Zero-field resonances treatment

Early treatments of powder patterns attempted to deal with the spatial distribution of resonant fields by analytical mathematics.9 This approach led to some valuable insights but the algebra is much too complex when non-axial hyperfine matrices are involved. Consider the simplest case a single resonance line without hyperfine structure. The resonant field is given by eqn (4.3). Features in the first derivative spectrum correspond to discontinuities or turning points in the absorption spectrum that arise when dB/dB or dB/dcp are zero ... 

The additional parameter r is adjusted from reaction to reaction it reflects the extent of the additional resonance contribution. A large r corresponds to a reaction wth a large resonance contribution, whereas when r goes to zero, the equation is identical with the original Hammett equation. This treatment does not attempt to separate resonance and field effects completely, since some resonance effects are included in the normal a, but instead corrects for increased resonance interaction. When there is direct conjugation with an electron-rich center, an equation analogous to Eq. 4.18 can be employed, substituent constant appropriate for such situations, is... 

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