In this paper we discuss the relevance of electron-nuclear double tran
sitions for the problem of the amplification of gamma rays. We conside
r the electron-nuclear coupled system of Eu-151 in a CaF2 lattice, in
the presence of a strong static magnetic field. We analyze the possibi
lity of pumping this electron-nuclear coupled system from the ground a
tomic electron Zeeman level to a higher electron level with the aid of
a microwave magnetic field, while the nucleus is simultaneously under
going a gamma-ray transition to the nuclear ground state. It is shown
that an overall inversion condition can be satisfied even in the absen
ce of inversion of the nuclear populations, provided that the ratio of
the electron populations in the excited and ground electron Zeeman st
ates has a sufficiently small value.