Relation between Superconductivity and Lattice Instability in theβ-W Compounds

Abstract

The temperature $T_0$, at which the shear modulus $\frac{1}{2}(C_{11}-C_{12})$ vanishes in the cubic phase and in the normal state of some high superconducting $\beta$-W compounds, is calculated as a function of the small number $Q$ of electrons (or holes) in the nearly empty (or nearly full) $d$ subband. The variation of $T_c$ with $Q$ was calculated in a previous paper. We show that $T_0$ is much more sensitive than $T_c$ to the value of $Q$, and thus to the exact chemical composition of the $A_{3}B$ phase. Moreover, we show that when $Q$ falls in some range of its value, a linear extrapolation at low temperatures cannot be used to obtain the value of $T_0$. [In other words, the temperature ${T_0}^{\prime }$ at which the extrapolated value of $\frac{1}{2}(C_{11}-C_{12})$ goes to zero may be somewhat different from the temperature $T_0$ at which $\frac{1}{2}(C_{11}-C_{12})$ is actually zero.] The failure of some samples to undergo the martensitic phase transition, when the extrapolated value of $\frac{1}{2}(C_{11}-C_{12})$ seems to go to zero in the normal state, may be explained in this way. Finally, we find that at room temperature $\frac{1}{2}(C_{11}-C_{12})$ is minimum for some small value of $Q$.