| # Scientific Sources — Case02 |
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| ## References |
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| 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) |
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| Calculation of multiphoton ionization amplitudes and cross sections of few-electron atoms |
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| ## Criterion-to-source crosswalk |
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| 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. |
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| | Criteria | Source locations, scientific use and limits | |
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| | 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. | |
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| ## Formula and convention notes |
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| 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. |
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| ## Scope of the source anchors |
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| 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. |
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