Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 99
time-domain and frequency-domain measurements are in
widespread use.
4.1.1. Meaning of the Lifetime or Decay Time
Prior to further discussion of lifetime measurements, it is
important to have an understanding of the meaning of the
lifetime τ. Suppose a sample containing the fluorophore is
excited with an infinitely sharp (δ-function) pulse of light.
This results in an initial population (n 0 ) of fluorophores in
the excited state. The excited-state population decays with
a rate Γ + k nr according to
(4.1)
where n(t) is the number of excited molecules at time t following
excitation, Γ is the emissive rate, and k nr is the nonradiative
decay rate. Emission is a random event, and each
excited fluorophore has the same probability of emitting in
a given period of time. This results in an exponential decay
of the excited state population, n(t) = n 0 exp(–t/τ).
In a fluorescence experiment we do not observe the
number of excited molecules, but rather fluorescence intensity,
which is proportional to n(t). Hence, eq. 4.1 can also be
written in terms of the time-dependent intensity I(t). Integration
of eq. 4.1 with the intensity substituted for the number
of molecules yields the usual expression for a single
exponential decay:
(4.2)
where I 0 is the intensity at time 0. The lifetime τ is the
inverse of the total decay rate, τ = (Γ + k nr ) –1 . In general, the
inverse of the lifetime is the sum of the rates which depopulate
the excited state. The fluorescence lifetime can be
determined from the slope of a plot of log I(t) versus t (Figure
4.1), but more commonly by fitting the data to assumed
decay models.
The lifetime is the average amount of time a fluorophore
remains in the excited state following excitation.
This can be seen by calculating the average time in the
excited state . This value is obtained by averaging t over
the intensity decay of the fluorophore:
dn(t)
dt (Γ k nr) n(t)
I(t) I 0 exp (t/τ)
∞
0 tI(t)dt
∞
0
∞
0
I(t) dt
∞
0
t exp (t/τ) dt
exp (t/τ)dt
(4.3)
The denominator is equal to τ. Following integration by
parts, one finds the numerator is equal to τ 2 . Hence for a
single exponential decay the average time a fluorophore
remains in the excited state is equal to the lifetime:
(4.4)
It is important to note that eq. 4.4 is not true for more
complex decay laws, such as multi- or non-exponential
decays. Using an assumed decay law, an average lifetime
can always be calculated using eq. 4.3. However, this average
lifetime can be a complex function of the parameters
describing the actual intensity decay (Section 17.2.1). For
this reason, caution is necessary in interpreting the average
lifetime.
Another important concept is that the lifetime is a statistical
average, and fluorophores emit randomly throughout
the decay. The fluorophores do not all emit at a time
delay equal to the lifetime. For a large number of fluorophores
some will emit quickly following the excitation,
and some will emit at times longer than the lifetime. This
time distribution of emitted photons is the intensity decay.
4.1.2. Phase and Modulation Lifetimes
The frequency-domain method will be described in more
detail in Chapter 5, but it is valuable to understand the basic
equations relating lifetimes to phase and modulation. The
modulation of the excitation is given by b/a, where a is the
average intensity and b is the peak-to-peak height of the
incident light (Figure 4.2). The modulation of the emission
is defined similarly, B/A, except using the intensities of the
emission (Figure 4.2). The modulation of the emission is
measured relative to the excitation, m = (B/A)/(b/a). While
m is actually a demodulation factor, it is usually called the
modulation. The other experimental observable is the phase
delay, called the phase angle (φ), which is usually measured
from the zero-crossing times of the modulated components.
The phase angle (φ) and the modulation (m) can be
employed to calculate the lifetime using
m
tan φ ωτ φ, τ φ ω 1 tan φ
1
√1 ω 2 τ 2 m
τ
, τm 1
ω
1 1/2
[ 1 ] 2 m
(4.5)
(4.6)