Speaker
Description
Channeling of fast charged particles in oriented crystals is usually amenable to classical treatment. Quantum effects, though, are known to exist for electrons with energies up to 100 MeV, manifesting themselves as a zone structure in the transverse energy spectrum. For positrons, the top of the potential barrier is narrower, so tunneling and above-barrier reflection is expected to occur at higher energies. In [1], such effects were studied for an idealized case of a sharp wedge potential, corresponding to a static atomic plane. The estimates indicated the significance of quantum effects for positrons or protons at 1 GeV energies. Thermal smearing of the continuous potential, however, makes the potential barrier intermediate between sharp wedge and smooth turnover, requiring a numerical study. In the present work, quantum transmission through such a barrier is calculated based on the generic representation [1] for probability of transmission through a symmetric potential $V(-x)= V(x)$,
$ T(E_\perp)=\frac{1}{1+J^{-2}\left[\frac{\partial}{\partial x}|\psi|^2\big|_{x=+0}\right]^2}, $
where
$ J=\frac{1}{i}\left(\psi^*_+\frac{\partial}{\partial x}\psi_+-\psi_+\frac{\partial}{\partial x}\psi^*_+\right) $
is real and proportional to the probability current, which for a real potential $V(x)$ is real and locally conserved: ${\partial}J/{\partial x}=0$.
The continuous potential is approximated by a thermally smeared linear wedge, involving a single spatial scale parameter [2]. Granted that away from the smearing region the potential is regarded as linear, the transmitted wave function assumes an Airy form. Adopting it as an initial value, we solve the Schr\"{o}dinger equation for a definite energy as an initial-value problem. The results for the transmission probability depending on the transverse energy and the smearing parameter indicate that for room temperature, quantum effects for 1 GeV positrons or protons can be noticeable, although the course of the transverse energy dependence of the transmission probability differs significantly from that for a static plane.
References
[1] M. V. Bondarenco, J. Phys. A: Math. Theor. 58 (2025) 415305; https://doi.org/10.1088/1751-8121/ae0202
[2] M. V. Bondarenco and N. S. Moskvitin, Phys. Rev. B 110 (2024) 205205;
https://doi.org/10.1103/PhysRevB.110.205205