In plasma physics, Langmuir waves are oscillations in the electron density,
which may or may not propagate, depending on the temperature.
Assuming no magnetic field ,
no ion motion (since ),
and therefore no ion-electron momentum transfer,
the two-fluid equations
tell us that:
These are the electron momentum and continuity equations.
We also need Gauss’ law:
We split , and into a base component
(subscript ) and a perturbation (subscript ):
Where the perturbations , and are very small,
and the equilibrium components , and
are assumed to satisfy:
We insert this decomposition into the electron continuity equation,
arguing that is small enough to neglect, leading to:
Likewise, we insert it into Gauss’ law,
and use the plasma’s quasi-neutrality to get:
Since we are looking for linear waves,
we make the following ansatz for the perturbations:
Inserting this into the continuity equation and Gauss’ law yields, respectively:
However, there are three unknowns , and ,
so one more equation is needed.
Cold Langmuir waves
We therefore turn to the electron momentum equation.
For now, let us assume that the electrons have no thermal motion,
i.e. the electron temperature , so that , leaving:
Inserting the decomposition then gives the following,
where we neglect
because is so small by assumption:
And then inserting our plane-wave ansatz yields
the third equation we were looking for:
Solving this system of three equations for
gives the following dispersion relation:
This result is known as the plasma frequency ,
and describes the frequency of cold Langmuir waves,
otherwise known as plasma oscillations:
Note that this is a dispersion relation ,
but that does not contain .
This means that cold Langmuir waves do not propagate:
the oscillation is stationary.
Warm Langmuir waves
Next, we generalize this result to nonzero ,
in which case the pressure is involved:
From the two-fluid thermodynamic equation of state,
we know that can be written as:
With this, insertion of our plane-wave ansatz
into the electron equation results in:
Which once again closes the system of three equations.
Solving for then gives:
Recognizing the first term as the plasma frequency ,
we therefore arrive at the Bohm-Gross dispersion relation
for warm Langmuir waves:
This expression is typically quoted for 1D oscillations,
in which case and :
Unlike for , these “warm” waves do propagate,
carrying information at group velocity ,
which, in the limit of large , is given by:
This is the root-mean-square velocity of the
Maxwell-Boltzmann speed distribution,
meaning that information travels at the thermal velocity for large .
- F.F. Chen,
Introduction to plasma physics and controlled fusion,
3rd edition, Springer.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,