Linearity response index

Up to this point, we have calculated the linear response of the medium, a polarization oscillating at the frequency m of the applied field. This polarization produces its own radiation field that interferes with the applied optical field. Two familiar effects result a change in tlie speed of the light wave and its attenuation as it propagates. These properties may be related directly to the linear susceptibility The index of... 

It is seen that, for a truly linear detector, the response index (a) will be unity and the experimentally determined value of (a) will be an accurate measure of the proximity of the response to strict linearity. It is also clear that (a) could be used to correct for any non-linearity that might occur in the detector and thus improve the accuracy of an analysis. 

It is seen that errors in the smaller component can be as great as 12.5% (1.25% absolute) when the response index is 0.94. Yet on examining the curve for a response index of 0.94 in figure 2 the non-linearity is scarcely apparent. When the response index is 1.05 the error is 9.5% (0.95% absolute) and again the poor linearity is not obvious in figure 2. As already stated, to obtain accurate results without employing a correction factor, the response index should lie between 0.98 and 1.02. Most LC detectors can be designed to meet this linearity criteria. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

Detection requirements in preparative-scale chromatography also differ from analytical erations where detectors are selected for their sensitivity. Sensitivity is not of overriding importance in preparative-scale chromatography the ability to accommodate large column flow rates and a wide linear response range are more useful. The sensitivity of the refractive index detector is usually quite adequate for prqtaratlve work but the ... 

The ideal HPLC detector should have the same characteristics as those required for GC detectors, i.e. rapid and reproducible response to solutes, a wide range of linear response, high sensitivity and stability of operation. No truly universal HPLC detector has yet been developed but the two most widely applicable types are those based on the absorption of UV or visible radiation by the solute species and those which monitor refractive index differences between solutes dissolved in the mobile phase and the pure mobile phase. Other detectors which are more selective in their response rely on such solute properties as fluorescence, electrical conductivity, diffusion currents (amperometric) and radioactivity. The characteristics of the various types of detector are summarized in Table 4.14. 

In this equation AN, ANs, and Av(r) are the changes with respect to each variable in the expansion x°(r, r ) is the linear response function at the reference ground state, and the other quantities have been already defined in the previous section the upper index 0 indicates that all reactivity indexes are evaluated at the reference state. 

Dealing with the (2D+T) problem of TE wave propagation, one has to take into account some additional terms responsible for light beam diffraction in the waveguide with linear refractive index profile nfc) ... 

Figure G1.8.4 An FID chromatogram of concentrated extract of Niagara grape juice drawn to display the data on a linear retention index scale where the y axis is flame ionization response (upper trace). Below it is the charm chromatogram, where the / axis is dilution value. By simply comparing the index of a peak with the data listed in the Flavornet (see Internet Resource), it is possible to determine which odorants have similar retention indices. Notice how large the methyl anthranilate peak is, whereas there is no convincing peak for p-damascenone in the FID chromatogram, even though both compounds have the same potency in the charm chromatogram.

Figure 15.3 Temperature dependences of the linear thermal expansion, Al/l [2], refractive index, n [3] and reciprocal dielectric permittivity, 1 /x (Samara, unpublished) for pmn showing deviations from linear response at a temperature (I d) much higher than the peak (Tm) in the dielectric susceptibility (from [14]).

However, the linear response of a dielectric to an applied field is an approximation the actual response is non-linear and is of the form indicated in Fig. 8.6. The electro-optic effect has its origins in this non-linearity, and the very large electric fields associated with high-intensity laser light lead to the non-linear optics technology discussed briefly in Section 8.1.4. Clearly the permittivity measured for small increments in field depends on the biasing field E0, from which it follows that the refractive index also depends on E0. The dependence can be expressed by the following polynomial ... 

This method for defining detector linearity is perfectly satisfactory and ensures a minimum linearity from the detector and consequently an acceptable quantitative accuracy. However, the specification is significantly looser than that given above and there is no means of correcting for any non-linearity that may exist as there is no correction factor given that is equivalent to the response index. It is strongly advised that the response index of all detectors (CiC and LC)... 

Thus the slope of the Log/Log curve will give the value of the response index (r). If the detector is truly linear, r = 1 i.e. the slope of the curve will be sin 7r/4 =1). Alternatively, if suitable software is available, the data can be curved fitted to a power function and the value of (r) extracted from the results. The same data can be employed to determine the linear range as defined by the ASTM E19 committee. In this case, however, a linear plot of detector output against solute concentration at the peak maximum should be used and the point where the line deviates from 45° by 5% determines the limit of the linear dynamic range. 

Thus if the logarithm of the detector output is plotted against time, then for a truly linear detector, a straight line will be produced having a slope (-QA )- If the detector has a response index of (r) and the... 

The Response Index - (r) - The response index of detector is a measure of its linearity and for a truly linear detector would take the value of unity. In practice the value of (r) should lie between 0.98 and 1.02. If (r) is known, quantitative results can be corrected for any nonlinearity. 

Linear Dynamic Range - (D ) - The linear dynamic range of a detector is that concentration range over which the detector response is linear within defined response index limits. It is also dimensionless and is taken as the ratio of the concentration at which the response index falls outside its defined limits, to the minimum detectable concentration or sensitivity. The linear dynamic range is important when the components of a mixture being analyzed cover a wide concentration range. 

This again emphasizes the need for an improved procedure for defining detector specifications. The linear dynamic range of the electron capture detector is again ill-defined by many manufacturers. In the DC mode the linear dynamic range is usually relatively small, perhaps two orders of magnitude, with the response index lying... 

The sensitivity should be given as that solute concentration that produces a signal equivalent to twice the noise. Such data allows a rational comparison between detectors. The linear dynamic range is also not precisely clear from the original publication but appears to be at least three orders of magnitude for a response index of (r) where 0.97 < r < 1.03, but this is an estimate from the data published. The... 

The out-of-balance signal caused by the presence of sample vapor in contact with the sensor is amplified and fed to a recorder or computer data acquisition system. For maximum sensitivity hydrogen should be used as the carrier gas, but to reduce fire hazards, helium can be used with very little compromise in sensitivity. The sensitivity of the katherometer is only about 10 g/ml (probably the least sensitive of all GC detectors) and has a linear dynamic range of about 500 (the response index being between 0.98 and 1.02). Although the least glamorous, this detector can be used in most GC analyses that utilize packed columns and where there is no limitation in sample availability. The device is simple, reliable, and rugged and, as already stated, relatively inexpensive. 

The detector was claimed to be moderately linear over a dynamic range of three orders of magnitude but values for the response index are not known. It is also not clear whether the associated electronics contained signal modifying circuitry or not. The disadvantages of this detector included erosion of the electrodes due to "spluttering", contamination of the electrodes from sample decomposition and the need for a well-controlled vacuum system. 

As a result of limited sensitivity and restricted linear dynamic range, the refractive index detector is often a "choice of last resort" and is used for those applications where, for one reason or another, all other detectors are inappropriate or impractical. However, the detector has one particular area of application for which it is unique and that is in the separation and analysis of polymers. In general, for those polymers that contain more than six monomer units, the refractive index is directly proportional to the concentration of the polymer and is practically independent of the molecular weight. Thus, a quantitative analysis of a polymer mixture can be obtained by the simple normalization of the peak areas in the chromatogram, there being no need for the use of individual response factors. The sensitivity of most RI detectors will be about 1 x 10 g/ml and the linear dynamic range around 1 x 10 to 2 X 10 4 g/ml with the response index (r) lying between 0.97 and 1.03. 

The detector response index has been featured as a means of defining linearity, but it can also be used to take into account any non-linearity that is present and appropriately modify the peak height or peak area calculations and thus improve quantitative accuracy. An example of two peaks constructed from Gaussian functions using response factors of 0.95 and 1.05 are shown in figure 5. Such values were considered to be outside those which would be acceptable for a detector to be defined as linear. 

A) is the linearity constant, and (r) is the numerical value of the response index. 

Now if the response index (r) is not unity, which will be the case for virtually all practical detectors, then the value of (r) should be taken into account in the expressions used to obtain quantitative results. Consider firstly the computer integration of the area. The area is calculated by summing all the signals from the detector over the period of the peak. Now, if (r l) then, from equation (2), the area that will incorporate the non-linearity will be... 

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