Magnetic Resonance — Question 2

Question 2

A resonance-frequency measurement determines a field and its uncertainty

A spin resonance occurs at $f=12.7800\,\mathrm{MHz}$ with independent frequency standard uncertainty $\sigma_f=.0020\,\mathrm{MHz}$. Treat $\gamma/(2\pi)=42.600\,\mathrm{MHz/T}$ as exact for this model.

1. Infer the field.

2. Propagate the frequency uncertainty and find the relative field uncertainty.

Question 2 - Solution

A resonance-frequency measurement determines a field and its uncertainty

(a) Method and calculation

$$f=\frac{\gamma}{2\pi}B\Rightarrow B=f/(\gamma/2\pi).$$$$\boxed{B=12.7800/42.600=.300000\,\mathrm T}.$$

(b) Interpretation and further calculation

$$\sigma_B=\left|\frac{dB}{df}\right|\sigma_f=\frac{.0020}{42.600}\,\mathrm T=\boxed{46.95\,\mu\mathrm T}.$$$$\boxed{\sigma_B/B=\sigma_f/f=1.565\times10^{-4}}.$$

Physical check

This uncertainty includes only the stated frequency uncertainty. Uncertainty in the gyromagnetic ratio or environmental field gradients would require additional terms.