Modern Physics

Kenneth S. Krane

Chapter 13

Nuclear Reactions and Applications - all with Video Answers

Educators

Chapter Questions

Problem 1

Fill in the missing particle in these reactions:
(a) ${ }^{4} \mathrm{He}+{ }^{14} \mathrm{~N} \rightarrow{ }^{17} \mathrm{O}+$
(c) ${ }^{27} \mathrm{Al}+{ }^{4} \mathrm{He} \rightarrow \mathrm{n}+$
(b) ${ }^{9} \mathrm{Be}+{ }^{4} \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+$
(d) ${ }^{12} \mathrm{C}+\rightarrow{ }^{13} \mathrm{~N}+\mathrm{n}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 2

In a certain nuclear reaction, outgoing protons are observed with energies $16.2 \mathrm{MeV}, 14.8 \mathrm{MeV}, 11.6 \mathrm{MeV}, 8.9 \mathrm{MeV},$ and $6.7 \mathrm{MeV}$. No energies higher than $16.2 \mathrm{MeV}$ are observed. Construct a level scheme of the product nucleus.

Eduard Sanchez

Numerade Educator

Problem 3

In order to determine the cross section for neutron capture, you are irradiating a thin gold foil, in the form of a circular disk of diameter $3.0 \mathrm{~mm}$ and thickness $1.81 \mu \mathrm{m},$ with neutrons to produce the reaction $\mathrm{n}+{ }^{197} \mathrm{Au} \rightarrow{ }^{198} \mathrm{Au}+\gamma . \mathrm{By}$ observing the outgoing gamma-ray photons in a detector, you determine that the gold decays at a rate of $5.37 \times 10^{6}$ per second. From an independent measurement, you have determined the neutron flux to be $7.25 \times 10^{10}$ neutrons $/ \mathrm{cm}^{2} / \mathrm{s}$. What value do you deduce for the cross section for this reaction?

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 4

The element cobalt is commonly used for measuring the intensity of neutron beams through the reaction $\mathrm{n}+{ }^{59} \mathrm{Co} \rightarrow$ ${ }^{60} \mathrm{Co}+\gamma .$ By observing the radioactive decay of ${ }^{60} \mathrm{Co},$ it is possible to deduce the rate at which it is produced in the reaction. The cross section for this reaction is $37.0 \mathrm{~b}$. A thin disk of Co-Al alloy has a diameter of $1.00 \mathrm{~cm}$ and a mass of $46 \mathrm{mg}$; the alloy contains $0.44 \%$ Co by weight. With neutrons spread uniformly over the surface of the foil, it is concluded that ${ }^{60} \mathrm{Co}$ is produced at the rate of $1.07 \times 10^{12}$ per second. What is the rate at which neutrons strike the target?

Eduard Sanchez

Numerade Educator

Problem 5

A beam of $20.0 \mu \mathrm{A}$ of protons is incident on $2.0 \mathrm{~cm}^{2}$ of a target of ${ }^{107} \mathrm{Ag}$ of thickness $4.5 \mu \mathrm{m}$ producing the reaction $\mathrm{p}+{ }^{107} \mathrm{Ag} \rightarrow{ }^{105} \mathrm{Cd}+3 \mathrm{n}$. Neutrons are observed at a rate of $8.5 \times 10^{6}$ per second. What is the cross section for this reaction at this proton energy?C

Eduard Sanchez

Numerade Educator

Problem 6

A beam of alpha particles is incident on a target of ${ }^{63} \mathrm{Cu}$ resulting in the reaction $\alpha+{ }^{63} \mathrm{Cu} \rightarrow{ }^{66} \mathrm{Ga}+\mathrm{n}$. Assume the cross section for the particular alpha energy to be $1.25 \mathrm{~b}$. The target is in the form of a foil, $2.5 \mu \mathrm{m}$ thick. The beam has a circular cross section of diameter $0.50 \mathrm{~cm}$ and a current of $7.5 \mu \mathrm{A}$. Find the rate of neutron emission.

Eduard Sanchez

Numerade Educator

Problem 7

A radioactive isotope of half-life $t_{1 / 2}$ is produced in a nuclear reaction. What fraction of the maximum possible activity is produced in an irradiation time of $(a) t_{1 / 2} ;$ (b) $2 t_{1 / 2}$; (c) $4 t_{1 / 2} ?$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 8

List five nuclear reactions, consisting of a light stable projectile nucleus (mass 4 or less) incident on a heavy stable target nucleus, that can produce the radioactive nucleus ${ }^{56} \mathrm{Co}$

Eduard Sanchez

Numerade Educator

Problem 9

Show that Eq. 13.5 is a solution to Eq. 13.4.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 10

The radioisotope ${ }^{15} \mathrm{O}\left(t_{1 / 2}=122 \mathrm{~s}\right)$ is used to measure respiratory function. Patients inhale the gas, which is made by irradiating nitrogen gas with deuterons $\left({ }^{2} \mathrm{H}\right)$. Consider a cubical cell measuring $1.24 \mathrm{~cm}$ on each edge, which holds nitrogen gas at a pressure of $2.25 \mathrm{~atm}$ and a temperature of $293 \mathrm{~K}$. One face of the cube is uniformly irradiated with a deuteron beam having a current of 2.05 A. At the chosen deuteron energy, the reaction cross section is 0.21 b. $(a)$ At what rate is ${ }^{15} \mathrm{O}$ produced in the cell? $(b)$ After an irradiation lasting for $60.0 \mathrm{~s}$, what is the activity of ${ }^{15} \mathrm{O}$ in the cell?

Eduard Sanchez

Numerade Educator

Problem 11

Neutron capture in sodium occurs with a cross section of $0.53 \mathrm{~b}$ and leads to radioactive ${ }^{24} \mathrm{Na}\left(t_{1 / 2}=15 \mathrm{~h}\right) .$ What is the activity that results when $1.0 \mu \mathrm{g}$ of $\mathrm{Na}$ is placed in a neutron flux of $2.5 \times 10^{13}$ neutrons $/ \mathrm{cm}^{2} / \mathrm{s}$ for $4.0 \mathrm{~h} ?$

Eduard Sanchez

Numerade Educator

Problem 12

Derive Eq. 13.14 from Eq. $13.13 .$

Manik Pulyani

Numerade Educator

Problem 13

Find the $Q$ value of the reactions:
(a) $\mathrm{p}+{ }^{55} \mathrm{Mn} \rightarrow{ }^{54} \mathrm{Fe}+2 \mathrm{n}$
(b) ${ }^{3} \mathrm{He}+{ }^{40} \mathrm{Ar} \rightarrow{ }^{41} \mathrm{~K}+{ }^{2} \mathrm{H}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 14

Find the $Q$ value of the reactions:
(a) ${ }^{6} \mathrm{Li}+\mathrm{n} \rightarrow{ }^{3} \mathrm{H}+{ }^{4} \mathrm{He}$
(b) $\mathrm{p}+{ }^{2} \mathrm{H} \rightarrow 2 \mathrm{p}+\mathrm{n}$
(c) ${ }^{7} \mathrm{Li}+{ }^{2} \mathrm{H} \rightarrow{ }^{8} \mathrm{Be}+\mathrm{n}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 15

In the reaction ${ }^{2} \mathrm{H}+{ }^{3} \mathrm{He} \rightarrow \mathrm{p}+{ }^{4} \mathrm{He},$ deuterons of energy $5.000 \mathrm{MeV}$ are incident on ${ }^{3}$ He at rest. Both the proton and the alpha particle are observed to travel along the same direction as the incident deuteron. Find the kinetic energies of the proton and the alpha particle.

Manik Pulyani

Numerade Educator

Problem 16

(a) What is the $Q$ value of the reaction $\mathrm{p}+{ }^{4} \mathrm{He} \rightarrow$ ${ }^{2} \mathrm{H}+{ }^{3} \mathrm{He} ?$ (b) What is the threshold energy for protons incident on ${ }^{4}$ He at rest? $(c)$ What is the threshold energy if ${ }^{4}$ He are incident on protons at rest?

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 17

(a) Find the $Q$ value of the fission decay ${ }^{254} \mathrm{Cf} \rightarrow{ }^{127} \mathrm{In}+{ }^{127} \mathrm{In},$ in which ${ }^{254} \mathrm{Cf}$ splits in half
(b) Find the $Q$ value for the more probable fission process ${ }^{254} \mathrm{Cf}_{156} \rightarrow{ }_{54}^{140} \mathrm{Xe}_{86}+{ }_{44}^{110} \mathrm{Ru}_{66}+4 \mathrm{n} .$ Masses are:
$m\left({ }^{127} \mathrm{In}\right)=126.917353 \mathrm{u}, \quad m\left({ }^{140} \mathrm{Xe}\right)=139.921641 \mathrm{u}$
$m\left({ }^{110} \mathrm{Ru}\right)=109.914136 \mathrm{u}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 18

Find the energy released in the fission of $1.00 \mathrm{~kg}$ of uranium that has been enriched to $3.0 \%$ in the isotope ${ }^{235} \mathrm{U}$.

Hubert Agamasu

Numerade Educator

Problem 19

We can understand why ${ }^{235} \mathrm{U}$ is readily fissionable, and ${ }^{238} \mathrm{U}$ is not, with the following calculation. ( $a$ ) Find the energy difference between ${ }^{235} \mathrm{U}+\mathrm{n}$ and ${ }^{236} \mathrm{U}$. We can regard this as the "excitation energy" of ${ }^{236} \mathrm{U} .$ ( ) Repeat for ${ }^{238} \mathrm{U}+\mathrm{n}$ and ${ }^{239} \mathrm{U} .(\mathrm{c})$ Comparing your results for $(a)$ and $(b),$ explain why ${ }^{235} \mathrm{U}$ will fission with very low energy neutrons, while ${ }^{238} \mathrm{U}$ requires fast neutrons of 1 to $2 \mathrm{MeV}$ of energy to fission.
( $d$ ) From a similar calculation, predict whether ${ }^{239}$ Pu requires low-energy or higher-energy neutrons to fission.

Eduard Sanchez

Numerade Educator

Problem 20

Find the $Q$ value (and therefore the energy released) in the fission reaction ${ }^{235} \mathrm{U}+\mathrm{n} \rightarrow{ }^{93} \mathrm{Rb}+{ }^{141} \mathrm{Cs}+2 \mathrm{n}$. Use
$m\left(9^{93} \mathrm{Rb}\right)=92.922042 \mathrm{u}$ and $m\left({ }^{141} \mathrm{Cs}\right)=140.920046 \mathrm{u}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 21

(a) Calculate the $Q$ value for the six reactions or decays of the carbon cycle of fusion. (b) By accounting for the electron masses, show that the total $Q$ value for the carbon cycle is identical with that of the proton-proton cycle.

Manik Pulyani

Numerade Educator

Problem 22

Show that the D-T fusion reaction releases $17.6 \mathrm{MeV}$ of energy.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 23

In the D-T fusion reaction, the kinetic energies of ${ }^{2} \mathrm{H}$ and ${ }^{3} \mathrm{H}$ are small, compared with typical nuclear binding energies. (Why?) Find the kinetic energy of the emitted neutron.

Eduard Sanchez

Numerade Educator

Problem 24

(a) If a tokamak fusion reactor were able to achieve a confinement time of $0.60 \mathrm{~s}$, what minimum particle density is required? $(b)$ If the reactor were able to achieve 10 times the density found in part $(a)$, what is the minimum plasma temperature required for ignition of a self-sustaining fusion reaction?

Eduard Sanchez

Numerade Educator

Problem 25

Find the energy released when three alpha particles combine to form ${ }^{12} \mathrm{C}$.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 26

To what temperature must helium gas be heated before the Coulomb barrier is overcome and fusion reactions begin?

Eduard Sanchez

Numerade Educator

Problem 27

Trace the path of the $s$ process from the stable isotope ${ }^{63} \mathrm{Cu}$ to the stable isotope ${ }^{75}$ As, showing the neutron capture and beta decay processes.

Manik Pulyani

Numerade Educator

Problem 28

Show how the $s$ process proceeds from stable ${ }^{81} \mathrm{Br}$ to stable ${ }^{95} \mathrm{Mo}$.

Manik Pulyani

Numerade Educator

Problem 29

An alpha particle of mass $m$ makes an elastic head-on collision with an atom of mass $M$ at rest. Show that the loss in kinetic energy of the alpha particle is given by Eq. $13.16 .$

Manik Pulyani

Numerade Educator

Problem 30

(a) Calculate the energy loss of a 2.50-MeV alpha particle after backscattering from an atom of copper, silver, and gold. Compare your calculated values with the peak energies in Figure $13.25 .$ (b) Calculate the expected energy difference between the peaks for the two isotopes of copper and also for the two isotopes of silver. Explain why the silver peaks are closer together than the copper peaks. Can you estimate the relative abundances of the two isotopes of copper from the figure?

Eduard Sanchez

Numerade Educator

Problem 30

A radioactive source is to be used to produce electrical power from the alpha decay of ${ }^{238} \operatorname{Pu}\left(t_{1 / 2}=88 \mathrm{y}\right) .(a)$ What is the $Q$ value for the decay? (b) Assuming $100 \%$ conversion efficiency, how much power could be obtained from the decay of $1.0 \mathrm{~g}$ of ${ }^{238} \mathrm{Pu} ?$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 31

A radioactive source is to be used to produce electrical power from the alpha decay of ${ }^{238} \mathrm{Pu}\left(t_{1 / 2}=88 \mathrm{y}\right) .(a)$ What is the $Q$ value for the decay? (b) Assuming $100 \%$ conversion efficiency, how much power could be obtained from the decay of $1.0 \mathrm{~g}$ of ${ }^{238} \mathrm{Pu}$ ?

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 32

A small sample of paint is placed in a neutron flux of $3.0 \times 10^{12}$ neutrons/cm $^{2} / \mathrm{s}$ for a period of $2.5 \mathrm{~min}$. At the end of that period the activity of the sample is found to include 105 decays/s of ${ }^{51} \mathrm{Ti}\left(t_{1 / 2}=5.8 \mathrm{~min}\right)$ and 12 decays/s ${ }^{60} \mathrm{Co}\left(t_{1 / 2}=5.27 \mathrm{y}\right) .$ Find the amount, in grams, of titanium and cobalt in the original sample. Use the following information: Cobalt is pure ${ }^{59} \mathrm{Co},$ which has a cross section of $19 \mathrm{~b}$; titanium is 5.25 percent ${ }^{50} \mathrm{Ti}$, which has a cross section of 0.14 b.

Manik Pulyani

Numerade Educator

Problem 33

A 2.0-mg sample of copper $\left(69 \%^{63} \mathrm{Cu}, 31 \%^{65} \mathrm{Cu}\right)$ is placed in a reactor where it is exposed to a neutron flux of $5.0 \times 10^{12}$ neutrons/cm $^{2} / \mathrm{s}$. After $10.0 \mathrm{~min}$ the resulting activities are $72 \mu \mathrm{Ci}$ of ${ }^{64} \mathrm{Cu}\left(t_{1 / 2}=12.7 \mathrm{~h}\right)$ and $1.30 \mathrm{mCi}$ of $^{66} \mathrm{Cu}\left(t_{1 / 2}=5.1 \mathrm{~min}\right) .$ Find the cross sections of ${ }^{63} \mathrm{Cu}$ and ${ }^{65} \mathrm{Cu}$.

Manik Pulyani

Numerade Educator

Problem 34

A beam of neutrons of intensity $I$ is incident on a thin slab of material of area $A,$ thickness $d x,$ density $\rho,$ and atomic weight $M .$ The neutron absorption cross section is $\sigma .$
(a) What is the loss in intensity $d I$ of this beam in passing through the material? (b) A beam of original intensity $I_{0}$ passes through a thickness $x$ of the material. Show that the intensity of the emerging beam is $I=I_{0} e^{-n \sigma x},$ where $n$ is the number of absorber nuclei per unit volume. ( $c$ ) Assume that the total cross section for neutrons incident on copper is 5.0 b. What fraction of the intensity of a neutron beam is lost after traveling through copper of thickness $1.0 \mathrm{~mm} ?$ $1.0 \mathrm{~cm} ? 1.0 \mathrm{~m} ?$

Manik Pulyani

Numerade Educator

Problem 35

A reaction in which two particles join to form a single excited nucleus, which then decays to its ground state by photon emission, is known as radiative capture. Find the energy of the gamma ray emitted in the radiative capture of an alpha particle by ${ }^{7} \mathrm{Li}$. Assume alpha particles of very small kinetic energy are incident on ${ }^{7} \mathrm{Li}$ at rest.

Eduard Sanchez

Numerade Educator

Problem 36

How much energy is required (in the form of gamma-ray photons) to break up ${ }^{7} \mathrm{Li}$ into ${ }^{3} \mathrm{H}+{ }^{4} \mathrm{He}$ ? This reaction is known as photodisintegration.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 37

The nucleus ${ }^{113} \mathrm{Cd}$ captures a thermal neutron $(K=$ $0.025 \mathrm{eV}$ ), producing ${ }^{114} \mathrm{Cd}$ in an excited state; the excited state of ${ }^{114} \mathrm{Cd}$ decays to the ground state by emitting a photon. Find the energy of the photon.

Manik Pulyani

Numerade Educator

Problem 38

When a neutron collides head-on with an atom at rest, the loss in its kinetic energy is given by Eq. $13.16 .$ ( $a$ ) What fraction of its energy will a neutron lose in a head-on collision with an atom of hydrogen, deuterium, or carbon?
(b) Consider a neutron with an initial energy of $2.0 \mathrm{MeV}$. How many head-on collisions must it make with carbon atoms for its energy to be reduced to the thermal range $(0.025 \mathrm{eV}) ?(c)$ Is the result of part $(b)$ an underestimate or an overestimate of the actual number of collisions necessary to "thermalize" the neutrons? Explain.

Manik Pulyani

Numerade Educator

Problem 39

Suppose we have $100.0 \mathrm{~cm}^{3}$ of water, which is $0.015 \%$ $\mathrm{D}_{2} \mathrm{O} .(a)$ Compute the energy that could be obtained if all the deuterium were consumed in the ${ }^{2} \mathrm{H}+{ }^{2} \mathrm{H} \rightarrow{ }^{3} \mathrm{H}+\mathrm{p}$ reaction. (b) As an alternative, compute the energy released if two-thirds of the deuterium were fused to form ${ }^{3} \mathrm{H},$ which is then combined with the remaining one-third in the $\mathrm{D}-\mathrm{T}$ reaction.

Manik Pulyani

Numerade Educator

Problem 40

(a) Find the $Q$ value of the reaction ${ }^{4} \mathrm{He}+{ }^{4} \mathrm{He} \rightarrow{ }^{8} \mathrm{Be}$
(b) In a gas of ${ }^{4} \mathrm{He}$ at a temperature of $10^{8} \mathrm{~K}$, estimate the relative amount of ${ }^{8}$ Be present.

Anand Jangid

Numerade Educator