Central Forces 2023 (3 years)
In a solid, a free electron doesn't see” a bare nuclear charge
since the nucleus is surrounded by a cloud of other electrons. The
nucleus will look like the Coulomb potential close-up, but be
screened” from far away. A common model for such problems is
described by the Yukawa or screened potential:
\begin{equation}
U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}}
\end{equation}
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Graph the potential, with and without the exponential term.
Describe how the Yukawa potential approximates the “real”
situation. In particular, describe the role of the parameter
\(\alpha\).
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Draw the effective potential for the two choices \(\alpha=10\) and
\(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of
\(\alpha\) is there the possibility of stable circular orbits?