  The total energy of the transverse spin-spiral wave as a function of the wave
vector for all 3$d$ transition metal atomic chains has been calculated within
{\it ab initio} density functional theory with generalized gradient
approximation. It is predicted that at the equilibrium bond length, the V, Mn,
and Fe chains have a stable spin spiral structure, whilst the magnetic ground
state of the Cr, Co and Ni chains remains to be collinear. Furthermore, all the
exchange interaction parameters of the 3$d$ transition metal chains are
evaluated by using the calculated energy dispersion relations of the
spin-spiral waves. Interestingly, it is found that the magnetic couplings in
the V, Mn and Cr chains are frustrated (i.e., the second near neighbor exchange
interaction is antiferromagnetic), and this leads to the formation of the
stable spin-spiral structure in these chains. The spin-wave stiffness constant
of these 3$d$ metal chains is also evaluated and is found to be smaller than
its counterpart in bulk and monolayer systems. The upper limit (in the order of
100 Kelvins) of the possible magnetic phase transition temperature in these
atomic chains is also estimated within the mean field approximation. The
electronic band structure of the spin-spiral structures have also been
calculated. It is hoped that the interesting findings here of the stable
spin-spiral structure and frustrated magnetic interaction in the 3$d$
transition metal chains would stimulate further theoretical and experimental
research in this field.
