Spin-Echo Studies of Chemical Exchange. II. Closed Formulas for Two Sites

Abstract

Closed expressions are derived for the decay in amplitude of successive echoes in nuclear magnetic resonance experiments for systems with two exchanging sites having different resonance frequencies but equal transverse relaxation times ``in the absence of exchange.'' The decay for a Carr—Purcell train of echoes is found to be the sum of two exponential terms and two ``pseudoexponential'' terms; the absolute values of the latter decay exponentially but the sign alternates from one echo to the next. For the case of two equally populated sites the decay simplifies to the sum of just one exponential and one ``pseudoexponential'' term. In most cases of experimental importance, a single exponential term is much larger than the rest and adequately describes the decay in echo amplitude. An iterative computer procedure is described which uses the expressions to extract information about the system from measurements of the apparent decay time constant (1/T<sub>2</sub>) as a function of 180° pulse separation (t<sub>cp</sub>). In favorable cases the method yields values not only of the lifetime between exchanges (1/2τ) but also of the chemical shift (δω) and the relaxation time in the absence of exchange (1/T<sub>2</sub><sup>0</sup>). Moreover, it is applicable to both slow and fast exchange, whereas use of the approximate equation derived by Luz and Meiboom leads to inaccurate values for relatively slow rates with large chemical shifts between the sites. Our experience shows that the spin‐echo technique can be very useful for the study of fast exchange processes, but even when closed expressions of appropriate accuracy are used the nature of these expressions is such that a high‐speed computer is required for fitting the experimental data, unless some ``outside'' information is available about δω and 1/T₂⁰.