Neutral collective excitation in fractional quantum Hall effect at Jain series

We have studied collective excitation almost all of the filling fraction, in the Jain series of positive flux attachment states. In this article we have presented excitation spectrum of all positive flux attachment states for a real two dimensional electron system to get the complete scenario. The fundamental roton energy becomes zero at the filling fractions associated with four pair of magnetic flux quanta attachment composite fermion, which suggest that there is a limitation of flux attachment to the electron to form a composite fermion in fractional quantum Hall effect. The roton energy and energy gap become zero at filling fraction ν=1/2, which shows that the gapless state ν=1/2 can be naturally obtained as an extrapolation of the collective excitation of the incompressible gaped states.