Abstract
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, R2,eff, is typically measured as a function of the pulse frequency, νCPMG. Here, an exact expression for how R2,eff varies with νCPMG 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 R2,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.
Highlights
Many chemical systems analysed by NMR spectroscopy spontaneously undergo dynamical changes that lead to variation in the isotropic chemical shift over time
Extensive efforts over recent years has resulted in a number of individually tailored CPMG experiments and associated labelling schemes to measure isotropic chemical shifts of excited states [18,19,20,21,22,23,24] and structural features such as bond vector orientations [25,26,27,28]
An exact solution describing how the effective transverse relaxation rate varies as a function of CPMG pulse frequency is presented (Eq (50), summarised in Appendix A)
Summary
Many chemical systems analysed by NMR spectroscopy spontaneously undergo dynamical changes that lead to variation in the isotropic chemical shift over time. Please cite this article in press as: A.J. Baldwin, An exact solution for R2,eff in CPMG experiments in the case of two site chemical exchange, J. Closed form solutions can provide greater insight into the physical principles behind experiments than numerical simulation Motivated by this principle, here, an exact solution for the effective transverse relaxation rate in a CPMG experiment, R2,eff, in the commonly encountered scenario of twosite exchange of in-phase magnetisation (Eq (50)) is derived. A single CPMG element is two concatenated echoes, which in the absence of relaxation and chemical exchange, returns transverse magnetisation to an identical state to which it started. It is these slowly relaxing terms that give rise to the characteristic increase in signal observed in a CPMG experiment
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