RF Interaction Representation

The quantum description of N coupled protons in Hilbert space is given by a spin Hamiltonian of dimension 2 equalling the number of direct product spin-i states. Two experimental tools have been used for the decoupling of spin interactions, RF irradiation and MAS. In the following, any discussion of sample spinning assumes MAS conditions. MAS effectively eliminates the CSA and DDfcetero interaction between protons and other spin-i nuclei. 

A system of coupled nuclei with spin- rotating at the magic-angle in an external magnetic field under RF irradiation has a Hamiltonian in the rotating frame of the form  

The forms of the first four terms are well known [14,89]. The isotropic chemical shift (Hiso) and chemical shift anisotropy (Hcsa) terms are [89] 

Explicit expressions for the G -parameters in terms of the Euler angles f2ab = aab,f ab,7ab) between the initial dipolar tensor frame and the rotor frame are also given in Appendix 1, as well as the distance dependence, 1 of the dipolar frequency coefficient ujab- The Gn,ab and gn,a coefficients satisfy 

The time independent isotropic spin-spin scalar coupling Hamiltonian is 

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At the end of this segment the operators for I = 1,2 return to themselves. This will repeat itself each time after two additional segments. Hence, during each single segment q, q = 0,1,..., fV - 1), during the time interval qrc/N < t < (< + l) rc/iV, the Tq operators in the RF interaction representation become ... 

The overall time dependence of To (t) during 0 < t < Xe in the RF interaction representation results in a product of two functions an odd function for and an even function for times a function FR mx,N). This... 

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