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.

[edit] Mathematical definition

Magic angle
Magic angle

The magic angle θm is

\theta_m = \rm{arccos}\frac{1}{\sqrt{3}} = \rm{arctan}\sqrt{2} \approx 54.7^\circ,

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

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

[edit] 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 θm to a strong external magnetic field, have zero dipolar coupling, D(θm)=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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