# Quadrupole MS: mass as a function of voltage

Every time I see an article about Quadrupole MS (e.g. here), they give formulas like:

$$a_x = \frac {8eU} {m r_0^2 \Omega^2}$$ and $$q_x = - \frac {4eV} {m r_0^2 \Omega^2}$$

I suppose the instrument changes voltage and keeps the rest constant, but I failed to find any article that would show direct formulas on how to calculate m/z from that. I could rearrange it like this:

$$\frac {m} {e} = \frac {8U} {m r_0^2 \Omega^2 a_x}$$ But this doesn't really help because $a_x$ itself is unknown and is derived from mass.

PS: this question is about Quadrupole MS though it looks to me that Ion Trap MS has very similar concepts and math.

• ?!? You don't seem to be asking about a Quadrupole Mass Analyzer but rather a Quadrupole Ion Trap. – MaxW Dec 1 '17 at 20:18
• My confusion is in part because you refer to Quadrupole MS which I infer to mean a Quadraople Mass Spectrometer. – MaxW Dec 2 '17 at 1:01
• My impression was that both Q Ion Trap and and Q Mass Analyzer (aka Filter?) use similar algorithms since all articles mention Mathieu equation. So I decided not to distinguish between them at the beginning. If I'm wrong, I'd greatly appreciate if you could explain the difference. At the moment though I'm mostly interested in the Q Mass Analyzer. As for MS - to me Spectrometer is a synonym of Analyzer (e.g. chem.libretexts.org/Core/Analytical_Chemistry/…). – Stanislav Bashkyrtsev Dec 2 '17 at 4:22
• And here is an example of source where they refer to the aforementioned equations in Quadrupole Mass Analyzer, and then discuss similar math in Ion Trap Analyzer: chromacademy.com/lms/sco36/Fundamental_LC-MS_Mass_Analysers.pdf – Stanislav Bashkyrtsev Dec 2 '17 at 8:13

These parameters arise from non-dimensionalization of the Mathieu equations for ion motion in exogenous, periodically varying electric fields (i.e. oscillating potentials). You can't calculate $$m/z$$ directly from $$a$$ and $$q$$. For a fixed $$U$$ and $$V$$, you can regard $$a$$ and $$q$$ as non-dimensional parameters that correspond to $$m/z$$. (Conversely, at fixed $$m/z$$ you can regard $$a$$ and $$q$$ as non-dimensional versions of the electrical potentials $$U$$ and $$V$$.) The reason that those parameters don't tell you everything is that you still have to solve the Mathieu equations.
An analogy might be to the Reynolds number for characterizing fluid flows. It, like $$a$$ and $$q$$, is a non-dimensional number that depends on properties of the molecules under study and on the system geometry. Knowing just the value of the Reynolds number is useful for understanding some of the general characteristics of a flow, but if you want to compute fluid motion, i.e. the velocity profile in a pipe, you still have to solve the Navier-Stokes equation.
For quadrupole mass analyzers, only certain regions of $$a$$-$$q$$ space result in stable ion trajectories (i.e. where ions don't collide with the surfaces of the four rods in the quadrupole). This map shows which values for $$a$$ and $$q$$ result in stable solutions to the Mathieu equations, and thus is called the Mathieu stability diagram. The same document you linked to shows an example of such a diagram:
Thus, to compute $$m/z$$, we need to impose the criterion that $$a$$ and $$q$$ are in a stable region of the stability diagram, more specifically in the region "A" shown in the diagram. By rough eyeballing, the "middle" of this diagram corresponds to about $$q = 0.75$$ and $$a = 0.07$$ or so. You can use either of those values to estimate $$m/z$$, but a more precise estimate, including line-widths etc., would require knowing both.