Absorption and Dispersion Spectra

Maximum induction current due to resonance between rf field and Larmor precession corresponds to maximum absorption of energy. Thus, the plot of the induction current in the v direction (n/2 ahead of the vector Bl i ) as a function of frequency is the NMR 

Before and after resonance, there are magnetization components of opposite sign 180° out of phase and in phase (+ u and - u direction) with the rf field B1 i (Fig. 1.8). At resonance, there is no magnetization in the u direction. If the receiver coil obtains the inductance current in phase with Bli (the u direction), a dispersion curve (Fig. 1.9) results, called the u mode. When the absorption or out of phase spectrum ( ) reaches its maximum (/ind (oj) = max.), the dispersion or in phase spectrum ( ) goes through zero and changes its sign, as illustrated in Fig. 1.9. 

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Absorption and dispersion spectra steady-state frequency-response... 

Whenever the same parameters are available from two different curves (e.g., wq aiid t from Figure 1 or Figure 4a), there is some mathematical relation between the curves. For the "linear" system we have considered (i.e., displacement is proportional to driving amplitude Fq) the time-domain and frequency-domain responses are connected by a Fourier transform. Similarly, absorption and dispersion spectra both yield the same information, and are related by a Hilbert transform (see Chapter 4). In this Chapter, we will next develop some simple Fourier transform properties for continuous curves such as Figures 1-4, and then show the advantages of applying similar relations to discrete data sets consisting of actual physical responses sampled at equally-spaced intervals. 

In the previous section, we established a correspondence between the transient time-domain response (exponentially damped cosine wave) to a sudden "impulse" excitation and the steady-state frequency-domain response (Lorentzian absorption and dispersion spectra) to a continuous excitation. The Fourier transform may be thought of as the mathematical recipe for going from the time-domain to the frequency-domain. In this section, we shall introduce the mathematical forms of the transforms, along with pictorial examples of several of the most important signal shapes. 

As noted in Chapter 1, spectroscopic "absorption" and "dispersion" signals each provide equivalent information, provided that the system is "linear" (i.e., that the response amplitude is proportional to the excitation amplitude, so that no "saturation" occurs). Historically, the absorption spectrum has been preferred over dispersion, because the absorption line shape is narrower and more symmetrical. However, the various Fourier transform spectrometer si produce digitized absorption and dispersion spectra with equal vertical scaling, each containing Falf the available signal information.2 Moreover, even when only the absorption-mode signal is available, the dispersionmode spectrum may be generated by a Hilbert transform.3... 

Figure 3 shows a family of DISPA plots for unresolved spectral doublets, in which the two component Lorentzians (each of the same area and width) are separated by up to about one-half the width of either component line. Also shown are the composite absorption and dispersion spectra from which the DISPA plots were constructed. Although each of the composite absorption spectra gives just a single unresolved peak, each of the DISPA plots shows a well-defined displacement outside and to the right of the reference circle. Moreover, the magnitude of the displacement is directly related to the magnitude of the doublet separation. 

Figure 4. Composite absorption and dispersion spectra (left) and...

Absorption and dispersion spectra (left) and corresponding DISPA plots (right) for a spectrum resulting from exchange between two Lorentzian lines of equal width and area, but different resonant frequency. Exchange rate constant k = 1.2/F, 1.4/F, and 1.6/F s-1 (proceeding from outermost to innermost solid curves), for line separation of 4 s"l. [Taken from ref. 4.]... 

Figure 12. Absorption and dispersion spectra (upper row) and corresponding DISPA plots (lower row) for Lorentzian line shape with S/N (measured for absorption-mode) of (A) 20 1, (B) 10 1, and (C) 5 1. [Taken from ref. 10.]...

This is a frequency-dependent (through wj(z) which depends on kj and hence , ) linear combination of the absorption and dispersion spectra. The pure absorption spectrum can be regained from Equation 52 by numerical methods if the sine Fourier transform is also available. 

C. H. Gohle et al.. Frequency comb vernier spectroscopy for broadband high resolution high sensitivity absorption and dispersion spectra. Phys. Rev Lett. 99, 263902 (2007)... 

One interesting new field in the area of optical spectroscopy is near-field scaiming optical microscopy, a teclmique that allows for the imaging of surfaces down to sub-micron resolution and for the detection and characterization of single molecules [, M]- Wlien applied to the study of surfaces, this approach is capable of identifying individual adsorbates, as in the case of oxazine molecules dispersed on a polymer film, illustrated in figure Bl.22,11 [82], Absorption and emission spectra of individual molecules can be obtamed with this teclmique as well, and time-dependent measurements can be used to follow the dynamics of surface processes. 

Peaks in homonuclear 2D /-resolved spectra have a phase-twisted line shape with equal 2D absorptive and dispersive contributions. If a 45° projection is performed on them, the overlap of positive and negative contributions will mutually cancel and the peaks will disappear. The spectra are therefore presented in the absolute-value mode. 

Figure 1.25. Absorption and fluorescence spectra of Py+ and Ox+ in zeolite L, measured in an aqueous dispersion. The different spectral overlap regions are shaded. 

The cross peaks in the 2D spectrum are a combination of absorption and dispersion lineshapes and consequently spectra are displayed in magnitude mode. 

Note that with a non-DQ-filtered, phase sensitive COSY experiment the cross peaks are again purely absorptive while diagonal peaks irrespective of the phase correction will have both absorptive and dispersive character. Unlike most other 2D spectra, it is therefore best to phase correct a non-DQ-filtered phase sensitive COSY spectrum while examining the cross rather than the diagonal peaks. 

The solution of eq. (2.11) is a complex function. FFT computation therefore yields both real and imaginary PFT NMR spectra, v(co) and i u(to), which are related to the absorption and dispersion modes of CW spectra. The two parts of the complex spectrum are usually stored in different blocks of the memory and can be displayed on an oscilloscope to aid in further data manipulations. 

The real and imaginary spectra obtained by Fourier transformation of FID signals are usually mixtures of the absorption and dispersion modes as shown in Fig. 2.13 (a). These phase errors mainly arise from frequency-independent maladjustments of the phase sensitive detector and from frequency-dependent factors such as the finite length of rf pulses, delays in the start of data acquisition, and phase shifts induced by filtering frequencies outside the spectral width A. 

There are two types of solute-solvent interactions which affect absorption and emission spectra. These are universal interaction and specific interaction. The universal interaction is due to the collective influence of the solvent as a dielectric medium and depends on the dielectric constant D and the refractive index n of the solvent. Thus large environmental perturbations may be caused by van der Waals dipolar or ionic fields in solution, liquids and in solids. The van der Waals interactions include (i) London dispersion force, (ii) induced dipole interactions, and (iii) dipole-dipole interactions. These are attractive interactions. The repulsive interactions are primarily derived from exchange forces (non bonded repulsion) as the elctrons of one molecule approach the filled orbitals of the neighbour. If the solute molecule has a dipole moment, it is expected to differ in various electronic energy states because of the differences in charge distribution. In polar solvents dipole-dipole inrteractions are important. 

Both absorption and emission spectra are somewhat blue shifted, which we attributed to the difference between the spectroscopic model (PAUMe) and the compound actually used in the experiments (PAdU). Furthermore, the dispersion interaction was neglected, mainly because the choice of the scaling parameter y Eq. (3-71) for states beyond the first excited state is cumbersome. Reducing the ionization energy of the solute by the excitation energy leads rapidly to a value smaller than zero, and hence to a positive dispersion interaction. In order to avoid this unphysical situation it is better to neglect the dispersion completely. Moreover, it is sometimes assumed that in semi-empirical wave functions electron correlation is accounted for because the parameters come from experiment. (The CIS... 

In this section the electronic structure of conjugated polymers is discussed. They form a special class of materials with particular types of excitations (such as the solitons) and properties, introduced briefly in Chapter 11. These problems are discussed here essentially in relation to the spectroscopic properties. The related but distinct subject of electrical conductivity is treated in Section IV. To set the scene, we first present some typical results visible absorption and emission spectra and resonance Raman spectra. We consider the theoretical issues in Section III.B, then return to the meaning of the experimental results in Section III.C. The interesting nonlinear optical properties of CPs will be considered in Section III.D. These sections are concerned with electronic states within the gap or near the band edges the structure (i.e., the dispersion relations) of valence and conduction bands is also of theoretical interest and is considered in Section III.E. 

Three of the experiments are completely new, and all make use of optical measurements. One involves a temperature study of the birefringence in a liquid crystal to determine the evolution of nematic order as one approaches the transition to an isotropic phase. The second uses dynamic laser light scattering from an aqueous dispersion of polystyrene spheres to determine the autocorrelation function that characterizes the size of these particles. The third is a study of the absorption and fluorescence spectra of CdSe nanocrystals (quantum dots) and involves modeling of these in terms of simple quantum mechanical concepts. 

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