Second harmonic generation — the conversion of two photons at frequency ω into one photon at frequency 2ω — is forbidden in materials with inversion symmetry. Diamond has inversion symmetry in its bulk. At its surface, the symmetry breaks. This makes optical SHG an inherently surface-sensitive probe: the signal comes from where the symmetry is broken, which is the surface.
But X-ray SHG involves a subtlety. At X-ray energies, the multipole expansion of the light-matter interaction includes not only electric dipole terms (which vanish in centrosymmetric bulk) but also electric quadrupole and magnetic dipole terms. These higher-order terms do not vanish in the bulk, even with inversion symmetry. The bulk contributes.
Schacher, Pascal, Lawler, and Schwartz (arXiv 2602.22497, February 2026) calculate how the balance between surface and bulk contributions changes with photon energy in diamond. Near the carbon K-edge at 285 eV, the SHG signal is surface-dominated — the core-level resonance enhances the dipolar surface response more than the quadrupolar bulk response. As the energy increases to 1000 eV, the bulk quadrupole begins to dominate. By 7000 eV, the surface contribution is negligible.
The probe depth is tunable. At 285 eV, X-ray SHG sees the first few atomic layers. At 7000 eV, it sees the interior. The same measurement technique, applied at different energies, interrogates different depths. No physical change to the sample is required — only a change in the photon energy.
The mechanism is the energy dependence of the multipole cross sections. Near a resonance (the K-edge), the dipolar response is enhanced by the resonant transition, and the surface dipole dominates because the bulk dipole is symmetry-forbidden. Far from resonance, the off-resonant quadrupole response — which grows with energy relative to the dipole — takes over, and the bulk wins.
Crystal orientation adds another axis. The (001) and (111) surfaces of diamond have different surface symmetries, producing different SHG tensor elements. The surface sensitivity window shifts depending on which face is probed.
The X-ray chooses its own depth. The surface is a frequency away from the bulk.