Two-level theory of second-order nonlinear X-ray response beyond the electric-dipole approximation
We develop a two-level theory of the second-order nonlinear X-ray response beyond the electric-dipole approximation, deriving the leading quadrupolar correction originating from interference with the dipolar pathway at the amplitude level. A compact scaling law links the correction to weighted linea...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/452122 |
| Acceso en línea: | https://hdl.handle.net/2117/452122 https://dx.doi.org/10.1021/acs.jpca.5c07630 |
| Access Level: | acceso abierto |
| Palabra clave: | Nonlinear optics X-ray spectroscopy Òptica no lineal Espectroscòpia de raigs X Àrees temàtiques de la UPC::Física::Electromagnetisme::Raigs X |
| Sumario: | We develop a two-level theory of the second-order nonlinear X-ray response beyond the electric-dipole approximation, deriving the leading quadrupolar correction originating from interference with the dipolar pathway at the amplitude level. A compact scaling law links the correction to weighted linear-response oscillator strengths, allowing parameter-free estimates across different core edges within the limits of the two-level description. For difference frequency resonant with the core transition, within the two-level description adopted here, the frequency dependence of the observable beyond-dipole correction is set by the electric dipole–quadrupole pathway through field gradients and is controlled by the dimensionless factor (¿1 + ¿2)2/O02 = (2r – 1)2 with r = ¿1/O0, while the underlying dipole–quadrupole interference occurs at the amplitude level and cancels in the isotropically averaged intensity, leaving a small positive quadratic correction whose magnitude is estimated from an isotropic linear-response oscillator-strength ratio. In particular, for a two-color scheme with ¿1 = 4O0 and ¿2 = 3O0, the quadrupolar contribution modifies the difference-frequency intensity by about 1.3% at the O K edge of CO and 5.5% at the S K edge of cysteine, consistent with the (2r – 1)2 dependence and the growth of f(2) with core energy. In liquids and gases, where the emitted difference-frequency field is strongly reabsorbed at the core edge, the relevant observable is the per-molecule nonlinear conversion efficiency obtained from orientationally averaged single-molecule emission. After isotropic averaging with linear polarizations, the dipole–quadrupole interference term vanishes by symmetry at the intensity level, so the observable correction arises solely from the surviving quadratic beyond-dipole contribution and follows a unified scaling. The same two-level structure carries over to sum-frequency generation with a reduced nondipole prefactor. The model targets molecules in the gas phase or solution and does not address oriented or crystalline systems. These results provide a practical rule for estimating beyond-dipole effects in second-order X-ray mixing and clarify when a dipole-only analysis becomes inadequate. |
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