  The Hall effect arises in a plasma when electrons are able to drift with the
magnetic field but ions cannot. In a fully-ionized plasma this occurs for
frequencies between the ion and electron cyclotron frequencies because of the
larger ion inertia. Typically this frequency range lies well above the
frequencies of interest (such as the dynamical frequency of the system under
consideration) and can be ignored. In a weakly-ionized medium, however, the
Hall effect arises through a different mechanism -- neutral collisions
preferentially decouple ions from the magnetic field. This typically occurs at
much lower frequencies and the Hall effect may play an important role in the
dynamics of weakly-ionised systems such as the Earth's ionosphere and
protoplanetary discs.
  To clarify the relationship between these mechanisms we develop an
approximate single-fluid description of a partially ionized plasma that becomes
exact in the fully-ionized and weakly-ionized limits. Our treatment includes
the effects of ohmic, ambipolar, and Hall diffusion. We show that the Hall
effect is relevant to the dynamics of a partially ionized medium when the
dynamical frequency exceeds the ratio of ion to bulk mass density times the
ion-cyclotron frequency, i.e. the Hall frequency. The corresponding length
scale is inversely proportional to the ion to bulk mass density ratio as well
as to the ion-Hall beta parameter.
