One can get a rough estimate of the energy released in fission by considering the above figure, which shows the potential energy of the two fragments. The energy released is roughly the potential energy at $x=x_0$, which, in turn, is roughly $k q_1 q_2 / x_0$. (Both of these approximations will actually be overestimates.) Taking $x_0 \sim 12 \mathrm{fm}$, estimate the energy released in the fission ${ }^{236} \mathrm{U} \rightarrow{ }^{92} \mathrm{Kr}+{ }^{144} \mathrm{Ba}$.

Reactions in which kinetic energy is lost $(\Delta K<0)$ are called endothermic and cannot occur below a minimum (threshold) initial energy. (Reactions in which kinetic energy is released are called exothermic.) Which of the following reactions are endothermic, and what are their threshold kinetic energies? (a) $\mathrm{n}+{ }^{208} \mathrm{~Pb} \rightarrow \mathrm{p}+{ }^{208} \mathrm{Tl}$ (b) $\mathrm{n}+{ }^{207} \mathrm{~Pb} \rightarrow \alpha+{ }^{204} \mathrm{Hg}$ (c) $\mathrm{n}+{ }^{187} \mathrm{Os} \rightarrow \mathrm{p}+{ }^{187} \mathrm{Re}$ (d) $\mathrm{n}+{ }^{197} \mathrm{Au} \rightarrow{ }^2 \mathrm{H}+{ }^{196} \mathrm{Pt}$

Use the laws of conservation of energy, charge, and nucleon number to decide which of the following processes are impossible. Give your reasons. (a) ${ }^{207} \mathrm{Po} \rightarrow{ }^{207} \mathrm{At}+\mathrm{e}^{-}+\bar{\nu}$ (b) ${ }^{213} \mathrm{Rn} \rightarrow{ }^{208} \mathrm{Po}+\alpha$ (c) ${ }^{212} \mathrm{Po} \rightarrow{ }^{208} \mathrm{~Pb}+\alpha$ (d) $\mathrm{p}+{ }^{14} \mathrm{~N} \rightarrow \alpha+{ }^{12} \mathrm{C}$ (e) $\mathrm{n}+{ }^{22} \mathrm{Na} \rightarrow \mathrm{p}+{ }^{22} \mathrm{Ne}$

Which of the following reactions violate the laws of conservation of charge or nucleon number (and are therefore impossible)? (a) $\mathrm{p}+{ }^{63} \mathrm{Cu} \rightarrow \mathrm{n}+{ }^{63} \mathrm{Ni}$ (b) $\mathrm{n}+{ }^{64} \mathrm{Zn} \rightarrow{ }^2 \mathrm{H}+{ }^{63} \mathrm{Cu}$ (c) $\mathrm{n}+{ }^{56} \mathrm{Fe} \rightarrow \alpha+{ }^{53} \mathrm{Mn}$ (d) $\mathrm{n}+{ }^{14} \mathrm{~N} \rightarrow \mathrm{p}+{ }^{13} \mathrm{C}$

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.)

The cross section for $0.01-\mathrm{MeV}$ neutrons to be captured by uranium 238 is $0.7$ barn. Two grams of ${ }^{238} \mathrm{U}$ is exposed to a flux of $3 \times 10^9$ neutrons per square meter (with energy $0.01 \mathrm{MeV}$ ). How many atoms of ${ }^{239} \mathrm{U}$ will be produced?

The isotope cobalt 60 has a decay constant $r=4.2 \times$ $10^{-1} \mathrm{~s}^{-1}$. Consider a pure sample of $1 \mu \mathrm{g}$ of ${ }^{60} \mathrm{Co}$. (a) How many atoms are in this sample? (b) How many ${ }^{60} \mathrm{Co}$ atoms will remain after one year? (c) What is the half-life of ${ }^{60} \mathrm{Co}$ ?

To find the mass of the neutron, Chadwick used the reaction

$$\alpha+{ }^{11} \mathrm{~B} \rightarrow \mathrm{n}+{ }^{14} \mathrm{~N}$$

The best available values for the masses of the three nuclei were $m_\alpha=4.00106 \mathrm{u}, m_{\mathrm{B}}=11.00825 \mathrm{u}$, and $m_{\mathrm{N}}=14.0042 \mathrm{u}$. The kinetic energies Chadwick measured as $K_\alpha=5.26 \mathrm{MeV}, K_{\mathrm{B}}=0, K_{\mathrm{n}}=3.26 \mathrm{MeV}$, and $K_{\mathrm{N}}=0.57 \mathrm{MeV}$. Using these data, find Chadwick's value for the mass of the neutron, and compare with the current value.

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.