greater than 2-3 for the best fit would probably signify the presence of several decaying exponentials. Our experience indicates that a coarser grid of values (equivalent to a point every 2 or 3 psec in Table I) might raise the minimum G to 5-6. Therefore any Gmingreater than 5-6 probably indicates that the chosen grid was too coarse or that the data cannot be fit accurately by a single exponential.
Table 11. Summary of Decay Time Results Lifetime by Lifetime by moments semilog plot (spoiled flash) Compound (fast flash) Eq.3 Eq.4 [Ru(bipy)slCl~ 4.92 f 0.05 4.80 f.0.05 4.79 0.33 Tris(2,2’-bipyridine)ruthenium(Il)chloride.
Lifetime by curve simulation (spoiled flash) 5.02 & 0.14
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
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photo tu be .anode
(a)
A-
T
phototube anode
(b)
During this discussion of all the various methods available for obtaining lifetimes of excited states, we have explicitly assumed that the phototube and its associated electronics could follow the light signals faithfully. We wish to point out, however, that Equation 5 is applicable even when the measuring system distorts the light signal appreciably. The final value obtained for 7 is unaffected by these distortions provided they are linear and occur equally in F(t) and D(t) (see Appendix A). Experimentally, one can assure equal effects of the measuring system on the signals, either by using matched detectors or by taking the flash and sample decays with the same detection system employing identical settings, Thus, the curve simulation and moment methods, which are all based on Equation 5, can be used to process information obtained from most practical circuits since the common distortions (phototube transit time spread and RC time constant) are linear. (Examples of nonlinear distortions are: signals from saturated amplifiers, currents from phototubes operated at excessive light levels, and outputs from operational amplifiers used outside of their proper ranges.) Caution must be exercised, however, when one employs the simple semilog methods on data from distorted signals. Considerable errors may be introduced in the results, or, sometimes, completely erroneous values can be obtained. (See next section.)
' 7 ; c = c, + c,+ CC + Co
=
1
Figure 5. (a) Sources of resistance and capacitance in a phototube circuit and (b) the RC equivalent circuit Rs = leakage resistance to ground; R L = load resistance; Ro = input resistance of oscilloscope; CA = capacitance of phototube anode; CW = capacitance of phototube wiring; CC = capacitance of cable connecting phototube to oscilloscope; CF = capacitance added externally for increased filtering of the signal; CO = input capacitance of oscilloscope; R = effective resistance (1/R = 1/Rs ~ / R L 1/Ro); C = effective capacitance (C = CA Cw
+
cc
+ CO).
+
+
+
The equations for both moment methods are derived from Equation 5 and are applicable where the latter equation applies (see Appendix A). They are also adaptable to evaluation by a desk calculator (2). The major disadvantage of the moment methods is that no warning of multiple exponential decays is given. A comparison of the lifetimes calculated by the two moment methods should, however, give some hint of multiple decaying species, if time-base jitter is known to be absent (see next paragraph). The curve-fitting technique and Moment Method I are both subject to error from jitter in the triggering circuit. If jitter occurs, the flash and the luminescence decays do not have the same time-base reference zero and neither Equation 3 nor 5 is correct. In contrast Moment Method I1 makes use of the radius of gyrations of the curves, and it can be shown (by a change of variable in Equation 4) that this quantity is independent of the choice of the zeros for the time bases (2). Hence, the presence of time-base jitter does not vitiate the usefulness of Moment Method 11. Whenever jitter cannot be avoided, Equation 4 yields the only easy way to extract lifetimes from the experimental data. Indeed, a comparison of the lifetimes calculated by Equations 3 and 4 can be used to detect time-base jitter. The limiting values for lifetimes obtainable by these techniques can be estimated from the magnitudes and reproducibilities of the parameters used in Equations 3 and 4. The center of gravity, CF,and the square of the radius of gyration, uFZ,for the spoiled flash used in this study are 16.3 =t0.3 psec and 72 f 2 psecz, respectively, Thus, with this flash, lifetimes as short as 1 psec (using Equations 3 and 5) and 2-3 psec (using Equation 4) can be estimated readily. These values are roughly an order of magnitude shorter than those obtained reliably from semilog plots. Use of a better (more reproducible but not shorter) flash would probably allow the measurement of even shorter 7 values. Signal averagers would extend the limits still further. Of course the errors inherent in very short lifetimes determined by a long flash could be very large, but still one could extract rough estimates of the decay times. 1014
RC TIME CONSTANT EFFECTS
It is common practice to increase the signal during a phototube measurement by increasing the size of the load resistor; an increase in signal-to-noise ratio is also realized by such a procedure due to the increased RC time constant of the system. An alternate way to improve the signal-to-noise ratio is to add capacitance in parallel with the load resistor. While the adoption of such procedures can produce significant gains in sensitivity and signal-to-noise ratio, it must be realized that the increased RC time constant of the system causes distortion of the signal. The important question thus arises: How large a load resistor and filter capacitor can be used on the photomultiplier output without generating significant errors in the measured values of sample decay times? A second practical question is: What criteria can be applied to the experimental parameters in order to decide the degree of sophistication in data handling needed in order to obtain reliable information from the experimental measurements? In this section we discuss the distortions introduced in signals by the RC time constant of practical detector circuits and give methods of handling the data and estimating the errors incurred by this factor under common experimental conditions. A typical phototube decay measuring circuit is given in Figure 5a showing the various resistances and sources of capacitance. For most phenomena slower than several nanoseconds this circuit can be replaced by the equivalent circuit of Figure 5b. In our development we assume the photomultiplier to be an ideal current source whose output current, Q(r), is directly proportional to the incident light intensity, The response H(r) of this electronic system is governed by a differential equation analogous to the one governing Equation 5. The required solution is given in Equation 8.
Q(t)
=
0 for t
50
where H(t) is the output voltage registered for any arbitrarily
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
shaped excitation current pulse Q(t) applied to the parallel combination of a resistance, R, and a capacitance, C. The different pulse shapes encountered for Q(t) are conveniently classified. They are:
A. The optical excitation pulse is rectangular with a duration much longer than the lifetime to be measured. B. The excitation pulse is a flash which has a duration much shorter than the lifetime to be measured. C. The flash pulse duration and the lifetime to be determined are comparable. Case A corresponds to the interruption of a continuous exciting beam by a mechanical shutter, a n arrangement commonly used for the measurement of phosphorescence lifetimes. Case B represents the usual flash technique for lifetime measurements, while Case C corresponds to the flash technique being pushed to its usable limits. Case A. A rectangular optical pulse of duration T is assumed to be zero before time 0 and after time T and to be constant between these limits. If T i s much greater than the sample decay time (in practice a pulse duration of 5 times the lifetime is usually sufficient), then the solution of Equation 8 yielding the shape of the oscilloscope signal after time T is given by Equation 9. H(t’) = L I R [,-t’/r - Z,-t’/zr 1 1-z (9) t’ 2 0, Z < 1, Z = RC/T, t’ = t - T, where Im is the maximum phototube current achieved during the illumination, and R is the effective resistance of the measuring circuit. This monotonically decreasing signal falls of€ more slowly than a true exponential. The semilog plot of H(r) is concave downward for short times and becomes progressively more linear with increasing time. (Note that for small values of 2, the signal level is directly proportional to the effective load resistance, R. Thus a considerable improvement in signal levels is frequently possible by increasing the effective load resistance.) In order to estimate the errors introduced by the analysis of this distorted signal using a semilog method, we have generated the signal expected from a compound having a decay time T when measured by a system having various RC values. Then, following the usual procedures for analyzing such data, we have constructed a semilog plot of the data and fit sections of the plot with a least-squares straight line. Two systematic, different methods of analyzing the generated decay curve were employed. Twenty data points were taken from
5 t’ 5 1.9 r in steps of 0.1 2 Z 5 r’ 5 2 Z + 1.9 r in steps of 0.1 r 0
(a)
(b) The results of these two calculations are given in Table I11 for different values of Z . For example, if a compound has a lifetime of 100 msec and the time constant of the measuring circuit is 20 msec ( Z 0.2), the error incurred in the measured lifetime by linearizing the first part of the curve would be -8% and by treating the second part only -2%, which shows the importance of waiting long enough (if noise does not interfere) to obtain accurate lifetimes from a decay measured under these conditions. For the larger errors indicated in Table I11 (errors >30 %) nonlinearities on the semilog plots (concave downward) are readily detected visually which emphasizes the desirability of making such plots of the data. Beyond 2 = 1 handling the resultant data by this method would not yield meaningful results since the signal eventually decays with the RC time constant of the circuit,
-
Table 111. RC Time Constant Errors in Lifetime Measurements Error* Method a Method b 0.05 0.9 0.1 0.10 2.4 0.4 0.15 4.7 0.8 0.20 7.9 1.4 0.25 12.0 2.3 0.30 16.7 3.3 0.35 22.2 4.6 0.40 28.1 6.0 0.45 34.6 8.7 0.50 41.3 9.5 0.55 48.4 11.5 0.60 55.7 13.6 0.65 63.2 15.8 0.70 70.8 18.1 0.75 78.6 20.6 0.80 86.5 23.2 0.85 94.5 25.9 0.90 >100.0 28.7 0.95 >100.0 31.5 a Z = (RCtime constant of circuit)/(lifetimeof sample). * See text for details.
Inspection of Table I11 leads to the useful rule: If a decay is monitored after interruption of the exciting beam by a mechanical shutter, the data can be analyzed by the simple semilog method without introducing appreciable error in the value of T , provided RC/T 5 0.1. A curve simulation method could also be used to obtain reliable T values under the experimental arrangement posited here even when 2 > 1. Synthetic decay curves could be generated from Equation 9 for a series of assumed decay constants until a best fit of the experimental data was obtained. Prior knowledge of the RC time constant of the system would be required. The best fit could be judged either analytically or visually. The method of nonlinear least squares ( 9 ) using Equation 9 could also be employed to estimate the lifetimes under these experimental conditions. An advantage to the nonlinear least-squares method is its capability to fit a curve with multiple exponentials in case several simultaneous decays are suspected. If the RC constant of the measuring system were chosen as a variable, the least-squares method could also determine its value along with the values for the parameters in the assumed combination of decaying exponentials. Both the synthetic curve method and the least-squares method just described fit the entire decay curve; thus they should turn up evidence for multiple exponential decays. They are applicable for all ranges of Z . The simple semilog method is only applicable to the situation for Z < 1 and could mask multiple decays, especially the presence of a decay much shorter than that of the predominant species. Case B. The flash pulse is assumed to be much shorter than the lifetime to be measured which results in a current pulse given by Q(r) = Zme-t/r. Substitution of this expression into Equation 8 yields the solution
(9) I. S. Sokolnikoff and R. M. Redheffer, “Mathematicsof Physics and Modern Engineering,” McGraw-Hill Book Co., New York, 1958; W. E. Deming, “Statistical Adjustment of Data,” Dover Publications, Inc., New York, 1943.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
1015
For this experimental situation, the decay rises to a maximum at a time, t,, given by tm
= r-
zlnz 2 - 1
The maximum is caused by the capacitance in the circuit which introduces a lag of the signal. This time lag should not be confused with an inter- or intramolecular energy transfer process which can give a curve of identical shape (10). The semilog plot past the maximum falls off more slowly than an exponential and only approaches one at long times. To estimate the errors resulting from evaluation of T by making semilog plots under these conditions, we have proceeded as in Case A and performed a linear least-squares fit on the semilog plots of two sets of 20 data points generated from Equation 10. The data sets for H(t) were chosen for tm tm
+22
5 t 5 t , + 1.9 7 in steps of 0.1 7 5 t 5 tm 2 Z + 1.9 r in steps of 0.1 r
+
(4 (b)
The relative errors are also given in Table 111. Again the semilog plots are concave downward indicating that further truncation of the data set could reduce the errors below those given here as long as the scatter was not too severe. Inspection of Table I11 leads to the following useful rule: If a photoluminescent sample is excited by a flash of duration much shorter than T , the data can be analyzed by the simple semilog method without introducing appreciable error in the value of T provided RC/r 5 0.1. A nonlinear least-squares (9)technique could also be employed to find the best value of the lifetime. The RC time constant in Equation 10 could be known or could be evaluated as an undetermined parameter in the least-squares process. As in Case A, synthetic decay curves could be generated and compared with the measured decay curves to obtain the best fit. Both the curve generation method and the nonlinear least-squares technique could reveal multiple decays. A linear least-squares analysis on the semilog plots might not supply such information, if 2 were too large. Still another option for extracting the decay time from data obtained under these assumed experimental conditions is available. One can employ the method of moments. The lifetimes could be obtained from Equations 3 and 4 where F(t) and D(t) are now the observed (distorted) signals from the flash and sample, respectively. Since the center of gravity, CF,and the radius of gyration, UP, of the flash decay signal are dictated entirely by the RC time factors of the detecting circuit and are, in fact, both equal to RC, then the lifetime could be obtained from Equations 12 and 13.
discussed in the section, “Evaluation of Decay Times.” Thus all three methods of that section may be employed for evaluating the lifetime. The question still arises, however: At what point does a semilog plot fail to give accurate lifetimes? We believe that the following conservative rule applies: If the signal from the flash decays exponentially with a lifetime less than one half the observed decay time of the sample, then R C / T = 0.1 introduces no more than a positive 10% systematic error. The data must have a good signal-to-noise ratio, and the slope must be evaluated after noticeable curvature is gone, however. For particular applications, the above error rule may not be specific enough, and a more detailed analysis could be performed. In such a case it would be necessary to generate a series of synthetic decay curves with different T ’ S using Equation 5 . The F(r) employed would either be the observed flash for a specific value of RC or a calculated curve derived from Equation 8, the true time dependence of the flash, and a selected value of RC. A data analysis of the type used in Cases A and B of this section would then be employed to estimate the errors incurred by various data handling techniques. In view of the simplicity of evaluating T ’ S directly by means of Equations 3, 4, and 5 , such a complex procedure would seldom be justified. ACKNOWLEDGMENT
We gratefully acknowledge the assistance of Robert W. Bushey who wrote the computer programs used in this work and also collected and reduced the lifetime data presented here. APPENDIX A
Equation 5 may be proved readily by solving a first-order differential equation if F(t) and D(t> represent the true time dependences of the flash and the luminescence decay, respectively; however, Equation 5 is also valid if F(t) and D(t) are both quantities observed on the same measuring system even when linear distortions of the signal by the apparatus are present. The generality of Equation 5 can be proved Diu the properties of the convolution integral. The convolution of two functions A and B is denoted by A * B and is defined by A*B=
L:ot
A(u)B(t - u)du
(A-1)
where the two functions are defined to be zero for t 5 0. The convolution satisfies the properties of commutivity and associativity (11). We employ the following definitions : = true time dependence of the flash g(t) = true time dependence of the luminescence decay when
f(t)
r =
[
- (G) 1H 2
- (RC)2]”*
excited by a flash of infinitely short duration true time dependence of the sample luminescence when excited by the actual Bash h(t) = observed response of the phototube readout combination to a flash of infinitely short duration F(t) = observed flash D(t) = observed luminescence decay d(t)
Because of the behavior of the RC term, Equation 13 would be preferred for 2 < 1 and Equation 12 for 2 > 1. This method does not reveal evidence for multiple exponential decays. Case C. The flash duration is comparable to the sample decay time. This condition is clearly the same as that (10) G. FriedIander, J. W. Kennedy, and J. M. Miller, “Nuclear and Radiochemistry,” John Wiley & Sons, New York, 1964; J. J. Freeman and G. A. Crosby, J. Phys. Chem., 67, 2717 (1963); M. Kleinerman and S. Choi, J. Chem. Phys., 49, 3901 (1968). 1016
=
By use of the impulse response property ( I 1 , 1 2 ) , the following (1 1 ) W. Kaplan, “Operational Methods for Linear Systems,” Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. (12) W. Hauser, “Introduction to the Principles of Mechanics,” Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 9, AUGUST 1970
r
results are obtained: =f
F(t) D(t)
=
d *h
=
*h
(A4
[(f * g) * hl
(A-3)
Since the convolution satisfies the commutative and associative laws, Equation A-3 may be rearranged to give D(t) = [ ( g * f ) * h l = F * g
=
[g*(f*h)l = g * F
(A-4)
This result shows that the observed luminescence decay, D(t), can be calculated from the observed flash, F(t), regardless of the type or extent of linear distortion by the measuring system (e.g., phototube transit time spread and RC time constant) as long as the functional form of g(t) is known. For a compound which decays by only first-order or pseudo first-order processes, g(t) is given by g(t) = Ke-t/'
(A-5)
Substitution of g(t) into Equation A-4 yields Equation 5.
=
+
2 D = K [ T ( ~ F ) 2r2('F)
lIot
(B-1)
=
uv
- Judu
(B-2)
For the computation of the moments the data points are assumed to be taken at even time intervals, and the data points are of the form [tc, W(tt)[i = 1 to N] with W(tl) = 0 at tl = 01. Assume that the decay curve is given by straight line interpolations between consecutive data points in the range 0 5 t I tN. Then the curve in the interval tn-' 5 t 5 t, is defined by
dt, v = -rZKe-t/'
.=Lo z=t
F(x)e2jr dx, du = [F(t)et/'
give
(: +
+ Bn
Bn = W(t,) - Mntn At = tz - ti
I t I tn
where Mn is the slope and B, is the t intercept for the linear fit in this interval. To define the entire curve from 0 I t I tN, the subscript n must run from 2 to N . For any times beyond f N the curve is assumed to be a decaying exponential given by
and the substitutions dv = Kte-']'
(B-6)
APPENDIX C
In-1
F(x)e"I' dx] dt
+ 2r3(OF)]
(B-5)
Combining Equations B-4, B-5, and B-6 in the appropriate ways gives Equations 3 and 4.
W(t) = M,t
Using the formula for integration by parts Judo
+ 7Y°F)1. OD = rK(OF)
Equations 3 and 4 may be proved by first expanding the moments of D(t) in terms of the moments of F(t). This may be done by substituting D(t) in terms of F(t) (Equation 5 ) into Equation 2 and integrating. For a specific example, consider the first moment of DO). t=
K[7('F)
By a similar development
APPENDIX B
Jt= ',It [Ke-'/' -
1
P-
W(t) = Ae-'lE;
t
2
tw
(C-2)
Using the defined functional form for W(r) and carrying out the necessary integrations leads to:
1)
- F(O)] = F(t)et/'
c
ow=-At2 j = 2 [W(t,) + W(t+l)] + AEe'-tN/E
(C-3)
(12+ I -
'D = -r2
11 X
[Ke-'/'
F(x)dx]}I'=t = O
$L
where the Mj's, B t s , and At are defined in (C-1). t 5 m , since The range of integration can be limited to 0 F(t) = 0 for t I 0. Note that the second term within the braces is just equal to D(t). As long as D(t) goes to zero faster at large times than t increases, the value of the expression within the braces evaluated at the limits will be zero. Since D(t) will normally fall off exponentially, this condition is satisfied. The expression for lD then becomes
RECEIVED for review March 23,1970. Accepted June 8,1970. J.N.D. was a National Science Foundation Predoctoral Fellow, 1966-68, and an American Chemical Society Petroleum Research Fund Fellow, 1968-69. Research was sponsored by AFOSR(SRC)-OAR, USAF Grant 68-1342.
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