Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 389
tions measured for each polarized component. In contrast to
the previous method, the instrument response functions are
not assumed to be identical for each polarization, and can
be rather different without affecting the validity of the procedure.
The calculated and measured polarized intensities are
then used to minimize χ R 2 based on the parameter values in
the intensity (α i and τ i ) and anisotropy decays (r 0i and θ i ):
χ 2 R 1
ν ∑n
t0
1
ν ∑n
t0
1
N ||(tk) Nc ||(tk) 2
σ 2 ||k
1
σ 2 k
N (t k) N c (t k) 2
(11.22)
Since the polarized intensity decays are used directly, the
weighting factors are given by
σ 2 ||k N ||(t k)
σ 2 k N (t k)
(11.23)
(11.24)
There is no need to propagate the weighting factors into the
sum and difference data. All the parameters are varied
simultaneously to obtain the best fit, so that the intensity
and anisotropy decay parameters are all optimized to match
the actual data.
Sometimes it is necessary to correct the polarized
intensity decays for background signals. The counts measured
for the blank sample with each polarizer position are
subtracted from the measured data for the same polarizer
position:
I ||(t k) I ||(t k) sample I ||(t k) background
I (t k) I (t k) sample I (t k) background
(11.25)
(11.26)
The weighting factor for the corrected data is given by the
sum of the weighting factors for the sample and for the
background. If the number of background counts is small,
this correction to the weighting factor can be ignored. It is
important to measure the background for both polarized
components, because the background can be different for
each component, particularly if scattered light reaches the
detector.
It is necessary to know the G factor in order to calculate
the intensity decay. The G factor can be obtained in the
usual manner (Chapter 10), in which the intensities are
measured with horizontally polarized excitation. The parallel
and perpendicular intensities are then measured for the
same period of time, assuming the excitation intensity is
constant. Another method is to measure the steady-state
anisotropy of the sample, which is then used to constrain
the total intensities of the polarized decays. The steady-state
anisotropy can be measured on a different instrument. The
G factor can be calculated using 10,12
G
(11.27)
where the sums are the total number of counts in the polarized
intensity decays. If the excitation intensity has
changed, or the counting time is different, these values need
to be corrected for the different experimental conditions.
11.2.3. Value of r 0
In the anisotropy decay analysis the value of r 0 can be considered
to be a known or unknown value. If the value of r 0
is known, the anisotropy decay law can be written as
r(t) r 0 ∑ j
(11.28)
where g j are the fractional amplitudes that decay with the
correlation times θ j . Since Σg j = 1.0, the use of a known r 0
value reduces the number of variable parameters by one. In
this type of analysis the time-zero anisotropy is forced to be
equal to r 0.
Alternatively, the total or time-zero anisotropy can be a
variable parameter. In this case,
r(t) ∑ j
1 r ∑ N ||(tk) 1 2r ∑ N(tk) g j exp (t/θ j)
r 0j exp (t/θ j)
(11.29)
where r 0j are the fractional anisotropies that decay with correlation
times θ j . When using this type of analysis it is
preferable to describe the time-zero anisotropy as r(0),
which is the recovered value of t = 0. If the anisotropy
decay contains fast components that are not resolved by the
instrument then the Σr 0j = r(0) is less than the fundamental
anisotropy r 0 .