Some mathematical properties of magnetic resonance line shapes

The present paper deals with some mathematical properties of Freed's method of calculating magnetic resonance line shapes in the slow tumbling region. It is first shown that this method is related to a moment expansion of the spectrum and a general expression for the spectral moments is given. A modification of Freed's approach of solving the spectrum of an axial Hamiltonian is then derived: Rather than truncating the infinite set of Freed's equation, we obtain an approximate solution of the discarded part (using the asymptotic values of the Clebsh-Gordan coefficients) which are substituted into the retained part of the set. This results in a significant improvement in the convergence of the spectrum and at the same time enables one to compute powder spectra (without motion). Finally the more complicated problem of a spectrum due to a nonaxial Hamiltonian is discussed. A method is derived to transform this ``two-dimensional'' problem into a one-dimensional one with lower order.