Term: magic angle
https://doi.org/10.1351/goldbook.MT07419

Definition:
Upon excitation of an 'isotropic' sample (assuming an ultra short excitation pulse) the relationship between the fluorescence intensity detected at a time t and through a polarization analyser oriented at an angle β with respect to the electric polarization of the exciting beam is given by \[I(t,\beta ) \propto  N(t)\left [ 1 + (3\, \text{cos}^{2}\, \beta - 1)R(t) \right ]\] where R(t) is the degree of alignment of the emitting transition dipole in the laboratory frame and N(t) is the excited-state population, both at time t. For β = 54.7° (the magic angle), the dipole-alignment contribution vanishes and I(t,β=54.7°) ∝ Nt.

Notes:
0) This concept also applies for time-resolved absorption measurements in cases in which @PT07461@ occurs because the detected species do not freely rotate fast enough to make the measurement @I03353@ within the time of the experiment.
1) Applies for steady-state measurements on fixed samples. In this case \[I(\beta ) \propto  N\left [ 1 + (3\, \text{cos}^{2}\, \beta - 1)R \right ]\] with I(β) the intensity of the effect observed at an analyser @A00346@ β with respect to the electric @P04712@ of the exciting beam, N the excited-state population at steady-state equilibrium, and R the degree of alignment of the @T06460@ of the excited molecular entity.
2) The term magic @A00346@ is also used in NMR @S05848@.

Related Terms:
1) photoselection (http://doi.org/10.1351/goldbook.PT07461).
2) isotropic (http://doi.org/10.1351/goldbook.I03353).
3) angle (http://doi.org/10.1351/goldbook.A00346).
4) polarization (http://doi.org/10.1351/goldbook.P04712).
5) transition (dipole) moment (http://doi.org/10.1351/goldbook.T06460).
6) spectroscopy (http://doi.org/10.1351/goldbook.S05848).
7) fluorescence (http://doi.org/10.1351/goldbook.F02453).

Source: PAC, 2007, 79, 293. 'Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)' on page 367 (https://doi.org/10.1351/pac200779030293)

Citation: 'magic angle' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. https://doi.org/10.1351/goldbook.MT07419

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