A hunter observes 100 crows settling randomly among the leaves of a large oak tree, where they are no longer visible. He fires 500 shots at random into the tree and hits 25 of the crows. If the area of the tree (as seen by the hunter) is $1200 \mathrm{ft}^2$, what is $\sigma$, the cross-sectional area of a single crow? (Notice how, in a classical experiment of this type, $\sigma$ is the actual cross-sectional area of the target.)

Before two nuclei can react, they must come within $1 \mathrm{fm}$ or so of one another, and because of their Coulomb repulsion, this requires a lot of energy. (a) Calculate the energy needed to bring a proton from infinity to a separation of $1 \mathrm{fm}$ from an alpha particle. $(1 \mathrm{fm}=$ distance from surface to surface. $)$ (b) Do the same for an alpha particle and a uranium nucleus. [Use the above eqution to estimate the radii involved.]

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 above equation and the data of Appendix D to predict $\Delta K$, the kinetic energy released in the reaction $\mathrm{p}+{ }^7 \mathrm{Li} \rightarrow \alpha+\alpha$.

In most nuclear reactions the total nucleon number is conserved; that is, $\sum A_{\text {initial }}=\sum A_{\text {final }}$, where $A=Z+N$ for each nucleus, as usual. Show that, except in reactions involving $\beta$ decay, both $\sum Z$ and $\sum N$ are separately conserved. [Hint: Total charge is always conserved, and except in $\beta$ decay, all the charge is carried by protons.]

Consider the $\alpha$ decay of a nucleus $\mathrm{X}$ at rest, $\mathrm{X} \rightarrow \mathrm{Y}+\alpha$, where $\mathrm{Y}$ denotes the offspring nucleus. (a) Prove that because the $\alpha$ is much lighter than Y, almost all the available kinetic energy goes to the $\alpha$; specifically, prove that

$$K_\alpha=\frac{m_{\mathrm{Y}}}{m_\alpha} K_{\mathrm{Y}}$$

[Hint: Use conservation of momentum and treat all particles nonrelativistically.] (b) In the $\alpha$ decay of a typical naturally radioactive nucleus, what percentage of the available kinetic energy does the alpha particle take? (c) Assuming that $\beta$ decay has the form $\mathrm{X} \rightarrow \mathrm{Y}+\mathrm{e}$, use the relation corresponding to (17.86) to find what percentage of the available kinetic energy goes to the electron. (Your answer here is really an upper bound, since some energy can go to the neutrino. It is also only approximate since the electron can be fairly relativistic.)

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.