“PKA → configurational change” process

CS Cong Su MT Mukesh Tripathi QY Qing-Bo Yan ZW Zegao Wang ZZ Zihan Zhang CH Christoph Hofer HW Haozhe Wang LB Leonardo Basile GS Gang Su MD Mingdong Dong JM Jannik C. Meyer JK Jani Kotakoski JK Jing Kong JI Juan-Carlos Idrobo TS Toma Susi JL Ju Li

This protocol is extracted from research article:

Engineering single-atom dynamics with electron irradiation

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Sci Adv**,
May 17, 2019;
DOI:
10.1126/sciadv.aav2252

Engineering single-atom dynamics with electron irradiation

Procedure

The total cross section of a dynamic process *i* can then be computed by integrating *Q* in Eq. 1 weighted by the outcome function *P _{i}* over the whole PKS$${\mathrm{\sigma}}_{i}({\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})=\int {d}^{3}\mathrm{\Gamma}\times {P}_{i}(\mathrm{\Gamma})\times Q(\mathrm{\Gamma};{\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})$$(3)where

In computer-controlled atomic engineering, in evaluating Eq. 3, although $Q(\mathrm{\Gamma};{\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})$ has many dependent variables and Eqs. 1 and 2 look complicated, they are analytical integrals and thus can be evaluated on the fly. *P _{i}*(Γ), however, is crystal and material dependent, and needs to be precomputed with expensive ab initio calculations, and tabulated or machine learned (

For simplicity, in the graphical illustrations in the main text, the “PKA → configurational change” dynamics are assumed to be deterministic, making *P _{i}*(Γ) either 0 or 1, without any smearing at the boundaries. This is reflected in Fig. 4C as the sharp boundaries of the PKS regions, where the probability of configurational outcome

To complicate the picture slightly, however, for a quantitative description of the outcomes, it has been shown that the precollisional momentum $\tilde{\mathrm{\Gamma}}$ of the PKA is significant and important (*32*, *39*), due to what we may conceptualize as a “Doppler amplification effect” on Γ. To illustrate this with an approximate example (see section S6 for details), the outgoing velocity, v, of a PKA with precollisional vibrational velocity, $\tilde{\mathit{v}}$, can be well approximated by $\mathit{v}\approx {\mathit{v}}_{0}+\tilde{\mathit{v}}$, where ${\mathit{v}}_{0}$ is the postcollisional velocity of a static PKA. Squaring the two sides yields the energy equation $E\approx {E}_{0}+M{\mathit{v}}_{0}\cdot \tilde{\mathit{v}}+\tilde{E}$. A small change in $\tilde{E}$ may result in up to ~10× change in *E* due to the second term $M{\mathit{v}}_{0}\cdot \tilde{\mathit{v}}$, since ${\mathit{v}}_{0}$ is significantly larger than $\tilde{\mathit{v}}$ (because ${\mathit{v}}_{0}$ corresponds to energy of 10 eV, whereas $\tilde{\mathit{v}}$ corresponds to energy of ~0.1 eV). Therefore, a change as small as 0.1 eV due to thermal and quantum zero-point fluctuations in the precollision nuclear kinetic energy can change the PKA postcollision kinetic energy by as much as 1 eV, which subsequently can significantly alter the outcome probabilities. In momentum space, it is shown that the in-plane vibration also contributes to the amplification effect (see section S6). This necessitates a careful integral treatment in Eq. 1, where the infinite thin-shelled differential cross section $q(\mathrm{\Gamma},\tilde{\mathrm{\Gamma}};{\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})$ will be smeared into a bowling pin–shaped probability density $Q(\mathrm{\Gamma};{\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})$ that depends on the precollisional velocity distribution (fig. S7). Postcollision, after a short period of τ_{E}, the PKS momentum distribution $Q(\mathrm{\Gamma};{\tilde{\mathrm{\Gamma}}}_{\mathrm{e}})$ will be convoluted with *P _{i}*(Γ), a crystal-dependent quantity that one can precompute with abMD that integrates the evolution of atom trajectories on the ground-state BO surface (since we are beyond τ

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