for A-\[1$\bar{1}$0\] direction. The spin moments decrease slightly and try to keep a constant as the chain length increases above 4. Our results show that the spin moments of Ni atoms on the reconstructed Au(110) surface are smaller than the value 0.675$\mu_{B}$ in crystal Ni having been calculated using the same methods in our work.

The magnetic order is an important aspect of material magnetism. Experimentally the long ferromagnetic order is found for Co chains stabilized above a finite temperature on Pt(997) surface [@Gambardella1]. Above the temperature, the long ferromagnetic order is destroyed and changed into the short ferromagnetic fragments. In our calculations, the short Ni chains are ferromagnetic. The Fig.\[fig6\] shows the ferromagnetic order of the surface supported Ni$_{5}$ chains for the magnetization for C-\[110\] and A-\[1$\bar{1}$0\] directions. The spin-polarizations of surface Gold atoms near the Ni chain are induced by the magnetic Ni atoms, which almost disappear for Gold atoms far from the Ni chain. We have also calculated the anti-ferromagnetic configurations, non-magnetic configurations and other non-collinear magnetic configurations and found their energies higher than that of the ferromagnetic configuration. Thus our calculations show that the Ni chains with ferromagnetic order are most energetically favorable. It is interesting 