Solid-State Interactions of Nuclear Spins

The dominant interactions of nuclei having spin I that are responsible for the broadening of solid-state NMR lines and for their characteristic line shapes are described by the total Hamiltonian  

The superposition of the different magnetic interactions complicates the interpretation and calculation of the resulting normalized line shape function,/(v). It is therefore advantageous to make use of the so-called second moment M2 as a measure of the linewidth of the solid-state NMR signals. The full-width at halfmaximum of an NMR signal in frequency units, also called the static linewidth, is 

The second moments of NMR signals in the observable spectral range can be calculated, provided that the local structure and the type of the internal magnetic interactions of the spin ensemble under study are known. According to the total Hamiltonian given in Eq. (1), it follows that 

h denotes Planck s constant h divided by 27i, and /zq is the permeability of vacuum. I and 5 are the nuclear spins of the resonating spins I and non-resonating spins S, respectively. Ni and Ns denote the numbers of resonating and nonresonating nuclei in the sample, respectively ys the magnetogyric ratio of the nonresonating spins S, and and ryj- the internuclear distances. 

In the case of spins I—i/l and assuming that the sample does not contain paramagnetic impurities and that other susceptibility effects can also be neglected, the relationship between v g and Vq is given by the following  

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