The total cross section of a dynamic process i can then be computed by integrating Q in Eq. 1 weighted by the outcome function Pi over the whole PKSσi(Γ˜e)=d3Γ×Pi(Γ)×Q(Γ;Γ˜e)(3)where d3Γ ≡ E2 sin θdEdθdφ is the PKS differential volume element for postcollisional PKA. The cross sections of different dynamic processes are functions of Γ˜e, indicating that the probabilities of different dynamics can be tuned by the energy of electron (E˜e) or by the incident angles (φ˜e,θ˜e) with respect to the sample, which can be tuned by tilting the beam or the sample. These are the primary control variables of atomic engineering, along with the selection of the PKA, and the electron beam profile which overlaps with the barn-scale areas centered on this PKA.

In computer-controlled atomic engineering, in evaluating Eq. 3, although Q(Γ;Γ˜e) has many dependent variables and Eqs. 1 and 2 look complicated, they are analytical integrals and thus can be evaluated on the fly. Pi(Γ), however, is crystal and material dependent, and needs to be precomputed with expensive ab initio calculations, and tabulated or machine learned (36) for efficient evaluation of Eq. 3.

For simplicity, in the graphical illustrations in the main text, the “PKA → configurational change” dynamics are assumed to be deterministic, making Pi(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 i is 1 within the boundary and is 0 everywhere else. On the other hand, the contour of δ[Ef(θ,φ,Γ˜;Γ˜e)] is an ovoid with infinitely thin shell in PKS. The electron cross section of certain configurational outcome, σi, can thus be visualized easily in this fictitious limit of no thermal or quantum uncertainties: The intersection areas between the ovoid and the c(Γ) = i regions represent the part of PKS space that can induce certain configurational change i, which is then integrated with dσ/dΩ to get the total cross section for each of them.

To complicate the picture slightly, however, for a quantitative description of the outcomes, it has been shown that the precollisional momentum Γ˜ 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, v˜, can be well approximated by vv0+v˜, where v0 is the postcollisional velocity of a static PKA. Squaring the two sides yields the energy equation EE0+Mv0v˜+E˜. A small change in E˜ may result in up to ~10× change in E due to the second term Mv0v˜, since v0 is significantly larger than v˜ (because v0 corresponds to energy of 10 eV, whereas 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(Γ,Γ˜;Γ˜e) will be smeared into a bowling pin–shaped probability density Q(Γ;Γ˜e) that depends on the precollisional velocity distribution (fig. S7). Postcollision, after a short period of τE, the PKS momentum distribution Q(Γ;Γ˜e) will be convoluted with Pi(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 τE). The overlap of Q(Γ;Γ˜e) (nuclear collisional kinematics) and Pi(Γ) (crystal structure–dependent transition probability) in PKS space gives the net rate of configuational change (→i), after which the correlated atomic momenta dephase and the momenta correlation information is lost, leaving only heat (which subsequently conducts away or radiates out with yet another, longer timescale). All these happen (or not) long before the next electron penetrates the system, and the system configuration evolves (ii′ → i″ → …) without carrying the detailed phase information about atomic momenta, so an uncorrelated probability distribution function of PKA momentum P˜(Γ˜) is all we need for characterizing this driven system for the next collision.

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