Electron spin resonance pulsed dipolar spectroscopy (PDS) has become popular in protein 3D structure analysis. PDS studies yield distance distributions between a pair or multiple pairs of spin probes attached to protein molecules, which can be used directly in structural studies or as constraints in theoretical predictions. Double-quantum coherence (DQC) is a highly sensitive and accurate PDS technique to study protein structures in the solid state and under physiologically relevant conditions. In this work, we have derived analytical expressions for the DQC signal for a system with N-dipolar coupled spin-1/2 particles in the solid state. The expressions are integrated over the relevant spatial parameters to obtain closed form DQC signal expressions. These expressions contain the concentration-dependent "instantaneous diffusion" and the background signal. For micromolar and lower concentrations, these effects are negligible. An approximate analysis is provided for cases of finite pulses. The expressions obtained in this work should improve the analysis of DQC experimental data significantly, and the analytical approach could be extended easily to a wide range of magnetic resonance phenomena.
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