Magnetic Resonance — Question 10

Question 10

A linear drive contains one resonant and one counter-rotating component

A transverse field is $\boldsymbol B_1(t)=B_a\cos(\omega t)\hat x$. Decompose it into two circularly rotating fields. In the rotating-wave approximation only the co-rotating term drives resonance, with Rabi angular frequency $\Omega=|\gamma|B_{\mathrm{co}}$.

1. Find each circular component and its amplitude.

2. Compare the inversion time with that from a purely co-rotating field of magnitude $B_a$.

Question 10 - Solution

A linear drive contains one resonant and one counter-rotating component

(a) Method and calculation

$$\boldsymbol B_\pm=\frac{B_a}{2}[\cos(\omega t)\hat x\pm\sin(\omega t)\hat y].$$$$\boldsymbol B_++\boldsymbol B_-=B_a\cos(\omega t)\hat x\Rightarrow\boxed{B_{\mathrm{co}}=B_a/2}.$$

(b) Interpretation and further calculation

$$\boxed{\Omega_{\mathrm{linear}}=|\gamma|B_a/2,\quad t_{\pi,\mathrm{linear}}=2\pi/(|\gamma|B_a)}.$$$$t_{\pi,\mathrm{circular}}=\pi/(|\gamma|B_a)\Rightarrow\boxed{t_{\pi,\mathrm{linear}}=2t_{\pi,\mathrm{circular}}}.$$

Physical check

The rotating-wave approximation requires a weak drive compared with the carrier frequency and near resonance. Which circular sense co-rotates depends on the sign of the gyromagnetic ratio.