Magic angle

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The magic angle is a precisely defined angle, the value of which is approximately 54.7°. The magic angle is a root of a second-order Legendre polynomial, [P_2(cos\theta)=0 \,], and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle. This property makes the magic angle of particular importance in solid-state NMR spectroscopy.

Mathematical definition

Magic angle

The magic angle θ_m is

[ \theta_m = \rm\frac} = \rm\sqrt \approx 54.7^\circ],

where arccos and arctan are the inverse cosine and tangent functions.

θ_m is the angle between the space diagonal of a cube and any of its three connecting edges, see image.

Magic angle and dipolar coupling

In nuclear magnetic resonance (NMR) spectroscopy, in a strong magnetic field, the dipolar coupling D depends on the orientation of the internuclear vector with the external magnetic field by

[D(\theta) \propto 3\cos^2\theta - 1]

Hence, two nuclei with a dipolar coupling vector at an angle of θ_m to a strong external magnetic field, have zero dipolar coupling, D(θ_m)=0. Magic angle spinning is a technique in solid-state NMR spectroscopy, which employs this principle to remove or reduce dipolar couplings, thereby increasing spectral resolution.

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