QB_PH_SEC9_010: Nuclear Properties & Nuclear Forces

Using Yukawa's meson exchange theory, estimate the mass (in $\text{MeV}/c^2$) of the mediating pion for a nuclear force of range $R = 1.4\text{ fm}$. Take $\hbar c = 197.3\text{ MeV}\cdot\text{fm}$. (Round off your answer to the nearest integer)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Yukawa Field Equation**: The static meson field satisfies Klein-Gordon equation in space: $\nabla^2 \phi - \left(\frac{m_\pi c}{\hbar}\right)^2 \phi = -g \delta(\vec{r})$ 2. **Yukawa Potential**: Yields the screened potential: $V(r) = -g^2 \frac{e^{-\mu r}}{r}$ where the screening parameter $\mu = \frac{m_\pi c}{\hbar} = \frac{1}{R}$. 3. **Range-to-Mass Relationship**: $R = \frac{\hbar}{m_\pi c} = \frac{\hbar c}{m_\pi c^2} \implies m_\pi c^2 = \frac{\hbar c}{R}$ 4. **Numerical Evaluation**: - $\hbar c = 197.3\text{ MeV}\cdot\text{fm}$ - Nuclear force range: $R = 1.4\text{ fm}$ $m_\pi c^2 = \frac{197.3\text{ MeV}\cdot\text{fm}}{1.4\text{ fm}} = 140.928\text{ MeV} \approx 141\text{ MeV}$ 5. **Official Range**: $[140, 142]$.