I don't think that you can apply the above equation for microscopic deceleration, because the drag force in a fluid is so-to-speak, a kind of statistical effect - if you look at fluid the description is usually done by averaging over many single particle motions.

In your case I would guess that you need to look at deceleration by inelastic collisions between your protons and other particles. If you neglect ionising collisions (which I guess can also happen with 25 keV protons, but just for the sake of simplicity omit them for the moment), you can use the equation for the resulting velocities given here:

http://en.wikipedia.org/wiki/Inelastic_collision

but be carefull as this is only for one collision process - so you have to include your mean free path into your equation - compared with the geometrical dimensions of your experiment this will give you the number of collisions at a certain pressure.

OK, accordingly that I would need the cross section area for such collision over O2 and N2.

Accordingly that the rejected particles must be: N1*N2*cs/A where N1=incident particles, N2=air particles, cs=inelastic cross section and A the area of the beam

Perhaps I should include some statistical coefficient due N2 particles are not aligned

Why elastic collisions are not considered?

Could it be used following cross section data obtained from Exfor? (I should interpolate downto 25keV)

You could also take into account elasticc collisions, but I guess you will have some kind of bremsstrahlung when your protons hit a target --> thus the kinetic energy of the two particles is not conserved.

Well, the free path for atmospheric pressure is 58nm accordingly the equation. Accordingly that it is impossible to accelerate ions in air due particle collisions. All in Si units:

L=free path in meters, Kb=1.38e-23 J/K, P in pascals, d=particle diameter in meters and T is temperature in degrees K. I used an average air molecule diameter 0.4nm

The formula is valid for maxwellian distribution of particles

I would not say that it's impossible - it depends on how much energy the protons can gain between two collisions.

The proton area can be neglected - it is just too small. The size of an atom is mainly determined by the electrons turning around the nucleus, and because of these, negative ion (anion) has larger diameter than atom and positive ion (cation) has less diameter. The proton has no surrounding electron so that its area is very small. In collision with air, the cross-section is determined by air molecule's diameter only.

To accelerate the proton in air is very hard - one has to have large impractical e-field (about 25kV / lamda, where lamda is free path). When collision happens the proton will change its direction and part of energy will be transferred to the target in case of elastic collision and to radiation in case of inelastic collision. Usually one has to use vacuum environment to accelerator the particles - most accelerators works this way.

I agree with Hanwu that vacuum is needed, I calculated 0.2 pascals for a 29mm free path only, but I am not sure that the proton size is its diameter because they would interact with target atom electrons at a distance similar to a neutral hydrogen atom

I agree with Johannes, it could be done depending on energy: railguns works at free air, but a lot of energy is wasted

Dear colleagues, I think that at so high temperature, inelastic collisions cannot decelerate the projectile. Only elastic collisions are important. The cross section can be calculated in approximate way using the classical theory and approximating the potential as a spherical one. We have used this approach to calculate transport collision integrals and we have used a phenomenological potential. See the following papers as an example. We limit the projectile energy, well below the keV. But I think it can be applied to higher energy. Probably the approximation of spherical potential is not valid because suppose that the rotation is rapid with respect to the projectile motion. However, the orientation of molecules met in the collision is random, therefore probably it will still works.

Classical transport collision integrals for a Lennard-Jones like phenomenological model potential

A Laricchiuta, G Colonna, D Bruno, R Celiberto, C Gorse, F Pirani, ...

Chemical Physics Letters 445 (4), 133-139

On the possibility of using model potentials for collision integral calculations of interest for planetary atmospheres

M Capitelli, D Cappelletti, G Colonna, C Gorse, A Laricchiuta, G Liuti, ...

Chemical Physics 338 (1), 62-68.

General numerical algorithm for classical collision integral calculation

G Colonna, A Laricchiuta

Computer Physics Communications 178 (11), 809-816