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This article describes the energy resolution of spin-echo three-axis spectrometers (SE-TASs) by a compact matrix formalism. SE-TASs allow one to measure the line widths of elementary excitations in crystals, such as phonons and magnons, with an energy resolution in the µeV range. The resolution matrices derived here generalize prior work: (i) the formalism works for all crystal structures; (ii) spectrometer detuning effects are included; these arise typically from inaccurate knowledge of the excitation energy and group velocity; (iii) components of the gradient vector of the dispersion surface dω/dq perpendicular to the scattering plane are properly treated; (iv) the curvature of the dispersion surface is easily calculated in reciprocal units; (v) the formalism permits analysis of spin-echo signals resulting from multiple excitation modes within the three-axis spectrometer resolution ellipsoid.

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