Science/Engineering/Technical forums

I happened to need the equation for this while working up a new design for a low energy drift-tube RF accelerator (nothing so fancy as an RFQ, just the simple version). Strangely, it doesn't exist in any of my physics books as such -- I guess you're supposed to know Newton's laws, and wade through all the unit conversions forced on us by those who never want to have to write pi or c in equations (ESU, EMU, SI and so forth). Since nearly all modern texts ignore the specification of what units they use, and often mix them, there's no way to find out from the text alone -- you need a sanity check, a practical case, else you're likely to be off by pi or c (or some simple factor of those).
As usual, Fredrick Terman's Radio engineers handbook has it all when it counts for a practical engineer. I thought I'd put it here in case I lose that page again (these old WWII books fall apart on you and are getting scarce used).

For electrons:

V = 5.97 x 107 x sqrt(E) cm per sec. E in plain old volts. Mass contained in the constant.

For other charged particles:

V = 5.97 107 sqrt(EnMe/m) cm per sec. E in volts, Me mass of electron, M mass of actual ion, n number of unit charges on the ion

(this is page 275 in my copy, for reference) Of course, this is for non relativistic situations.

People wanting to mess with charged particles and magnets might also find this page scan interesting. Sorry 'bout the focus here, I have just one good copy of this and I'm not going to cut it up to lay flat. Once again, real units!

Here is *mostly* the same information from Spangenberg/Terman about 5 years later. Kinda splitting hairs, but the numbers have changed slightly as the electron mass was determined more accurately. They're probably different again now. The one thing worth noting though is that the numbers only work up to 30kV potentials. I presume the 30kV limit only applies to electrons because of relativistic effects?

I tend to view electrons as going relativistic around 1 kV, let alone 30 kV! But it depends
on the accuracy you need.

The 'exact' formula for lab-frame velocity is;
v=SQRT(c^2-[c^3/({E/m}+c^2)]^2)

[use SI units - so 'E' is in Joules, the voltage conversion being x 1.6E-19 J/V, m in kg and c in m/s].

OK
Andreas did this calc recently and for our implanters Ions are travelling at around 200 km/sec to 600 km /sec for voltages of 5 to 85 kV all depending on mass of ion, charge state, (and which way the wind is blowing).

but non of us are considering electrons ---these are pesky after ions are produced --unless you want to neutralise charge.

To not count out electrons --they are required to get our ions in the first place --a bit like the male of some species --not needed after that

all for now

Interestingly (ish) for light element fusion, all closing velocities are between 1 Mm/s and 10 Mm/s.

Thanks for the update, Joe. Looks like electrons started the drive to obesity after the war, eh?
As John says, electrons are mostly a pain, unless making an ion source. I just grabbed that source of the equation because it's normal units (yeah, measure electrons in kG - that's practical ), and it's easy to convert for our ions. Electrons do get noticeably relativistic very easily, so the correction there is useful (Terman has it too), but in general, ions won't get close enough to need the correction in our world. The drift tube design has some "slop" due to phase bunching, at any rate -- and a little adjustment to F or V can bring a properly scaled design into resonance.

Now the next evolution of a practical and useful equation to characterize some of this would be one that gave a transit time from a standing start for some charged particle in a given field over a distance. Even with the faster electrons, it's an issue in vacuum tubes over a certain frequency -- diodes can even go to negative resistance due to that and space charge for example. This would be nice to know to get transit times of ions in a fusor -- that would be a large clue as to what frequency you might want to "tickle" a fusor at to enhance, or drive, recirculation, for example. I've come up with some characteristic frequencies in measurements here (there is more than one) -- which don't make sense if you assume the ions see the full applied field. So in this case, if you had that math, you could work backwards and find the actual field they see and learn something no one seems to have measured and reported on.

For time of flight over a distance s, the eqn of motion is s=1/2{at^2}

As the force on a partice of charge q, is Eq, so its acceleration is its mass, m, times acceleration = Eq, or Eq=ma, a=Eq/m

So time of flight is;

t=sqrt(2.s.m/Eq), s in metres, E in V/m, q in Coulombs, m in kg

.. I think that's right for a uniform field?!



EXAMPLE:

Let's work with deuterons, then. Only one q of charge. And I'll work it into centimeters, which makes it a bit more 'meaningful' on our scales.

The time, t, in microseconds, taken for a deuteron to cross a potential, V, between electrodes, D centimeters apart, is;

t = 2.D SQRT[ 1/V ]

Thanks for the very useful thread - this is something that I really struggle with - the maths is bad enough without all the unit conversions, or worse, units 'assumed'. I have been trying to write an ion simulation, in order to try and model approximate trajectories of a small number of ions in electric and magnetic fields. It all came unstuck when I tried to model the electric field density distribution on a non-symmetrical conductor. In the image, red is magnetic field, green is electric field and yellow is 'overlap'. This run is of a cyclotron type configuration, and is obviously simplified - I have other attempts using more accurate non-homogenous fields. Real world numbers and examples are much appreciated!

The generalisation of the above formula - the time taken between electrodes, assuming uniform field;

The time t, in microseconds, taken for an initially stationary particle to cross a potential of V volts between electrodes D centimeters apart, is;

t = D.(SQRT[ 2A/Vq])

A = Particle's atomic mass
q = Particle's unit charge

Good one Chris. This is what's needed to get the feel for gating something like an ion source (along with the other equation). It would take some finite time from acceleration voltage applied to ions leaving the source, and it should be fairly obvious that you can't cheat that time very hard at all. This is the key to the puzzle of transit times in electron tubes as well. With ions, we have all the same effects, only a lot slower.