Solution to the Anderson exchange narrowing model for I > [formula omitted

Abstract

The Anderson exchange and motional narrowing model of a resonance spectrum was solved for two distinct cases having nuclear spin values of I = 1 2 , 1, 3 2 , 2, and 5 2 . In the absence of exchange interactions the spectrum consists of n = 2 I + 1 resolved lines of equal intensity and separated by the frequency 2ω 0. Case (a) corresponds to equal transition probability to all other lines of the spectrum, while allowed transitions for case (b) occur only between adjacent lines. For both cases the initial effect of increasing the magnitude of the exchange interaction ω e is the broadening and drawing together of the spectral lines. At higher exchange frequencies the spectrum is dominated by a single exchange narrowed component while the other n−1 lines are broadened beyond detection. For case (a) exchange narrowing occurs when ( ω e > 2 ( n−1) ω 0 and for case (b) it occurs when ω e > 2( n−1) 2 ω 0. In the limit of strong exchange all of the hyperfine components collapse to the spectral center of gravity for case (b), as contrasted with case (a) in which only the inner (M I = 0 or ± 1 2 ) components actually collapse to the center. Values of the ratio ω e /ω 0 at each hyperfine component's point of collapse to the center were determined. The variations of the line positions, widths and relative intensities were calculated as a function of the exchange frequency. Asymptotic formulae were derived for the limits of large and small exchange.