Background (isotropic screening): Consider an overall quasi neutral plasma. An ion/electron in a dense plasma attracts other opposite-charged particles and repels those with the same charge, thereby creating a "cloud" of net opposite charge around itself. It is known that microscopic cloud shields the particle's own charge, causing its Coulomb field to fall off exponentially. In short, the particles in the medium interact via a screened version of the, more fundamental, Coulomb interaction, see e.g. Wikipedia.

The standard derivation of this "screened interaction" is in terms of the screened Poisson equation, which can arise from different approaches to the problem, like the Thomas-Fermi approximation, Lindhard theory, or the treatment of Debye, see e.g. this paper and these notes.

In any case, the screening length $\lambda$ defines a screened potential (Yukawa potential) between the particles in the plasma:

$$ V(r) \propto r^{-1} \, e^{-r/ \lambda} $$

In this case, the screening length $\lambda $ is just a single number (the "cloud" is spherical)... but what if there is a strong enough magnetic field? Now we have a preferred direction, the one of the magnetic field, and also the "cloud" could be affected.

Question: The derivations of $\lambda$ I am aware of seem not to consider the possible presence of a macroscopic magnetic field. I suspect that, in the presence of a magnetic field that breaks the isotropy, we should have 2 different screening lengths in the plane orthogonal to the magnetic field and in the direction of the magnetic field: the polarization cloud is not spherical and the screened Coulomb interaction is not invariant under rotations, e.g.

$$ V \propto r^{-1} \, e^{-{r}_{1} /\lambda_1 - r_2/ \lambda_2} $$

where $r_{1,2}$ is the length of the projection of $r$ parallel or orthogonal to the magnetic field direction and $\lambda_{1,2}$ can depend on the magnetic field strength (when there is no macroscopic magnetic field, then $\lambda_{1}=\lambda_{2}$).

Is there something like an "anisotropic" version of the screened Yukawa potential? This should define the effective interaction potential between particles in, say, an overall neutral plasma of electrons and positive ions (or protons) in a strong external magnetic field. References are welcome.

Edit: After the comments and the forst answer, it still seems that we should find two different screening lengths (parallel and orthogonal to the magnetic field direction). In fact, the electrons will tend to "move in circles" orthogonally to the magnetic field, which may influence the screening.

  • $\begingroup$ An observer outside the plasma will see the bare excess charge due to the test particle inserted into the plasma, so the potential must fall of like $1/r$ for sufficiently large radii. So the Yukawa potential is not applicable here. On the other hand, if the plasma is quasi-neutral, there is no field at all outside the voilume. $\endgroup$
    – Thomas
    May 27, 2022 at 17:29
  • $\begingroup$ @Thomas, the Yukawa potential is a "renormalized" effective interaction for particles within the (overall charge-neutral) plasma. I am asking if the presence of a magnetic field impacts this effective interaction. $\endgroup$
    – Quillo
    May 27, 2022 at 17:47
  • $\begingroup$ The time averaged field in a homogeneous charge neutral plasma is zero everywhere. $\endgroup$
    – Thomas
    May 27, 2022 at 20:37
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  • $\begingroup$ @Thomas are you sure? How about MHD? $\endgroup$
    – Quillo
    May 27, 2022 at 23:41

1 Answer 1


I'm not an expert, but I think that there is no net effect, in part due to the Bohr-Van Leeuwen theorem.

First of all, a static magnetic field does not influence the thermal charge fluctuations which cause the screening, so the potential should not change.

You could think however, that the addition of an extra charge could somehow generate a magnetic field, which is where the mentioned theorem comes to the rescue. Since classically you can't have any magnetisation, you should not get a magnetic version of Yukawa potential.

Hope this helps.

  • $\begingroup$ Thank you for the hint. I thought that maybe we should find two different screening lengths (parallel and orthogonal to the magnetic field direction). In fact, the electrons will tend to "move in circles" orthogonally to the magnetic field, which may influence the calculation of the screening length. Regarding the theorem you are invoking: en.wikipedia.org/wiki/Bohr%E2%80%93Van_Leeuwen_theorem (for the interested reader) $\endgroup$
    – Quillo
    May 27, 2022 at 15:45

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