Spectral Resolution Methods

In some investigations it may be important to extract the pure component spectra (S) and the associated concentration (C) for the different species present within a mixture. A generalized solution to Eq. (4) has an infinite number of solutions available, while analysis methods such as PCA produce abstract factors that are linear combinations of the pure component spectra and concentrations [Eq. (15)]. An additional transformation step is required to rotate these abstract factors into the pure component spectra and pure concentration profiles. Two specific examples that allow this transformation step are presented below. 

The alternative matrix (C or S) is then estimated (C or S), followed by applying the constraints to that specific matrix (C or S) 

The estimated and constrained concentration and pure component spectra matrices are then used to estimate the data matrix, allowing the error to be evaluated and checked for convergence. If convergence is not reached the process is repeated using C as the initial input into Eq. (43). 

Recall that the data matrix can be described as the product of the concentration and the pure component spectra [Eq. (42)], but that the solution to this relationship is not trivial or unique. It was demonstrated by Kubista that if two proportional data sets can be obtained for a sample, then an unambiguous unique solution (i.e., the single correct solution) is obtained through a self-modeling analysis of the spectral data sets. These two proportional data sets A and B can be formulated as 

Note that we have elected to use the nomenclature A and B instead of the previously used D [Eq. (4)] to denote the different data matrices, and to clearly distinguish these matrices as proportional data sets. In addition, many of the previous DECRA derivations decompose the data matrices as A = CS where the transpose is utilized because in those descriptions the pure component spectral matrix is in a column representation (wfreq comp)) d is the transpose of the definitions we presented in Section 2.1. Rearrangement of Eq. (46) leads to the generalized eigenvalue problem 

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For the example in Fig. 2, the Fourier transformed NMR spectra (variables or descriptors being intensity as a function of frequency) were utilized for the creation of the data matrix D. It should be noted that many different descriptors can be used to create D, with the descriptor selection depending on the analysis method and the information to be extracted. For example, in the spectral resolution methods (Section 6), the desired end result is the determination of the true or pure component spectra and relative concentrations present within the samples or mixtures [Eq. (4)]. For this case, the unmodified real spectra Ij co) are commonly used for the chemometric analysis. In contrast, for the non-supervised and supervised methods described in Sections 3 and 4, the classification of a sample into different categories is the desired outcome. For these types of non-supervised and supervised methods the original NMR spectrum can manipulated or transformed to produce new descriptors including... 

The trade-off between the simplicity of atmosphere pressure work, spectral resolution, method selectivity and sensitivity strongly favours it for process analysis compared with even the modest vacuum pumping requirements of MMW spectrometry at the usual pressures of tens of Pa. Where the balance occurs will... 

This chapter develops and applies a new spectral resolution method that permits insight into the dynamic structure of the process. It can be used for the analysis and retrofit of existing designs and also as a means to help develop new process designs. 

Some "fast methods" such as single-scan experiments,93 94 Hadamard95 and projection-reconstruction techniques96 take a few seconds to produce 2D spectra, but suffer from a lack of spectral resolution. As mentioned... 

A significant development in the FRET imaging field has been the systematic implementation of spectral resolution [15-20], including D-A population analysis [8, 19] (see also Chapter 8), often in the context of single-molecule determinations [21-26] see focus issue on this subject, Nature Methods, June 2008. Invariably, photobleaching phenomena [21, 27, 28] intervene either as a hindrance (that can be minimized, CLEM [29]) or a facilitation of the FRET determination [1, 30], The equally important issue of background suppression or compensation can be achieved by novel means photon-free (bio)chemical instead of photonic excitation... 

Nonresonance Raman spectra of the alternating LB films were measured by a total reflection method shown in Figure 23. The films were deposited on quartz prisms. The s-polarized beam of 647.1 nm from a Kr laser was incident upon the interface between the quartz and film at an angle of 45° from the quarz side, and totally reflected. Raman line scattered from the film in the direction of 45° from the surface was measured through a Spex Triplemate by a Photometries PM512 CCD detector with 512x512 pixels operated at -125 °C. The spectral resolution was about 5 cm 1. 

We note in passing that the spectral method can be regarded as a special case of FD with L = 1, for which the uncertainty principle dictates that the spectral resolution is inversely proportional to the propagation length. In FD, the spectral resolution is enhanced beyond the uncertainty principle, because of the diagonalization of the Hamiltonian in the subspace spanned by multiple filtered vectors. 

Although this book is devoted to molecular fluorescence in condensed phases, it is worth mentioning the relevance of fluorescence spectroscopy in supersonic jets (Ito et al., 1988). A gas expanded through an orifice from a high-pressure region into a vacuum is cooled by the well-known Joule-Thomson effect. During expansion, collisions between the gas molecules lead to a dramatic decrease in their translational velocities. Translational temperatures of 1 K or less can be attained in this way. The supersonic jet technique is an alternative low-temperature approach to the solid-phase methods described in Section 3.5.2 all of them have a common aim of improving the spectral resolution. 

In 2-substituted adamantanes 25 both types of 8-positioned carbon atoms (8syn and 8 ) exist within one molecule (Scheme 37). Early measurements with limited spectral resolution (176) did not differentiate between their signals. Later (124,244), differences of up to 0.7 ppm were detected, and application of various independent methods, including addition of lanthanide shift reagents (245), determination of longitudinal relaxation times T, (246), and evaluation of deuterium... 

The complexity of the solvable structure strongly depends on the spectral resolution of the diffraction method in use. Structures with about 60 atoms in the asymmetric unit were solved from powder data combining synchrotron X-ray diffraction with refinement from neutron diffraction data from the same material (Morris et al. 1994 Admans 2000). About half of that complexity can be achieved with good laboratory X-ray diffractometers (Masciocchi et al. 1996 Kariuki et al. 1999). Neutron diffraction data can better be used for structure refinement than for structure determination, for the same reason. 

In kinetic analysis of complex reactions, 210, 382 fluorescence decay rate distributions, 210, 357 implementation in Laplace de-convolution noniterative method, 210, 293 in multiexponential decays, 210, 296 partial global analysis by simulated annealing methods, 210, 365 spectral resolution, 210, 299. 

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