Question

The nuclear force between two neutrons in a
nucleus is described by the Yukawa potential
U(r) = — Uo ro/r exp (-r/ro)
where r is the distance between neutrons
and Uo and ro are constants. Determine the
force F(r) = — V U(r).

Answer 1

Answer on Question #78048 Physics / Other

The nuclear force between two neutrons in a nucleus is described by the Yukawa potential $U(r) = -U_0r_0 / r\exp (-r / r_0)$ where $r$ is the distance between neutrons and $U_{0}$ and $r_0$ are constants. Determine the force $\mathbf{F}(\mathbf{r}) = -\nabla U(r)$.

Solution:

The force between two neutrons in a nucleus

$$\begin{array}{l} \mathbf {F} (r) = - \nabla U (r) = U _ {0} r _ {0} \nabla \left[ \frac {\exp \left(- \frac {r}{r _ {0}}\right)}{r} \right] = U _ {0} r _ {0} \left[ - \frac {\exp \left(- \frac {r}{r _ {0}}\right)}{r _ {0} r} \hat {\mathbf {r}} - \frac {\exp \left(- \frac {r}{r _ {0}}\right)}{r ^ {2}} \hat {\mathbf {r}} \right] \\ = - U _ {0} \exp \left(- \frac {r}{r _ {0}}\right) \left[ \frac {1}{r} + \frac {r _ {0}}{r ^ {2}} \right] \hat {\mathbf {r}}, \end{array}$$

where $\hat{\mathbf{r}} = \mathbf{r} / r$

Answer: $\mathbf{F}(r) = -U_0\exp \left(-\frac{r}{r_0}\right)\left[\frac{1}{r} +\frac{r_0}{r^2}\right]\hat{\mathbf{r}}.$

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