Linear mass bias model

Ri/2 — R1/3R3/2 and rxii = rxiiriji Substituting this into the linear mass bias model, we obtain ... 

Inherent to all mass bias models stemming from Eq. (5.7) is the built-in variable (discrimination exponent) that distinguishes the various mass discrimination phenomena. Hence we have linear, exponential, equilibrium, power, and other discrimination models. This variable, in turn, is often used to identify the presence of a particular discrimination model [32, 41], Consequently, considerable effort has been spent in extracting the numerical value of the mass bias discrimination... 

Like the traditional mass bias correction approaches, the double-spike method also relies on the choice of the mass bias model. The original formalism of the double spikes employed the linear mass bias law and, although double-spike calibration equations adapted for the exponential mass bias discrimination are available, linear models are still often used owing to their simplicity (see, for example, [50-52]). The caveat here is that erroneous results can be obtained when a linear correction is applied to data that do not follow such behavior. This is illustrated below. 

To date, interrogation of the efficacy of mass bias correction models has largely resorted to attempts to determine the value of the discrimination exponent. In such experiments, the slope of the log-linear two-isotope ratio regression is used, which, in turn, leads to the discrimination exponent by solving the following expression [43-46] ... 

This is the model equation for the calibration of isotope amount ratios based on the log-linear temporal isotope amount ratio regression. Note that a and b are perfectly correlated (p = +1) if Rtp < 1 (inRup < 0) and perfectly anti-correlated (p = —1) if Rk/i > 1 (InRfe/ > 0). It is important to stress that this calibration method is fundamentally different from the conventional mass bias correction la vs. Since the regression model does not invoke the principle of time-mass separation, it does not need either the discrimination exponent or the equality of the discrimination functions [17]. 

What is more important, this calibration model is not derived from either the exponential or Russell s laws as is commonly perceived (and originally presented) [15, 32], Rather, it only requires the mass spectrometer response be linear [Eqs. (5.27) and (5.28)]. It is the interpretation of the slope and the intercept that can lead to the reliance on the exponential mass bias correction or even erroneous results. Consider, for example, the substitution of Eq. (5.34) in Eq. (5.32) ... 

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