Proton Spin−Lattice Relaxation of Tunneling Methyl Groups:  Calculation of the Time Dependent Correlation Functions

Abstract

A theory of magnetic nuclear relaxation, providing a calculation of the correlation functions of complex motion of methyl groups is presented. The complex motion consists of jumps over the barrier (classical motion) and jumps through the barrier (tunneling). The Schrödinger equation has been applied in the calculation of the rate constant of tunneling jumps through the barrier. The equations for the spectral densities (ω), where ω = (ω I ± ωT), and ω = (m ω I), where m = 1 or 2, ω I is the resonance angular frequency, and ωT is the angular frequency of the tunnel splitting, are derived. These spectral densities concern the motion of spin pairs inside methyl groups (“intra”) and outside methyl groups (“inter”). The calculated spectral densities are applied to analyze the temperature dependencies of the spin−lattice relaxation rate in solids containing methyl groups. A wide regime of temperatures from 0 K up to the melting point is considered. The temperature at which the tunneling process ceases is discussed. The theory proposed explains the different temperature dependencies of T 1 for CH3COOK obtained in the experiments caused by small amounts of CH3COOH or water impurities. The theoretical equations derived in this paper are compared to those known in the literature.

NMR relaxation study of methyl groups in solids from low to high temperatures

“…The values of T 1(intra) determined by the methyl group hindered rotation (Eqs. [12], [14], [38], and [39]) are also closely related both to i and to the tunneling splitting frequency T . The tunneling splitting frequency T is different in different materials and varies from kHz to GHz.…”

“…The use of Eq. [26], explicitly, for determination of the rate constant of tunneling was proposed in (14):…”

“…The line represents the best fit of Eqs. [14] and [63] to the data. The arrow shows the temperature T tun ϭ E H /C p (see Table 1).…”

Complex dynamics of 1.3.5-trimethylbenzene-2.4.6-D3 studied by proton spin–lattice NMR relaxation and second moment of NMR line

“…The jumps over the barrier are characterized by Arrhenius correlation time, while the tunnelling jumps can be characterized by the Schrödinger correlation time [16,18]. The correlation time of C 3 jumps over the potential barrier follows the Arrhenius dependence:…”

“…The tunnelling jumps through the barrier and jumps over the barrier cause the transport of mass between the same equilibrium sites, but they are based on different correlation times, therefore in the spectral density calculations they have to be treated as components of a complex motion. Following the Woessner theoretical assumptions such equations for the total spectral density of complex motion have been derived [16,18] and have been applied in present paper. The long values of T 1 at lowest temperatures will be explained and correlated with the reduced values of the second moment in this temperature regime.…”