# Scientific Sources — Case02

## References

Andrej Mihelič and Martin Horvat. 2021. *Calculation of multiphoton ionization amplitudes and cross sections of few-electron atoms*. *Physical Review A*, 103, 043108. [Source](https://doi.org/10.1103/PhysRevA.103.043108)

Calculation of multiphoton ionization amplitudes and cross sections of few-electron atoms

## Criterion-to-source crosswalk

Article page identifiers below are journal pages, not PDF viewer indices. Supplemental pages are explicitly marked. The crosswalk distinguishes direct source facts, mathematical entailments and task-defined evidence duties. Original images and PDFs are referenced only; none are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Mihelic and Horvat (2021), article pp.043108-3-4, Eqs.(19),(20),(28)-(30), velocity-form paragraph following Eq.(21): ordered unit-coefficient dipole-response hierarchy. |
| F2, O3, I3, M3, M4, M5 | Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory. |
| F3, M7 | Mihelic and Horvat (2021), article pp.043108-2-5, Eqs.(11)-(15),(26),(27),(31) and SI-unit paragraph: Coulomb amplitude normalization and absolute order-dependent generalized cross sections. |
| O1, I2, M2 | Mihelic and Horvat (2021), article p.043108-5, Fig.2(a), Eq.(31) and first Sec.III.A paragraph: hydrogen two-photon s/d partial cross sections and their arithmetic total; E1 channel parity extends to the higher orders in Fig.2(b,c). |
| O2, O4, I1, M1 | Mihelic and Horvat (2021), article p.043108-5, Fig.2(b,c) and Sec.III.A threshold paragraphs: intermediate openings near 6.8/13.6 eV for K=3 and 4.5/6.8/13.6 eV for K=4, with off-pole high-energy decline. |
| M6, V1 | Mihelic and Horvat (2021), article pp.043108-2-5, Eq.(9), paragraph preceding Eq.(12), Fig.1 caption, Sec.III.A pole limitation: extraction inside unscaled asymptotic space; exact bound poles are outside the unmodified real-energy perturbative prescription. |
| M8, V2 | Mihelic and Horvat (2021), article pp.043108-4-5, Eqs.(23)-(31), Fig.1 and Sec.III.A discussion of radial extent/collocation resolution; p.043108-9, Sec.IV limits the extraction to single-electron ejection. Evidence synthesis and triggered alternative tests are task-defined, not prescribed source solver settings. |

## Formula and convention notes

For F2-F3/M4-M7, article pp.043108-3-5, Eqs.(14),(23)-(31), define
$$H_\ell(k;r)=e^{-i\eta_\ell(k)}[F_\ell(k;r)+iG_\ell(k;r)],\quad
\widehat P_\ell(r)\sim\sum_q A_{\ell q}H_\ell(k_q;r),\quad k_q=\sqrt{2(E_0+j_q\omega)}>0.$$
With the article's outgoing Coulomb convention and final index $n$,
$$B_\ell=i^{-\ell+1}\sqrt{\frac{k_n}{2\pi}}A_{\ell n},\qquad
\sigma^{(K)}_\ell=2\pi\left(\frac{2\pi}{c\omega}\right)^K|B_\ell|^2.$$
The SI conversion factor is $a_0^{2K}t_0^{K-1}$, giving area$^K$ time$^{K-1}$. The canonical extraction uses $M_{pq}=W\{F_\ell(k_n;r),H_\ell(k_q;r)\}_{r_p}$ and $b_p=W\{F_\ell(k_n;r),\widehat P_\ell(r)\}_{r_p}$, with $W\{f,g\}=fg'-f'g$ and $r_p<R_0$ in the asymptotic region. A common complex direct fit or equivalent surface relation is accepted when it isolates the same absolutely normalized coefficient.

## Scope of the source anchors

The locators identify scientific objects and mechanisms, not compulsory figure layouts, solver settings, exact grids, pointwise tolerances or a fixed evidence package. Canonical examples may be replaced by functionally equivalent evidence subject to the rubric. Source readback verifies the candidate's science; it cannot supply missing candidate presentation, identification or inference. Reporting, applicability/N/A and descriptive Research Closure duties remain those stated in the rubric and task.