Magnetic Resonance — Question 1

Question 1

Electron and proton resonances probe very different frequency bands

Use the magnitudes $|\gamma_e|/(2\pi)=28.0\,\mathrm{GHz/T}$ and $\gamma_p/(2\pi)=42.6\,\mathrm{MHz/T}$. A spin-half Zeeman gap is $\Delta E=\hbar|\gamma|B$. Both species experience $B=1.20\,\mathrm T$.

1. Find each resonance frequency.

2. Find the ratio of their photon energies and the proton field required to resonate at the electron frequency found in part (a).

Question 1 - Solution

Electron and proton resonances probe very different frequency bands

(a) Method and calculation

$$hf=\hbar|\gamma|B\Rightarrow f=|\gamma|B/(2\pi).$$$$\boxed{f_e=33.6\,\mathrm{GHz},\quad f_p=51.12\,\mathrm{MHz}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\gamma,e}/E_{\gamma,p}=f_e/f_p=657.3}.$$$$\boxed{B_{p,\mathrm{required}}=33600/42.6=788.7\,\mathrm T}.$$

Physical check

The comparison uses the same field but different gyromagnetic ratios. The very large inferred proton field is an algebraic comparison, not an assumed laboratory operating condition.