When Rutherford first observed the reaction , he assumed that the alpha particle simply knocked a proton out of the ${ }^{14} \mathrm{~N}$ in the process $\alpha+{ }^{14} \mathrm{~N} \rightarrow \alpha+\mathrm{p}+{ }^{13} \mathrm{C}$. Given that the incident alpha particles had about $5.6 \mathrm{MeV}$, was this reaction possible?

Use the data in Appendix D to write down the whole of the radioactive series that begins with ${ }^{235} \mathrm{U}$. Draw the series on a plot of $Z$ against $N$ as in the above figure.

Use the data in Appendix $\mathrm{D}$ to write down the whole of the radioactive series that begins with ${ }^{238} \mathrm{U}$. Draw the series on a plot of $Z$ against $N$ as in the above figure.

(a) Use the data in Appendix D to find the decay constant and mean life of uranium 232. (b) Do the same for polonium 214.

(a) Use the data in Appendix D to calculate the total kinetic energy released in the $\alpha$ decay of ${ }^{220} \mathrm{Rn}$. (b) Find the total kinetic energy released in the $\beta$ decay of ${ }^{220} \mathrm{Rn}$. Is this decay possible? (c) Answer the same questions for ${ }^{212} \mathrm{Bi}$. (d) What is the relevance of your answers in consideration of the above figure ?

The supernova explosion that is believed to have preceded the formation of our solar system created large quantities of radioactive nuclei. There are reasons to think that the isotopes ${ }^{235} \mathrm{U}$ and ${ }^{238} \mathrm{U}$ would have been created in roughly equal numbers. That is, the ratio $\mathrm{N}_{238} / \mathrm{N}_{235}$ was close to 1 at $t=0$ (the time of the explosion). (a) Show that at any later time

$$\frac{N_{238}}{N_{235}}=\exp \left[\left(r_{235}-r_{238}\right) t\right]$$

where $r_{235}$ and $r_{238}$ denote the decay constants of ${ }^{235} \mathrm{U}$ and ${ }^{238} \mathrm{U}$. (b) Use the data in Appendix D to find $r_{235}$ and $r_{238}$. (c) Use the observed abundances of ${ }^{235} \mathrm{U}$ and ${ }^{238} \mathrm{U}$ on earth today to estimate how long ago the supernova explosion occurred.

Use the data in Appendix D to find the kinetic energy released in each of the following conceivable decays:

$$\begin{aligned} { }^{228} \mathrm{Th} & \rightarrow{ }^{227} \mathrm{Th}+\mathrm{n} \\ & \rightarrow{ }^{227} \mathrm{Ac}+\mathrm{p} \\ & \rightarrow{ }^{228} \mathrm{~Pa}+\mathrm{e}^{-}+\bar{\nu} \\ & \rightarrow{ }^{224} \mathrm{Ra}+\alpha \\ & \rightarrow{ }^{216} \mathrm{Po}+{ }^{12} \mathrm{C} \end{aligned}$$

Which of these processes is possible? (Note: You will find that the last one is possible. However, its probability is so small compared to $\alpha$ decay that it has not been observed.)

(a) Use the semiempirical binding-energy formula (16.30) to write an expression for the mass of a nucleus in terms of $Z$ and $A$. (b) Now consider a set of isobars with $A$ odd. Use your mass formula to show that a plot of mass against $Z$, as in the above figure, should fit a parabola $\left(m=\alpha Z^2+\beta Z+\gamma\right)$. (c) Now consider a set of isobars with $A$ even. Show that because of the pairing energy, the masses will alternate between two parabolas, one for the nuclei with $Z$ even and the other for $Z$ odd. (d) Draw the plot of $m$ against $Z$, and explain why one can have two stable isobars if $A$ is even. (e) Find two examples of this phenomenon in Appendix D.

If we neglect the neutrino's mass, the total kinetic energy, $K$, released in the $\beta^{-}$decay $(Z, N) \rightarrow$ $(Z+1, N-1)+e^{-}+\bar{\nu}$ is given by (17.22). In particular, the criterion that $\beta^{-}$decay is possible is just that $K$ must be positive, or

$$m_{\text {atom }}(Z, N) \geq m_{\text {atom }}(Z+1, N-1) \quad \text { (for } \beta^{-} \text {decay) }$$

In other words, one has only to compare the atomic masses involved to see if $\beta^{-}$decay is possible. (a) Find an expression for the total kinetic energy released in the $\beta^{+}$decay

$$(Z, N) \rightarrow(Z-1, N+1)+e^{+}+\nu$$

and show that the condition for this process to be possible is

$$m_{\text {atom }}(Z, N) \geq m_{\text {atom }}(Z-1, N+1)+2 m_e$$

$$\text { (for } \beta^{+} \text {decay) }$$

(b) Do the same for the electron capture

$$(Z, N)+e^{-} \rightarrow(Z-1, N+1)+\nu \quad(17.85)$$

assuming that the original two particles are at rest, and show that the condition for electron capture is the same as (17.84) except that the term $2 m_e$ is missing. (c) The mass of ${ }^{231} \mathrm{U}$ is $231.03626 \mathrm{u}$, while that of ${ }^{231} \mathrm{~Pa}$ is $231.03588 \mathrm{u}$. Show that ${ }^{231} \mathrm{U}$ can decay by electron capture but not by $\beta^{+}$emission.

Write down an equation [similar to the above Eqution] to describe the $\beta$ decay of each of the following nuclei: (a) ${ }^{60} \mathrm{Co}$, (b) ${ }^3 \mathrm{H}$ (tritium), (c) ${ }^{15} \mathrm{O}$, and (d) the decay by electron capture of ${ }^{231} \mathrm{U}$.

Calculate the total kinetic energy released in the $\beta$ decay of a single free neutron.

Use the data in Appendix D to find the total kinetic energy released in the $\beta^{-}$decay of ${ }^{14} \mathrm{C}$.

The decay constant for a certain nucleus is $r=0.010 \mathrm{~min}^{-1}$. At $t=0$ a sample contains $10^6$ of these nuclei. (a) How many of these nuclei will remain after an hour? (b) What is the total rate of decays, $R(t)$, at $t=0$ ? (c) What is $R(t)$, after an hour?

The above equation, which gives the kinetic energy released in $\beta$ decay, was derived on the assumption that there are only two final particles (the electron and the residual nucleus, but no neutrino). Prove, nevertheless, that correctly predicts the total kinetic energy available to all three final particles, given that the neutrino's mass is negligible in this context.