Spectrum Representation

Fig. 5 The influence of the Franck-Condon principle on the appearance of the absorption band in a diatomic molecule where the equilibrium internuclear separations are (a) identical in the ground and excited states, (b) smaller in the excited state, and (c) greater in the excited state. The spectra representations (d), (e), and (f) correspond to the situations depicted in (a), (b), and (c), respectively. The numbers in (a) represent the vibrational quantum numbers in the ground and excited states and in (d), (e), and (f) the transitions between these sublevels.

The porous electrode theory was developed by several authors for dc conditions [185-188], bnt the theory is usually applied in the ac regime [92,100,101,189-199], where mainly small signal frequency-resolved techniques are used, the best example of which are ac theory and impedance spectra representation, introdnced in the previons section. The porous theory was first described by de Levi [92], who assumed that the interfacial impedance is independent of the distance within the pores to obtain an analytical solution. Becanse the dc potential decreases as a fnnction of depth, this corresponds to the assnmption that the faradaic impedance is independent of potential or that the porons model may only be applied in the absence of dc cnrrent. In snch a context, the effect of the transport and reaction phenomena and the capacitance effects on the pores of nanostructured electrodes are equally important, i.e., the effects associated with the capacitance of the ionic donble layer at the electrode/electrolyte-solntion interface. For instance, with regard to energy storage devices, the desirable specifications for energy density and power density, etc., are related to capacitance effects. It is a known fact that energy density decreases as the power density increases. This is true for EDLC or supercapacitors as well as for secondary batteries and fnel cells, particnlarly due to the distributed nature of the pores... 

Moreover, using the spectra representation of the response matrix, we can write... 

A remarkable improvement in the visualization of the signal can be obtained by displaying the 33S power spectrum,22 although this is feasible only when a single signal is present (information about the phases and intensities of peaks are lost in the power spectrum representation). 

Fig. 11. An overview of the observed D2O transitions with respect to the translational band of liquid D2O shows that the 142.8 cm band lies well within the translational band of the liquid. The first inset shows a stick spectrum representation of the 142.8 cm ( 20)3 band and the second inset is a scan of the R2(2) transition, representative of the strongest observed rovibrational transitions. ...

Both causes of spectral enhancement act in the case of external reflection from transparent or low-absorbing substrates. The positive geometric effect is demonstrated in Fig. 1.15. The negative effect due to the spectrum representation is the most pronounced in the p-polarized spectra measured at [Pg.55]

Electrostatic potential map of the HsO ion. In the rainbow color spectrum representation, the most electron-rich region is red and the most electron-poor region is blue. 

Soimd Spectrum Representation of a sound in terms of the amount of vibration at a each individual frequency. Usually presented as a graph of amplitude (plotted vertically) versus frequency (plotted horizontally). 

In order to compare elastic design spectrum (demand) and capacity curve, first the capacity curve is transformed into an energy equivalent elastic-perfectly plastic system, then the ADRS (Acceleration Displacement Response Spectrum) representation is used, as shown in Figure 2. 

Mark, W.D., A Power Spectrum Representation for Nonstationary Random Vibration", Random Vibration - Status and Recent Developments, S.H. Crandall Festschrift, Ed. by Elishakoff, I. and Lyon R.H., Elsevier, 211-240, 1985. 

NVID, (TP13) used for the normal spectrum representation is set by RV3 and... 

We have alluded to the comrection between the molecular PES and the spectroscopic Hamiltonian. These are two very different representations of the molecular Hamiltonian, yet both are supposed to describe the same molecular dynamics. Furthemrore, the PES often is obtained via ab initio quairtum mechanical calculations while the spectroscopic Hamiltonian is most often obtained by an empirical fit to an experimental spectrum. Is there a direct link between these two seemingly very different ways of apprehending the molecular Hamiltonian and dynamics And if so, how consistent are these two distinct ways of viewing the molecule ... 

Mass spectrum. A spectrum obtained when ions (usually in a beam) are separated according to the mass-to-charge (m/z) ratios of the ionic species present. The mass-spectrum plot is a graphical representation of m/z versus measured abundance information. 

Figure 1 Representation of a typical density of eiectron states for a metal having X K and Z core levels (top) and REELS spectrum expected from metal shown in top panel (bottom).

Several features of ISS quantitative analysis should be noted. First of all, the relative sensitivities for the elements increase monotonically with mass. Essentially none of the other surface spectroscopies exhibit this simplicity. Because of this simple relationship, it is possible to mathematically manipulate the entire ISS spectrum such that the signal intensity is a direct quantitative representation of the surface. This is illustrated in Figure 5, which shows a depth profile of clean electrical connector pins. Atomic concentration can be read roughly as atomic percent direcdy from the approximate scale at the left. 

As you pay homage to the diversity of behavior that emerges across the spectrum of all possible representations of all possible systems, you inevitably conclude that (1) there can be no objectively privileged system for which the emergent structures are real, and suc.h that all other structures, for all other derived systems, are less real, and (2) reality, or the emergence and identity of particular sets of objects and their interactions, is wholly dependent on the arbitrary dynamical labels that prescribe a particular system. In short, reality lo.ses its objectivity, and takes on a more tentative, ineffably relative facade. You begin to wonder if the best that you can do to get a hold on objective reality is to look for whatever remains fixed -i.e. what is universal - within the space of all possible representations, all possible rules, and all possible emergent structures. 

A mass spectrum is a graphic representation of the ions observed by the mass spectrometer over a specified range of m/z values. The output is in the form of an x,y plot in which the x-axis is the mass-to-charge scale and the y-axis is the intensity scale. If an ion is observed at an m/z value, a line is drawn representing the response of the detector to that ionic species. The mass spectrum will contain peaks that represent fragment ions as well as the molecular ion (see Figure 1.3). Interpretation of a mass spectrum identifies, confirms, or determines the quantity of a specific compound. 

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

FT is essentially a mathematical treatment of harmonic signals that resolved the information gathered in the time domain into a representation of the measured material property in the frequency domain, as a spectrum of harmonic components. If the response of the material was strictly linear, then the torque signal would be a simple sinusoid and the torque spectrum reduced to a single peak at the applied frequency, for instance 1 Hz, in the case of the experiments displayed in the figure. A nonlinear response is thus characterized by a number of additional peaks at odd multiples of the... 

Fig. 14.—Schematic Representation of the Fragmentation Observed in the Positive F.a.b.-Mass Spectrum of a Permethylated Ganglioside Isolated from Granulocytes. [Other glyco-sphingolipids fragment in a similar way. Major cleavages are shown with solid lines, and minor cleavages with dotted lines. The masses of ions resulting from cleavages (a), (b), and (c) define the type of sphingosine and the type of fatty acid. In this example, (a) is 548, (b) is [M + H] minus 238, and (c) is [M + H] minus 533.]...

Mass spectrum of neon (a) actual appearance, (b) bar graph representation. 

Schematic representation of an apparatus that measures the absorption spectrum of a gaseous element. The gas in the tube absorbs light at specific wavelengths, called lines, so the intensity of transmitted light is low at these particular wavelengths.

Figure 3.4 Schematic representation of the steps involved in obtaining a two-dimensional NMR spectrum. (A) Many FIDs are recorded with incremented values of the evolution time and stored. (B) Each of the FIDs is subjected to Fourier transformation to give a corresponding number of spectra. The data are transposed in such a manner that the spectra are arranged behind one another so that each peak is seen to undergo a sinusoidal modulation with A second series of Fourier transformations is carried out across these columns of peaks to produce the two-dimensional plot shown in (C).

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

Two-dimensional NMR spectra are normally presented as contour plots (Fig. 3.11a), in which the peaks appear as contours. Although the peaks can be readily visualized by such an overhead view, the relative intensities of the signals and the structures of the multiplets are less readily perceived. Such information can be easily obtained by plotting slices (cross-sections) across rows or columns at different points along the Fi or axes. Stacked plots (Fig. 3.11b) are pleasing esthetically, since they provide a pseudo-3D representation of the spectrum. But except for providing information about noise and artifacts, they offer no advantage over contour plots. Finally, the projection spectra mentioned in the previous section may also be recorded. 

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