The optimal amount of dysprosium in the highly magnetostrictive rare-earth compounds ${\mathrm{Tb}}_{1-x}{\mathrm{Dy}}_x{\mathrm{Fe}}_2$ for room-temperature applications has long been known to be $x=0.73$ (Terfenol-D). Here, we derive this value from first principles by calculating the easy magnetization direction and magnetostriction as a function of composition and temperature. We use crystal-field coefficients obtained within density-functional theory to construct phenomenological anisotropy and magnetoelastic constants. The temperature dependence of these constants is obtained from disordered-local-moment calculations of the rare-earth magnetic order parameter. Our calculations find the critical $\mathrm{Dy}$ concentration required to switch the magnetization direction at room temperature to be $x_c=0.78$, with magnetostrictions ${\lambda }_{111}=2700$ and ${\lambda }_{100}=-430$ ppm, close to the Terfenol-D values.

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• Magnetic anisotropy
• Magnetoelastic effect
• Magnetostriction
• Transition-metal rare-earth alloys
• Crystal-field theory
• Density functional theory development