Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Ernst angle

The Ernst angle is the flip angle (a.k.a. "tip" or "nutation" angle) for a particular spin that gives the maximal signal in the least amount of time when signal averaging over many transients. This relationship was developed by Richard Ernst, winner of the 1991 Nobel Prize in Chemistry.

The following equation relates the Ernst angle, theta, to the experimental interpulse delay, d1; the duration of the free induction decay, aka "acquisition time", or "at"; and the longitudinal relaxation time of the spin in question, T1:

$$cos(\theta)=e^{-(d1+at)/T_1}$$

For example, if one wishes to get the best signal from a resonance with T1=3 sec, and one wishes to use d1=1sec and at=2sec, the optimal tip angle is 68 degrees.

This relationship is especially important in Magnetic Resonance Imaging (MRI), where interscan delays (d1) and acquisition times (at) are often short relative to the signal's T1 value. In MRI, there is typically just one resonance being observed - H2O - and the T1 of H2O depends on its local environment. Note that d1 and at together may be referred to collectively as TR in the MRI community.