Function resolution

Determination of concentration profiles from the raw data can be more complicated when protons are used as the incident particles. The energy loss ( dE/ die) is smaller for protons and straggling effects are more important. The observed profile A (AJ)) is a convolution of the actual concentration profile C x with a depth resolution function (x, Eq), which broadens with increasing x roughly as Jx- Hence, resolution deteriorates with depth. However, near-surface resolution for resonant profiling may be on the order of tens of A. 

Adequate resolution of the components of a mixture in the shortest possible time is nearly always a principal goal. Establishing the optimum conditions by trial and error is inefficient and relies heavily on the expertise of the analyst. The development of computer-controlled HPLC systems has enabled systematic automated optimization techniques, based on statistical experimental design and mathematical resolution functions, to be exploited. The basic choices of column (stationary phase) and detector are made first followed by an investigation of the mobile phase composition and possibly other parameters. This can be done manually but computer-controlled optimization has the advantage of releasing the analyst for other... 

The formulas obtained in section 2 can be used to estimate a sensitivity of oscillation experiments to different structures in the density profile. Apparently, an integration over the neutrino energy leads to averaging of oscillations and therefore to lost of the sensitivity. To estimate this effect we introduce the energy resolution function f(E, E) and perform averaging of the probability ... 

Let us take for simplicity the box like resolution function f(E, E), so that... 

The process of obtaining homochiral product from a racemate is known as kinetic resolution. Kinetic resolution functions by the transformation of two enantiomers of a racemic mixture at different rates. The objective is to effect a change in the physical properties of one enantiomer to such an extent that the resulting product is readily separable from the other. The technique suffers from the inherent inability to access >50 % of the desired enantiomer unless the unwanted enantiomer can be racemized and recycled or inverted. 

Before comparing with experiment, however, the theoretical results at an energy of 40 meV (equivalent to room temperature) were convoluted with the energy resolution function of the detector used for the measurement. This procedure was adopted because deconvolution of the experimental data was found to be numerically unstable. The convoluted theoretical data were then normalized to the experimental data at zero Doppler shift to yield the results shown. The agreement between the convoluted theoretical results and experiment is extraordinarily good, extending as it does over more than three orders of magnitude. These results also reveal... 

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

Figure 6.3 Schematic of the resolution function in the reciprocal space. Two vectors ki and kd represent the incident and the diffracted wave vectors and H represents the scattering vector. The divergence of the incident X-ray and the acceptance window of the diffracted beam side optic are represented as Suj and 6(29).

Figure 8.5 Typical in-plane distributions of diffuse x-ray intensity around the PbTiOs 304 peak, for various film thicknesses and temperatures. Darker shade indicates higher intensity (log scale). Elongation of central Bragg peak is due to asymmetric resolution function.

Okerberg ES, Wu J, Zhang B et al (2005) High-resolution functional proteomics by active-site peptide profiling. Proc Natl Acad Sci USA 102 4996-5001... 

The non-linearity of the retention lines is apparent from this figure. The response lines have been drawn for two different criteria the normalized resolution product r (drawn line eqn.4.19) and the product resolution function FIRS (dashed line eqn.4.18). It is seen that the product resolution criterion would in fact have guided us to a completely different optimum at a composition of 24.1% methanol and 25.2% THF. The chromatogram that we would have obtained at this composition is shown in figure 5.33h. Clearly, this chromatogram is less attractive than the one of figure 5.33g. Obviously, the normalized resolution product is to be preferred to the resolution product itself (see the discussion in section 4.3.2). 

The CCDs for the direct measurement of the antiprotonic X-rays were installed in the second bore hole of the cyclotron trap close to the stop volume. Thus, a few per mille of the full solid angle were covered. The relative efficiency and the in-beam resolution function were obtained in a separate measurement from the saturated X-ray transitions in pN. Two different types of CCDs were used (i) MOS CCDs with a typical depletion depth of about 30 pm [22,23] and (ii) the prototype of a high-rate X-ray detector based on a fully depleted (290 pm) pn-CCD [24],... 

Some people are confused by the difference between Fourier filters and linear smoothing and resolution functions. In fact, both methods are equivalent and are related... 

Fig. 9. X-ray and UV photoelectron spectra of the conduction bands of Au and a theoretical density of states curve. For purposes of comparison the theoretical curve includes broadening with a 0.6 eV resolution function. Ref. (32)...

The resolution function and short lifetimes remain rather constant throughout the porosity range. The longer lifetimes increase steadily with porosity up to about 50% porogen load. At larger loads the lifetimes remain similar while their intensities drop. 

More information can be extracted when the data are deconvoluted from the experimental resolution and the backscattering component and separate lifetimes are extracted with PATFIT to obtain the experimental resolution function and MELT to obtain lifetime distributions as shown in Fig 7.19 for the case of 23% and 80% porogen load. The probabilities were scaled to the peak value for the component at 0.5 ns and are enhanced by factors of 10 and 200 for lifetimes larger than 2 ns and 7 ns respectively. 

The analysis of the positron annihilation lifetime spectra is a very important aspect of using the PAL techniques to analyze polymers. Without proper data analysis interpretation of data might be misleading and important scientific information will be lost. In PAL studies of polymers the PAL spectrum can be analyzed in two ways (1) a finite lifetime analysis or (2) continuous lifetime analysis. In the finite lifetime analysis the PAL spectra is resolved into a finite number of negative exponentials decays. The experimental data y(t) is expressed as a convoluted expression (by a symbol ) of the instalment resolution function R(t) and a finite number (n) of negative exponentials ... 

In order to take advantage of the full information contained in the AMOC data we use a two-dimensional fitting procedure A two-dimensional model function representing the number of counts as a function of positron age and energy of the annihilation quanta is fitted to the raw AMOC relief without prior data reduction. On the age axis, each positron state is represented by an exponential decay function convoluted with the time resolution function of... 

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