# Velocity distribution in ion source (electron bombardment) for Bainbridge mass spectrometer

Consider the following schematics of a Bainbridge mass spectrometer (Source: http://www.schoolphysics.co.uk/age16-19/Atomic%20physics/Atomic%20structure%20and%20ions/text/Mass_spectrometer/images/1.png)

Suppose the gas contains two sorts of atoms (for example two isotopes of one element) with different masses. The electric force between the electrodes of in the gas chamber which accelerates them is $qE$, where $E$ is the electric field and $q$ the charge of the ions. For simplicity consider the case that you have one ion of sort A and one of type B each with equal charge q, but different masses $m_A$ and $m_B$ with $m_A > m_B$. Then the velocity of the $A$ before entering the Wien-filter will be smaller than hat of atom $B$ (because of it's mass). Furthermore suppose that the Wien filter only lets pass particles with velocity $v_A$, which is by assumtion the same as the velocity of our atom $A$. Because $B$ is faster than $A$, ions of type $A$ will pass the Wien-filter, but ions of type $B$ will not.

But then the magnetic field after the Wien-filter would be useless, because only ions of type $A$ would pass the filter.

So I guess that there must be another reason why the velocity distributions of type $A$ and $B$ atoms overlap after leaving the ion source such that both types $A$ and $B$ could reach the magnetic field after the Wien-filter.

Why is this the case? How can one quantitatively estimate how large the difference of $m_A$ and $m_B$ may be such that the velocity distributions of $A$ and $B$ overlap? Can you give me a quantitaive example of $A$ and $B$ and the concrete velocity distributions from experiments?

I am also looking for good references where those questions are discussed.

• @Julia The existing answer seems pretty good to me. Can you elaborate a bit more about the points you feel are missing? I could see just a couple: (1) the velocities with which particles enter the Wien filter will depend on their mass (same accelerating potential): how does this affect things; (2) what are typical values for potential / magnetic field for a working mass spectrometer; (3) what is a good book describing these. Is that what you need, or is there more? The more specific your question, the better your chance that your bounty will be well spent. – Floris Jun 11 '15 at 3:22
• The main points are: -Get a realistic example with numbers -In particular the velocity distributions before entering the Wien filter -I am not sure if this type of spectrometer is really used with an electron bombardement source (rather a source which has a much broader velocity distribution); I don't know what is really used in this setup. -Good specific references are missing ... – Julia Jun 11 '15 at 19:13
• Since I saw the bounty get posted, I've tried a few times to see if there was any more information I could find to improve my answer. I can think it can be improved, but I don't know enough about mass spec machines to answer @Julia's questions. It looks like the only way to get those specs is to request them from the manufacturers, and I don't have enough experience with mass spec to know what questions to ask (or even if the manufacturer would know some of the answers). – Colin McFaul Jun 12 '15 at 21:44

The Lorentz force on a given particle in the Wien filter is $\vec{F} = q(\vec{v}\times\vec{B} + \vec{E})$. Assuming the velocities and the magnetic and electric field are all mutually perpendicular, and using your notation, this gives $v_A B = -E$ for a particle that passes straight through the filter with no deflection. So if we assume all particles are travelling in the same direction, we can let a general speed be $v = v_A + dv$. The force on a particle with such a speed is then $\vec{F} = q(\vec{v_A}\times\vec{B} + \vec{E}) + q\vec{dv}\times\vec{B} = qB(dv)$.
Say that the filter has a length $l$ in the direction of the particles' travel, and that the slit has a width $d$. Then applying the kinematics to a particle of mass $m$, we get that any particle with a speed such that $$\boxed{dv \leq \left(\frac{2v^2m}{l^2qB}\right)d}$$ will be allowed through the filter into main region of the spectrometer. You might be able to derive a relation between $v$ and $m$ based on the physics of the ion creation, but this works in general.
So why would you even want to do this two-step sifting process, anyway? The point of the initial filter is to narrow the range of particles that you're looking at. You then use the main part of the spectrometer to look more precisely at that set of particles. One typical use of mass spectrometry is to measure the abundance of different isotopes of a given element. In that case, the isotopes are your particles $A$ and $B$, and you want both of them to pass through the filter. You filter everything else, then optimize the main part of the spectrometer to distinguish between those two particles.