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, $P\_2(cos\backslash theta)=0\; \backslash ,$, 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

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The magic angle θ_m is

$\backslash theta\_m\; =\; \backslash rm\{arccos\}\backslash frac\{1\}\{\backslash sqrt\{3\}\}\; =\; \backslash rm\{arctan\}\backslash sqrt\{2\}\; \backslash approx\; 54.7^\backslash circ$,

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

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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

$D(\backslash theta)\; \backslash propto\; 3\backslash cos^2\backslash theta\; -\; 1$

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.

Materials science | Spectroscopy | Chemical physics