Chapter Questions
Give the proper isotopic symbols for $(a)$ the isotope of fluorine with mass number $19 ;(b)$ an isotope of gold with 120 neutrons; $(c)$ an isotope of mass number 107 with 60 neutrons.
Tin has more stable isotopes than any other element; they have mass numbers $114,115,116,117,118,119,120,122,$124. Give the symbols for these isotopes.
There are four stable isotopes with odd $N=Z$. Give their isotopic symbols.
(a) Compute the Coulomb repulsion energy between two nuclei of ${ }^{16} \mathrm{O}$ that just touch at their surfaces.(b) Do the same for two nuclei of ${ }^{228} \mathrm{U}$
Find the nuclear radius of $(a)^{197} \mathrm{Au} ;(b)^{4} \mathrm{He} ;(c)^{20} \mathrm{Ne}$
The ionic radius of lead is $0.180 \mathrm{nm}$. Compute the fraction of the volume of a lead atom occupied by a nucleus of ${ }^{208} \mathrm{Pb}$
Find the total binding energy and the binding energy per nucleon for $(a)^{208} \mathrm{Pb} ;(b)^{133} \mathrm{Cs} ;(c)^{90} \mathrm{Zr} ;(d)^{59} \mathrm{Co}$
Find the total binding energy and the binding energy per nucleon for $(a)^{4} \mathrm{He}$$(c)^{40} \mathrm{Ca} ;(d)^{55} \mathrm{Mn}$$(b)^{20} \mathrm{Ne}$
Calculate the total nuclear binding energy of ${ }^{3} \mathrm{He}$ and ${ }^{3} \mathrm{H}$. Account for any difference by considering the Coulomb interaction of the extra proton of ${ }^{3}$ He.
Find the neutron separation energy of $(a)^{17} \mathrm{O} ;(b)^{7} \mathrm{Li}$$(c)^{57} \mathrm{Fe}$
Find the proton separation energy of $(a)^{4} \mathrm{He} ;(b)^{12} \mathrm{C}$$(c)^{40} \mathrm{Ca}$
The nucleus ${ }^{13} \mathrm{C}$ has 6 protons and 7 neutrons, and the nucleus ${ }^{13} \mathrm{N}$ has 7 protons and 6 neutrons. Calculate the binding energy per nucleon for each of these nuclei and on that basis predict which one will decay into the other.
The nuclear attractive force must turn into a repulsion at very small distances to keep the nucleons from crowding too close together. What is the mass of an exchanged particle that will contribute to the repulsion at separations of $0.25 \mathrm{fm} ?$
The weak interaction (the force responsible for beta decay) is produced by an exchanged particle with a mass of roughly 80 GeV. What is the range of this force?
Determine the depth of the proton and neutron potential energy wells for $(a)^{16} \mathrm{O} ;(b)^{235} \mathrm{U}$
The two-neutron separation energies of ${ }^{160}$ Dy and 164 Dy are, respectively, 15.4 MeV and 13.9 MeV, and the twoproton separation energies of ${ }^{158}$ Dy and ${ }^{162}$ Dy are, respectively, 12.4 MeV and 14.8 MeV. From these data alone, determine whether alpha decay is energetically allowed for ${ }^{160} \mathrm{Dy}$ and ${ }^{164} \mathrm{Dy}$
What fraction of the original number of nuclei present in a sample will remain after ( $a$ ) two half-lives; ( $b$ ) four half-lives; $(c)$ 10 half-lives?
A certain sample of a radioactive material decays at a rate of 548 per second at $t=0 .$ At $t=48 \mathrm{min}$, the counting rate has fallen to 213 per second. $(a)$ What is the half-life of the radioactivity? $(b)$ What is its decay constant? $(c)$ What will be the decay rate at $t=125 \mathrm{min} ?$
What is the decay probability per second per nucleus of a substance with a half-life of $5.0 \mathrm{h} ?$
Tritium, the hydrogen isotope of mass $3,$ has a half-life of $12.3 \mathrm{y} .$ What fraction of the tritium atoms remains in a sample after 50.0 y?
Suppose we have a sample containing $2.00 \mathrm{mCi}$ of radioactive $^{131} \mathrm{I}\left(t_{1 / 2}=8.04 \mathrm{d}\right)$(a) How many decays per second occur in the sample?(b) How many decays per second will occur in the sample after 4 weeks?
Ordinary potassium contains $0.012 \%$ of the naturally occurring radioactive isotope ${ }^{40} \mathrm{K}$, which has a half-life of $1.3 \times 10^{9}$ y. $(a)$ What is the activity of $1.0 \mathrm{kg}$ of potassium?(b) What would have been the fraction of ${ }^{40} \mathrm{K}$ in natural potassium $4.5 \times 10^{9} \mathrm{y}$ ago?
Derive Eq. 12.23 from Eqs 12.21 and 12.22
For which of the following nuclei is alpha decay permit$\operatorname{ted} ?(a)^{210} \mathrm{Bi}(b)^{203} \mathrm{Hg}(c)^{211} \mathrm{At}$
Find the kinetic energy of the alpha particle emitted in the decay of ${ }^{234} \mathrm{U}$.
\text { Derive } \mathrm{Eq} \times 12.29,12.33, \text { and } 12.36
Find the maximum kinetic energy of the electrons emitted in the negative beta decay of $"$ Be.
${ }^{75}$ Se decays by electron capture to ${ }^{75}$ As. Find the energy of the emitted neutrino.
${ }^{15}$ O decays to ${ }^{15}$ N by positron beta decay. (a) What is the $Q$ value for this decay? (b) What is the maximum kinetic energy of the positrons?
The nucleus ${ }^{198} \mathrm{Hg}$ has excited states at 0.412 and 1.088 MeV. Following the beta decay of ${ }^{198}$ Au to ${ }^{198} \mathrm{Hg}$, three gamma rays are emitted. Find the energies of these threegamma rays.
Compare the recoil energy of a nucleus around mass 200 that emits $(a)$ a 5.0 -MeV alpha particle and $(b)$ a $5.0-$ MeVgamma ray.
A certain nucleus has the following sequence of rotational states $E_{L}$ (energies in $\mathrm{keV}$ ): $E_{0}=0, E_{1}=100.1$ $E_{2}=300.9, E_{3}=603.6,$ and $E_{4}=1010.0 .$ Assumingthat the emitted gamma rays occur only from changes of one or two units in the rotational quantum number, find all possible photon cnergics that can be emitted from these states. Sketch the excited states, showing the allowed transitions.
The radioactive decay of ${ }^{232}$ Th leads eventually to stable 208 Pb. A certain rock is examined and found to contain $3.65 \mathrm{g}$ of ${ }^{232} \mathrm{Th}$ and $0.75 \mathrm{g}$ of ${ }^{208} \mathrm{Pb},$ Assuming all of the Pb was produced in the decay of $T h,$ what is the age of the rock?
The $4 n$ radioactive decay series begins with ${ }^{222} \mathrm{Th}$ and ends with ${ }_{82}^{208} \mathrm{Pb} .(a)$ How many alpha decays are in the chain? (See Question 22 ) $(b)$ How many beta decays?(c) How much energy is released in the complete chain?(d) What is the radioactive power produced by $1.00 \mathrm{kg}$ of $232 \mathrm{Th}\left(t_{1 / 2}=1.40 \times 10^{10} \mathrm{y}\right) ?$
A piece of wood from a recently cut tree shows $12.4^{14} \mathrm{C}$ decays per minute. A sample of the same size from a tree cut thousands of years ago shows 3.5 decays per minute. What is the age of this sample?
Figure 12.2 suggests that the Rutherford scattering formula fails for $60^{\circ}$ scattering when $K$ is about 28 MeV. Use the results derived in Chapter 6 to find the closest distance between alpha particle and nucleus for this case, and compare with the nuclear radius of 208 Pb. Suggest a possible reason for any discrepancy.
Assuming the nucleus to diffract like a circular disk, use the data shown in Figure 12.3 to find the nuclear radius for ${ }^{12} \mathrm{C}$ and ${ }^{16} \mathrm{O}$. How does changing the electron energy from 360 MeV to 420 MeV affect the deduced radius for ${ }^{16}$ O? (Hint: Use the extreme relativistic approximation from Chapter 2 to relate the electron's energy and momentum to find its de Broglie wavelength.)
A radiation detector is in the form of a circular disk of diameter $3.0 \mathrm{cm} .$ It is held $25 \mathrm{cm}$ from a source of radiation, where it records 1250 counts per second. Assuming that the detector records every radiation incident upon it, find the activity of the sample (in curies).
What is the activity of a container holding $125 \mathrm{cm}^{3}$ of tritium $\left({ }^{3} \mathrm{H}, t_{1 / 2}=12.3 \mathrm{y}\right)$ at a pressure of $5.0 \times 10^{5} \mathrm{Pa}$(about 5 atm $)$ at $T=300$ K?
With a radioactive sample originally of $N_{0}$ atoms, we could measure the mean, or average, lifetime $\tau$ of a nucleus by measuring the number $N_{1}$ that live for a time $t_{1}$ and then decay, the number $N_{2}$ that decay after $t_{2}$ andso on:$$\tau=\frac{1}{N_{0}}\left(N_{1} t_{1}+N_{2} t_{2}+\cdots\right)$$(a) Show that this is equivalent to $\tau=\lambda \int_{0}^{\infty} e^{-\lambda t} t d t$(b) Show that $\tau=1 / \lambda$. (c) Is $\tau$ longer or shorter than $t_{1 / 2} ?$
Complete the following decays:$(a)^{27} \mathrm{Si} \rightarrow \frac{27}{\mathrm{Al}}+$$(b)^{74} \mathrm{As} \rightarrow{ }^{74} \mathrm{Se}+$$(c)^{228} \mathrm{U} \rightarrow \alpha+$$(d)^{93} \mathrm{Mo}+\mathrm{e}^{-} \rightarrow$$(e)^{131} \mathrm{I} \rightarrow{ }^{131} \mathrm{Xe}+$
${ }^{239} \mathrm{Pu}$ decays by alpha emission with a half-life of $2.41 \times$ $10^{4}$ y. Compute the power output, in watts, which could be obtained from $1.00 \mathrm{g}$ of ${ }^{239} \mathrm{Pu}$.
${ }^{228} \mathrm{Th}$ alpha decays to an excited state of ${ }^{22}{ }^{4} \mathrm{Ra}$, which in turn decays to the ground state with the emission of a 217-keV photon. Find the kinetic energy of the alpha particle.
By replacing the Coulomb barrier in alpha decay with a flat barrier (see Figure 12.16 ) of thickness $L=\frac{1}{2}\left(R^{\prime}-R\right)$, equal to half the thickness of the Coulomb barrier that the alpha particle must penetrate, and height $U_{0}=$ $\frac{1}{2}\left(U_{\mathrm{B}}+K_{\mathrm{a}}\right),$ equal to half the height of the Coulomb barrier above the energy of the alpha particle, estimate the decay half-lives for $^{232}$ Th and ${ }^{218} \mathrm{Th}$ and compare with the measured values given in Table $12.2 .$ (Hint: In calculating the speed of the alpha particle inside the nucleus, assume that the well depth is 30 MeV.) Although the results of this rough calculation do not agree well with the measured values, the calculation does indicate how barrier penetration is responsible for the enormous range of observed half-lives. How would you refine the calculation to obtain better agreement with the measured values?
(a) Using the same replacements described in Problem 44 , estimate the decay probability of ${ }^{226} \mathrm{Ra}$ for alpha emission and for ${ }^{14} \mathrm{C}$ emission. (See Examples 12.9 and $12.10 .$ )(b) Using the results of part $(a),$ estimate the number of ${ }^{14} \mathrm{Cemitted}$ relative to the number of alpha particles emitted by a source of 226 Ra.
Compute the recoil proton kinetic energy in neutron beta decay (a) when the electron has its maximum energy and (b) when the neutrino has its maximum energy.
In the beta decay of ${ }^{24} \mathrm{Na}$, an electron is observed with a kinetic energy of 2.15 MeV. What is the energy of the accompanying neutrino?
The first excited state of 57 Fe decays to the ground state with the emission of a $14.4-\mathrm{keV}$ photon in a mean lifetime of 141 ns. $(a)$ What is the width $\Delta E$ of the state?(b) What is the recoil kinetic energy of an atom of ${ }^{3}$ Fe that emits a 14.4-keV photon? (c) If the kinetic energy of recoil is made negligible by placing the atoms in a solid lattice, resonant absorptions will occur. What velocity is required to Doppler shift the emitted photon so that resonance does not occur?
What is the probability of a ${ }^{14} \mathrm{C}$ atom in atmospheric $\mathrm{CO}_{2}$ decaying in your lungs during a single breath? The atmosphere is about $0.03 \% \mathrm{CO}_{2}$. Assume you take in about 0.5 L of air in each breath and exhale it 3.5 s later.
(a) Calculate the $Q$ value for the negative beta decayble. (b) In deriving the expressions for the beta decay, we neglected the electron binding energy, as we also did in going from Eq. 12.3 to Eq. 12.4 . Derive an expression for the beta decay $Q$ value (starting with the difference in the nuclear masses) that includes the difference in the total electron binding cnergy of the Dy and Ho atoms.(c) For Dy and Ho, the difference in binding energy is $B_{1 \mathrm{lo}}-B_{\mathrm{Dy}}=12.9 \mathrm{keV} .$ What is the new $\mathrm{Q}$ value includ-ing this correction? Does this make the decay possible?(d) Would the decay be possible if instead of a neutral atom of Dy we used instead a fully ionized Dy (stripped of all its electrons)? Would it be possible if instead of a free particle the beta decay electron was injected directly into a bound state in Ho? (The K electron binding energy in Ho is $65.1 \mathrm{keV} .$ )
The nucleus ${ }^{2} \mathrm{H}$ has no rotational excited states. Justify this observation with a calculation of the energy that the first rotational excited state would have.
It is theoretically possible for some "stable" nuclei to decay by emitting simultancously two electrons in a process called double beta decay. In analogy with Eq. 12.28 , the decay is $\frac{d}{z} X_{N} \rightarrow z+{ }_{2}^{A} X^{\prime \prime}+2 e^{-}+2 \bar{v}$ and it proceedseven when the beta decay of $\mathrm{X}$ to the intermediate nucleus $z_{+1}^{A} \mathrm{X}^{\prime}$ is not permitted. That is the process does not consist of two successive beta decays the two electrons are emitted in a single decay process.(a) Show that the $Q$ value for this process is given by $Q=\left[m\left({ }^{4} \mathrm{X}\right)-m\left({ }^{4} \mathrm{X}^{\prime \prime}\right)\right] c^{2} .$ (b) For which of the followingstable nuclei is double beta decay possible: $^{82} \mathrm{Se},{ }^{120} \mathrm{Sn}$, $130 \mathrm{Te}, 132 \mathrm{Xe} ?$
${ }^{209}$ Bi is normally regarded as a stable nucleus, but in 2004 researchers observed its alpha decay. (a) What is the decay O value?(b) Using data from Table $12.2,$ make a log-log plot of the alpha decay half-life against the $Q$ value and make a rough estimate of the expected half-life for ${ }^{209}$ Bi. (The barrier penetration probability also depends on the Z of the nucleus, but that dependence is very weak compared with the dependence on the $Q$ value and we can ignore it for a rough estimate.) (c) The researchers used a sample of $93 \mathrm{g}$ of $\mathrm{Bi}$ and observed 128 decays in 5 days. What was their deduced half-life?
Neutral atoms of ${ }^{52}$ Fe decay by a mixture of $\beta^{+}$ and electron capture with a half-life of $8.275 \pm 0.008 \mathrm{h}$. The half-life of fully ionized $^{52}$ Fe, however, is $12.5 \mathrm{h}$ with an experimental uncertainty that encompasses the range from $11.3 \mathrm{h}$ to $14.0 \mathrm{h} .$ For neutral atoms of ${ }^{52} \mathrm{Fe},$ what are the relative amounts of $\beta^{+}$ and electron capture decays and what is their range of uncertainty?