An exact solution for R(2,)(eff) in CPMG experiments in the case of two site chemical exchange

The Carr–Purcell–Meiboom–Gill (CPMG) experiment is widely used to quantitatively analyse the effects of chemical exchange on NMR spectra. In a CPMG experiment, the effective transverse relaxation rate, R(2,)(eff), is typically measured as a function of the pulse frequency, ν(CPMG). Here, an exact expression for how R(2,)(eff) varies with ν(CPMG) is derived for the commonly encountered scenario of two-site chemical exchange of in-phase magnetisation. This result, summarised in Appendix A, generalises a frequently used equation derived by Carver and Richards, published in 1972. The expression enables more rapid analysis of CPMG data by both speeding up calculation of R(2,)(eff) over numerical methods by a factor of ca. 130, and yields exact derivatives for use in data analysis. Moreover, the derivation provides insight into the physical principles behind the experiment.