Factor spectrum

Case history Frequency of illness, provocative factors, spectrum of symptomatology... 

Sometimes the interpretation of the canonical variates cannot be done with the coefficients bij only. A tool for interpretation is the so-called factor spectrum with intensities bi,. In the case of a binary mixture, the positive part of the factor spectrum can be interpreted as the differences in mass spectrum of one component and the negative part as the differences in the spectrum of the other component. This can be illustrated for the factor spectrum for a mixture of pectin and methylated pectin [76a]. The first canonical variates contained m/z = 31, 32 (coming from methanol) and m/z = 85 attributed to a lactone ion also formed from the methylated pectin. The low DM factor spectrum contains five major masses common to furaldehyde. This spectrum is shown in Figure 5.5.8. 

Figure 5.5.8. Factor spectrum for the first canonical variate for pectin related to the degree of methylation (DM). The upper spectrum shows high DM values while the lower spectrum shows low DM.

A correlation between two variables can be either positive or negative. The interpretation of the factors in terms of patterns in the samples is the most difficult part and not always possible. However, a particular combination of m/z values in a mass spectrum can be indicative of the presence of a chemical compound in the sample. A tool for interpretation, similar to that utilized to interpret canonical variates, is the factor spectrum with intensities Si bij (where Si is the standard deviation of each column in the data matrix X). This spectrum, plotting the values Sj bij at m/z, shows the part of the intensity change described by the factor. In other words, such a spectrum will show those masses that contribute the most to the discrimination of the samples. Because the loadings b j generally can be either positive or negative, the factor spectrum exhibits positive and negative intensities. 

Fig. 22 Monitoring of ethylene polymerization over 3f/TiBA (Al/Cr = 4) by in-situ ESR spectroscopy (from 220 to 350 K) (a) 220 K, (b) 270 K, (c) 290 K, and (d) 350 K. For determination of the g factor spectrum recorded with TEMPO (g = 2.0058), the three lines of TEMPO are marked with asterisks...

XPS measurements were carried out on an AXIS NOVA photoelectron spectrometer (Kratos Analytical, Manchester, UK). The surface atomic concentration was determined firom peak areas using sensitivity factors. Spectrum background was subtracted according to Shirley. The XPS peaks of the Ti species were analyzed by spectra deconvolution software (CasaXPS-Vision 2, Kratos Analytical, UK). 

Influence of the SDOF oscillator inelastic characteristics. One of the earliest proposed q j-fx-T expression that satisfied the functitmal limitations of the reduction factor spectrum by Newmark and Hall (1973) was the exponential function proposed by Nassar and KrawinMer (1991). In their statistical study, they cmisidered 36 records to establish the attenuatimi of the ground motion parameters and 15 records from Western US earthquakes on firm-to-medium stiffness soils (plus 10 records from the Whittier Narrows earthquake, for verification) to establish q. Two hysteretic shapes were considered to... 

Another problem often encountered in absorbance subtraction is the preferential orientation of one of the samples relative to the other. Unfortunately, there is no general method or algorithm to eliminate this orientation problem because each band has its own characteristic dichroic behavior. The only solution is to prepare the samples with reproducible or random orientation. Unless this is accomplished, three-dimensional tilted sample methods must be used. The three-dimensional tilted sample method removes the orientation of the sample and generates a pure structure-factor spectrum prior to absorbance subtraction. 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The above fomuilae for the absorption spectrum can be applied, with minor modifications, to other one-photon spectroscopies, for example, emission spectroscopy, photoionization spectroscopy and photodetachment spectroscopy (photoionization of a negative ion). For stimulated emission spectroscopy, the factor of fflj is simply replaced by cOg, the stimulated light frequency however, for spontaneous emission... 

Note the presence of the ra prefactor in the absorption spectrum, as in equation (Al.6.87) again its origm is essentially the faster rate of the change of the phase of higher frequency light, which in turn is related to a higher rate of energy absorption. The equivalence between the other factors in equation (Al.6.110) and equation (Al.6.87) under linear response will now be established. 

Spp(A) which is the liquid structure factor discussed earlier in section A2.2.5.2. The density fluctuation spectrum is... 

A8, which leads to D, = 1/(2A8). The factor of two arises because a minimum of two data points per period are needed to sample a sinusoidal wavefonn. Naturally, the broadband light source will detennine the actual content of the spectrum, but it is important that the step size be small enough to acconunodate the highest frequency components of the source, otherwise they... 

A microwave pulse from a tunable oscillator is injected into the cavity by an anteima, and creates a coherent superposition of rotational states. In the absence of collisions, this superposition emits a free-mduction decay signal, which is detected with an anteima-coupled microwave mixer similar to those used in molecular astrophysics. The data are collected in the time domain and Fourier transfomied to yield the spectrum whose bandwidth is detemimed by the quality factor of the cavity. Hence, such instruments are called Fourier transfomi microwave (FTMW) spectrometers (or Flygare-Balle spectrometers, after the inventors). FTMW instruments are extraordinarily sensitive, and can be used to examine a wide range of stable molecules as well as highly transient or reactive species such as hydrogen-bonded or refractory clusters [29, 30]. 

MW frequency of 10 Hz. There are various considerations that influence the choice of the radiation frequency. Higher frequencies, which require higher magnetic fields, give inlierently greater sensitivity by virtue of a more favourable Boltzmaim factor (see equation (b 1.15.11)). However, several factors place limits on the frequency employed, so that frequencies in the MW region of the electromagnetic spectrum remain favoured. One limitation is the sample size at frequencies around 40 GHz the dimensions of a typical... 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

Table VIII. The compounds selected are as typical as possible, but it must be remembered that there are many environmental factors that produce changes in the location of the absorption bands. These displacements are usually of the order of a few mp., but in some cases they are so great as to move the absorption band into a completely different region of the spectrum.

These results provide so-called "selection rules" because they limit the L and M values of the final rotational state, given the L, M values of the initial rotational state. In the figure shown below, the L = L + 1 absorption spectrum of NO at 120 °K is given. The intensities of the various peaks are related to the populations of the lower-energy rotational states which are, in turn, proportional to (2 L + 1) exp(- L (L +1) h /STi IkT). Also included in the intensities are so-called line strength factors that are proportional to the squares of the quantities ... 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

The factor tG(t) is called the relaxation spectrum and is given the symbol H(r). 

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