# Scientific Sources — Case30

## References

A. W. Bray, S. Eckart, and A. S. Kheifets, Keldysh-Rutherford Model for the Attoclock, Physical Review Letters 121, 123201 (2018).

[Source](https://doi.org/10.1103/PhysRevLett.121.123201)

Scattering-angle identities, weak-field assumptions, equal-binding screened potentials, screening-dependent response, and the limits of the classical interpretation. Quantum and CTMC results provide context; they are not required candidate calculations or pointwise targets.

### Relevant locations

- pp. 123201-1–2, Eqs. (1)–(6), Fig. 1

- p. 123201-3, Eq. (9), Fig. 3

- pp. 123201-3–4, Fig. 4 and adjacent screening/applicability discussion

- pp. 123201-4–5, Fig. 5 and conclusion; quantum/real-atom context only

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Bray, Eckart and Kheifets (2018), pp. 123201-1–2, Eqs. (1), (2), (4)–(6), Fig. 1; intensity convention on p. 123201-3, Eq. (9). |
| F2 | Bray et al. (2018), pp. 123201-2–4, Eqs. (2)–(6), (9), Figs. 3–4. Reconstructible submission and branch-separation requirements follow the stated examination task, not a source-specific file convention. |
| F3 | Bray et al. (2018), pp. 123201-3–4, Fig. 4 discussion: each screened target retains hydrogen binding energy $I_p=0.5$. |
| F4 | Bray et al. (2018), pp. 123201-1–2, Eqs. (1), (3) and physical-root discussion; omitted correction acknowledged adjacent to Eq. (3). |
| O1 | Bray et al. (2018), pp. 123201-2–3, Eqs. (2), (4)–(6), Fig. 3 (classical branches only). |
| O2 | Bray et al. (2018), pp. 123201-3–4, Fig. 4 and adjacent equal-binding/range-response discussion (robust structure, not pointwise matching). |
| O3 | Bray et al. (2018), p. 123201-4, Fig. 4 comparison and classical-applicability discussion. |
| I1 | Bray et al. (2018), pp. 123201-2–3, Eqs. (5)–(6), Fig. 3 and intensity-scaling discussion. |
| I2 | Bray et al. (2018), p. 123201-4, Fig. 4 and adjacent tunnel-width versus screening-length explanation. |
| I3 | Bray et al. (2018), p. 123201-4, Fig. 4 discussion of wavelength/range competition; p. 123201-2, Eq. (4) and weak-field assumptions. Equivalent scale arguments are accepted by the examination criteria. |
| M1 | Bray et al. (2018), p. 123201-2, Eqs. (2), (4)–(6) and kinetic/potential scale explanation. |
| M2 | Bray et al. (2018), pp. 123201-2, 123201-4, Eq. (3), Fig. 4 and exit-width/range competition. |
| M3 | Bray et al. (2018), p. 123201-4, Fig. 4 and conclusion: smooth transition across screening, not a universal inference from $\lambda=1$. |
| M4 | Bray et al. (2018), p. 123201-4, Fig. 4 classical-versus-quantum applicability discussion; p. 123201-2, Eq. (4) assumptions. Functionally equivalent validity arguments are examination acceptance routes. |
| M5 | Bray et al. (2018), p. 123201-4, Fig. 4 range dependence and conclusion. |
| M6 | Bray et al. (2018), pp. 123201-1–2 and 123201-4–5, model assumptions, Fig. 4 discussion and conclusion. |
| M7 | Bray et al. (2018), p. 123201-1 abstract and pp. 123201-4–5 conclusion; Fig. 5 is context for target-specific testing, not required CTMC reproduction. |
| V1 | Bray et al. (2018), pp. 123201-1–2, Eqs. (1)–(4), and p. 123201-4, Fig. 4; risk-triggered award and N/A treatment are examination scoring rules, not claims made by the article. |
| V2 | Bray et al. (2018), p. 123201-2, Eq. (4), barrier-suppression and weak-field assumptions; p. 123201-4, Fig. 4 quantum-applicability discussion, Fig. 5 and conclusion. Risk-triggered scoring is an examination rule. |
| V3 | Bray et al. (2018), pp. 123201-3–5, TDSE/iSURF description, Figs. 4–5 and conclusion; examination task explicitly excludes regenerating TDSE/iSURF and CTMC. |

## Formula and convention notes

The commissioned observable is the outgoing half-deflection of post-release central-potential scattering. For an attractive potential, $\theta=2\int_{r_0}^{\infty}\frac{\rho/r^2}{\sqrt{1-(\rho/r)^2-2V(r)/v_\infty^2}}\,dr-\pi$, with r0 the physical closest-approach root; $\theta_A=\theta/2$, with a declared consistent sign convention.

For Coulomb scattering $\tan(\theta/2)=Z/(v_\infty^2\rho)$, $v_\infty=A_0=E_0/\omega$, and $\theta_A=\arctan[Z/(A_0^2\rho)]$. The full deflection is not the attoclock offset. $\omega=0.057$, $I_p=0.5$ and $I=2E_0^2$ are the stated atomic-unit conventions.

With $\rho\approx I_p/E_0$ and the small-angle approximation, $\theta_A\approx\omega^2Z/(E_0 I_p)$, hence $I^{-1/2}$. The kinetic scale $v_\infty^2/2$ grows as I; the Coulomb potential scale $Z/\rho$ grows as $I^{1/2}$, yielding the linked scaling. Finite-angle or exact-exit departures must be distinguished from a false observable.

The exact Coulomb outer exit solves $Z/r+E_0r=I_p$: $r_{\mathrm{out}}=I_p/(2E_0)+\sqrt{I_p^2/(4E_0^2)-Z/E_0}$. The double-root barrier threshold is $E_{\mathrm{BSI}}=I_p^2/(4Z)$, equal to 0.0625 for the Coulomb reference. Low gamma alone does not establish the separate sub-barrier or weak-field condition; this Coulomb threshold is not a universal screened-target threshold.

For the leading screened approximation, $\tan(\theta/2)=Z_\lambda e^{-1/z_0}/(v_\infty^2\rho)$, with z0 the physical root of $1-(\rho/\lambda)^2z^2-(d/\lambda)z e^{-1/z}=0$ and $d=2Z_\lambda/v_\infty^2$. This approximation is an accepted route, not the only allowed implementation; direct classical integration is also accepted.

The required family includes the Coulomb limit, $\lambda=1$ and enough candidate-chosen intermediate lengths to test transfer. No fixed intermediate list, charge table, intensity grid, radial discretization, solver, tolerance, figure count, panel order, plotting range or source curve style is prescribed.

## Scope of the source anchors

Scattering-angle identities, weak-field assumptions, equal-binding screened potentials, screening-dependent response, and the limits of the classical interpretation. Quantum and CTMC results provide context; they are not required candidate calculations or pointwise targets.