Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 133
the numerator does not matter. The autocorrelation plots
have half as many datapoints as the original data set.
One example of an autocorrelation plot was seen in
Figure 4.27, where data for NATA were presented for measurements
versus scattered light and versus a lifetime reference.
This instrument showed a minor color effect, which
resulted in some systematic deviations between N c (t k ) and
N(t k ). The systematic differences are barely visible in the
direct plot of the deviations (middle left). The autocorrelation
plot (upper left) allowed the deviations to be visualized
as positive and negative correlations in adjacent or distant
channels, respectively. For closely spaced channels the
deviations are likely to both be the same sign. For more distant
channels the deviations are likely to be of opposite
signs. These systematic errors were eliminated by the use of
a lifetime reference, as seen by the flat autocorrelation plot
(upper right panel).
4.10. ANALYSIS OF MULTI-EXPONENTIAL
DECAYS
4.10.1. p-Terphenyl and Indole:Two Widely
Spaced Lifetimes
An understanding of time-domain data analysis is facilitated
by examination of representative data. As an example we
chose a mixture of p-terphenyl and indole, which individually
display single exponential decays of 0.93 and 3.58 ns,
respectively. For the time-domain measurements a mixture
of p-terphenyl and indole was observed at 330 nm, where
both species emit (Figure 4.46). TCSPC data for this mix-
Figure 4.46. Emission spectra of p-terphenyl (p-T), indole (In) and of
the mixture. Excitation was at 292 nm, from a frequency-doubled
R6G dye laser. The emission at 330 nm was isolated with a monochromator.
From [187].
Figure 4.47. Time-domain intensity decay of a two-component mixture
of indole and p-terphenyl. The dashed line shows the one-decaytime
fit, and the solid line the two-decay-time fit. From [187].
ture are shown in Figure 4.47. The presence of two decay
times is evident from curvature in the plot of log N(t) versus
time. The time-dependent data could not be fit to a single
decay time, as seen by the mismatch of the calculated
convolution integral (dashed) with the data (dots).
The lower panels show the deviations (D k ) or differences
between the measured and calculated data:
D k I(t k) I c(t k)
√I(t k)
(4.24)
The weighted residual (W.Res.) or deviations plots are used
because it is easier to see the differences between I(t k ) and
I c (t k ) in these plots than in a plot of log I(t k ) versus t k . Also,
the residuals are weighted according to the standard deviation
of each datapoint. For a good fit these values are
expected to randomly distribute around zero, with a mean
value near unity.
4.10.2. Comparison of χ R 2 Values: F Statistic
How can one compare the values of χ R 2 for two fits? This
can be done using the F statistic, which is a ratio of χ R 2 val-