Removal of Dipolar Broadening of Nuclear Magnetic Resonance Spectra of Solids by Specimen Rotation

Abstract

THE nuclear magnetic resonance spectrum of a solid rotated at high speed consists of a narrowed central line and a set of side-spectra spaced at integral multiples of the rotation rate of either side^1,2. The interaction energy between each pair on nuclear magnets in the solid contains the angular factor (3 cos²θ – 1), where θ is the angle between the internuclear vector and the applied field. When the solid rotates about an axis making an angle α with the applied field, the angular factor may be expressed³ as the sum of its mean value ½ (3 cos² α – 1) (3 cos² γ – 1), where γ is the angle between the internuclear vector and the axis of rotation, and two time-dependent terms, (3/2) sin 2α sin 2γ cos ω_r t, and (3/2) sin² α sin² γ cos 2 ω_r t, where ω_r is the angular frequency of rotation. The two time-dependent terms lead to the formation of the side-spectra. It should be noted that when α is 90°, as in our earlier experiments^1,2, the first of these terms is absent, so that side-spectra occur in this case at even multiples of ω_r only ; for general α, the side-spectra occur at odd and even multiples of ω_r. The form of the mean value of the angular factor for each nuclear pair leads to the conclusion that the central spectrum should have the shape found for the static crystal when the applied field is directed along the axis of rotation, but reduced in width by the factor ½ (3 cos² α – 1), and having a second moment⁴ reduced by a factor ¼ (3 cos² α – 1)². Thus, in particular, when α has the value cos⁻¹ (1/√3) = 54° 44′, the dipolar broadening of the central line should be removed⁵. With a speed of rotation large compared with the static line-width, the side-spectra will be displaced far from the centre and will be weak, leaving only a sharp central line broadened by residual non-dipolar causes.