Note on sideband intensities in one-dimensional magic angle spinning nuclear magnetic resonance

It is well known that in the NMR spectra of solid samples spinning at the magic angle centrebands and sidebands occur. The centrebands are found at the isotropic value of the chemical shift and the sidebands are found at integral multiples of the spinning frequency as long as the spinning frequency does not exceed the width of the chemical shift anisotropy pattern. From the intensities of centrebands and sidebands it is possible to extract information about the chemical shift tensor. In fact, from an incomplete set of sidebands the principal values of the chemical shift tensor can be determined. This was first demonstrated by Herzfeld and Berger (J. Herzfeld and A. E. Berger, J. Chem. Phys., 73, 6021 (1980)). We present a simplified derivation of the expression for the sideband intensities in the NMR spectra of solid samples spinning at the magic angle as given by Herzfeld and Berger. The manipulations with series of Bessel functions are replaced by an application of elementary Fourier series theory. We furthermore show that expressions for the moments of the intensity distribution can be derived.