Spectral quantities

Spectral quantities may also be defined with respect to frequency v, or wavelength 2 see spectral radiant energy density above. 

Modified electron-gas procedures also yield quite accurate results for equilibrium and dynamic properties of MgO, if the effect of the Madelung potential upon the oxide ion charge density is properly incorporated, as in the potential-induced breathing model (Mehl et al., 1986, and references therein). As described previously, this approach yields less information than that from a band calculation (e.g., no spectral quantities are obtained), but it is extremely rapid computationally and can readily be extended to low-symmetry situations, so as to evaluate properties such as shear moduli in addition to the bulk moduli readily obtained from the... 

A frequently used expression equivalent to Eq. 81, but based on directly obtained spectral quantities (peak absorption frequency, Vmax/cm , band width, Avi/x/cm , and molar absorptivity, mol ), with Tp in cm and / da in A, is given... 

It is important to note that no motion having a period in excess of L/v can be reproduced in the simulations, where L is the length of the simulation box and is a velocity of sound in the medium.In addition, use of periodic boundary conditions together with a single structural unit cell as the simulation box restricts the calculation of spectral quantities to those at the center of the Brillouin zone the periodic boundary conditions force atoms in all images of the simulation box to vibrate in-phase, that is, the definition of a motion at the center of Brillouin zone. When comparing results of the calculations with the experimental spectra, one must also bear in mind that the model used in the calculations implies a perfect crystal structure, whereas experiments are usually done with microcrystals having defects. 

Directional spectral quantities. These describe the directional and wavelength distribution of the radiative energy in a detailed manner. They are of fundamental meaning, but are very difficult to determine experimentally or theoretically. This is why we frequently employ radiation quantities that only include one effect, either the dependence on the wavelength or the direction. 

Hemispherical spectral quantities average the radiation into all directions of the hemisphere over a surface element and so are only dependent on the wavelength. 

We will now investigate how the emitted radiation d is distributed over the spectrum of wavelengths and the directions in the hemisphere. This requires the introduction of a special distribution function, the spectral intensity Lx. It is a directional spectral quantity, with which the wavelength and directiondistribution of the radiant energy is described in detail. 

The spectral intensity Lx(X,j3, ip,T) describes the distribution of the emitted radiation flow over the wavelength spectrum and the solid angles of the hemisphere (directional spectral quantity). 

The hemispherical spectral emissive power Mx(X,T) covers the wavelength dependency of the radiated energy in the entire hemisphere (hemispherical spectral quantity). 

It belongs to the hemispherical spectral quantities. Integration of (5.26) over all wavelengths leads to... 

Figure 5. (A2) Various spectral quantities related to liquid water in the mid-IR region. Experimental values of log(/Q(i )//(i )) are shown in the case of an absorption set-np with a 1 p,m thick sample and of an ATR cell with an immersed portion of the crystal about 3 cm long (41). The optical constants n and k are also displayed together with the imaginary dielectric constants e".

By applying Fermi s golden rule, Forster derived a very important relation between the critical transfer distance R0 and experimentally accessible spectral quantities (Equation 2.35),° 67,68 namely the luminescence quantum yield of the donor in the absence of acceptor A, orientation factor, k, the average refractive index of the medium in the region of spectral overlap, n, and the spectral overlap integral, J. The quantities J and k will be defined below. Equation 2.35 yields remarkably consistent values for the distance between donor and acceptor chromophores D and A, when this distance is known. FRET is, therefore, widely applied to determine the distance between markers D and A that are attached to biopolymers, for example, whose tertiary structure is not known and thus... 

Spin densities (p) are theoretical quantities, defined as the sum of the squared atomic orbital coefficients in the nonbonding semi-occupied molecular orbital (SOMO) of the radical species (Hiickel theory). For monoradical species, the spin density is connected to the experimental EPR hyperfine coupling constant a through the McConnell equation [38]. This relation provides the opportunity to test the spin density dependence of the D parameter [Eq. (8)] for the cyclopentane-1,3-diyl triplet diradicals 10 by comparing them with the known experimental hyperfine coupling constants (ap) of the corresponding substituted cumyl radicals 14 [39]. The good semiquadratic correlation (Fig. 9) between these two EPR spectral quantities demonstrates unequivocally that the localized triplet 1,3-diradicals 9-11 constitute an excellent model system to assess electronic substituent effects on the spin density in cumyl-type monoradicals. 

The spectral irradiance (and analog of other spectral quantities) is given by ... 

Such analytical variable interchanging is of great importance because it allows the transcription from one spectral quantity to another, depending on the conceptual/computational needs and issues involved. For example, the spectral energy density per unit wavelength, is successively obtained from Eq. (1.24a/b)... 

Scores indicate new positions of the points (observed spectra) on the coordinate axes corresponding to the loading vectors. It should be noted that the variation of spectral quantity is fully recorded in the scores. The number of scores on each loading vector is exactly the same as the number of points in the multivariate space. This plays an important role in principal component regression (PCR), described in the next section. 

Spectral absorption (transmission) lines are not monochromatic, due to which physical values characterizing transitions of the molecular system from one quantum state to another are also energetically diffused. Therefore, any spectral quantity F (absorption cross section, absorption coefficient, Einstein coefficients, and others) can be of three types F, is the spectral value, Fq is the maximum value corresponding to the frequency Hq, and F = 6F dn is the integral value for the spectral line. The integral and spectral values are related by the following relationship ... 

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