Frequency-domain intensity decay

Fig. 19. Frequency-domain intensity decay of oxytocin at 20 C. The symbols ( ) represent the data, the solid line the best three-exponential fit, and the dashed line the best one-exponential fit. The lower panels show the deviations between the data and the calculated values for the one (a) and three ( ) decay time models. The values of Xr 377, 5.9 and 2.1 for the 1, 2 and 3 exponential fits, respectively.

A. Least-Squares Analysis of Frequency-Domain Intensity Decays... 

BIOCHEMICAL EXAMPLB OF FREQUENCY-DOMAIN INTENSITY DECAYS... 

Figure 5.30. Frequency-domain intensity decays of SPQ in the presence of 0, lO, 40, and 70mM chloride. From Ref. 82.

Figure 18 shows frequency-domain intensity decays for the free-space emission (top) and the surface plasmon-coupled emission (bottom). Overall, the lifetimes of SPCE (bottom) and free-space do not differ significantly. This was an unexpected result which we do not fully understand. We carefully considered possible artifacts and the effects of sample geometry, but can only conclude that our experiments indicate that the component of SPCE that we observe occurs without a substantial change in lifetime. At present we do not understand the origin of this discrepancy. 

Figure 18. Frequency-domain intensity decays of SlOl in PVA. Top, free-space emission. Bottom, surface plasmon-coupled emission. Adopted from [30].

At present, two main streams of techniques exist for the measurement of fluorescence lifetimes, time domain based methods, and frequency domain methods. In the frequency domain, the fluorescence lifetime is derived from the phase shift and demodulation of the fluorescent light with respect to the phase and the modulation depth of a modulated excitation source. Measurements in the time domain are generally performed by recording the fluorescence intensity decay after exciting the specimen with a short excitation pulse. 

The Fourier transform (FT) relates the function of time to one of frequency—that is, the time and frequency domains. The output of the NMR spectrometer is a sinusoidal wave that decays with time, varies as a function of time and is therefore in the time domain. Its initial intensity is proportional to Mz and therefore to the number of nuclei giving the signal. Its frequency is a measure of the chemical shift and its rate of decay is related to T2. Fourier transformation of the FID gives a function whose intensity varies as a function of frequency and is therefore in the frequency domain. 

Fig. 10.1. The top panel shows the free induction decay (FID) acquired for a sample of strychnine (1) at an observation frequency of 500 MHz. The spectrum was digitized with 16 K points and an acquisition time of 2 s. Fourier transforming the data from the time domain to the frequency domain yields the spectrum of strychnine presented as intensity versus frequency shown in the bottom panel.

There are two widely used methods for measuring fluorescence lifetimes, the time-domain and frequency-domain or phase-modulation methods. The basic principles of time-domain fluorometry are described in Chapter 1, Vol.l of this series(34) and those of frequency-domain in Chapter 5, Vol. 1 of this series.<35) Good accounts of time-resolved measurements using these methods are also given elsewhere/36,37) It is common to represent intensity decays of varying complexity in terms of the multiexponential model... 

Obviously, it is difficult to find a schematic representation for a compound absorbing 10 different frequencies. In such a case, M0 can be dissociated into many vectors, each of which precesses around the field with its own frequency (Fig. 9.7 shows a simplified situation). As the system returns to equilibrium, which can take several seconds, the instrument records a complex signal due to the combination of the different frequencies present, and the intensity of the signal decays exponentially with time (Fig. 9.9). This damped interferogram, called free induction decay (FID), contains at each instant information on the frequencies of the nuclei that have attained resonance. Using Fourier transform, this signal can be transformed from the time domain into the frequency domain to give the classical spectrum. 

Fourier transform NMR spectroscopy, on the other hand, permits rapid scanning of the sample so that the NMR spectrum can be obtained within a few seconds. FT-NMR experiments are performed by subjecting the sample to a very intense, broad-band, Hl pulse that causes all of the examined nuclei to undergo transitions. As the excited nuclei relax to their equilibrium state, their relaxation-decay pattern is recorded. A Fourier transform is performed upon this relaxation-decay pattern to provide the NMR spectra. The relaxation-decay pattern, which is in the time domain, is transformed into the typical NMR spectrum, the frequency domain. The time required to apply the Hl pulse, allow the nuclei to return to equilibrium, and have the computer perform the Fourier transforms on the relaxation-decay pattern often is only a few seconds. Thus, compared to a CW NMR experiment, the time can be reduced by a factor of 1000-fold or more by using the FT-NMR technique. 

The raw data or FID is a series of intensity values collected as a function of time time-domain data. A single proton signal, for example, would give a simple sine wave in time with a particular frequency corresponding to the chemical shift of that proton. This signal dies out gradually as the protons recover from the pulse and relax. To convert this time-domain data into a spectrum, we perform a mathematical calculation called the Fourier transform (FT), which essentially looks at the sine wave and analyzes it to determine the frequency. This frequency then appears as a peak in the spectrum, which is a plot in frequency domain of the same data (Fig. 3.27). If there are many different types of protons with different chemical shifts, the FID will be a complex sum of a number of decaying sine waves with different frequencies and amplitudes. The FT extracts the information about each of the frequencies ... 

Fluorescence anisotropy decay can also be measured by frequency-domain methods. In this approach, the polarized fluorescence intensities 7 (oo) and 7j (oo) are measured as a function of the modulation frequency of the polarized excitation beam. Even more information about frequency-domain anisotropy measurement and analysis can be found in the monograph by Lakowicz (see Eurther Reading). 

A free induction decay is a display of observed signal intensity as a function of time. To be able to interpret the response of the spins this time domain data has to be transformed into frequency domain data, the spectrum. 

A dynamical mechanism is specified by answering the classic questions about the time-evolution of an isolated system where does it start, how fast does it leave, what causes it to leave, where does it go next, and why does it go there Answers to all of these questions are obtainable by constructing an appropriate Heff model following the techniques presented in Chapters 1-8. The parameters that define the Heff may equally well be defined by fitting the information contained in a frequency domain (transition frequencies, intensities, and linewidths) or a time domain (arrival times of the wavepacket at a specified location in coordinate or state space, decay rates, transfer rates, and recurrence times and amplitudes) experiment. 

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