Principles of Fluorescence Spectroscopy

180 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
Figure 5.29. χ R 2 surfaces for the two-component mixture of p-terphenyl
and indole. The horizontal arrows show the asymptotic standard
errors (ASE), and the dashed lines are at the appropriate F χ values.
From [116].
dom (ν = m – p). It is useful to consider some representative
values for the ratio of χ R 2 values. For the two-component
mixture of p-terphenyl and indole there are 29 frequencies
and 58 datapoints. The two-decay-time model has three
variable parameters. The F statistic for p = 3 and m = 60 can
be found from Table 4.4 and is 1.19. Hence, the F P value
used for the confidence interval of each parameter is 1.06.
As described in Section 4.10.3, there is some disagreement
in the statistics literature as to the exact equation for finding
the confidence intervals. We will use eq. 5.24, but its correctness
has not yet been proven.
The χ R 2 surfaces for the two-component mixture are
shown in Figure 5.29. The confidence intervals (CI) are
determined from the intercept of the χ R 2 surfaces with the
χ R 2 ratio appropriate for the number of parameters and
degrees of freedom. The arrows in Figure 5.29 show the
asymptotic standard errors. The ASEs are about twofold
smaller than the confidence intervals. The decay times in
this mixture are widely spaced. For more closely spaced
decay times it becomes even more important to consider
parameter correlation in the calculation of confidence intervals,
and the ASEs can grossly underestimate the true confidence
intervals.
5.7.2. Resolution of Two Closely Spaced Lifetimes
The resolution of multi-exponential decays becomes more
difficult as the decay times become more closely spaced. It
was previously noted that two decay times spaced by a factor
of 1.4 represents the practical resolution limit for double
exponential decay. 12 It is instructive to examine data for
such a mixture because the analysis illustrates the difficulties
encountered at the limits of resolution.
To illustrate a sample with two closely spaced decay
times we have chosen the mixture of anthranilic acid (AA,
τ = 8.5 ns) and 2-aminopurine (2-AP, τ = 11.3 ns). This may
seem to be an easy resolution, but it is difficult to resolve
decay times which are less than two-fold different. Emission
spectra are shown in Figure 5.30. Frequency-domain
data for the individual fluorophores and for the mixture are
Figure 5.30. Emission spectra of a two-component mixture of
anthranilic acid (AA) and 2-aminopurine (2-AP). Also shown are the
amplitudes recovered from the global analysis (Figure 5.33). From
[116].