Nuclear Physics and Nuclear Medicine

Abstract

In this chapter we consider some of the properties of radioactive nuclei and their use for medical imaging and for cancer therapy. We begin by reviewing different mechanisms of radioactive decay, such as alpha, beta, and gamma decay, internal conversion, electron capture, and positron emission. It is important to know the dose to the patient from a nuclear medicine procedure, and we describe the standard technique for calculating it. Imaging using radiopharmaceuticals based on isotopes such as technetium-99m is analyzed, including both single photon emission computed tomography and positron emission tomography. We end with a description of radon and its health implications.

Problems

17.1.1 Section  17.1

Problem 1.

In the 1940s, a pressing question in biology was whether DNA or protein was responsible for the transmission of genetic information. A simple system to study this is a bacteriophage, a virus that injects a substance into Escherichia coli, thereby transforming the bacteria’s genetic material. Design an experiment using radioactive tracers that could determine whether DNA or protein was the injected substance. Hint: DNA contains many phosphorus atoms but no sulfur, whereas protein has many sulfur atoms but no phosphorus. Alfred Hershey and Martha Chase performed such an experiment in 1952.

Problem 2.

An alpha particle is fired directly at a stationary aluminum nucleus. Assume the only interaction is the electrostatic repulsion between the alpha particle and the nucleus, and the nucleus is so heavy that it is stationary. Calculate the distance of their closest approach as a function of the initial kinetic energy of the alpha particle. This calculation is consistent with Ernest Rutherford ’s famous alpha particle scattering formula for energies lower than 3 MeV, but deviates from his formula for energies higher than 3 MeV. If the alpha particle enters the nucleus, the nuclear force dominates and the formula no longer applies. Estimate the radius of the aluminum nucleus.

Problem 3.

The best current (2010) value for the mass of the proton is 1.007276467 u. The mass of the electron is $5.485799095\times 10^{-4}\,$u. The BE of the electron in the hydrogen atom is 13.6 eV. Calculate the mass of the neutral hydrogen atom.

Problem 4.

Solve Eq. 17.1 for the kinetic energy, $t$. Show that when $v\ll c$, it reduces to the familiar $T=m_{0}v^{2}/2$.

Problem 5.

The rest energy of the $_{74}^{184}$W nucleus is 171303 MeV. The average binding energies of the electrons in each shell are

$$ \begin{tabular} [c]{lll}Shell & Number of electrons & BE per electron (eV)\\ $K$ & \multicolumn{1}{r}{$2$} & \multicolumn{1}{r}{69,525}\\ $L$ & \multicolumn{1}{r}{$8$} & \multicolumn{1}{r}{11,015}\\ $M$ & \multicolumn{1}{r}{$18$} & \multicolumn{1}{r}{2125}\\ $N$ & \multicolumn{1}{r}{$32$} & \multicolumn{1}{r}{213}\\ $O$ & \multicolumn{1}{r}{$12$} & \multicolumn{1}{r}{49}\\ $P$ & \multicolumn{1}{r}{$2$} & \multicolumn{1}{r}{$\approx6$} \end{tabular} \ $$

Calculate the atomic rest energy of tungsten.

17.1.2 Section  17.5

Problem 6.

Refer to Figs. 17.2 and 17.5. Uranium splits roughly in half when it undergoes nuclear fission. Will the fission fragments decay by $\beta ^{+}$ or $\beta ^{-}$ emission?

Problem 7.

The following nuclei of mass 15 are known: $_{6}^{15}$C, $_{7}^{15}$N, and $_{8}^{15}$O. Of these, $^{15}$N is stable. How do the others decay?

Problem 8.

Look up the decay schemes of the following isotopes (for example, in the Handbook of Chemistry and Physics, CRC Press or at www.nndc.bnl.gov/). Comment on their possible medical usefulness: $^{3}$H, $^{15}$O, $^{13}$N, $^{18}$F, $^{22}$Na, $^{68}$Ga, $^{64}$Cu, $^{11}$C, $^{123}$I, and $^{56}$Ni.

Problem 9.

Look up the half lives of the isotopes in Fig. 17.6 (for instance in the Handbook of Chemistry and Physics. CRC Press or at www.nndc.bnl.gov/). Relate qualitatively the half life to the position of the isotope on the parabola.

17.1.3 Section  17.6

Problem 10.

Calculate the conversion factor $K$ of Eq. 17.21b.

17.1.4 Section  17.6.1

Problem 11.

Show that 1 $\mu$Ci h${}=1.332\times 10^{8}$ disintegrations or Bq s.

Problem 12.

Obtain a numerical value for the residence time for $^{99\text {m}}$Tc-sulfur colloid in the liver if 85 % of the drug injected is trapped in the liver and remains there until it decays.

Problem 13.

Derive Eqs. 17.39– 17.41.

Problem 14.

Calculate numerical solutions of Eqs. 17.41 and 17.43 and plot them on semilog paper. Use $\lambda =2,\lambda _{1}=0.5,\lambda _{2}=3$.

Problem 15.

Eq. 17.41 is not valid if $\lambda _{1}=\lambda _{2}$. In that case, try a solution of the form $N_{2}=Bte^{-\alpha t}$ where $\alpha$ is to be determined, and obtain a solution.

Problem 16.

Derive Eqs. 17.42 and 17.43.

Problem 17.

The biological half-life of iodine in the thyroid is about 25 days. $^{125}$I has a half-life of 60 days. $^{132}$I has a half-life of 2.3 h. Find the effective half-life in each case.

Problem 18.

For Sect. 17.6.1.4, with $\lambda =0.05~$h$^{-1},\lambda _{1}=1~$h$^{-1}$, and $\lambda _{2}=0.1~$h$^{-1}$, find $\tilde {A}_{1}$ and $\tilde {A}_{2}$ in terms of the initial activity $A_{0}$ and in terms of the initial number of nuclei $N_{0}$.

Problem 19.

$N_{0}$ radioactive nuclei with physical decay constant $\lambda$ are injected in a patient at $t=0$. The nuclei move into the kidney at a rate $\lambda _{1}$, so that the number in the rest of the body falls exponentially: $N(t)=N_{0}e^{-(\lambda +\lambda _{1})t}$. Suppose that the nuclei remain in the kidney for a time $t$ before moving out in the urine. (This is a crude model for the radioactive nuclei being filtered into the glomerulus and then passing through the tubules before going to the bladder.)

• (a) Calculate the cumulated activity and the residence time in the kidney by finding the total number of nuclei entering the kidney and multiplying by the probability that a nucleus decays during the time $t$ that it is in the kidney.

• (b) Calculate the cumulated activity and residence time in the bladder, assuming that the patient does not void.

Problem 20.

Suppose that at $t=0$, $^{99\text {m}}$Tc with an activity of 370 kBq enters a patient’s bladder and stays there for 2 h, at which time the patient voids, eliminating all of it. What is the cumulated activity? What is the cumulated activity if the time is 4 h?

Problem 21.

Suppose that the $^{99\text {m}}$Tc of the previous problem does not enter the bladder abruptly at $t=0$, but that it accumulates linearly with time. At the end of 2 h the activity is 370 kBq and the patient voids, eliminating all of it. What is the cumulated activity?

Problem 22.

A radioactive substance has half-life $T_{1/2}$. It is excreted from the body with biological half-life $T_{1}$. $N_{0}$ radioactive nuclei are introduced in the body at $t=0$. Find the total number that decay inside the body.

Problem 23.

The fractional distribution function $\alpha _{h}$ is the fraction of the total activity that is in organ $h$: $\alpha _{h}(t)=A_{h}(t)/A(t)=A_{h}(t)/A_{0}e^{-\lambda t}$.

• (a) Show that $\tau _{h}=\int _{0}^{\infty }\alpha _{h}(t)e^{-\lambda t}dt$.

• (b) Calculate $\alpha _{1}(t)$ and $\alpha _{2}(t)$ for Eqs. 17.39 and 17.41 and show that integration of these expressions leads to Eqs. 17.43.

Problem 24.

Suppose that the fractional distribution function (defined in the previous problem) is $\alpha (t)=1,\ t<T$; $\alpha (t)=b,\ t>T$; $(b<1)$. Find the residence time. This is a simple model for the situation where a bolus (a fixed amount in a short time) of some substance passes through an organ once and is then distributed uniformly in the blood.

Problem 25.

The distribution function $q_{h}(t)$ is defined to be the activity in organ $h$ corrected for radioactive decay to a reference time. If the correction is from time $t$ to time $0$, find an expression for $q_{h}(t)$ in terms of $A_{h}(t)$.

Problem 26.

The “official” definition of the fractional distribution function $\alpha _{h}(t)$ is the ratio of the distribution function $q_{h}(t)$ produced by a bolus administration to the patient, divided by the activity $A_{0}$ in the bolus. Show that this is equivalent to the definition in Problem 23.

Problem 27.

Show that if the uptake in a compartment is not instantaneous but exponential, with subsequent exponential decay, the cumulated activity is $\tilde {A}=1.443A_{0}(T_{e}T_{ue}/T_{u})$, where $T_{e}$ is the effective half-life for excretion, and $T_{ue}=T_{u}T_{1/2}/(T_{u}+T_{1/2})$. Hint: see Eq. 17.42.

17.1.5 Section  17.6.2

Problem 28.

Rearrange the data of Fig. 17.4. Find the total $\Delta$ for emission of photons below 30 keV and charged particles. Rank the radiations in the order they contribute to the dose.

Problem 29.

Nitrogen-13 has a half-life of 10 min. All of the disintegrations emit a positron with end point energy 1.0 MeV (average energy 0.488 MeV). There is no electron capture. Make a table of radiations that must be considered for calculating the absorbed dose and determine $E_{i}$ and $\Delta _{i}$ for each one.

Problem 30.

A patient swallows $3.5\times 10^{9}~$Bq of $^{131}$I. The half-life of the iodine is 8 days. Ten min later the patient vomits all of it. If none had yet left the stomach and all was vomited, determine the cumulated activity and residence time in the stomach.

17.1.6 Section  17.6.3

Problem 31.

Derive Eq. 17.57 by substituting Eqs. 17.56 and 17.54 in Eq. 17.54. You will also have to justify and use Eq. 17.58.

Problem 32.

The body consists of two regions. Region 1 has mass $m_{1}$ and cumulated activity $\tilde {A}_{1}$. It is completely surrounded by region 2 of mass $m_{2}$ and cumulated activity $\tilde {A}_{2}=\tilde {A}_{0}-\tilde {A}_{1}$. We can say that the mass of the total body is $m_{TB}=m_{1}+m_{2}=m_{1}+m_{\text {RB}}$. A single radiation is emitted with disintegration energy $\Delta$. The radiation is nonpenetrating so that

$$ \phi(1\leftarrow1)=\phi(2\leftarrow2)=1, $$

$$ \phi(1\leftarrow2)=\phi(2\leftarrow1)=0. $$

• (a) What are $\phi ($TB$\leftarrow 1)$ and $\phi ($TB$\leftarrow 2)$?

• (b) What are the corresponding values of $\phi$ and $S$?

• (c) Show that directly from the definition, Eq. 17.54

  $$ D_{1}=\tilde{A}_{1}\Delta/m_{1}, $$
  $$ D_{2}=D_{\text{RB}}=\tilde{A}_{2}\Delta/m_{2}, $$
  $$ D_{\text{TB}}=\tilde{A}_{0}\Delta/(m_{1}+m_{2}) $$

• (d) Calculate $\tilde {A}_{u}$ and $\tilde {A}_{1}^{\ast }$.

• (e) What is $S(1\leftarrow$TB$)$? Remember that $\phi$ is calculated for activity uniformly distributed within the source region.

• (f) Calculate the dose to region 1 using Eq. 17.57 and show that it agrees with (c).

• (g) Evaluate $S(1\leftarrow$RB$)$ using Eq. 17.58 and show that it agrees with $S(1\leftarrow 2)$.

Problem 33.

The body consists of two regions. Region 1 has mass $m_{1}$ and cumulated activity $\tilde {A}_{1}$. It is completely surrounded by region 2 of mass $m_{2}$ and cumulated activity $\tilde {A}_{2}$. A single radiation is emitted with disintegration energy $\Delta$. The characteristics of the radiation are such that

$$ \phi(1\leftarrow1)+\phi(2\leftarrow1)=1, $$

$$ \phi(1\leftarrow2)+\phi(2\leftarrow2)+\phi(0\leftarrow2)=1, $$

where $\phi (0\leftarrow 2)$ represents energy from region 2 that has escaped from the body. Obtain expressions for the dose to each region and the whole body dose.

Problem 34.

Consider the decay of a parent at rate $\lambda _{1}$ to an offspring that decays with rate $\lambda _{2}$.

• (a) Write a differential equation for the amount of offspring present.

• (b) Solve the equation.

• (c) Discuss the solution when $\lambda _{2}>\lambda _{1}$.

• (d) Discuss the solution when $\lambda _{2}<\lambda _{1}$.

• (e) Plot the solution for a technetium generator that is eluted every 24 h.

Problem 35.

$N_{0}$ nuclei of $^{99\text {m}}$Tc are injected into the body. What is the maximum activity for the decay of the metastable state? When does the maximum activity for decay of the ground state occur if no Tc atoms are excreted? What is the ratio of the maximum metastable state activity to the maximum ground-state activity?

Problem 36.

If 1 $\mu$Ci of $^{99\text {m}}$Tc is injected in the blood and stays there, relate the activity in a sample drawn time $t$ later to the volume of the sample and the total blood volume. If the gamma rays are detected with 100 % efficiency, what will be the counting rate for a 10-ml sample of blood if the blood volume is 5 l? (Using non-SI units was intentional.)

Problem 37.

Assume that aggregated human albumin is in the form of microspheres. A typical dose of albumin microspheres is 0.5 mg of microspheres containing 80 MBq of $^{99\text {m}}$Tc and 15 $\mu$g of tin. There are $1.85\times 10^{6}$ microspheres per mg.

• (a) How many $^{99\text {m}}$Tc atoms are there per microsphere?

• (b) How many tin atoms per microsphere?

• (c) How many technetium atoms per tin atom?

• (d) What fraction of the surface of a microsphere is covered by tin? Assume the sphere has a density of $10^{3}~$kg m$^{-3}$.

Problem 38.

It is estimated that the total capillary surface area in the lung is 90 m$^{2}$. Assume each capillary has 50 segments, each 10 $\mu$m long, and a radius of 5 $\mu$m.

• (a) How many capillaries are there in the lung?

• (b) There are about $3\times 10^{8}$ alveoli in both lungs. How many capillaries per alveolus are there?

• (c) An alveolus is 150–300 $\mu$m in diameter. Are the above answers consistent?

• (d) A typical dose of albumin microspheres is 0.5 mg with an average diameter of 25 $\mu$m. There are $1.85\times 10^{6}$ spheres per mg. What fraction of the capillaries are blocked if there is good mixing?

17.1.7 Section  17.6.4

Problem 39.

Look up the decay schemes and half-lives for $^{123}$I and $^{131}$I. Explain why $^{123}$I is used to image the thyroid and $^{131}$I is used to treat thyroid cancer.

Problem 40.

Identify all the isotopes in Fig. 17.7 using the $^A_Z$Symbol notation. What are the stable isotopes? What isotope can decay by both $\beta ^-$ and $\beta ^+$ emission?

Problem 41.

The half-life of $^{99\text {m}}$Tc is 6.0 h. The half-life of $^{131}$I is 8.07 day. Assume that the same initial activity of each is given to a patient and that all of the substance remains within the body.

• (a) Find the ratio of the cumulated activity for the two isotopes.

• (b) $^{99\text {m}}$Tc emits 0.141-MeV photons. For each decay of $^{131}$I the most important radiations are 0.89 $\beta ^{-}$ of average energy 0.192 MeV and 0.81 photons of 0.365 MeV. If all of the decay energy were absorbed in the body, what would be the ratio of doses for the same initial activity?

Problem 42.

A patient is given an isotope that spreads uniformly through the lungs. It emits a single radiation: a $\gamma$ ray of energy 50 keV. There are no internal-conversion electrons. The cumulated activity is 40 GBq s. Find the absorbed dose in the liver ($m$ = 1.83 kg).

Problem 43.

The decay of $^{99\text {m}}$Tc can be approximated by lumping all of the decays into two categories:

$$ \begin{tabular} [c]{p{1.5in}p{0.7in}p{1in}}Radiation & $E_{i}$ (MeV) & $\Delta_{i}$ (J)\\ & & \\[-10pt]$\gamma$ & $0.14$ & $2\times10^{-14}$\\ Electrons and soft x rays & & $2.76\times10^{-15}$ \end{tabular} \ \ $$

Sulfur colloid labeled with 100 MBq of $^{99\text {m}}$Tc is given to a patient and is taken up immediately by the liver. Assume it stays there. Find the dose to the liver, spleen, and whole body. Use the following information:

$$ \begin{tabular} [c]{p{1in}p{0.8in}p{0.8in}} \multicolumn{3}{c}{Absorbed fraction for a source in the liver}\\ Target organ & Mass (kg) & $E(\gamma)=$\\ & & $0.14$~MeV \\ \hline & & \\[-10pt]Liver & $1.833$ & $0.161$ \\ Spleen & $0.176$ & $0.000629$\\ Whole body & $70.0$ & $0.431$ \\ & & \\[-10pt] \end{tabular} $$

Problem 44.

An ionization type smoke detector contains 4.4 $\mu$Ci of $^{241}$Am. This isotope emits $\alpha$ particles (which we will ignore) and a 60-keV $\gamma$ ray, for which $n=0.36$. The half-life is 458 yr.

• (a) How many moles of $^{241}$Am are in the source?

• (b) Ignoring attenuation, backscatter, and buildup in any surrounding material (such as the cover of the smoke detector), what is the absorbed dose in a small sample of muscle located 2 m away, if the muscle is under the detector for 8 h per day for 1 year?

Problem 45.

One mCi of a radioactive substance lodges permanently in a patients lungs. The substance emits a single 80-keV $\gamma$ ray. It has a half-life of 12 h. Find the cumulated activity and the dose to the liver (mass 1833 g).

Problem 46.

The dose calculation for microspheres in the lung was an oversimplification because technetium leaches off the spheres. The footnote in Sect. 17.6.4 lists some more realistic residence times. If none of the technetium is excreted from the body, the sum of all the residence times will still be 8.7 h. Assume that the residence time in the lungs is 4.3 h and the residence time in the rest of the body is 4.4 h.

• (a) Show that $\tilde {A}_{u}=4.46\times 3600\times A_{0}$ and $\tilde {A}_{\text {lung}}^{\ast }=4.24\times 3600\times A_{0}$.

• (b) For a source distributed uniformly throughout the total body, the absorbed fractions for 140-keV photons are $\phi ($lung$\leftarrow$TB$)=0.0053$, $\phi ($TB$\leftarrow$TB$)=0.3572$. Split the radiation into penetrating and nonpenetrating components:

  $$ \begin{aligned} S(\text{lung}\leftarrow\text{TB})& =(\phi_{\text{nonpen}}\Delta_{\text{nonpen}} \\& +\phi_{\text{penetrating}}\Delta_{\text{penetrating}})/m_{\text{lung}}. \end{aligned} $$

  Remember that for activity uniformly distributed in the total body, $\phi ($lung$\leftarrow$TB$)=m_{\text {lung}}/m_{\text {TB}}$ and use some of the information in Table 17.3 to show that

  $$ S(\text{lung}\leftarrow\text{TB})=1.463\times10^{-16}\,\text{J}\,\text{kg}^{-1}, $$
  $$ S(\text{TB}\leftarrow\text{TB})=1.414\times10^{-16}\,\text{J}\,\text{kg}^{-1}. $$

• (c) Calculate the dose to the lungs and the total body dose for an initial activity of 37 MBq. Compare the values to those in Table 17.3.

17.1.8 Section  17.8

Problem 47.

Nuclear counting follows Poisson statistics. Show that for a fixed average counting rate $R$ (counts per second) the standard deviation of a sum of $N$ measurements each of length $t$ is the same as a single measurement of duration $NT$. (Hint: You will first have to consider the situation where one measures $y=x_{1}+x_{2}+\cdots$ and find the variance of $y$ in terms of the variances of the $x_{i}$ when there is no correlation between the $x_{i}$.)

Problem 48.

The interaction of a photon in a nuclear detector (an “event”) initiates a process in the detector that lasts for a certain length of time. A second event occurring within a time $\tau$ of the first event is not recorded as a separate event. Suppose that the true counting rate is $R_{t}$. A counting rate $R_{o}$ is observed.

• (a) A nonparalyzable counting system is “dead” for a time $\tau$ after each recorded event. Additional events that occur during this dead time are not recorded but do not prolong the dead time. Show that $R_{t}=R_{o}(1-R_{o}\tau )$ and $R_{o}=R_{t}/(1+R_{t}\tau )$.

• (b) A paralyzable counting system is unable to record a second event unless a time $\tau$ has passed since the last event. In other words, an event occurring during the dead time is not only not recorded, it prolongs the dead time. Show that in this case $R_{o}=R_{t}e^{-R_{t}\tau }$. (Hint: Use the Poisson distribution of Appendix J to find the fraction of events separated by a time greater than $\tau$. The probability that the next event occurs between $t$ and $t+dt$ is the probability of no event during time $t$ multiplied by the probability of an event during $dt$.)

• (c) Plot $R_{o}$ vs $R_{t}$ for the two cases when $\tau$ is fixed. The easiest way to do this is to plot $R_{o}\tau$ vs $R_{t}\tau$.

Problem 49.

Two channels of a collimator for a gamma camera are shown in cross section, along with the path of a photon that encounters the minimum thickness of collimator septum (wall).

• (a) Show that if $(d+t)/l\ll 1$, then $w/t=l/(2d+t)$.

• (b) If transmission through the septum is to be less than 5 %, what is the relationship between $t$, $D$, $l$, and $\mu$? Evaluate this for $^{99\text {m}}$Tc and for a positron emitter.

Problem 50.

Photons from a point source a distance $B$ below a collimator pass through channels out to a distance $A$ from the perpendicular to the collimator passing through the source.

• (a) Find an expression for $A$ in terms of $B$, $D$, and $l$.

• (b) Assume that $A$ is related to the spatial frequency $K$ for which the modulation transfer function $($MTF$)=0.5$ in Fig. 17.18 by $a=K/k$, where $K$ is a constant. Calculate the thickness $l$ of the collimator.

Problem 51.

The collimator efficiency of a gamma camera is defined to be the fraction of the $\gamma$ rays emitted isotropically by a point source that pass through the collimator into the scintillator.

• (a) Consider a circular channel of diameter $D$ in the collimator directly over the source. Show that the fraction of the photons striking the scintillator after passing through that channel is $d^{2}/16(l+b)^{2}$. (Assume that any which strike the septum are lost).

• (b) Use the result of the previous problem to estimate the number of channels through which at least some photons from the point source pass. Assume that the fraction of collimator area that is occupied by channels rather than lead is $\left [ d/(d+t)\right ] ^{2}$.

• (c) Calculate the geometric efficiency $g$ assuming that all channels that pass any photons have the same efficiency as the one on the perpendicular from the source. Show that it is of the form

  $$ g=K^{2}\left( \frac{d}{l}\right) ^{2}\left( \frac{d}{d+t}\right) ^{2} $$

  and evaluate $K$. More detailed calculations show that $K$ is about 0.24 for a hexagonal array of round holes and 0.26 for hexagonal holesAQ1.¹¹

• (d) How does the detector efficiency relate to the collimator resolution?

17.1.9 Section  17.9

Problem 52.

• (a) Derive Eq. 17.60 from Eq. 17.59.

• (b) Calculate the limit of Eq. 17.60 when there is no attenuation.

Problem 53.

The attenuation distortion for SPECT can be reduced by making measurements on opposite sides of the patient and taking the geometric mean. The geometric mean of variables $x_{1}$ and $x_{2}$ is $(x_{1}x_{2})^{1/2}$. Calculate the geometric mean of two SPECT measurements on opposite sides of the patient. Ignore possible $1/r^{2}$ effects.

Problem 54.

Consider a radioactive source having a uniform activity per unit volume $A_{V}$ and the square geometry shown below.

• (a) Calculate the projection $F(x)$ including the effects of attenuation with coefficient $\mu$.

• (b) Plot $F(x)$ for $\mu =0$ and for $\mu R=3$.

17.1.10 Section  17.10

Problem 55.

Suppose that $A$ positrons are emitted from a point per second. They come to rest and annihilate within a short distance of their source. When a positron annihilates, two photons are emitted in opposite directions. Two photon detectors are set up on opposite sides of the source. The source is distance $r_{1}$ from the first detector, of area $S_{1}$, and $r_{2}$ from the second detector of area $S_{2}$. The area $S_{2}$ is large enough so that the second photon will definitely enter detector 2 if the first photon enters detector 1. Assume that both detectors count with 100 % efficiency.

• (a) Show that the number of counts in the first detector would be $2AS_1 /4\uppi r_1^2$ if there were no attenuation between source and detector, and that it is $(2AS_1/4\uppi r_1^2)e^{-\upmu a_1}$ if attenuation in a thickness $a_{1}$ of the body is considered.

• (b) Detector 2 detects the second photon for every photon that strikes detector 1. Assuming a uniform attenuation coefficient and body thickness $a_{2}$, find an expression for the number of events in which both photons are detected.

Problem 56.

Positron emission tomography relies on simultaneous detection of the back-to-back annihilation gamma rays (a coincidence). In addition to true coincidences, there can be “scatter coincidences” in which annihilation photons coming from a point that is not on the line between the two detectors enter both detectors. There can also be “random coincidences” which arise from photons from completely independent decays that occur nearly simultaneously. Consider a ring of detectors around a patient. Make three drawings showing true coincidences, scatter coincidences and random coincidences.

17.1.11 Section  17.12

Problem 57.

The half-life of $^{235}$U is $7\times 10^{8}\,$yr. The age of the earth is 4.5 billion years. What fraction of the $^{235}$U that existed on the earth when it was first formed is present now?

Problem 58.

There are three naturally-occurring decay series beginning with three long-lived isotopes: $^{238}$U (Figs. 17.26 and 17.27), $^{235}$U, and $^{232}$Th. The $^{232}$Th series begins with the $\alpha$ decay of $^{232}$Th (half life $=1.4\times 10^{10}\,$yr) to nucleus $A$ which undergoes $\beta ^{-}$ decay to nucleus $B$ which undergoes $\beta ^{-}$ decay to nucleus $C$ which undergoes $\alpha$ decay to nucleus $D$ which undergoes $\alpha$ decay to nucleus $E$, etc. Make a chart like Fig. 17.26 showing the first five steps in the series, and identify the five nuclei $A$–$E$.

Problem 59.

One way to determine the age of biological remains is carbon-14 dating. The common isotope of carbon is stable $^{12}$C. The rare isotope $^{14}$C decays with a half-life of 5370 yr. $^{14}$C is constantly created in the atmosphere by cosmic rays. The equilibrium between production and decay results in about 1 of every $10^{12}$ atoms of carbon in the atmosphere being $^{14}$C, mostly as part of a CO$_{2}$ molecule. As long as the organism is alive, the ratio of $^{12}$C to $^{14}$C in the body is the same as in the atmosphere. Once the organism dies, it no longer incorporates $^{14}$C from the atmosphere, and the number of $^{14}$C nuclei begins to decrease. Suppose the remains of an organism have one $^{14}$C for every $10^{13}$ $^{12}$C nuclei. How long has it been since the organism died?

Problem 60

Consider a fictitious two-step decay series analogous to the more complex series shown in Fig. 17.26. The series starts with isotope $A$ which decays at rate $\lambda _{1}$ to isotope $B$. Isotope $B$ decays to isotope $C$ with rate $\lambda _{2}$. Isotope $C$ is stable.

• (a) Derive the differential equation governing the number of nuclei $N_{A}$, $N_{B}$, and $N_{C}$. Where else in this chapter have you seen the same equations?

• (b) Solve the differential equations using the initial conditions $N_{A}(0)=N,$ $N_{B}(0)=N_{C}(0)=0$. Make sure your solutions make sense for $t\rightarrow 0$ and $t\rightarrow \infty$.

• (c) Find $N_{B}/N_{A}$ in the limit $\lambda _{2}\gg \lambda _{1}$. Ignore short times. Also find activities $A_{A}$ and $A_{B}$. Explain physically how such a small number of nuclei $N_{B}$ can contribute so much to the total activity.