A Mass measurement in a Penning trap

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The discussion centers on measuring the mass of a molecular ion in a Penning trap, highlighting the impact of polarization on mass measurements. The author derives an effective mass formula, incorporating an induced electric dipole, which aligns with existing research findings. They explore the implications of assuming a fully polarized molecule, leading to a different treatment of the dipole moment in the Lagrangian. The challenge arises in interpreting the term related to the intrinsic dipole moment when factoring in the effects of polarization. The author seeks clarification on how to accurately account for this term in the context of mass measurement.
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Hello! My question is motivated by this paper (also attached below). They are measuring the mass of a molecular ion in a Penning trap, and they are able to see a difference due to the fact that the molecule gets polarized (the motion is classical and non-relativistic). I was able to derive their result, for an induced electric dipole, using the Lagrangian:

$$L = mv^2/2 + \alpha E^2/2$$
where ##\alpha## is the polarization and ##E## is the electric field. If we use the fact that ##E = vB## we can see from the form of the Lagrangian, if we take the derivative with respect to ##v##:

$$\frac{\partial L}{\partial v} = mv + \alpha v B^2 = (m+\alpha B)v$$
From this we get an effective mass of:

$$m+\alpha B^2$$
which is consistent with their result. However, I was wondering, if we assume that the molecule is highly (or fully) polarized and not just weakly, instead of ##\alpha E^2/2## we have simply ##d E##, where ##d## is the intrinsic dipole moment of the molecule. However, assuming ##d## is constant, which is (very close to being) true for a fully polarized molecule, we have ##d E = dvB## which gives:

$$\frac{\partial L}{\partial v} = mv + dB$$
now we can't factor out ##v## anymore and thus it's not clear anymore how to count ##dB## towards the mass of the molecule. However, intuitively, I would expect that the higher the polarization, the higher the shift in the measured mass. What am I doing wrong, or how should I interpret the ##dB## term in this case? Thank you!
 
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