General EPR intensity expression

In Chapter 6 we presented an expression for the transition probability (or intensity, amplitude) of field-swept spectra from randomly oriented simple 5=1/2 systems (Equation 6.4), and we could perhaps tacitly assume (as is generally done in the bioEPR literature) that the expression also holds for effective S = 1/2 systems, such as for the high-spin subspectra defined by the rhombograms discussed in Chapter 5. But what about parallel-mode spectra And how do we compute intensities in complex situations like for systems in the B S B B intermediate-field regime Clearly, we need a more generic approach towards intensity calculations. 

We have seen that a spin Hamiltonian in combination with its associated spin wavefunctions defines an energy matrix, which can always be diagonalized to obtain all the real energy sublevels of the spin manifold. Furthermore, the diagonaliza-tion also affords a new set of spin wavefunctions that are a basis for the diagonal matrix, and which are linear combinations of the initial set of spin functions. The coefficients in these linear combinations can be used to calculate the transition probabilities of all transitions within the spin manifold. 

In this section we will derive two general expressions for the transition probability one for regular perpendicular-mode EPR (Bl 1B) and one for parallel-mode EPR CBj IIB). The two expressions are related in the sense that they also provide the correct ratio of intensities (perpendicular over parallel) for data obtained with a single, dual mode resonator. The expressions are derived here, and not just given, because all expressions thus far published in the EPR literature contain small inconsistencies and/or errors. 

Our starting point is Fermi s golden rule for transition to a single state 

The k s are the direction cosines of Bx with respect to the xyz-axes system. Equations 8.14 and 8.15 combine into an expression with a system-dependent prefactor and a part that depends on the specific spin system under consideration, 

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