Gaussian curves derivation

However, the quantal type of relationship with dose will also show a sigmoid curve when appropriately plotted. In this case, the curve derives from a frequency distribution (Fig. 2.5), which is the familiar Gaussian curve. 

Fig. 5. Scans of the pump-probe time delay between the laser pump pulse and the x-ray probe of the transient X-ray absorption signal of [Run(bpy)3]2+ (the x-ray energy is set at the minimum of the signal in fig. 20.b) over several tens of nsec (a) and in the psec time domain (b). The latter represents the cross-correlation signal between the 200 fs laser pulse and the 70-80 ps long x-ray pulse (indicated by the Gaussian-shaped derivative of the fit curve to the data).

To attribute absorption bands to the individual Co2+ sites, second derivative mode analysis (not shown here) and decomposition of the spectra to the Gaussian curves was used (details of methods see in Refs [5,7,10], The decomposition of the VIS absorption of the... 

Figure 8. Fitting of two Gaussian curves with the curve difference between the sulfur dioxide evolution from PTO of the Provence coal and of the Low Temperature Ash derived from it.

In our example, all the uncertainties have the same magnitude. This restriction is not necessary to derive the equation for a Gaussian curve. 

The shape of the dose-response curve depends on a number of factors, but it is basically derived from the familiar Gaussian curve (figure 2,4), which describes a normal... 

Fig. 6.7 Allowed intra-ligand transitions from x to V -type ligand orbitals for tris-chelate complexes with D3 symmetry. The circular dichroism has a lower right-circularly polarized (rep) band and an upper left-circularly polarized Ocp) band. This gives the CD spectmm the appearance of the first derivative of a Gaussian curve, with a negative part at longer wavelength and a positive part at shorter wavelength...

The spectrum of this substance has the shape of an asymmetric Gaussian curve. Only efficient derivative modules are able to resolve the signal in d" into two main peaks (Fig. 4-4). 

DERIVATION SUMMARY The Harmonic Approximation. The harmonic approximation yields a solution to the vibrational energy, Eq. 8.17, which predicts that the vibrational quantum states should be evenly spaced by a vibrational constant = h Zk/fjL, where k is the force constant and /z is the reduced mass. The wavefunctions are Gaussian curves multiplied by polynomials, with the order of the polynomial (and the number of nodes in the wavefimction) equal to the vibrational quantum number v, which can be any whole number. 

Some caution must be exercised in derivative spectroscopy, as artifacts arise due to the interaction of the side-wings (375). Furthermore, in the higher order spectra, the amplitude of a band does not reflect its amplitude in the original zero-order spectrum. Resolution of a particular band also depends upon the band shape, with Lorentzian curves resolving better than Gaussian curves. 

It may be noted that eqs (3.52) and (3.53) are identical for chains with r nl Figure 3.19 compares the stress-strain curves derived under the two conditions (eqs (3.50) and (3.51)). The strong upturn in stress experimentally verified can only be obtained by the non-Gaussian equation. X-ray diffraction on natural rubber has shown that the initial upturn in the load-extension curve is a genuine non-Gaussian effect, unrelated to crystallization. At higher extension... 

Braslau and coworkers [83] synthesized cyclic PSTY through the combination of nitroxide-mediated radical polymerization (NMRP) and CuAAC click reaction. The synthesis procedure was relatively complex compared with other strategies. 1 - [4-(Chloromethyl)phenyl] ethyl alkoxyamine was used to mediate the styrene polymerization, followed by successive azidation and oxidative cleavage with ammonium cerium(IY) nitrite in the presence of propargyl alcohol. The azide and alkyne groups were then introduced to each end of the polymer. Finally, the cyclization reaction was carried out in toluene with CuBr and PMDETA as catalyst at 100°C (Scheme 35). The cyclization results showed about 64% click product, as derived from Gaussian curve fitting. 

If we consider an absorption band showing a normal (Gaussian) distribution [Fig. 17.13(a)], we find [Figs. (b) and (d)] that the first- and third-derivative plots are disperse functions that are unlike the original curve, but they can be used to fix accurately the wavelength of maximum absorption, Amax (point M in the diagram). 

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

The line shapes of the resonance absorption curves were rather well described by a Gaussian shape, as is shown in Fig. 14. Two parameters are necessary in fitting a recorder derivative with a Gaussian shape function the maximum value of the derivative signal (dx"/3 f)m. and the width between points of maximum slope of the absorption, which are... 

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