= Window-capture sensitivity to mass offset
<Window-capture sensitivity to mass offset> describes how the <extracted-ion window capture fraction> changes when an ion peak shifts relative to a fixed <extracted ion mass window>. The shift is in the reported mass coordinate; it can change the selected signal even if the target's total peak area is constant.
For a normalized peak shape $p(m-\delta)$ with offset $\delta$ and window boundaries $a,b$, write
$$
F(\delta)=\int_a^b p(m-\delta)\,dm.
$$
Differentiation gives
$$
F'(\delta)=p(a-\delta)-p(b-\delta).
$$
The first-order response is determined by the signal density at the two boundaries. A symmetric peak centered in a symmetric window has equal boundary densities and hence zero first derivative. Zero first derivative does not mean perfect immunity: finite offsets can still reduce capture.
For an original <Gaussian function> model with <standard deviation> $\sigma$ and half-width $w$, let $\phi$ and $\Phi$ be the <standard normal density> and <standard normal cumulative distribution function>. Then
$$
F(\delta)=\Phi\!\left(\frac{w-\delta}{\sigma}\right)-\Phi\!\left(\frac{-w-\delta}{\sigma}\right),\qquad
F''(0)=-\frac{2w}{\sigma^3}\phi(w/\sigma).
$$
For small offsets the leading loss is consequently quadratic:
$$
F(\delta)\approx F(0)-\frac{w\phi(w/\sigma)}{\sigma^3}\delta^2.
$$
With $w=2\sigma$, the exact capture is 0.9545 at zero offset and 0.9270 at $\delta=0.5\sigma$. The quadratic approximation gives 0.9275, close for this stipulated displacement. At the boundary $\delta=2\sigma$, capture is about 0.5000 and a small-offset approximation is no longer appropriate.
An experimental check can re-extract the same recorded spectra with deliberately shifted windows while retaining the same time limits and <extracted-ion chromatogram baseline>. This isolates a software-selection effect. IUPAC's selected-mass profile definition supplies the measurement setting, rather than a claim that all instruments have Gaussian peaks.<Extracted ion profile in IUPAC Gold Book>[Entry 12410, selected mass-coordinate signal.]{ref} Wider windows may reduce target losses but admit interfering ions; <mass resolving power> and <instrument calibration> remain separate instrumental considerations.
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