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Appearance energies are particularly import when it comes down to mass spectrometers that use electron impact bombardment. They typically denote the quantum energy value for fragmenting the molecule into two other molecules, for instance the fragmentation sequence $CH_4 + e(21.3eV) \rightarrow CH_3 + H^+ $ is given by NIST Webbook.

I understand that these appearance energies are typically found by using a power law fit on data collected at low energy electron bombardment sweeps. The equation used for the fitting typically being,

$$\begin{cases} I = 0 & x < AE\\ I = A(E-AE)^p & x\geq AE \end{cases}$$

Sometimes researchers use,

$$\begin{cases} I = 0 & x < AE_1\\ I = A(E-AE_1)^p & AE_1 \geq x \leq AE_2 \\ I = A(E-AE_1)^{p_1} + B(E-AE_2)^{p_2}& x >AE_2 \end{cases}$$

Which results in two appearance energies for the particular fragment investigated. If the appearance energy is quantum mechanical by nature then shouldn't there only be one possible energy value that could produce that fragment.

It has been suggested to me that the multiple appearance energies can be attributed to different 'channels' of electron interaction with the molecule(ie: an electron bombardment of Butanol [$C_4H_{10}O$] can produce an $H^+$ by breaking the $H-O$ bond or $H-C$ bond). However, if this were the case why are all appearance energies of larger molecules not multi-leveled (ie: n number of AEs where n is the number of ways to fragment the larger molecule into the fragment of interest)?

If any of my above understanding is incorrect please tell me, and if possible please explain to me how and why we are able to have two appearance energies for certain fragments.

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