To improve the chance that the neutrons produced in fission will themselves induce further fissions, it is important to slow them down. A moderator is a substance with which neutrons can collide and lose energy. It must be chosen so that the neutrons lose as much energy as possible (but are not actually captured). In practice the best process for reducing the neutrons' energy is an elastic collision. Consider a head-on elastic collision between a neutron (mass $\mathrm{m}$ ) and a nucleus (mass $M$ ). (a) Use conservation of energy and momentum (nonrelativistic) to derive an expression for the neutron's fractional loss of energy, $\Delta K / K$, as a function of the mass ratio $\mu=m / M$. (b) Show that the loss is maximum when the masses are equal $(\mu=1)$. (c) This result suggests that hydrogen - in water, for example - would be the best moderator. Unluckily, ordinary hydrogen, ${ }^1 \mathrm{H}$, has the disadvantage that it can capture neutrons readily to form deuterium. Two practical alternatives with masses at least fairly close to the neutron mass are deuterium (in "heavy" water) and carbon (in the form of graphite). What is $\Delta K / K$ for these two moderators?

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 ${ }^{236} \mathrm{U} \rightarrow{ }^{92} \mathrm{Kr}+{ }^{144} \mathrm{Ba}$.

Use the masses given in Appendix D to find the energy released in the fusion reaction and the fission .

One of the ways that Chadwick estimated the neutron mass was to direct neutrons (all of the same velocity) at two different targets ( $A$ and $B$, say). The neutrons struck the target nuclei and caused them to recoil. (a) If we denote the maximum recoil speeds (which occur in head-on collisions) by $v_A$ and $v_B$, prove that

$$m_n=\frac{m_B v_B-m_A v_A}{v_A-v_B}$$

(use nonrelativistic mechanics). (b) Chadwick used hydrogen and nitrogen for his targets, and his measured speeds were $v_{\mathrm{H}}=3.3 \times 10^7 \mathrm{~m} / \mathrm{s}$ and $v_{\mathrm{N}}=4.7 \times$ $10^6 \mathrm{~m} / \mathrm{s}$. What was his value for $\mathrm{m}_{\mathrm{n}}$ based on these data?

When the Joliot-Curies observed the reaction $\alpha+{ }^9 \mathrm{Be} \rightarrow \mathrm{n}+{ }^{12} \mathrm{C}$, they suggested (erroneously) that the neutral particle was a photon. Chadwick's main argument against this concerned the large momentum (up to $100 \mathrm{MeV} / \mathrm{c}$ ) of the protons that these supposed photons ejected from wax. Assuming that it was photons that ejected the protons and given that a proton gets the largest momentum in a head-on collision, do the following: (a) To a fair approximation, the photon would bounce back with its energy unchanged; in this approximation, what is the incident photon energy required to eject a proton with momentum $100 \mathrm{MeV} / c$ ? (b) Using conservation of energy and momentum, redo this calculation without the approximation of part (a).

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?