cases/case02/sources.md

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

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 $$H_{\mathrm{\ell }}(k;r)=e^{-i{\eta }_{\mathrm{\ell }}(k)}[F_{\mathrm{\ell }}(k;r)+i{G}_{\mathrm{\ell }}(k;r)],{\widehat{P}}_{\mathrm{\ell }}(r)\sim \sum _q{A}_{\mathrm{\ell }q}{H}_{\mathrm{\ell }}(k_q;r),k_q=\sqrt{2(E_0+j_q\omega )}>0.H\_\backslash ell(k;r)=e^\{-i\backslash eta\_\backslash ell(k)\}[F\_\backslash ell(k;r)+iG\_\backslash ell(k;r)],\backslash quad\; \backslash widehat\; P\_\backslash ell(r)\backslash sim\backslash sum\_q\; A\_\{\backslash ell\; q\}H\_\backslash ell(k\_q;r),\backslash quad\; k\_q=\backslash sqrt\{2(E\_0+j\_q\backslash omega)\}>0.$$ With the article's outgoing Coulomb convention and final index $n$, $$B_{\mathrm{\ell }}=i^{-\mathrm{\ell }+1}\sqrt{\frac{k_n}{2\pi }}{A}_{\mathrm{\ell }n},{\sigma }_{\mathrm{\ell }}^{(K)}=2\pi {\left(\frac{2\pi }{c\omega }\right)}^K\mathrm{\mid }{B}_{\mathrm{\ell }}{\mathrm{\mid }}^2\mathrm{.}B\_\backslash ell=i^\{-\backslash ell+1\}\backslash sqrt\{\backslash frac\{k\_n\}\{2\backslash pi\}\}A\_\{\backslash ell\; n\},\backslash qquad\; \backslash sigma^\{(K)\}\_\backslash ell=2\backslash pi\backslash left(\backslash frac\{2\backslash pi\}\{c\backslash omega\}\backslash right)^K|B\_\backslash ell|^2.$$ 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.