Inelastic neutron scattering and dynamical structure factors

Use momentum transfer $\mathbf q=\mathbf k_i-\mathbf k_f$ and sample energy gain $E=E_i-E_f$. Positive E means neutron energy loss. Let $A_{\mathbf q}^\dagger=A_{-\mathbf q}$ be an extensive Fourier component of the chosen sample operator, with N equivalent sites. Define the unsymmetrized structure factor per site and per energy by the correlation below. Its spectral representation has positive-energy transitions from initially occupied lower-energy states to higher-energy states, weighted by their thermal populations.

Integrating over all E gives the equal-time correlation divided by N. A spectrum defined per angular frequency instead differs by a factor of hbar, so normalization cannot be inferred from the symbol S alone. For a scalar Hermitian density the spectral weights are nonnegative; magnetic scattering involves a tensor of spin correlations and a transverse polarization projection. Elastic or disconnected contributions also need to be retained or explicitly subtracted when comparing integrated sum rules.

Interchanging the initial and final states in a thermal spectral sum gives detailed balance with both momentum and energy reversed. At positive sample energy gain, the reverse process is suppressed because it requires a thermally populated excited state. Replacing minus q by plus q requires an additional symmetry such as inversion for the chosen channel; it must not be done silently in a nonreciprocal magnetic material. Dividing out a Bose factor does not manufacture information outside the measured kinematic range.

Define the retarded response to a perturbation minus field times the conjugate operator with the positive-i commutator convention below, and Fourier transform with the same positive exponential. Its dissipative imaginary part is positive at positive energy in a passive scalar channel. The spectral sum then produces the displayed fluctuation–dissipation factor. Other authors choose a negative-i Green function or integrate frequency rather than energy; their signs and hbar factors must be translated before their formulas are combined with these definitions.

For identical nuclei with real coherent scattering length b, neglecting isotope and spin incoherence, a Born single-scattering nuclear cross section is proportional to the density structure factor, with an outgoing-to-incoming wave-vector ratio. Magnetic neutrons instead probe spin components perpendicular to momentum transfer, multiplied by magnetic form factors and the appropriate magnetic coupling. Ordinary nonresonant X rays predominantly probe charge density with electronic form factors and polarization factors. Resonant X-ray channels can carry other effective operators, so none of these signals is universally interchangeable.

Detector efficiency, monitor normalization, sample mass, absorption, multiple scattering, backgrounds, and resolution must be handled before absolute spectral weights are compared. A finite-resolution elastic line can leak into low-energy inelastic intensity. Momentum-integrated sum rules require sufficient reciprocal-space and energy coverage, not merely one strong scan. Compare measured counts to a resolution-convolved cross section; a fitted dispersion alone does not establish the nature, uniqueness, or fractionalization of the excitation carrying that spectral weight.