Absorption and Dispersion

When an electromagnetic wave passes through a medium with refractive index n, not only the waoe amplitude decreases (absorption) but also the phase velocity changes from its value c in vacuum to v = c/n (dispersion). The refractive index n = n(w) depends on the frequency w of the E.M. wave. 

The classical model describing the atomic electrons as damped harmonic oscillators which are forced to oscillate by the electric field E(w) of the 

The conclusions of this classical model can be transferred to real molecules in a relatively simple way by introducing the concept of oscillator strength. 

At first we shall discuss this classical model. Although the quantum mechanical treatment will be only briefly outlined, this should still allow the reader to understand more advanced presentations [2.6,11]. 

The Bloch equations can be solved analytically under the condition of slow passage, for which the time derivatives of Eq. 2.48 are assumed to be zero to create a steady state. The nuclear induction can be shown to consist of two components, absorption, which is 90° out of phase with B, and has a Lorentzian line shape, and dispersion, which is in phase with B,. The shapes of these signals are shown in Fig. 2.10. By appropriate electronic means (see Section 3.3), we can select either of these two signals, usually the absorption mode. 

The Jones Matrix M 6) for any polarizing element, turned by an angle 0 against its original position Af(0) is obtained by the product 

For example a linear horizontal polarizer turned by an angle 6 is described by the Jones matrix 

A A/2-plate with its fast axis in the x-direction is rotated by an angle 9 around the z-axis (direction of light propagation). If linear horizontal polarized light passes through the device the output light is 

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Figure 3.6 The first set of Fourier transformations across <2 yields signals in V2, with absorption and dispersion compronents corresponding to real and imaginary parts. The second FT across /, yields signals in V, with absorption (i.e., real) and dispersion (i.e., imaginary) components quadrants (a), (b), (c), and (d) represent four different combinations of real and imaginary components and four different line shapes. These line shaptes normally are visible in phase-sensitive 2D plots.

The phase-twisted peak shapes (or mixed absorption-dispersion peak shape) is shown in Fig. 3.9. Such peak shapes arise by the overlapping of the absorptive and dispersive contributions in the peak. The center of the peak contains mainly the absorptive component, while as we move away from the center there is an increasing dispersive component. Such mixed phases in peaks reduce the signal-to-noise ratio complicated interference effects can arise when such lines lie close to one another. Overlap between positive regions of two different peaks can mutually reinforce the lines (constructive interference), while overlap between positive and negative lobes can mutually cancel the signals in the region of overlap (destructive interference). 

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. 

E Herzfeld and T. R. Litovitz, Absorption and Dispersion of Ultrasonic Waves, Academic Press, New York, 1959. 

K. C. Allen and H. J. Liebe. Tropospheric Absorption and Dispersion of Millimeter and Submillimeter Waves IEEE Trans. Antennas Propagat., Vol. 31, pp. 221-223, January 1983. 

Figure 6 illustrates a block diagram of a crossed-coil variable frequency spectrometer and associated electromagnet. A calibrator circuit 66) is useful for intensity calibration of absorption and dispersion mode signals. A calibrator circuit for the Pound-Knight type of spectrometer is also used... 

Hodgson, J. N. Optical absorption and dispersion in solids. London Chapman and Hall 1970. 

Hodgson, J. N., 1970. Optical Absorption and Dispersion in Solids, Chapman Hall, London. 

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. 

The lineshape function which describes the absorption and dispersion modes of an unsaturated, steady-state NMR spectrum is proportional to the Fourier transform of the function MxID(t) (24, 25, 99)... 

Contrary to the point by point approach the diagonalization method consists of the generation of an entire lineshape function in one step. (13, 14, 57-60) Time-consuming calculations are carried out only once. The resulting set of complex numbers can be used for a simple calculation of the lineshape (absorption and dispersion modes) at any desired point on the frequency axis. Thus, the complex matrix from equation (147) can be diagonalized by a similarity transformation using an co-independent complex matrix W ... 

With this particular example of a located, invariable charge model, Barriol used a method that would be frequently used in his laboratory, particularly in the many studies on the Onsager model to work on a very simple model and to adjust it punctually for one case or another. Other authors calculated the atomic polarizability of a molecule according to a dynamic model based on absorption and dispersion infrared measurements. But the problem is to determine the charge value participating effectively in polarization. [36] Barriol, for his part, did work on the simple model of located, invariable charges, with very disputable hypotheses Things are certainly not like this, but there are some difficulties to find a more elaborated model, with which it would be possible to do calculations. [37]... 

Hylenex Hyaluronidase Increase absorption and dispersion of other injected drugs... 

Despite the difficulty imposed by the attenuation and dispersion of the atmosphere on subpicosecond THz pulses, there has been some effort to extract spectroscopic signatures of explosives, such as RDX, from standoff ranges up to 30 m [91], However, the measured spectral absorption peak of RDX artificially broadens, due to the absorption and dispersion in the atmosphere, as the distance to the target increases. This broadening at standoff distances suggests that spectroscopic identification of explosives such as RDX might be problematic because the apparent spectral shape cannot be directly compared to a spectral standard curve. To circumvent the atmospheric attenuation and dispersion... 

After you Fourier transform your FID, you get a frequency-domain spectrum with peaks, but the shape of the peaks may not be what you expected. Some peaks may be upside down, whereas others may have a dispersive (half up-half down) lineshape (Fig. 3.36). The shape of the peak in the spectrum (+ or — absorptive, + or — dispersive) depends on the starting point of the sine function in the time-domain FID (0° or 180°, 90° or —90°). The starting point of a sinusoidal function is called its phase. Phase errors come in all possible angles, including those intermediate between absorptive and dispersive (Fig. 3.37). The spectrum has to be phase corrected ( phased ) after the Fourier transform to obtain the... 

Recall that the raw NMR data (FID) consists of two numbers for each data point one real value and one imaginary value. After the Fourier transform, there are also two numbers for each frequency point one real and one imaginary. In a perfect world, the real spectrum would be in pure absorptive mode (normal peak shape) and the imaginary spectrum would be in pure dispersive (up/down) mode. In reality, each spectrum is a mixture of absorptive and dispersive modes, and the proportions of each can vary with chemical shift (usually in a linear... 

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