  During the previous decade, K.L. Jensen et. al. developed a general
analytical model that successfully describes electron emission from metals both
in the field and thermionic regimes, as well as in the transition region. In
that development, the standard image corrected triangular potential barrier was
used. This barrier model is valid only for planar surfaces and therefore cannot
be used in general for modern nanometric emitters. In a recent publication the
authors showed that the standard Fowler-Nordheim theory can be generalized for
highly curved emitters if a quadratic term is included to the potential model.
In this paper we extend this generalization for high temperatures and include
both the thermal and intermediate regimes. This is achieved by applying the
general method developed by Jensen to the quadratic barrier model of our
previous publication. We obtain results that are in good agreement with fully
numerical calculations for radii $R>4nm$, while our calculated current density
differs by a factor up to 27 from the one predicted by the Jensen's standard
General-Thermal-Field (GTF) equation. Our extended GTF equation has application
to modern sharp electron sources, beam simulation models and vacuum breakdown
theory.
