Principles of Fluorescence Spectroscopy

140 TIME-DOMAIN LIFETIME MEASUREMENTS
the deviations are randomly distributed around zero and the
values of χ R 2 are near unity.
The intensity decay for indole in Figure 4.54 illustrates
the need to consider the convolution integral when using an
MCP PMT, even with a 4.41-ns decay time. At long times
the plot of log I(t k ) versus time becomes nonlinear even
though there is only a single lifetime. This effect is most
visible for indole, with the shortest lifetime of 4.41 ns. This
long tail on the intensity decay is due to continued excitation
from the tail of the impulse response function and possibly
some background emission. If one did not consider
convolution, and calculated the decay times only from the
slopes, then one would reach the erroneous conclusion that
the indole sample displayed a second long decay time.
Now consider similar data for a mixture of the three
fluorophores. The decay times range threefold from 4 to 12
ns, but this is a difficult resolution. Examination of Figure
4.55 shows that the single exponential fit (dashes) appears
to provide a reasonable fit to the data. However, the failure
of this model is easily seen in the deviations, which are
much larger than unity and not randomly distributed on the
time axis (lowest panel). The failure of the single exponential
model can also be seen from the value of χ R 2 = 26.45,
which according to Table 4.2 allows the single exponential
model to be rejected with high certainty. To be more specific,
there is a less than a 0.1% chance (P < 0.001) that this
value of χ R 2 could be the result of random error in the data.
The situation is less clear with a double exponential fit.
In this case the fitted curve overlaps the data (not shown),
χ R 2 = 1.22, and the deviations are nearly random. According
to the χ R 2 data (Table 4.2), there is only a 2% chance
that χ R 2 = 1.22 could result from statistical uncertainties in
the data. In practice, such values of χ R 2 are often encountered
owing to systematic errors in the data. For comparison,
the systematic errors in Figure 4.27 resulted in an elevation
of χ R 2 to a similar value. In an actual experiment we
do not know beforehand if the decay is a double, triple, or
non-exponential decay. One should always accept the simplest
model that accounts for the data, so we would be
tempted to accept the double exponential model because of
the weak evidence for the third decay time.
An improved fit was obtained using the triple exponential
model, χ R 2 = 0.93, and the deviations are slightly more
random than the two-decay-time fit. It is important to
understand that such a result indicates the data are consistent
with three decay times, but does not prove the decay is
a triple exponential. By least-squares analysis one cannot
exclude other more complex models, and can only state that
Figure 4.55. TCSPC data for a mixture of indole (In), anthranilic acid
(AA), and 2-aminopurine (AP). From [187].
a particular model is adequate to explain the data. In this
case, the data are consistent with the three-exponential
model, but the analysis does not exclude the presence of a
fourth decay time.
The two- and three-decay-time fits can also be compared
using the ratio of χ R 2 values. For this mixture the ratio
of χ R 2 values was 1.31. This value can be compared with the
probability of this ratio occurring due to random deviations
in the data, which is between 5 and 10% (Table 4.3). Hence,
there is a relatively low probability of finding this reduction
in χ R 2 (1.22 to 0.93) unless the data actually contained three
decay times, or was described by some model other than a
two-decay-time model. Stated alternatively, there is a 90 to
95% probability that the two-decay-time model is not an
adequate description of the sample.