Magnetism and Electrons — Question 9

Question 9

Magnetic resonance and mechanical torque infer the same spin ratio

A localized spin moment has gyromagnetic magnitude gamma. Its resonance angular frequency is omega in field B. A prepared expectation vector has transverse angular-momentum magnitude Sperp.

1. Infer gamma and the transverse moment magnitude from the resonance measurement.

2. Find the torque magnitude and verify that it gives the rate of transverse-vector change required by precession.

Question 9 - Solution

Magnetic resonance and mechanical torque infer the same spin ratio

(a) Method and calculation

$$\omega=\gamma B\Rightarrow\boxed{\gamma=\omega/B}.$$$$\boxed{\mu_\perp=\gamma S_\perp=\omega S_\perp/B}.$$

(b) Interpretation and further calculation

$$|\boldsymbol\tau|=\mu_\perp B=\boxed{\omega S_\perp}.$$$$|d\mathbf S/dt|=|\boldsymbol\Omega\times\mathbf S|=\omega S_\perp=|\boldsymbol\tau|.$$

Physical check

Angular frequency must be used; substituting cyclic frequency without the factor 2pi would understate the gyromagnetic ratio.