  Plasma ions heating (especially minor heavy ions preferential heating) in
fast solar wind and solar corona is an open question in space physics. However,
Alfv\'en waves have been always considered as a candidate of energy source for
corona heating. In this paper, by using a two-dimensional (2-D) hybrid
simulation model in a low beta electron-proton-alpha plasma system, we have
investigated the relationships between plasma ions heating and power spectra
evolution of density and magnetic field fluctuations excited from the
parametric instabilities of initial pump Alfv\'en waves with an incoherent
spectrum at different propagation angles theta_k0B0 (an oblique angle between
the initial pump wave vector k0 and the background magnetic field B0). It is
found that, the wave-wave coupling as well as wave-particle interaction play
key roles in ions heating, and an Alfv\'en spectrum with small propagation
angle (e.g. theta_k0B0=15degree) can most effectively heat alpha particles in
perpendicular direction as well as in parallel direction for both proton and
alpha particle than the case of a monochromatic Alfv\'en wave or an Alfv\'en
spectrum with larger propagation angle.
