Magic angle
The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, P2(cos θ) = 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 magic angle spinning solid-state NMR spectroscopy. In magnetic resonance imaging, structures with ordered collagen, such as tendons and ligaments, oriented at the magic angle may appear hyperintense in some sequences, this is called the magic angle artifact or effect.
The magic angle θm is
where arccos and arctan are the inverse cosine and tangent functions respectively.
θm is the angle between the space diagonal of a cube and any of its three connecting edges, see image.
Magic angle θ is also half of the opening angle formed when a cube is rotated from its space diagonal axis, which may be represented as arccos −1/3 or 2 arctan √2 radians ≈ 109.4712°. This double magic angle is directly related to Tetrahedral molecular geometry and is the angle from one vertex to the exact center of the tetrahedron (i.e., the edge central angle also known as the tetrahedral angle).
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