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　The relativistic instabilities of electromagnetic ion cyclotron waves driven by
MeV ions are studied analytically and numerically. The instabilities are reactive
and come from the coupling of slow ions’ first-order pole and fast ions’ second-order
pole of the dielectric function. This essential extra mechanism is due to relativistic
effect. Owing to the wave magnetic field, novel characteristics such as Alfv´enic
behavior and instability transition are discovered and illuminated in detail. The
wave magnetic field contributes to the nonresonant plasma dielectric which affects
the instability conditions and scaling laws. A negative harmonic cyclotron frequency
mismatch between the fast and slow ions is required for driving a cubic (and a coupled
quadratic) instability; the cube (square) root scaling of the peak growth rate makes
the relativistic effect more important than classical mechanism, especially for low
fast ion density and Lorentz factor being close to unity. For the cubic instability,
there is a threshold (ceiling) on the slow ion temperature and density (the external
magnetic field and the fast ion energy); the Alfv´en velocity is required to be low.
This Alfv´enic behavior is interesting in physics and important for its applications.
The case of fast protons in thermal deuterons is numerically studied and compared
with the analytical results. When the slow ion temperature or density (the
external magnetic field or the fast ion energy) is increased (reduced) to about twice
(half) the threshold (ceiling), the same growth rate peak transits from the cubic
instability to the coupled quadratic instability and a new cubic instability branch
appears. The instability transition is an interesting new phenomenon for instability.

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