Peak-valley ratio definitions

Three definitions of peak-valley ratios are illustrated in figure 4.2. All of them express the extent of separation as some measure of the depth of the valley between two peaks divided by some measure of the peak height. The first criterion (P) measures the depth of the valley relative to the interpolated peak height as shown in figure 4.2.a. The corresponding expression is ... 

Figure 4.2 Three definitions for peak-valley ratios as elemental criteria to quantify the extent of separation between a pair of adjacent peaks in a chromatogram, (a) Peak-valley ratio (P eqn.4.3) according to Kaiser, (b) median peak-valley ratio (Pm eqn.4.4) according to Schupp and (c) (opposite page) the valley-to-top ratio (P eqn.4.5) according to Christophe. 

The peak-valley ratios vary from zero for separations where no valley can be detected, to unity for complete separation. It ought to be noticed that a P value equal to zero does not necessarily imply that two solutes elute with exactly the same retention time (or k value). There is a threshold separation below which the presence of two individual bands in one peak only leads to peak broadening or deformation, without the occurrence of a valley. In these cases Rs values are indeed not equal to zero, because by definition (eqn.1.14) Rs is proportional to the difference in retention times. 

For Gaussian peaks of equal height the value of the peak-valley ratio (then the same according to all three definitions) can readily be expressed in terms of Rs. This can be done by relating the parameters f, g and v (see figure 4.2) to the parameters that describe a Gaussian peak (trand h). For the first of a pair of Gaussian peaks (peak A) we can write... 

A comparison of various elemental criteria has been reported by Knoll and Midgett [412] and by Debets et al. [413]. Figure 4.4 shows the variation of some of the criteria for the separation of pairs of chromatographic peaks as a function of the time difference between the peak tops (At = t2 — t,). By definition, Rs (and hence S) varies linearly with At. The peak-valley ratios (P) and the fractional overlap both increase rapidly with increasing At at first, but level off towards At 4 ct to reach the limiting value of 1. At high values of At, Rs and S will keep increasing, while the other criteria will not. 

Let two peaks of equal height in a mass spectrum at masses m and m, Am, be separated by a valley which at its lowest point is just 10% of the height of either peak. For similar peaks at a mass exceeding m, let the height of the valley at its lowest point be more (by any amount) than 10% of either peak. Then the resolution (10% valley definition) is m/Am. The ratio m/Am should be given for a number of values of m [4], Comment. This is a typical example of the confusion regarding the definition of the term resolution. Here resolution is used instead of the more appropriate phrase mass resolving power (which is the inverse of resolution). 

Resolution the ratio m/Sm where m and m + Sm are the relative masses of the two ions that yield neighbouring peaks with a valley depth x % of the weakest peak s intensity. In the commercial description of mass spectrometers, x = 50 is normally used. However, 10 % valley has been largely used in the past. Another definition entails using for Sm the width of an isolated peak at x % of its maximum. 

Two neighboring peaks are assumed to be sufficiently separated when the valley separating their maxima has decreased to 10% of their intensity. Hence, this is known as 10% valley definition of resolution, Rim,- The 10% valley conditions are fulfilled if the peak width at 5% relative height equals the mass difference of the corresponding ions, because then the 5% contribution of each peak to the same point of the m/z axis adds up to 10% (Fig. 3.13). With the advent of linear quadru-pole analyzers the full width at half maximum (FWHM) definition of resolution became widespread especially among instruments manufacturers. It is also commonly used for time-of-flight and quadrupole ion trap mass analyzers. With Gaussian peak shapes, the ratio of / fwhm to Rim, is 1.8. The resolution for a pair of peaks at different m/z and its practical implications are illustrated below (Fig. 3.14). 

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