Anharmonic phonons and the isotope effect in superconductivity

Anharmonic interionic potentials are examined in an Einstein model to study the unusual isotope-effect exponents for the high-Tc oxides. The mass dependences of the electron-phonon coupling constant and the average phonon frequency 2 are computed from weighted sums over the oscillator levels. The isotope-effect exponent is depressed below 1/2 by either a double-well potential or a potential with positive quadratic and quartic parts. Numerical solutions of Schrödinger's equation for double-well potentials produce 's in the range 1.5-4 for a material with a vanishing isotope-effect parameter . However, low phonon frequencies limit Tc to roughly 15 K. A negative quartic perturbation to a harmonic well can increase above 1/2. In the extreme-strong-coupling limit, is 1/2, regardless of anharmonicity.