QB_PH_SEC6_027: Lasers and Quantum Optics

In a Proton NMR spectrometer operating with a static magnetic field of $B_0 = 7.05\text{ T}$, what is the resonance Larmor frequency $\nu$ (in MHz)? Take the proton gyromagnetic ratio $\gamma = 2.675 \times 10^8\text{ rad}\cdot\text{s}^{-1}\text{T}^{-1}$. (Round off to the nearest integer)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Larmor Precession Angular Frequency**: $\omega_0 = \gamma B_0$ 2. **Cyclical Frequency Conversion**: $\nu = \frac{\omega_0}{2\pi} = \frac{\gamma B_0}{2\pi}$ 3. **Numerical Computation**: - $\gamma = 2.675198 \times 10^8\text{ rad/s}\cdot\text{T}$ - $B_0 = 7.05\text{ T}$ $\nu = \frac{(2.675198 \times 10^8)(7.05)}{2 \times 3.14159} = \frac{1.88601 \times 10^9}{6.28318} = 300.16 \times 10^6\text{ Hz} = 300\text{ MHz}$ 4. **Official Integer Answer**: 300.