I am not sure if this can be solved in an exact way analytically, but you can try to make a reasonable estimation as follows:

The two components of the mean-free path (mfp) normal to the B-field lines should not be altered very much and you know the mean free path without B-field (or can calculate it). In the case of strong B-fields, which you have, the gyro radius will be much smaller than the non-magnetic mfp.

So, you can calculate the total path length along your field lines between two collisions. The length of field line is equal to the length of mfp in the non-magentic case. So you have to add the length of your helical path due to gyrational motion. This you can do with help of you knowledge of your average kinetic energy of your particle (i.e. Te). During one gyrative motion of 2*Pi*rgyr the particle will travel a little bit along your B-field line.

When you have calculated your total path length between two collions in the manner above, you can just make an average over the two mean free paths normal to the magnetic field line and the one contribution along the B-field line.

That should give you a reasonable result, I guess.

The shape of the path taken by a charged particle won't affect the distance it can travel between collisions. The mean free path remains the same.

Reference https://www.physicsforums.com/threads/the-mean-free-path-and-the-magnetic-field.411482/

@Vladimir Kolobov, of course the shape of path influences the length of way between two points (if not, you would have an infinite number of shortest distances between these two points and not only the one given by a straight line).

Even this would be the case, the B-field would have still an influence as it also changes plasma confinement properties (usually rising the electron density) - which then alters the mfp (when there are more targets to hit in a given volume, collisions will be more frequently).

However I got interested in the question and did some literature research - usually in a strong B-field it is common to substitute the mfp with the gyro radius (see, for example, A. Dinklage et. al. Plasma Physics: Confinement, Transport and Collective Effects, Springer 2005).

This makes absolute sense as strong collective motion along the B-field lines will greatly reduce collisions in this direction, whereas normal to the field lines the number of collisions will be vastly enhanced.

I am not an expert in this kind of problems. I think that if the gyro radius is large with respect to the mean distance between heavy particles Vladimir is right.  I am thinking of an interesting possible situation.What happens if the gyro radius of an electron is much smaller than the distance between particles? I think that in this case, if we consider the length of the electron path as integral of vdt, I think that the distance between two collisions will be much longer in presence of magnetic field than in its absence. We should try to verify this case with a simple test particle Monte Carlo or as first approximation is is possible too considering a regular grid of colliders with the proper section. 

@Gianpiero:

In the case of such strong B-fields the pressure has to be very low in order to make the gyroradius larger than the mean distance between heavy particles. Furthermore the electrons have different cross sections than the heavy species.

In your situation you can use the approximation I gave above as this is basically exactly what my simple model does.

The mfp lambda is defined in the kinetic equation as

df/dt=vf/lambda

where f is the particle distribution function, df/dt  is its full time derivative along trajectory,

v is velocity. The trajectory does not  need to be a straight  line,

it could be Larmor radius (in magnetic field) or a more complex line (when electron bounces in an electrostatic trap).

I was thinking exactly to Vladimir's definition when I propose my example. If the Larmor radius is much smaller that the mean distance between the free particles, the mean free path becomes much longer than in absence of a magnetic field. That is why Johannes talked Moreover this definition presents some problems, because mfp depends on the velocity, while in a plasma it should be a global property. In the Vladimir's definition is equivalent (considering one term in the Boltzmann collision integral) to 1/(N sigma) where N is the density of collision centers and sigma is the (elastic/total) cross section, neglecting/weighting the motion of collision centers. 

One more comment.

The MFP is a characteristic of scattering media, which is calculated using differential collision cross section. It depends on electron energy but does not depend on specifics of electron trajectory or the values of electric or  magnetic fields.

The question you asked is a mixed bag. The Larmor radius for a 10 keV electron is .1 mm (10-4 m). The mean free path is dependent on the gas density and the energy of the electron. Collisions is how an electron crosses a magnetic field, as in a cold cathode or Penning vacuum gauge. The atomic cross section of some atoms will vary with the magnetic field (slightly). Where are you actually trying to go?

Dear all,

just to clearify terms. A magnetized plasma requires that at least the elctrons can make a full gyro-turn before they collide. if this condition is not fullfilled the plasma is rather isotropic an the influence of the magnetic field for the particle motion is rather unimportant.

The I think we all agree that collisions are independent of the direction of motion, just depend on the relative velocities of the involved two particles. The it does not matter that one motion is parallel to the magnetic while the other is a rotation around the field line.

Then we have the collision frequencies you can see in all text books of

nu_ee=n e^4 ln(Lambda)/(2 pi epsilon^2 m_e^2 v_th,e^3) for electron-electron collisions,

nu_ii=n e^4  ln(Lambda)/(2 pi epsilon^2 m_i^2 v_th,i^3) for ion-ion collisions,

n_ei = nu_ee(1-1/(2(v_th,e/v_th,i)^2)) for electron at ion collisions,

nu_ie=nu_ii 4/(3 sqrt(pi))(v_th,i)/v_th,e for ion at electron collisions

here epsilon is the dielectric constant, and Lambda the Coulomb logarithm - see text books.

Now, for magnetized plasmas you are usually interested in the mfp parallel to B =>

mfp = nu_xy v_parallel (depending on the collisions you are after). This also shows the typical effect of the strong magnetic field. The strong magnetic field often allows you to simplyfy you equations by separating parallel and perpendicular motion/transport etc.

Best regards,

Werner