  The microbunching instability usually exists in the LINAC of a free electron
laser (FEL) facility. In many cases, the longitudinal space charge (LSC) is a
dominant factor that generates the instability. For the highly bright electron
beams, the plasma effect is found to be non-trivial in the development of the
instability. In this paper, starting from the Vlasov and Poisson equations in
the multiple-dimensional phase space, we perform the straightforward analysis
of the microbunching instability based on the explicit formula of the
longitudinal electric field introduced by the density perturbation in the
longitudinal direction, in such a way to be highly comparable to the
well-developed method for higher energy beams. This method generally applies in
both the cases with and without acceleration and independent of lattice
components. The results show that for a electron beam with small transverse
emittance at low energies, which is always the case in the injector of a free
electron laser device, the plasma effect results in the oscillation of the
longitudinal electric field in the modified plasma frequency that depends on
the transverse size of the beam, and the Landau damping effect in the
longitudinal electric field due to the uncorrelated longitudinal velocity
spread during the beam transportation. These two effects both play important
roles in the development of the instability. As the result, the energy
modulation driven by the LSC impedance differs from the regular value
significantly and the discrepancy leads to the noticeable change of the final
gain of the instability.
