Spin-Echo NMR Studies of Chemical Exchange. IV. Intramolecular Exchange of a Coupled AB System

Abstract

A mathematical and experimental study of the spin-echo amplitudes in a Carr—Purcell train of echoes is reported for a coupled AB system undergoing intramolecular exchange. The density-matrix approach was used to derive a set of four coupled recursion relations for the amplitudes of successive echoes. It seems unlikely that the set of equations can be reduced to a simple analytical function applicable to the general case, so a program was written to investigate the properties of the system numerically with a high-speed computer. Such an analysis predicts that when the exchange rate 1/2τ is small compared with the coupling constant A in rad sec−1, the amplitude of the echo train is modulated. The modulation frequency depends on A, 1/2τ, and on the chemical shift δω and the rf pulse separation tcp. The modulation diminishes as the exchange rate increases, and it disappears when 1/2τ≳3A. In the latter case, the train of echoes decays exponentially with an apparent decay constant 1/T2 which is described by the semiclassical equations derived previously for the uncoupled AB system. These predictions have been verified by 19F spin-echo experiments at 16.70 and 25.27 Mc/sec on 1,1-difluorocyclohexane in which axial and equatorial fluorines are interchanged by the chair—chair isomerization. The condition 1/2τ≳3A is satisfied at temperatures above about −40°C. The complex, modulated echo trains observed at lower temperatures are very sensitive to instrumental instabilities and adjustments, and no serious attempts were made to use them for determinations of 1/2τ, A, or δω. However, exchange rates were obtained from the exponential decays observed at temperatures from −40° to 42°C, and the results are compared with those from a previous high-resolution study. Exchange rates were determined to much higher temperatures in the spin-echo experiments because they are unaffected by the H–F coupling, but neglect of the coupling in the high-resolution line-shape analysis leads to erroneous results once the coalesced linewidth approaches 〈JHF〉ex.