Cluster-variation calculation for random-field systems: Application to hydrogen in niobium alloys

The cluster-variation method is applied to random-field lattice systems and specifically used to model the disorder-disorder phase transition of hydrogen in niobium-molybdenum and niobium-vanadium alloys. A small concentration of molybdenum or vanadium in the niobium lattice is treated as adding at each hydrogen site a random energy with a known probability distribution. Pairwise interactions between hydrogen atoms are included out to the first fifty shells on the bcc tetrahedral interstitial lattice, allowing for the effect of site blocking for the first three shells. The results show the small and large depression in the critical temperature for ${\mathrm{Nb}}_{1\mathrm{-}\mathrm{y}}$ ${\mathrm{V}}_{\mathrm{y}}$ ${\mathrm{H}}_{\mathrm{x}}$ and ${\mathrm{Nb}}_{1\mathrm{-}\mathrm{y}}$ ${\mathrm{Mo}}_{\mathrm{y}}$ ${\mathrm{H}}_{\mathrm{x}}$, respectively, with increasing concentration of V or Mo, as observed in experiments. Comparison is made with Monte Carlo calculations of other workers with use of the same interaction parameters, and the effects of variations in the random-energy distribution are described.