# A Small & large argument expansion of plasma dispersion function

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1. Nov 26, 2018

### sreerajt

In plasma physics we have what is known as plasma dispersion function. There are two approximation under which this function can be expanded: when the argument is less than 1, we can use power series expansion and when the argument is greater than 1 we can have asymptotic expression.

My question is, given any plasma system, how do we know that under which approximation we have to expand the plasma dispersion function? How does this change when one introduces magnetic fields into it?

2. Nov 30, 2018

### jasonRF

There is no easy answer to this. If the phenomenon you are studying can be (at least partially) described with fluid theory, it is often helpful to do the fluid theory calculation to gain physical insight to help determine what approximations might make sense in the kinetic theory calculation. Sometimes you just have to make a choice and see where the calculation leads you, then investigate the answer you get to see if it is of interest. If it isn't, then try the other approximation. But in general, if you are trying to understand a particular phenomenon or situation there is no guarantee that either of these expansions will be particularly useful and you may need to resort to numerically solving the actual dispersion relation.

When you add a zeroth order magnetic field then things get more complicated. You end up with a dielectric tensor, where each element contains an infinite series of plasma dispersion functions and modified Bessel functions (at least if you have a Gaussian zeroth-order distribution function). This tensor is then part of the dispersion relation. There are some analytical approximations that can be useful to gain insight into some important phenomena (see any textbook that covers plasma kinetic theory), but in general you numerically solve the dispersion relation. 20+ years ago there was open source code called WHAMP (Waves in Homogeneous Anisotropic Magnetized Plasmas) that you could get to do the numerics for you; I believe it came out of one of the Scandinavian groups, but I have been out of this field for a long time so cannot remember much. However, if I recall correctly WHAMP did require the user to provide a starting "guess", which you would need to get either from one of the large/small argument expansions, or (if appropriate) from a fluid theory calculation. I certainly remember doing a lot of analysis to even get WHAMP started on the dispersion surface I was interested in. Once it found part of the surface, you can then have it follow the surface into the regimes where the analytical approximations do not hold.

Please don't get discouraged if you struggle with this - kinetic theory is complicated and the messy mathematics can obscure the physics - so it is not easy to do this stuff. Even after a graduate course in kinetic theory of plasmas and a subsequent course on plasma waves (that included waves requiring kinetic theory to understand) I didn't find it easy when needing to use it during my research. In grad school it was comforting to hear well-established faculty and research scientists arguing at the blackboard about just this sort of stuff when working on research problems.

Jason

Last edited: Nov 30, 2018
3. Dec 1, 2018

### sreerajt

Thanks you for the reply. Seems like I am not the only one who has this doubt ...