<b>Theory of Transitions from Large to Nearly-Small Polarons, with Application to Zr-Doped Superconducting SrTi</b><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold">O</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

Expressions for the matrix elements of the Hamiltonian and overlap integrals between large-polaron and nearly-small-polaron states are given in terms of various large- and nearly-small-polaron parameters. The dependence of the mass of the ground state on the bare-electron mass in a region of transition between large- and nearly-small-polaron states is found. Parameters of the theory are evaluated using the continuum-polarization model of electron-phonon interactions. Modifications due to anisotropy and the presence of many longitudinal polar modes of lattice vibration are discussed briefly. Making use of a recent analysis by Reik of free-carrier optical-absorption results in SrTi${\mathrm{O}}_{3}$ using small-polaron theory, it is shown that the rate of mass rise deducible from superconductivity data on Zr-doped SrTi${\mathrm{O}}_{3}$ can be explained by the theory if the bare transverse mass ${m}_{\mathrm{bt}}$ is taken to be ${m}_{\mathrm{bt}}\ensuremath{\simeq}0.6{m}_{e}$, where ${m}_{e}$ is the free-electron mass, and if small-polaron binding energies are assumed to be larger by a factor of about 1.4 than their values in the continuum-polarization model. The dependence on Zr concentration of the ratio of phonon-induced to Coulomb intervalley electron-electron interactions deducible from the superconductivity data is discussed within the framework of the model.

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