Spin — Rotation Interaction

Abstract

The line width arising from the rotational modulation of anisotropic interaction tensors has the general form^1–3 introduced in Chapter X (see X.10.1):

$$\Delta = \left( {2/g{{\rm{\mu }}_{\rm{B}}}\surd 3} \right){\rm{T}}_2^{ - 1} = {\rm{\alpha ' + \beta }}{{\rm{m}}_{\rm{I}}} + {\rm{\gamma m}}_{\rm{I}}^2 + {\rm{\delta m}}_{\rm{I}}^3 + {\rm{\varepsilon m}}_{\rm{I}}^4$$

((1))

where α’ arises from the modulation of the g and hyperfine anisotropies. A careful test of this formula³ indicated that α’ does not always account for the m_I-independent contribution to the line width and that there is in some systems a residual line width, α“, which is the deviation of α’ from the experimental value, a The viscosity and temperature dependence of β, γ, δ, and ε was found to be quite adequately accounted for by the rotational Debye correlation time (and all the coefficients α’, β,… ε varied as η /T) but at low values of η/T it was found that α_ex varied in a peculiar manner: when α’ was extracted the residual line width varied as T/η. The variation with η/T of α_ex, α’, and α“ for vanadyl acetylacetonate in toluene is illustrated in Fig.1³. This kind of behaviour (α“ ∝ T/η) points to the importance of a

spin-rotation interaction as the relaxation modulation because in n.m.r. such a T/ η dependence has been widely observed^4–26. It is plausible that the line-width should be proportional to T/ η because a perturbation of the type (C/ ħ)J · S will lead to a spectral density of the form (C/ ħ)² <J²> τ _J , where τ_J is an angular momentum correlation time; the average value of J² is proportional to T, and the correlation time τ_J might be expected to be proportional to 1/ η because it depends inversely on the strength of the intermolecular forces, and so the overall dependence is T/ η .