Chapter 30

Uranium- 238 undergoes alpha decay as follows: $$ { }_{92}^{238} \mathrm{U} \quad \rightarrow \quad{ }_{90}^{234} \mathrm{Th}+\quad{ }_{2}^{4} \mathrm{He} $$ $$ (238.050786 u) \quad(234.043583 u) \quad(4.002603 u) $$ (a) Would you expect the \(Q\) value to be (1) positive, (2) negative, or (3) zero? Why? (b) Find the \(Q\) value.

Find the threshold energy for the following reaction: $$ { }_{8}^{16} \mathrm{O}+{ }_{0}^{1} \mathrm{n} \quad \rightarrow \quad{ }_{6}^{13} \mathrm{C} \quad+\quad{ }_{2}^{4} \mathrm{He} $$ $$ \begin{array}{llll} (15.994915 u) & (1.008665 u) & (13.003355 u) & (4.002603 u) \end{array} $$

Find the threshold energy for the following reaction: $$ { }_{2}^{3} \mathrm{He}+{ }_{0}^{1} \mathrm{n} \quad \rightarrow \quad{ }_{1}^{2} \mathrm{H} \quad+\quad{ }_{1}^{2} \mathrm{H} $$ $$ \begin{array}{llll} (3.016029 u) & (1.008665 u) & (2.014102 u) & (2.014102 u) \end{array} $$

Is the given reaction endoergic or exoergic? Prove your answer by determining the \(Q\) value. $$ { }_{3}^{7} \mathrm{Li}+{ }_{1}^{1} \mathrm{H} \quad \rightarrow \quad{ }_{2}^{4} \mathrm{He} \quad+\quad{ }_{2}^{4} \mathrm{He} $$ $$ \begin{array}{llll} (7.016005 u) & (1.007825 u) & (4.002603 u) & (4.002603 u) \end{array} $$

Determine the \(Q\) value of the following reaction: $$ { }_{4}^{9} \mathrm{Be}+{ }_{2}^{4} \mathrm{He} \quad \rightarrow \quad{ }_{6}^{12} \mathrm{C} \quad+\quad{ }^{1} \mathrm{n} $$ $$ \begin{array}{llll} (9.012183 u) & (4.002603 u) & (12.000000 u) & (1.008665 u) \end{array} $$

\({ }^{226}\) Ra decays and emits a 4.706 -MeV alpha particle. Find the kinetic energy of the recoiling daughter nucleus from the decay of a stationary radium- 226 nucleus.

(a) In power reactors, using water as a moderator ("neutron slower") works well, because the proton and neutron have nearly the same mass. Explain why this is true. [Hint: Consider an elastic head-on collision of objects of equal mass.] (b) From your reasoning in part (a), it follows that in a head-on elastic collision, we might expect a neutron to lose \(a l l\) of its kinetic energy in one collision, whereas for an "almost miss," it might be expected to lose essentially none. Let's therefore assume that, on average, the neutron loses \(50 \%\) of its kinetic energy during each proton collision. Estimate how many collisions are needed to reduce a \(2.0-\mathrm{MeV}\) neutron to a neutron with a kinetic energy of only \(0.02 \mathrm{eV}\) (approximately "thermal"

In a decay process the \(\beta\) has a kinetic energy of \(0.65 \mathrm{MeV}\), and the neutrino energy is \(0.25 \mathrm{MeV}\). Neglecting daughter recoil, find the disintegration energy.

Show that the disintegration energy for \(\beta^{-}\) decay is \(Q=\left(m_{\mathrm{P}}-m_{\mathrm{D}}-m_{\mathrm{e}}\right) c^{2}=\left(M_{\mathrm{P}}-M_{\mathrm{D}}\right) c^{2},\) where the \(m^{\prime}\) s represent the masses of the parent and daughter nuclei and the \(M^{\prime}\) s represent the masses of the neutral atoms. [Hint: The number of electrons before and after are the same. Why?]

Show that the disintegration energy for \(\beta^{+}\) decay is \(Q=\left(m_{\mathrm{P}}-m_{\mathrm{D}}-m_{\mathrm{e}}\right) c^{2}=\left(M_{\mathrm{P}}-M_{\mathrm{D}}-2 m_{\mathrm{e}}\right) c^{2},\) where the \(m^{\prime}\) s represent the masses of the parent and daughter nuclei and the \(M^{\prime}\) s represent the masses of the neutral atoms. [Hint: Count the number of electrons both before and after the decay. They are not the same.]

(a) The quark combination for a antineutron is (1) \(\bar{u} \overline{d d}\), (2) uud, (3) \(\overline{u u} \overline{\bar{d}}\) (4) ddd. (b) Prove that your answer to (a) gives the correct electric charge for the antineutron. part

(a) Show that the neutral pion cannot be composed solely of any pair of quarks in which one is an up quark (or an anti-up quark) and one is a down quark (or an antidown quark). (b) According to quantum theory, the \(\pi^{\circ}\) can be thought of as a sum (superposition) of two different pairs of quarks/antiquarks. Each pair would be either both up or both down quarks. What are these two pairs?

Suppose the grand unified theory (GUT) was correct and the half-life of a proton was \(1.2 \times 10^{35} \mathrm{y}\). Estimate the decay rate of a liter of water in decays per second and curies. How does this compare to a small (by laboratory standards) radioactive source of one microcurie?

Show that the \(Q\) value for electron capture (EC) is given by \(Q=\left(m_{\mathrm{P}}+m_{\mathrm{e}}-m_{\mathrm{D}}\right) c^{2}=\left(M_{\mathrm{P}}-M_{\mathrm{D}}\right) c^{2},\) where the \(m^{\prime}\) s represent the masses of the parent and daughter nuclei and the \(M^{\prime}\) s represent the masses of the neutral atoms.

Determine the \(Q\) value and the threshold energy for: $$ { }_{8}^{16} \mathrm{O}+{ }_{0}^{1} \mathrm{n} \quad \rightarrow \quad{ }_{6}^{13} \mathrm{C} \quad+\quad{ }_{2}^{4} \mathrm{He} $$ $$ \begin{array}{llll} (15.994915 u) & (1.008665 u) & (13.003355 u) & (4.002603 u) \end{array} $$