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This paper describes our experimental investigation on a correlation between saturation magnetostriction and damping constant in 50-nm-thick (Ni-Fe)1-xMx (M=Pt , Au, Pd, and Cr; 0 <; x <; 0.12) films. The change in a with x is closely related with that of λs for every M: when M=Pt , Au, and Pd, a decreases almost monotonically with increasing λs, while it increases in (Ni-Fe)1-xCrx films. These results give clear evidence that a correlates with λs in (Ni-Fe)1-xMx films. These results also reveal that a can be controlled effectively by transition metal dopants. In addition, it should be pointed out that when adjusting a of magnetically soft films, particular attention must be paid to considerable change of magnetoelastic properties that leads to deterioration of magnetic softness.