ESEEM of disordered systems in frequency domain: Analytical formulae in the case of nuclear spin 1/2 and weak hyperfine interaction

The analytical expressions for the spectral density of the dead time free electron spin echo envelope modulation (ESEEM) signal of disordered system are obtained for a paramagnetic center with nuclear spin 1/2 and weak axially symmetric hyperfine interaction. The spectral density is given by the Fourier transformation of the ESE signal averaged over all orientations. The order of the two linear operations may be changed. Fourier transformation of the nonaveraged ESE signal supplies us with the sum of the Dirac δ -functions. Averaging of such a spectrum is a rather trivial operation leading to the spectral densities in the final form.