Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 247
Figure 7.16. Time-resolved emission spectra of Patman-labeled
DPPC vesicles. Top, 8°C; bottom, 46°C. Time-resolved spectra are
shown at 0.2 (!), 2 ($"$) and 20 ns ($D$). Revised from [26].
Copyright © 1984, with permission from Elsevier Science.
ν cg(t)
∞
I’(ν,t) ν dν
0
∞
0
I’(ν,t) dν
(7.5)
where I'(ν,t) represents the number of photons per
wavenumber interval. These are the intensity decays as normalized
in eq. 7.4, but on the wavenumber scale. The data
are typically collected for selected wavelengths, and the
center of gravity in kK (= 10 3 cm -1 ) is calculated using
νcg(t) 10,000 ∑ λ I’(λ,t) λ1
∑ λ I’(λ,t)
(7.6)
Note that the integral in eq. 7.5 is over the emission spectrum
(ν), and not over time. The TRES at any instant in time
are used to calculate ν cg (t) at the chosen time. The calculated
center of gravity is typically an approximation since the
time-resolved data are not collected at all wavelengths.
Also, a vigorous calculation of the center of gravity requires
use of the corrected emission spectra on the wavenumber
scale. Equation 7.6 is simply an expression which uses the
available data (I'(λ,t)) to obtain an approximate value of
ν cg (t).
The time-dependent emission centers of gravity are
shown in Figure 7.17 (top panel). It is apparent that the
extent of relaxation is greater at 46EC than at 8EC. The rate
of relaxation is somewhat faster at the higher temperature.
If desired, the values of ν cg (t) versus time can be fit to
multi-exponential (eq. 7.15, below) or other non-exponential
decay laws for ν cg (t).
The time-dependent spectral half width ∆ν(t) (cm –1 )
can be used to reveal whether the spectral relaxation is best
described by a continuous or two-step model. This half
width can be defined in various ways. One method is to use
a function comparable to a standard deviation. In this case
∆ν(t) can be defined as
∆ν(t) 2
(7.7)
For calculation of ∆ν(t) one uses the TRES calculated for a
chosen time t and integrates eq. 7.7 across the emission
spectrum. For data collected at various wavelengths in
nanometers the value of ∆ν(t) in kK is given by
∆ν(t) 2
∑ λ
∞
0 (ν ν cg(t)) 2 I’ (ν,t) dν
10,000/λ ν cg(t) 2 I’ (λ,t)
∑ λ
∞
0 I’(ν,t)dν
I’(λ,t)
(7.8)
Figure 7.17. Time-resolved emission center of gravity (top) and spectral
half width (bottom) of Patman-labeled DPPC vesicles. Revised
from [26]. Copyright © 1984, with permission from Elsevier Science.